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E X P L O R A T O R Y

AND

C O N F I R M A T O R Y

F C T O R

N L Y S IS

Understanding

oncepts

and

Applications

B R U C E T H O M P S O N

American Psychological Association • Washington DC



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Library

of ongress Cataloging in Publication Data

Thompson, Bruce , 9 5 —

Exploratory

and

conf i rmatory factor analysis

:

u n d e r s t an d i n g

concepts

and

ap p l i c a t i o n s

/

Bruce Thompson .—1s t

cd .

p. cm.

Includes bibl iographical references .

ISBN 1-59147-093-5

1.

Fac to r ana lys i s—Textbooks . 1. Ti t le .

BF3 9 . 2 . F3 2 T4 8 2 0 0 4

150 ' .1 '5195354—dc22

3 596

British L ibrary Cataloguing in Publication Data

A C 1P record is av ai la ble f rom the B ri t ish

Library .

Printed in

th e United States

o f

Am erica

First Edition



CONTENTS

Preface ix

Chapter 1 . Introduction to Factor Analysis 3

Purposes of

Factor Analysis

4

Two

M a j o r Classes

of

Factor Analysis

5

Organizat ion

of the

Book

7

Chapte r 2 . Foundational

Concepts

9

Heuris t ic

Example 9

Two

Wor lds

of

Statistics

11

Some Basic Matrix Algebra Concepts

12

More Heurist ic Results

15

Sample Size Considerat ions

in EFA 24

Major

Concepts

25

Chapte r

3.

Exp loratory Factor An alysis Decision Sequence

27

Matr ix

of

A ssociation

Coefficients 28

N u m b e r

of Factors 31

Factor Extraction Method

36

Factor

Rotation 38

Factor Scores 44

Major

Concepts

47

Ch apter 4. Com ponents Versus Factors Co ntroversies 49

Measurement

Error Versus Sampling Error

50

Pr incipal Components Versus Pr incipal

Ax es Factors 53

Major Concepts 55



Chapter 5. Comparisons of Some

Extraction, Rotation,

and

Score M ethods 57

Factor Scores in Different Ext rac t ion Methods 57

Structure Coeff ic ients

i n a

Components

Context

61

Communal i ty Coef f ic ien t s a s R

Values 61

Major

Concepts

64

Chap ter 6. Obl iqu e Ro tat ions and High er-Order Factors 67

M a x i m i z i n g Sample

F it

Versus Repl icabi l i ty

69

Higher-Order Factors 72

Major Concepts 81

Chapter 7. Six Two-Mode Techniques 83

Cattell 's "Data Box"

83

Q-Technique 86

Major Concepts

91

Chapte r 8 . Conf i rmatory Rota t ion and Factor

In te rpre ta t ion

Issues

93

"Best-Fit" Ro tat ion Ex am ple 94

Factor Interpre ta t ion Considerat ions 96

M a j o r

Concepts 98

Chap ter 9. Intern al Repl icabi l i ty Ana lyses 99

EFA Cross-Validation 101

EFA Boots t rap

102

Major

Concepts 106

Chapter 10. Co nfirm atory Factor Ana lysis Decision Sequence ... 109

CFA

Versus

EFA 110

Preanalysis

Decisions 114

Postanalysis Decisions

127

M aj o r

Concepts 131

Chapte r 11. Some Confi rmatory Factor Analys is

Interpre ta t ion Principles 133

Model Ident i f icat ion Choices 134

Different Es t imat ion

Theories

138

CFA Pat tern and Structure Coefficients 138

Comparisons

of

Model Fits

142

Higher -O rder M ode ls 145

M a j o r Concepts 147

vi CONTENTS



Chapter 12. Testing Mode l Invar iance 153

Same M ode l , Different Est imates 153

S am e M ode l , Equa l Es t imates 155

Same M ode l , Test Invar iance of Param ete r s 159

M a jo r Concepts 159

A ppe nd ix

A:

L ibQU A L + ™ D a ta

163

A ppe nd ix

B :

SPSS S yn t ax

to

E xe cu t e

a

"Best-Fit"

Fac tor Rota t ion 169

Notation 177

Glossary

179

References

183

I nde x 191

About the A u t h o r 195

CONTENTS

vii



PREFACE

Investigation

of the

s tructure under lying var iables

(o r

people,

or

t ime)

has intrigued social scientists since th e early origins of psychology. Conduct -

in g one s

first

factor analysis can yield a sense of awe regarding th e power

of these methods

to

inform judgment regarding

the

dimensions underly-

ing constructs.

This book presents the impor tant concepts required for implement ing

two disciplines of factor analysis: exploratory factor analysis (EFA) and

confirmatory factor analysis (CF A). T he book may be uniq ue in its

effort

to present both analyses within the single rub ric of the general l inear model.

Throughout th e book canons of best factor analytic practice are presented

and

explained.

The book has

been

wr i t ten to str ike a happy medium between accuracy

and completeness versus overwhelm ing technical complexi ty .

A n

ac tua l

data set, randomly drawn

from a

large-scale international study (cf.

Cook

6k Thompson, 2001) involving faculty and graduate student perceptions of

academ ic libraries, is presented in Appe ndix A. Throughout the book differ-

ent combinations of

these

variables and participants are used to il lustrate

EFA

and CFA

applications.

Real u nde rstand ing of s tat is t ical concepts comes wit h applying these

methods

i n

one s

o w n

research.

The

next-best learning opportunity involves

analyzing

real data provided

by

others.

You are

encouraged

to

replicate

the

analyses

reported

in the

book

and to

conduct methodological var iat ion s

on

the reported analyses. You also may find it useful to conduct complementary

analyses to

those reported here (e.g.,

if 11

variables from

100

faculty

are

analyzed, conduct

th e

same analysis using data from

the 100

graduate

s tudent s ) .



There are also other data sets that have a long history of use in

heuristics

in the factor analytic context . For exam ple, the d ata reported hy

Holz inger

and Swineford (1939) are widely reported. Readers are encouraged

to access and analyze these and other data sets.

To improve the instruct iona l effect iveness of the book, some liberties

have been taken in using Am erican Psychological Association (A PA ) style.

For

example,

to

d is t inguish between score world

an d

area world statistics,

area world statistics are often presented with percentage signs, even across

the rows of a single table, for emphasis and cla rity. And more t han th e

usual num ber of decim al places may be reported, to help the reader com pare

reports in the book with the related reports in computer outputs , or to

dis t inguish

one block of results from ano ther by using different numbers of

decimal places

for

different results.

I

appreciate

in

this regard

th e

l a t i tude

afforded

me by the A PA

produc-

tion

staff.

I also app reciate the o pp ortu nit y to wr ite the book in my own

voice. This book has been wr itten as if I were speaking direc tly to yo u, the

student ,

in my own

office

or

classroom. Fortunately,

I

have

a

very

b ig

office

I

also apprec iate th e thoughtful suggestions of Randy A rnau , Univer s ity

of Southern M ississippi, Susan Crom well, Texas

A& M

Univers i ty ,

and

Xitao Fan, University of Virginia, on a

draft

of the book. Notwithstanding

their suggestions, of course I remain responsible for the book in its final form.

PREFACE



E X P L O R T O R Y

AND

O N F I R M T O R Y

F C T O R N L Y S I S



1

INTRODUCTION TO

FACTOR ANALYS IS

The early 1900s were exciting times fo r social scientists. Psychology

was

emerging

as a

formal

discipline.

In

France, Alfred Binet created

a

measure

of in tel lectual performance that w as the forerun ner to modern

IQ tests. But from the very outs et there were heated controversies

regarding

definition s of intelligence and related questions abou t how to

measure intell igence.

Some psychologists took the vie w that in tel l igence was a s ingle general

abi l i ty .

Those taking th is

view

presumed

that

ind iv idua l s who w ere

very

bright in one area of performa nce tended to perform

wel l

across the full

range of in tel le ctu al tasks. This cam e to be called the general , or G, theory

of

intell igence . Other psychologists argued that individuals may be prof icient

at one

task,

bu t

that

expertise

in one

area

has

l i t t le

or no

predict ive abi l i ty

as regards other areas of cognitive performance. These latter psychologists

argued that intell igence involved specific factors that correlated min ima l ly

with

each

other.

However, in these early years psychologists who wished to test their

dive rgen t

views

had few

analy t ic tools w ith which

to do so. To

address this

void, Spearman (1904) conceptual ized methods that today we term factor

analysis.

In the intervening decades, numerous statistical developments

mean that today

the

analyst confronts

an

elaborate array

of

choices wh en

using

these methods.



Of

course,

for

many decades factor analysis

was

infrequently

used.

In

the era of statistical analysis via hand-cranked calculators, the computations

of factor analysis were u s u a l l y overwhelming. In part this was also because

responsible researchers conducting analyses prior

to the

advent

of

modern

computers felt obligated

to

calculate their results repeatedly, until

at

least

two

analyses

of the

data yielded

the

same results

Happily,

today the

v e r y ,

very complicated calculations of modern factor

analysis can be accomplished on a microcomputer in only moments, and

in

a

user-friendly

point-and-click environment.

As a

consequence, today

factor analytic reports comprise

as

many

as 18% to 27% of the

articles

in

some

journals (Fabrigar, Wegener, MacCallum, &

Strahan,

1999; Russell, 2002).

PURPOSES OF

FACTOR

ANALYSIS

Factor analytic methods can be used for various purposes, three of

which

are

noted

here.

First, when

a

measure

and its

associated scoring keys

have been developed,

factor

analysis can be used to inform

evaluations

of

score

validity.

V a l i d i t y questions focus

on

whether scores measure "the correct some-

thing." Measures producing scores that are completely random measure

"nothing" (Thompson, 2003).

Such

tools yield scores that are

p e r f ec t l y

unreliable. We never develop tests to measure nothing, because we have

readily available tools, such as dice and roulette wheels, that we can use to

obtain random scores. Obviously,

for

scores intended

to

measure something

(e.g.,

intelligence, self-concept),

the

scores must

be

reliable.

But r e l i a b i l i t y

is

only a necessary—not a sufficient—condition for validity.

V a l i d i t y questions involve issues such

as,

"Does

the

tool produce scores

that seem to measure the intended dimensions?" and "Are items intended

only

to

measure

a

given dimension actually measuring

and

only measuring

that dimension?" These are exactly the kinds of questions that factor analysis

w as created to confront. So it is not

s u r p r i s i n g

that historically "construct

v a l i d i t y

has

[even]

been

spoken

of as ...

'factorial validity'

"

(Nunnally,

1978, p.

111).

Joy Guilford's discussion some

50

years

ago

illustrates this application:

Val id i ty , in my opin ion is of two

kinds .

. . . The factorial validity

[italics

added] of a test is given by its . . . [factor weights on] m e a n i n g f u l ,

common,

reference

factors.

This

is the k ind of va l id i t y

t ha t

is really

m e a n t w h e n th e

ques t ion

is

asked

"Does

this test

measu re

w h a t

it is

supposed to measure? A mor e p e r t i n en t

ques t ion

should he "What

does this

test

measure?

The answer then should be in t e r m s of factors.

EXPLORATORY AND CONFIRMATORY

FACTOR

ANALYSIS



... 1 pred ic t a t i me when a n y test a u t hor wil l b e expected to present

in forma t ion rega rd ing

t he

factor

compos i t ion of his [sic] tests. (Gui l fo rd ,

1946,

pp.

428, 437-438)

Although modern v a l i d i t y language does not recognize a type of v a l i d i t y

called "factorial validity," factor analysis nevertheless

is

h e l p f u l

in

addressing

construct v a l i d i t y questions.

Thus, Gorsuch (1983)

has

noted that

"a

prime

use of

factor analysis

has

heen

in the development of

both

the operational constructs for an area

and the

operational representatives

for the

theoretical constructs"

(p.

350).

And Nunnally (1978) suggested that "factor analysis is intimately involved

with questions

of

validity.

. . .

Factor

a n a l y s i s is at the

heart

of the

measure-

ment of psychological constructs" (pp. 1 1 2 — 1 1 3 ) .

Second, factor analysis

can be

used

to

develop theory

regarding th e

nature

of

constructs At least historically,

e m p i r i c a l l y

driven theory development

was sometimes held

in

high regard.

In

some

of

these applications, numerous

d i f f e r en t measures are administered to various samples, and the results of

factor analysis then are used to

specify

construct dimensions. This applica-

tion,

unlike the first one, is inductive.

A

classic example

is the

structure

of

intellect model developed

by

Guilford (1967). Guilford administered dozens

of

tests

of

cognitive abilities

over

years to

derive

his

theory,

in

which

he

posited that intelligence consists

of more than

100

d i f f e r en t abilities that

are

independent

of

each

other.

Third, factor analysis can be used to summarize relationships in the form

of

a

more parsimonious

set of

factor

scores

that

ca n

then

be

used

in

subsequent

analyses (e.g., analysis of variance, regression, or descriptive discriminant

analysis).

In this application, unlike the first two, the factor analysis is only

an intermediate step in i n q u i r y , and not the final analysis. Using

f e w e r

variables in substantive a n a l y s i s tends to conserve

degrees

of freedom and

improve power against Type

II

error. Furthermore, factor scores

can be

computed

in

such

a

manner that

the

unreliable variance

in the

original

variables

tends

to be

discarded

once the

original scores

are

reexpressed

in

a smaller set of factor scores.

TWO M A J O R CLASSES OF FACTOR

ANALYSIS

There are actually two discrete classes of factor analysis: exploratory

factor

analysis (EFA) and confirmatory factor analysis (CFA). Analyses such

as

those originally proposed

by

Spearman (1904) have

now come to be

called exploratory factor analysis.

In

EFA,

the

researcher

may not

have

any

specif ic

expectations

regarding

the

number

or the

nature

of

underlying

constructs

or

factors. Even

if the

researcher

has

such expectations

(as in a

INTRODUCTION TO

FACTOR

ANALYSIS



val id i ty inv es t iga t ion ) ,

EFA

does

not

require

the

researcher

to

declare these

expectations, and the analysis is not influe nce d by these expectations.

Confi rmatory factor

analysis

methods were developed much more re-

cent ly (cf. Joreskog, 1969). These analyses requ ire that the researcher mu st

have

specific

expectat ions regarding

(a) the

n u m b e r

of

factors,

(b)

wh i ch

variables reflect g iven factors,

and (c)

whether

th e

factors

are

correlated .

CFA expl ic i t ly and directly tests the fit of factor models.

Researchers without theories cannot use CFA, but researchers with

theories usua l ly find CFA more

useful

tha n EFA. Confi rmatory factor

analysis

is more useful in the presence of theory because (a) the theory is di rect ly

tested

by the

analysis

and (b) the

degree

of

model

fit can be

quan t i f ied

in

var ious

ways.

Of course, some researchers begin an

inquiry

wi t h

specific

theoretical expectations, but after invoking CFA find that their exp ectat ions

were wi ldly wrong, and may

then

revert to E FA procedu res .

Alth ou gh factor analysis has been ch aracteriz ed as one of the m ost

powerful methods yet for reduc ing var iab le complex i ty to greater s implici ty

(Kerlinger, 1979, p. 180) and as the furthes t logical developm ent and

re igning queen of the correlat ion al methods (Ca t tel l , 1978, p . 4) , the

methods have been harshly crit icized by others (cf. Armstrong, 1967). For

example , with respect to research in the area of com m un ication , Cron khite

a n d L i s k a (1980) noted:

A p p a r e n t l y , it is so easy to f ind semantic scales which

seem

re leva n t to

[ information] sources , so easy to name or descr ibe potential/hypothetical

sources,

so

easy

to capture college

s tuden ts

to use the scales to rate the

sources , so easy to s u b m i t

those

ra t ings to fac tor ana lys is , so much fun

to name the

fac tors when

one's research

assistant re tu rns

with the

computer

p r i n t o u t , and so rewa rd ing to have a guaranteed publication

w i t h no

fe a r

of nonsignif icant resu l ts

that

researchers, once exposed to

the pleasures of the factor analytic

approach,

r a p i d l y become addicted

to it . (p. 1 0 2 )

However, these cr i t ic isms apply more

to EFA

than

to

CFA, because

the only danger in EFA would he that factors might not be in terpretable ,

in

which case

th e

resul ts

may be

regarded

as

nega t ive .

But CFA

p resumes

the invocation of specific expectations, w ith the attend an t p ossibili ty that

the predetermined theory in

fact

does not fi t , in which case the researcher

m ay be

unable

to

exp la in

the

relat ionships among

the

var iab les be ing ana-

lyzed. Both EFA and CFA remain useful today, and our selection between

the two classes of factor analysis general ly depends on wh ether w e have

specific theory regarding data s t ructure.

But even researchers who only conduct research in the presence of

such theory

mus t

unders tand EFA. Mastery

of CFA

requ i res und ers tand ing

of EFA, which

was the

his torical precursor

t o

CFA. Indeed,

a

f u n d a m e n t a l

EXPLORATORY AND

CONFIRMA TORY FACTOR ANALYSIS



purpose of this hook is to teach the reader what is common and what is

different across these tw o classes of factor analysis .

Understa nding EFA also may facili tate deeper insight

into

other statis-

t ica l

analyses .

A n

important idea

in

statistics

is the not ion of the

general

l inear model (GLM;

cf.

Bagozzi, Fornell,

&

Larcker, 1981; Cohen, 1968;

Kn a p p , 1978). Central

to the GLM are

realizations tha t a ll statistical analyses

are correlat ional and can yield effect sizes analogous t o r and apply weights

to measured variables to obtain scores on the composite variables tha t are

actual ly

t he

focus

o f all

these analyses (see Thom pson, 1991).

In

fact , because

all

parametric methods (e.g. , analysis of variance, regression, multivariate

analysis

of

var iance)

are

special cases

of

canonical correlation analysis,

and one kind of EFA is always im plic it ly invoked in canonical analysis

(Thompson, 1984,

pp.

11-13),

it can be

shown that

an EFA is an

impl ic i t

part of all parametric statistical tests. And as Gorsuch (2003) recently

pointed out , no factor analysis is ever completely exploratory (p . 143) .

ORGANIZATION OF THE BOOK

The book is organized to cover EFA first, followed by CFA. The

concepts und erlying EFA generally have c ounterpa rts in CFA. In fact, a

major

focus of the hook is com m unic ating the l inkages between EFA and

CFA.

In

general,

if a

statistical issue

is

important

in

EFA,

the

same issue

is

impor tan t

in

CFA.

For

example ,

jus t

as EFA

pat tern

and

s t ruc tu re

coeffi-

cients are

both

important in interpretation when factors a re correlated, the

same sets of

coefficients

are impo rtan t when factors are correlated in CFA.

Chapter 2 covers some fou nda tiona l statistica l topics, such as score

world

versus area

w orld

statistics,

and

basic m atrix algebra concepts. Chapters

3 through 9 elaborate EFA decisions and a naly tic choices. Finally , chapters

10 through

12

cover

CFA

methods.

Each of the chapters concludes with a section called M ajor Concepts."

These sections

are not

in tended

to summar ize all the key

points

of the

chapters. Instead, these sections emphasize major concerns tha t are i n s u f f i

ciently recognized in contemporary practice, or that otherwise might get

lost

in the

details

of

chapter coverage.

INTRODUCTION TO FACTOR

ANALYSIS



2

FOUNDATIONAL CONCEPTS

A simple heurist ic example of exploratory factor analysis (EFA) may

m a k e the discussion of some founda tional concepts m ore concrete. Along

the way a few elem entary ideas involvin g a type of ma them atics called

matrix

algebra

will be introdu ced. It is som etimes jo kin gly said

that

matr ix

algebra invariably induces psychosis. It is t rue

that

matr ix a lgebra in some

respects

is

very diffe rent from

th e

simple algebra taught

in

high school.

A nd

knowing the mechanics of matrix algebra thankfully is

not

necessary for

unders tanding factor analys is .

But we do

need

to

know that some basic

concepts of matrix algebra (such as when matrices can be multiplied and

how

div i s ion

is

performed wi th matr ices)

are

important , even though

the

mechanics of the process are unimportant .

HEURISTIC EX AMPLE

Table

2 .1

presents hyp othetica l data in volv ing seven students ' rat ings

of

me reg ardin g how strongly th ey ag ree (9 = mo st agree) versus disagree

(1 =

most

disagree) that I am

Handsom e, Bea utiful , U gly, Bril l iant , Smart ,

and Dumb. Because these var iables a re di rect ly measured, w i thout applying

any

weights (e.g., regression

(3

weights)

to the

scores,

these

variables

are

called synonymously measured

or observed

variables.

As we

shall soon see, scores that

are

obtained

by

applying weights

to

th e

observed scores

are

synonymously called composite, latent,

or

synthetic

variables.

A ll

analyses within

the

general l inear model (G L M ) yield scores



TABLE 2.1

Heuristic

EFA

Data

Student/

Statistic

Barbara

Deborah

Jan

Kelly

Murray

Susan

Wendy

Mean

S

Measured variable

Handsome

6

8

9

5

4

7

3

6.00

2.16

Beautiful

5

7

8

4

3

6

2

5.00

2.16

Ugly

4

2

1

5

6

3

7

4.00

2.16

Brilliant

8

7

9

9

9

7

7

8.00

1.00

Smart

6

5

7

7

7

5

5

6.00

1.00

Dumb

2

3

1

1

1

3

3

2.00

1.00

fo r eve ry pa r t i c ipan t on at least one composi te var iable , and often on several

composi te var iables .

When we conduct a fac tor analys is , we are explor ing the relationships

among measured var iables and t r y ing to de te rmine w he the r these relation-

ships can be summar ized in a smal le r number of latent constructs. Several

different statistics

can be

used

to

summar ize

the

re la t ionships among

the

variables (e.g. , Pearson correlation coefficients, Spearman's rho coeff ic ients) .

Here we use Pearson product-moment correlation coefficients. Table 2.2

presents the ma t r ix of Pearson correlation coefficien ts for all the pairwise

com bination s of these six variables.

The Table 2.1 data set involves re la t ively few pa r t i c ipan ts (n = 7) and

re la t ive ly few va riable s ( i> = 6) . These features make the data set easy to

unders tand. However , the data and the resul ts are unreal is t ic . Real EFA

problems involve more par t ic ipants

and

more variables. Factor analysis

is

usually

less re levant

when w e

have

so few

variables, because

w e

gain li t t le

by sum ma r iz ing re la t ionships among so few measured var iables in a smal ler

set of la tent var iables .

TABLE 2.2

Bivariate Correlation Matrix

Variable

Variable

Handsome

Beautiful

Ugly

Brilliant

Smart

Dumb

Handsome

1.0

1.0

-1.0

.0

.0

.0

Beautiful

1.0

1.0

-1.0

.0

.0

.0

Ugly

-1.0

-1.0

1.0

.0

.0

.0

Brilliant

.0

.0

.0

1.0

1.0

-1.0

Smart

.0

.0

.0

1.0

1.0

-1.0

Dumb

.0

.0

.0

-1.0

-1.0

1.0

10

E XPLO RA TO RY A N D C O N F I R M A T O R Y F A CT O R

ANALY S I S



Furthermore, we would rarely confront a correlation m atr ix, such as

that in Table

2 . 2 ,

in which the

correlation coefficients

are

only -1.0, +1 .0,

or 0.0. N or would we

tend

to see relationships fo r which th e latent constructs

are so

obvious

from

a

s imple examinat ion

of the

correlation matrix even

without performing any statistical analyses.

An examination of the Table 2.2 correlations suggests that two factors

(i.e.,

latent variab les) underlie the covariations in the Table 2 .1 data. Three

patterns

are

obvious

for the

heuristic example.

First, one

factor involves perceptions

of my

physical attractiveness.

Three

variables have r values

that are 1.0 (or

100%): Handsome,

Beautiful,

and Ugly, although scores on Ugly are scaled in the opposite direction

from

scores on Handsome and

Beautiful. Second,

another factor involv es

perceptions

of my

intellectual prowess.

Three

variables have

r

values

that

are

1.0 (or

100%): Brilliant, Smart,

and

Dumb, although scores

on

Dumb

are scaled in the opposite direction from scores on Brilliant and Smart.

Third, th e fact that all pairwise combinations of r values are 0.0 (or

0 % > )

for combinations involving one measured variable from one set (e.g., Hand-

some) and one measured variable from the other set (e.g., Brilliant) suggests

that

the two latent variables also are uncorrelated with each

other.

TWO W O R L D S OF STATISTICS

All statistics can be conceptualized as existing in or describing dynam ics

of

one of two "worlds": the

score

world or the

area

world. All statistics in

the score world are in an unsq uared me tric, as are the scores them selves.

For example, if we have d ata on how many dollars our classmates have on

their persons,

each

measured var iab le score is in the me tric, dollars. Stati stics

in the score world have implicit superscripts of 1. Example statistics in this

world are the mean, the median, the standard deviation (SD), and the

correlation

coefficient

( r ) .

If the mean is 7, the mean is 7

dollars.

If the SD is 3, the SD is 3

dollars.

The weights (e.g., regression (3 [beta] we ights , regression

b

weights)

applied

to

measu red variables

to

obtain scores

on

composite variables

are

also

in the score world (o r they couldn't be applied to the measured variables ).

The

statistics

in the

area world

are in a

squared metric.

For

example,

i f the scores are mea sured in dollars, and the v arian ce is 9, the variance

(SD

2

) is 9 squared dollars (not 9 dollars). M any of the statistics in this world

have explicit su perscripts of 2 (e.g., r

2

, R

2

) or names

that

communicate

that

these statistics

are

squared valu es (e.g.,

the sum of

squares,

the

mean square).

Other

statistics

in

this world

do not

have either superscripts

of 2 or

names

clearly

indicating

that

they

are in the

score world (e.g.,

th e

variance,

th e

covariance).

FOUNDATIONAL CONCEPTS 11



TABLE 2.3

Unsquared and Squared Factor Weights

Factor Squared values

Measured variable

Handsome

Beautiful

Ugly

Brilliant

Smart

Dumb

1.0

1.0

-1.0

.0

.0

.0

II

.0

.0

.0

1.0

1.0

-1.0

100

100

100

0

0

0

II

0

0

0

100

100

100

Any

statistics

we

compute

as

ratios

of one

area world stat ist ic

to

another

area

world stat ist ic

are

also themselves

in the

atea world .

For

example,

both

R

2

and the analysis of variance (ANOVA) effect size T |

2

(eta-squared ) are

computed

as:

sum of

squares E X P L A I N E D

su m

of squaresjoTA L

Rel iab i l i ty

coefficients (e.g., Cronbach's a) are also in the area world,

because they

are

computed

as

ratios

of

reliable

variance or

rel iable

sum of

squares divided by total score v arian ce or the total sum of squares. R eliab i l i ty

statistics

are in the

area world ,

even

though

they

can be

neg ative (e.g.,

-.3), and indeed even though they can be less than -1.0 (e.g., -7.0;

Thomp-

son, 2003).

Table

2.3

presents

the

unsquared factor weights (analogous

to

regres-

sion

(3

weigh t s )

for the

Table

2 .2

correlat ion mat r ix (and

the

Table

2 .1

data used

to

create

the correlation

ma t r i x ) .

Table 2.3

also presents

the

squared valu es associated wi th this analysis .

To

help readers keep straight

which

stat ist ics are in the area versus the score world, in many cases I

present these statistics

as

percen tages (e.g.,

r = 1.0 =

100%).

The Table 2.3 results from the factor analysis of the Table 2.2 correla-

tion

mat r ix confi rm

our

expec tat ions.

The

Table

2.3

mat r ix (and

the

Table

2.1 data

and the

Table

2.2 correlation

matrix) suggests that

two

uncorrelated

factors

(i .e.,

my

perceived physical at t ract iveness

and my

perceived intel lec-

tua l

abi l i t ies) underl ie the six measured variables.

S OME BASIC M ATRIX ALGEBRA CONCEPTS

As noted

previously ,

it is not

important

to

know

how to do

mat r ix

algebra (which

is

for tuna te, because

as

noted previously this knowledge

can

12

EXPLORATORY

A N D

C O N F I R MA T O R Y F A C TO R ANALYSIS



cause severe psychological distress). Readers who desire an understanding

of the mechanics of matrix algebra can consult any number of hooks (e.g.,

Cooley & Lohnes, 1971, pp. 15-20; Tatsuoka, 1971, chap. 2). But it is

important

to know some matrix algebra

concepts,

so that one can

have

some understanding

of

what

is

being done

in

certain analyses

Different books

use di f ferent

notations

to

indicate

that a

matrix

is

being discussed. In this book, capital letters in bold are used to represent

a matrix (e.g.,

X, R).

There

are six

matrix algebra concepts that must

be

mastered.

These

are discussed below.

Rank

In this book we

p r imar i l y

consider matrices

that

are two-dimensional

(i.e., number of rows > 1, and number of columns > 1). Matrix rank character-

izes how

many rows

and

columns

a

particular matrix has.

When we specify

rank, we always report the number of rows first, and the number of columns

second, using subscripts.

For

example,

the

measured variable scores reported

in Table 2.1 could be represented as: X

7x6

. And the bivariate correlation

mat r ix

presented in Table 2 . 2 i s

R(

1X

6.

Transpose

For

some purposes

we can

reorganize

the

data

in a

matrix such that

the

columns

in the

original matrix become

the

rows

in a

second matrix,

and the

rows

in the

original matrix become

the

columns

in the

second

m a t r i x . Such a new matrix is called a transpose. We designate a transpose

by putting a single quotation or tick mark on the matrix. For example, if

w e have a matrix, Y ^

X

2 , the transpose of this m a t r i x is ¥ 2 x 3 ' -

Consider

the following matrix, ¥3

x

i\

1

4

2 5

3

6.

The

transpose

of

this matrix,

¥2

x

/, is:

1 2 3

4 5 6 .

Conformability

In regular (nonmatrix) algebra, any two numbers can be multiplied

t imes each other

(e.g., 4x9 = 36; 2 x 8 = 16). In matrix algebra, however,

matrices can be multiplied times each other only if a certain condition is

met: The number of columns in the

left

of two matrices being mu l t ip l i ed

FOUNDATIONAL

CONCEPTS

13



mus t equa l

t he

num b e r

of

rows

in the

right matrix being multiplied. Matrices

me eting this condit ion (arid eligible for m ultip licatio n) are said to be con-

formable.

F or

example ,

the Table 2 .1

m a t r i x ,

X y

X

5 can be

pos tmul t ipl ied

by the Table 2.2

correlation

matrix, R 6

X

6 , because 6 (columns in X) equals

6

( the number

of

rows

in

R .

When w e

m ul t iply matr ices ,

th e

rank

of the

result

o r

product matrices

wil l equal the num ber of rows in the lef tmost matr ix , and the num ber of

columns

in the

produc t ma t r ix will equa l

th e

num b e r

of

columns

in the

r ightmost m atr ix . For example , the matr ices Zy

x 6

and W

6

x

2

are conform able.

The product of these two matrices would have a rank of 7 by 2:

Z y

x 6

W

6 x 2

= 0

7 x 2

.

Symmetric Matrix

Some "square" matr ices (number of rows equal number of columns)

are symmetric, meaning

that

every

i,j

e lement

of the

matr ix equals

th e

corresponding j,i e lement of the same matr ix. For example , in the Table

2 .2

corre la t ion matr ix,

the

3 1 e l ement

in R

equ als -1.0 eq uals

the

1 3

e lement

in R.

Correlation m atr ices

a re

a lways sym metr ic .

A

related property

of symmetric matrices, which are a lways conformable to themselves, is that

a

sym metr ic m atr ix a lways equals

i ts own

transpose,

by

definit ion (i .e. ,

Rfi x 6

=

R-6 x 6 •

Identity Matrix

In regular algebra,

a ny

number t imes

1

equals

th e

o r ig ina l num ber

that

w as m ul t ip l i ed times

1

(e.g.,

9x1 =9 ; 15x 1 =

15) .

In

matrix algebra

there is a

m a t r i x , ca l led

the

identity

matr ix

( I) , which when mul t ipl ied t imes

any m atrix , equ als the init ial matrix . For exam ple, let 's say we have ¥ 3 x 2 :

1

4

2 5

3

6.

For this example, to be conformable the ident i ty matr ix must

have

two rows. The ide nti ty ma trix is always sym me tric. Th e identi ty m atrix

always has

ones

on the

"diagonal" running

from the

upper

left

corner down

to the lower right corner, and zeroes in all

off-diagonal

locations. So for

this

example I

2 x 2

would be:

1 0

0 1.

For

this example

¥ 3

x

71

2 x 2

(i.e.,

Y

pos tmul t ipl ied

by I)

yields

a

product

matrix that contains exactly

the

same numbers that

are in

Y }

x 2

itself.

T he

J 4

EXPLORATORY

A N D C O N F I R M A T O R Y F A CT O R ANALYSIS



identity matrix

is a

critical matrix algebra concept, because

as we see

next

I is invoked in order to be able to do matrix division.

Matrix Inverse

For symmetric matrices,

one

could state

a

matrix equation such

as

R ( j

x

6 R 6 x 6

=

l e x 6 . The matrix, R < s

x(

," ' ( i .e . , a matrix with a-1 superscript),

is called an inverse. Not every symmetric matrix has an inverse. If a matrix

eligible

for

inversion (e.g.,

a

symmetric matrix) does

not

have

an

inverse

fo r a

given data set,

the

matrix

is

said

to be

"ill conditioned."

Some symmetric matrices, such as some correlation matrices, do not

have an inverse, because the numbers in the matrix cannot yield an R"

1

such that

R R' = I. If this

happens

in factor analysis,

this

may be due to

sample

size

being too small.

Some factor analyses

require

that

an

inverse

be

computed

in

order

to

compute

the

factor solution.

When

using factor analyses requiring matrix

inversion, if a matr ix is ill conditioned, the solution simply cannot be

computed using

the

particular technique requiring

the

inversion. Another

technique must be selected, or sample

size

must be increased.

In regular algebra, division is straightforward. For example, to compute

the

F C A L C U L A T E D

m

ANOVA

or in

multiple regression,

we

compute

F =

[ S u m of Squares E X P L A I N E D / ^ / E X P L A I N E D ] / (Sum of Squares U N E X P L A I N E D /

^ /U N E X P L A I N E D -

But in

matrix algebra

the way

that we do

division

is by

postmultiplying

the

numerator matrix times the inverse of the matrix with which we wish t o divide.

For example, th e division of 62

x

2 by W 2

x

2 is accomplished using the form ula :

62

x

2 W 2

x

2

=

^2 x 2 . (This is actually on e basis fo r obtaining F C A L C U L A T E D

fo r a multivariate

ana lys i s

of variance [MANOVA] and is directly analogous

to the

univariate

ANOVA

computation, except

for the use of

matrix

algebra.)

Division is used a lot in univariate statistics. Division is also used a

lot in multivariate statistics, including factor analysis. Clearly, access to an

inverse is usually critical for most factor analytic applications.

M O R E

HEURISTIC

RESULTS

Throughout the GLM, weights are applied to the scores on the mea-

sured variables

to

obtain scores

on

composite variables.

In the

GLM, weights

are

invoked (a) to compute scores on the

latent

variables or (b) to interpret

what

the

composite variables represent. Logically, because

the

weights

are

used

to

obtain

the

scores

on the

composite variable,

the

weights must

be

FOUNDAT1ONAL

CONCEPTS

15



of

some assistance

in

understanding

the

nature

of the

latent variables

(i.e.,

help in interpreting what latent variables represent).

For

example,

in

mul t ip le regression these weights

are

often called

j3

(beta)

weights. A set of weights for the predictors in regression is called the

regression

equation.

These

P weights in regression are also sometimes called

standardized regression

c o e f f i c i e n t s because they are applied to the measured

predictor variables

in

their t-score form (i.e.,

th e

measured variables have

been

transformed so

that

th eir means equ al zero and the ir standard deviations

and variance s equal one). It is actually a misn om er to call the weigh ts

"standardized,"

because

th e

weights

are a

constant

for a

given data set. Thus,

th e

weight

for the

f i r s t predictor

itself

cannot

be

standardized.

If p = .5, this weight is applied to the f i r s t predictor z scores of

everybody in the sample. We

cannot

apply the formula ^ = ( X j - X ) / SD to

a given weight. If everybody is assigned a weight of Pi = .5, SDpi = 0,

then

th e division is m athem at ical ly impermiss ible . To be technically correct, we

should call such weights "weights applied

to the

standardized measured

variables" rather

than

"standardized weights."

Perhaps

we are

performing

a

regression using

tw o

predictor variables.

Steve has z\ = -1 and ^2 = .5. The form of the regression equation is:

For a given data set , Pi might equal 2.0, and P? might equal — 1 . 2 .

Note

that ,

as is the case thr oug hou t the G LM , weights are not necessarily

correlation coefficients (see Courville & Thompson, 2001).

Thus,

as in this

example,

it is

possible that none

of the

weights will

lie in the

range between

-1 and +1. For Steve, Y

S T

E V E = 2 .0 (0 .7 ) + -1.2(0.5) = 1.4 + -0.6 = 0.8.

In

regression,

these Y

scores

are the

composite, latent,

or

synthetic variables.

Pattern

Coefficients

In

factor analysis,

pattern

c o e f f i c i e n t s

are the

weights applied

to the

measured variables

to

obtain scores

on the

factor analysis latent variables

(called

factor

scores) . As n oted , these w eights are analog ous to the P weights

in

multiple regression. These weights

are

also analogous

to the

standardized

discriminant function coefficients in descriptive discriminant analysis (Hu-

berty,

1994)

and the

stan dardized canonical function

coefficients in

canoni-

cal correlation analysis (Thompson, 1984, 2 00 0a).

T he

factor pattern coefficients (P y

x

F )

a re

computed part ly

to

reexpress

th e variance represented in the correlation matrix that is being analyzed,

and from wh ich the factors (i.e., the sets of pattern coefficients) are extracted.

T he

factors

are

extracted

so

that

the first

factor

can

reproduce

th e

most

variance

in the

matr ix being analyzed,

th e

second factor reproduces

th e

16 E XPLO RA TO RY A N D C O N F I R M A T O R Y

FACTOR

A N A L Y S I S



second most variance, and so forth. The ability of one or more factors to

reproduce the ma trix being analyzed is qu antified by the reproduced m atrix

of associations, such as the reproduced inter var iable correlation m atr ix

( R y x V

+

) - This

is the "how f u l l is the glass" perspective.

The

ability

of the

factors

to

reproduce

th e

matrix being analyzed

can

also be qu antified by com puting the ma trix

that

is left after a given number

of

facto rs have been extracted . This is called the resid ual m atr ix of associa-

tions, such as the residu al inter var iab le correlation ma trix ( R

V x V

~ ) -

This

is

the

converse, "how empty

is the

glass" perspective.

If the factor pat tern coefficients perfectly reproduced the matrix of

associations (here

an

intervariable Pearson correlation coefficient m atri x) ,

then R y

x

v ~ would consist entirely

of

zeroes, indicating that

there is no

information or variance

left

in this m atrix. An d if the factor pat tern

coeffi-

cients perfectly reproduced th e matr ix of associations,

then

th e entries in

RV x v

+

would exactly match the entries in Ry

x

v.

The reproduced intervariable correlation matrix (Ryxv

+

) is com-

puted as:

Note

that

th e

pattern coefficient matrix

and its

transpose

are

conformable,

and the product matrix has the rank

v

by

v

(the same rank as the original

intervariable correlation matrix).

Possible Numbers of

Factors

In

regression there

is

only

a

single "equation" (set

o f

weights,

[3

weights)

in a given analysis. In fac tor analy sis the sets of weights (i.e., patte rn

coefficients) are called /actors rather than equations, mainly just to be incon-

sistent and to confuse th e graduate students. The number of factors worth

keeping can range between one and the number of variables.

Let's

go

back

to our

example

of

ratings

by

seven participants

of me as

regards six measured variables. If every entry in the interv ariab le correlation

matrix was either a +1 or a -1, the r

2

between every pair of measured

variables

would

be

100%. This would imply that

a

single factor underlay

th e ratings. W e would extract one factor (Pg

x

|) consisting exclusively of

negative

or

plus ones.

A nd

this single factor w ould perfectly reproduce

th e

original

R

6 x

6 mat r ix .

Technically,

there

would be five addit ional factors, each consisting

exclusively of zeroes, meaning that they each contain and reproduce no

information. But we would never bother with such factors. Indeed, we

usually do not bother even w ith factors that reproduce some, but only a

FOUNDAT10NAL CONCEPTS I 7



l i t t le information,

if we

deem

th e

amount

of

informat ion

or

variabil i ty

reproduced to not be noteworthy.

At the other extreme,

if

somehow all the ra tings were perfectly uncorre-

lated, then e very off-diagonal element

of

R

6 x 6

would

be a

zero (and R ^x

5

=

I

6x6

).

Now,

no two

variables could

be

combined

to

create

a

factor (i.e.,

each

measured variable would define

its own

factor).

There

would

be six

factors.

Each factor would have one +1 value, and the rema ining five entries

would be

zeroes.

A t this second extrem e (per fectly uncorrelated measured variab les),

RV x v

+

would exactly match th e entries in Ry

x

v- In fact , whenever we

extract all possible factors (i.e., the number of factors equals the number

of measured var iab les) , the pattern coefficients will perfectly reproduce the

original matrix of associations being analyzed.

Factor

Structure

Coefficients

All GLM analyses are correlational and yield weights (e.g., factor

pattern coefficie nts) that are applied to m easured variables to obtain scores

on composite va riables.

These

weights are sometim es correlation coefficients,

and

sometimes they are not (cf. Courville & Thompson, 2001;

Thompson,

1984, pp . 2 2 -23 ) .

Researchers can also comp ute biva riate correlation

coefficients

between

measured variables an d their composite variables (e.g., regression Y scores,

factor

scores, discriminant function scores). These correlations across various

GLM analyses are always called structure coefficients.

Throughout the G LM consult ing the struc ture coefficients is almost al-

ways essential to correctly interpreting results. For example, in regression

Courvil le a nd

Thompson

(2001) documented the misinterpretations that ca n

occur when only p weights a re consulted in interpretation (also see Cooley

&Lohnes , 1971, p. 55; D unla p& La ndis , 1998; Thompson &B orrello , 1985).

For

descriptive disc rimina nt analysis, Hu berty (1994) noted that "con-

struct definit ion and struc ture dime nsion [and not hit rates] constitute th e

focus [italics added]

of a

descriptive discriminant analysis"

(p .

206).

The

importance

of

structure

coefficients

in

canonical correlation analysis

is

widely

recognized (cf. Cohen

&

Cohen, 1983,

p.

456; Levine, 1977,

p.

20) .

For

exam ple, M eredith (1964) noted, "If the variables within each set are

moderately intercorrelated th e p ossibil i ty of interpreting th e canonical vari-

ates by inspection of the appropriate regression weights [func tion coefficients]

is

practically nil"

(p .

55) .

Structure coefficients

are

equal ly important

in

factor analysis (cf. Gra-

ham, Guthrie , & Thompson , 2003; Thompson , 1997). Thus, Gorsuch (1983)

emphasized

that a

basic

[italics added]

matrix

fo r

interpreting

th e

factors

is th e

factor s t ructure"

(p .

2 0 7 ) .

18

EXPLORATORY A N D

C O N F I R M A T O R Y F A CT O R ANALYSIS



In the GLM, sometimes the weights are correlation coefficients. F or

example , in regression, if the predictor variables are perfectly uncorrelated,

the P weight for a given predictor equals that variable 's correlation with

the outcome var iab le (Yi) .

In

factor analysis,

the

s t ruc ture

coefficients ( S v

x

p )

can

be

computed

using the equation

P v x F

Rp 'xF

S vxFi

where R

F x F

is the

interfactor corre la t ion mat r ix (not

the v by v

intervar iab le

correlation matrix). When

the

factors

are

perfect ly uncorre la ted,

by defini-

tion R p x F

=

I p x F -

an

J therefore

P y x F

=

S V X F -

In other words, whenever the factors are perfectly uncorrelated the

pattern coefficients are also structure coefficients (and therefore must fall

with in

th e

range — 1

to

+ 1 ) ,

and in

this case

are

more appropriately termed

pattern/structure coefficients.

And wh en factors are first extrac ted, the factors

are

always

perfectly uncorre la ted.

"Loadings"

It is not uncom mon in factor analytic reports to see authors referring

to coefficients as "loadings." The problem is that this term is inherently

ambiguous. In some reports authors

refer

to pattern

coefficients

as "loadings."

In

other

reports authors

refer

to structure coefficients as "loadings." In some

reports

it is

impossible

to

discern wh at coefficients

th e

authors

are

descr ibing.

And in

some reports

th e

pa t te rn

and the

s tructure coefficients

are not

equal but authors

refer

to both sets of coefficients as "loadings." These

ambigui t ies

cause confusion tha t impedes clear

scientific

d ialogue. Therefore,

some journals (see Thompson

&

D aniel, 1996) form ally discourage

the use

of this vag ue colloq uialism . For the same reasons, the term loading is not

used

here.

Communality Coefficients

For the Table 2 .3 factors, because the factors are perf ectly u ncorrelated ,

the unsquared coefficients reproduced in Table 2 .4 are pat tern/structu re

coefficients.

Recall that Pearson correlation coefficients (a) characterize

the linear relationship between two in terv ally scaled varia bles but (b) the

coefficients

themselves are not inte rva lly scaled.

That

is, an r of 1.0 is no t

twice as

large

as an r of

0.5.

T o

m a ke

th e

correlations intervally scaled,

so

that

we can say how

much larger one r is versus another r, we m us t first square the rs before w e

compare them. Thus, the r of 1.0 is four times larger than the r of 0.5,

FOUNDATIONAL

CONCEPTS

19



TABLE

2.4

Pattern/Structure Communality Coefficients

and

Eigenvalues

Factor

Measured variable

Handsome

Beautiful

Ugly

Brilliant

Smart

Dumb

Sum of

squared column values

1.0

1.0

-1.0

.0

.0

.0

3.0

II

.0

.0

.0

1.0

1.0

-1.0

3.0

1

00

100

100

100

100

1 00

Note

The eigenvalues (3.0 and 3.0) are in a squared, area-world metric, even though they are not pre-

sented as percentages.

because

r

2

=

l.O

2

=

100%

is fou r

times larger than

r" =

0.5""

=

25%. Because

the Table 2.4 structure coefficients are correlation coefficients, they too

must

be squared before they are compared.

One set of manipulations involves taking the squared structure coeffi-

cients for the uncorrelated factors (the formula is d i f fe rent for correlated

factors) and

adding

the

squared values along

the

rows across

the

columns

(e.g., 100%

+ 0% =

100%).

The

resulting coefficients, called

communality

coefficients h

2

) , are area-world statistics. Because these factors are uncorre-

lated, the variance each measured variable shares with a factor is unique

or nonoverlapping, and thus the

h

value for a measured variable indicates

ho w much of th e variance in a measured variable the

factors

as a se t can reproduce .

If a measured variable had a communality coefficient close to 0%, this

would

mean

that this variable is not being represented within the factors.

If the

researcher desires

the

variable

to be

represented

in the

factors,

it may

be

necessary

to

extract additional factors.

An

equally valuable perspective takes

a

converse view

of the

interpreta-

tion of the communality coefficients. The h value for a measured variable

reflects

ho w

much

of the

variance

of a

given measured

variable was

useful

in

delineating the

factors

a s a

set.

There

is yet another

perspective from which communality coefficients

can be viewed. The communality coefficient for a variable is a "lower bound"

estimate of the re l iabi l i ty of the scores on the variable. This means that if

a variable has an h

2

of 50%, we believe that the reliability of the scores on

the variable is no lower than .5 (and, of course, may be higher).

In

some unusual cases communality coefficients greater than 100%

occur. These are called Heywood cases (Heywood, 1931). Such solutions

are

statistically "inadmissible,"

and

they indicate fundamental problems

s imi la r

to

estimating variances

to be

negative.

20

E XPLO RA TO RY A N D C O N F I R M A T O R Y

FACTOR

ANALYSIS



Table 2 .4

presents

th e

communal i ty coeff ic ients

fo r each of the six

measured

variables in the heurist ic example. Each of these is 100%. This

reflects th e fact that for the present example, if we reexpress th e data

presented in Table 2.1 in the form of scores on the two factors presented

in

Table

2.3,

we

lose

no

informat ion whatsoever

Eigenvalues

For the Table 2 .3

factors,

if we add the

squared s t ruc ture coeff ic ients

down

th e

columns across

th e

rows,

we get a

very important

set of

squared,

area-world stat is t ics called e igenva lues (X ). As is the case e lsewhere in statis-

tics, important concepts are given multiple names, to confuse everybody;

natural ly it is the best use of effort to be confusin g to focus such atten tion

primari ly on the most important concepts, by giving them the most synony-

mous names . Another synony m ous term for eigenvalues is characteristic roots.

T he eigenvalues fo r this s imple heurist ic example are 3.0 and 3.0, as

reported in Table 2 .4 . (N orm al ly e igenvalues descend in valu e , but for this

heurist ic example

the first two

eigenvalues were exactly equal.) Eigenvalues

are always indices of the amount of information represented in some multi-

variate result . In some m ul t iva r ia te analyses e igenvalues are mul t iva r i a t e

squared correlation coefficients . Clearly, 'given that

the first two

e igenvalues

associated with

th e

factor an alysis were

both

greater than 1.0,

in EFA

eigenvalues

are

not

squared correlation coefficients .

T he fo l lowing

four

s ta tements apply to e igenvalues in an EFA:

1. The n u m b e r of e igenvalues equals th e n u m b e r of measured

variables being analyzed.

2. The sum of the

e igenvalues equals

th e

n u m b e r

of

m easured

variables.

3. An

e igenvalue divided

by the

n u m b e r

of

measured var iables

indicates th e proport ion of informat ion in the m a t r ix of associ-

ations being analyzed that

a

given factor reproduces.

4. The sum of the

e igenvalues

for the

ext racted factors divid ed

by th e num b e r of measured variables indicates th e proportion

of the inform at ion in the m atr ix being analyzed that the factors

as

a set

reproduce.

For th e current example th e number of measured variables is six. The refore,

there are six eigenvalues associated with the Table 2 .2 corre la t ion m atr ix.

Because the sum ot the e igenv alues for the cu rren t exam ple is 6.0, give n

that

the first two

e igenvalues

are 3.0 and

3.0,

the

remaining e igenvalues

mus t be 0.0, 0.0, 0.0, and 0.0.

Because 3.0 / 6 equals 0.5, the first e igenvalue indicates

that

Factor I

reproduces

0 .5 (or

5 0 % )

of the

informat ion conta ined

in the

Table

2 .2

FOUNDATJONAL CONCEPTS 21



HANDSOME

BEAUTIFUL

UGLY

BRILLIANT

SMART

DUMB

DSOME

BEAUTIFUL

UGLY

BRILLIANT SMART

DUMB

0

0

0

0

0

0

0

0

0

0

0

0 0

0 0 0

This result reflects the fact

that

fo r

these

heurist ic data the two factors

alone reproduce

all the

information contained within

the

original

Tahle

2 .2

correlation matr ix.

SAMPLE SIZE

CONSIDERATIONS

IN

EXPLORATORY FACTOR A N A L Y S IS

Sample size

affects

the precision of all statistical estimates, inc lud ing

those

made in EFA. V arious researchers have proposed rules of t hum h fo r

sample size m i n i m u m s that

are a

funct ion

of the

ra t io

of the

num b e r

of

people

to the

num b e r

of

measured var iables .

T he

recommended ratios often

are wi th in th e range of 10 to 20 people per measured variable. Gorsuch

(1983) suggested

that

"an absolute minimum ratio is five individuals to

every variab le, but not less t h an 100 in div idu als for any an alys is" (p. 332) .

However, some

Monte

Carlo

s imulat ion research (Gua dagnol i

&

Ve l i -

cer, 1988) suggests tha t th e most cri t ical issue is how saturated th e factors

are

by the


They

suggested

that

replicable factors tend

to be

es t imated

if :

1. factors

are

each defined

by

four

or

more measured variables

with structure coefficients each greater than |.6|, regardless o f

sample size; or

2 .

factor s are each defined with 10 or more s tru ctu re coefficients

each around |.4|,

if

sample size

is

greater

than

150;

or

3.

sample

size

is at

least 300.

MacCal lum, Widaman, Zhang,

and

Hong (1999) found that sample

sizes

as low as 60 accu rately reproduced population pattern coefficients if comm u-

nalities were all .60 or greater. If h values were arou nd .50, sample

sizes

of

100 to 200 were required.

O f

course, when

it comes to

m athe ma t ical ly complex analyses such

as EFA, m ore

is

always better. This pr inciple

is

par t icular ly re levant , because

one may not know prior to conduc t ing the analysis what the s a tura t ion

pat tern

of the

factors will

be, and so

rules revolving aroun d saturation

2 4


A N D


FACTOR

A N A LYSIS



3

EXPLORATORY FACTOR ANALYSIS

DECISION SEQUENCE

Exploratory

factor analysis (EFA) in actuality involves a linear se-

quence of decisions each involving a menu of several available choices.

Given that there are five major decisions, with many choices at each decision

point,

the

number

of different analysis

combinations that

are available to

the researcher is quite large. These five decisions, discussed in more detail

in the remainder of the present chapter, address the following questions:

1. Which matrix of association coefficients should be analyzed ?

2. How

many factors should

be

extracted?

3.

Which method should be used to extract the

factors?

4 -

H ow

should

th e

factors

be

rotated?

5. How

should factor scores

be

computed

if

factor scores

are

of

interest?

In EFA, using more than one set of analytic choices is usually best practice.

A s Gorsuch (1983)

wisely

suggested, Factor th e data by several

different

analytic procedures and hold sacred only those factors that appear across

all the

procedures used

(p.

330).

This chapter presents the EFA decision sequence. Not

every

choice

at each decision point is described. But the most common selections are

presented.

And in

addition

to

covering

the

most commonly encountered

choices, th e presentation is

sufficiently

broad to give th e reader a

flavor

of

27



TABLE 3.1

SPSS Variable Names and Variable Labels for the Appendix A Data

Variable SP SS variable label

PER1 I

Willingness

to

help users'

PER2

'1

Giving users

individual

attention'

PER 3 '1 Employees who deal with users in a caring fashion'

PER4

'1

Employees

who are

consistently courteous'

PER5 '2 A haven fo r quiet an d solitude'

PER 6 '2 A meditative place'

PER7 '2 A

contemplative environment'

PER8

'2

Space that facilitates quiet study'

PER9 '3 Com prehensive print collections'

PER10 '3 Complete runs of journal titles'

PER11

'3

Interdisciplinary library needs being addressed'

PER12

'3

Timely document delivery/interlibrary loan'

th e

complexi ty

of the choices.

Exploratory factor analysis ,

to pu t it

mi ld ly ,

is

not any one

s ingle analys is—EFA encompasses

a

broad

family of choices.

The sm a l l , hut real , data set presented in A p p e n d i x A wi l l he used to

make this discussion

concrete. The

da ta were randomly sampled

from the

L i h Q U A L + ™

s tu dy

of

user perceptions

of

serv ice qu al i ty

at

academic l ibrar-

ie s

in the

Uni t ed

States and Canada

(cf.

Cook &. Thompson,

2001;

Thomp-

son,

Cook, &

Heath, 2001; Thompson,

Cook, & Thompson,

2002) . Appen-

d ix

A presents the ra t ings provided by 100 g ra d u a t e s tu den t s and 100

faculty

from

one LibQUAL+ s t u dy . Table 3.1 presents the 12 measured var iables

sampled from

the

data set.

MATRIX

OF A S S OC IATION COEFFICIENTS

The scores on measured var iables are not the actual basis for E F A .

Ins tead , these da t a are used to compute some matr ix of biv ariate associat ions

am ong the m easured v ariables. I t is the m atrix of associat ions that is analyzed

within EFA. Indeed, g iven

the

m a t r ix

o f

associations (e.g.,

R H

X | I

) for a

data set ,

all the

steps

of the

factor analysis (except

for the

c om pu t a t ion

of

factor

scores) can be

accomplished

even

without

access to

original data

(e.g.,

X i o c x i i ) .

Because

th e

factors

are

ext rac ted

from

som e m a t r ix

of

associat ions,

the factors themselves are sensit ive only to

those

dynamics captured in a

given s ta t is t ic m easur ing b ivar ia te re lat ionships .

If the

chosen

stat ist ic only

evalua tes

the

orderings created

by the

m easured var iables , ignor ing d is tances ,

then the

factors

wi l l

also

be

exc lus ively order-dr iven.

If the

statistic chosen

to characterize associat ion is in f l u enc ed b y

both

re la t ionships and score

variabili t ies,

then

the

factors

in

turn wil l also

be a

fu nc t ion

of all

these

28 E XPL OR AT OR Y AND

C O N F I R M A T O R Y F A CT O R ANALYSIS



dynamics . And for the

sam e data set,

different

factors

m ay

emerge depending

on the selection of the matr ix of associations to he analyzed.

The

Pearson produ ct-m om ent hiv aria te correlation m atrix

is the ma-

trix

of associations most

commonly used

in

EFA.

In fact, in

most statist ical

packages

th e

default

(used unless

th e

user overrides

th e

default choice)

matr ix

of associations used in EFA is the Pearson correlation matrix.

But

there

ar e

quite

a

n u m b e r

of

other available choices.

Different

statistics

that characterize relationships are sensi tive to different aspects of

the data. Different relationship s tatistics also presu m e that different levels

of

scale underlie

th e

data.

For example, th e Pearson r requires that da ta mus t he interv ally scaled.

Spearman's rho,

on the other hand,

presumes only that data

are at

least

ordinally scaled.

But if

data

are

intervally scaled,

the

researcher could

use

either Pearson or Spearman

coefficients

as the basis of the ana lysis. In fact,

Spearman's

rho is

Pearson's

r

between

tw o

variables

once

interval data have

been converted into ranks with

no

ties.

For example, presume a data set with interval scores of three people

on two variab les:

Part ic ipant X Y

Barbara 1 1

Deborah

2 2

Jan

3 96

The

Pearson correlation coefficient (r

x

y )

can be

c o mp u t e d

as:

r

XY

=

COVxv

/

(S D

X

*

S D

Y

) ,

where

the

covariance

of X and Y is

c o mp u t e d

as :

C O V

X Y

=

( 2 ( X , - X ) ( Y

i

- Y ) ) / n - l ,

or

equiv alent ly (because

x, = X; — X and y, = Y, — Y) as:

COVxv

=

( I x y , ) / n

-1.

For these init ial data, we obtain:

Part ic ipant X X x Y Y y xy

Barbara 1 2.0

-1.0

1

33.0 -32.0 32.0

Deborah

2 2.0 0.0 2

33.0 -31.0

0.0

Jan

3 2.0 1.0 96 33.0 63.0 63.0

Sum

6.00

99.00

95.00

Mean 2.00

33.00

SD 1.00 54 .56.

DECISION SEQUENCE 9



So the covariance is

95.00

/ 2 = 47.50. And the Pearson r equals:

47.50

/

( 1 . 0 0 x 5 4 . 5 6 )

47.50 /

54.56

= .87.

If

these interv ally scaled data are conv erted into ranks, the only change

is that Jan's score

of 96 on Y

hecomes

3

w h e n

the

scores

are all

converted

into ranks. Now we can c o mp u t e th e Spearman's rh o using th e Pearson

r

formula:

Part icipant

Barbara

Deborah

J a n

Sum

M e a n

SD

X

1

2

3

6.00

2.00

1.00

X

2.0

2.0

2.0

X

-1.0

0.0

1.0

Y

1

2

3

6.00

2.00

1.00

Y

2.0

2.0

2.0

y

-1.0

0.0

1.0

xy

1.0

0.0

1.0

2.00

The co va rian ce is 2.00 / 2 = 1.00. A nd the Pearson r equals:

L O O /

(1.00

x

1.00)

l . O O /

1.00

=

1.00.

In

effect,

w hether computed on ordinal or on interv al data, Spearman's

rh o

addresses

the

qu est ion,

"Do the two

v ariables order

th e

people

in

exact ly

th e

same order?" Pearson's r evaluates this question also,

but

takes into

account

as

wel l

th e

distances between

th e

ordered scores. Spearman's

rh o

presumes

that

no

such informat ion

is

present

in

da ta

(or

ignores this

informa-

t ion) , which is w hy rho can be used w h e n both variables are ordin ally scaled.

These

two m atrices of association are two of the

choices

that researchers

can use in

extracting factors.

But a

third choice

is the

mat r ix

of

c ovariances

itself. In fact, th e covariance matr ix is the most common choice as the

matrix of

associations investigated

in

confirmatory factor

analysis

(CFA).

From the

formula

T

X

Y

= CO VX Y / ($ DX * S DY ), we can see

that

rX Y

will

equal

CO V

X

Y

if either (a) r (or the covar iance) is zero, or (b) the two

standard

deviat ions are reciprocals of each other (e.g., 1 and 1, .5 and 2).

In all

other cases,

r

x

y ?t /

CO V

XY

In many contexts th e covariance is used primari ly as an intermediate

calculat ion in obtaining th e correlation coefficient, and not to describe

relationship or

association.

The

covariance

is used infrequently

because,

un l i ke r, the covariance does not have a defini t ive range of possible

values.

For

example, consider

th e

following combinat ions

of r and SD

values:

30 E X P L O R A T O R Y AND C O NFI RM A TO RY FA CTO R A NA LY SI S



r

XY

* SD

X

* SD

Y

=

CO V

X Y

1.00

* 100 * 100 =

10000

0.50

* 100 * 200 = 10000

0.01

*

1000

*

1000

= 10000.

Clearly the covariance can equal a giv en valu e und er a wid e array of

circumstances

The covariance is joi ntl y influenced by three aspects of the two vari-

ables: th e correlation between the two variables, th e var iabi l i ty of the

first variable, and the variability of the second variable. Therefore, when

exploratory factors are extracted from a covariance matrix, some factors

m ay be a function of correlations, whereas others may be more a function

of score spreadoutness. Sometimes we want our factors to be sensitive to

an array of aspects of the scores. But at

other

times we may prefer all the

factors

to be sensitive only to a single aspect of our data.

N U M B E R O F FACTORS

A

critical decision

in any EFA is

determining

how

many factors

to

retain. There are numerous strategies for making this decision. In general,

several strategies should be used with the hope that different approaches

to making this decision will corroborate each other. Although empirical

evidence

can

inform

this

jud gm ent, these decisions

are in the final

analysis

matters of exactly that: judg m ent. Zw ick and V elicer (1986) presented what

is

the most authoritative empirical evaluation of how well these various

strategies work.

Statistical Significance Tests

Statistical significanc e tests

d ue to

Bartlett (1950)

can be

used

in

either

of tw o

ways.

First, the correlation matrix can be tested to evaluate whether

the matrix is an identity matrix. If this null hypothesis that the correlation

matrix

is an identity ma trix

cannot

be rejected,

then

factors

cannot

sensibly

be extracted

from

th e matr ix . Second,

after

each factor is extracted, with

factors being extracted successively so as to contain progressively less and

less inform ation or variance, the residu al correlation m atrix can b e analyze d

to evaluate whether any information remains (i .e. , whether the residual

matrix

is essentially a n identi ty ma tr ix) .

The problem with these applications is the same general problem that

applies

in all

statis tica l significance tests (see Thompson, 1996). Statistical

significance

is driven in large part by sample

size.

Because researchers gener-

ally

use EFA

only with reasonably large samples, even trivial correlations

DECISION SEQUENCE 31



Series'

r-

o>

i-

Factor

Figure 3.1. Scree plot

for

first

100

cases

and 11

variables presented

in

Appendix

A.

recognize and retain. Trivial factors, however, are analogous to scree, and

should he left hehind in the extraction process.

A scree plot graphs eigenvalue magnitudes on the vertical access, with

eigenvalue numbers const i tut ing th e horizontal axis. T he eigenvalues are

plotted as asterisks w ithin the graph, and suc cessive valu es are connec ted

by a line. Factor extraction should be stopped at the point where there is

an "elbow," or leveling of the plot. This visual approach, not invoking

statistical significance, is sometimes called a "pencil test," because a pencil

can be laid on the rightmost portion of the relevant graphic to determine

where th e elbow or flattening seems to occur.

Statistical

pack ages prov ide the scree plot on requ est. Of c ou rse, be-

cause

this strategy requires judgment, different researchers given the same

plot m ay d isagree regarding what number of factors should be retained.

Figure

3.1 presents the scree plot

associated with

the

analysis

of the

first 100 cases of data for the first 11 va riab les listed in A p p e n d ix A . These

data are perceptions of 100 grad ua te studen ts regarding

library

service quality

on the first 11

variables listed

in

Table 3.1.

This

plot suggests that three

factors shou ld be extracted. Em piric al varia tions on this "visual scree" test

use regression analyses to evaluate objectively where th e elbow in the plot

occurs (Nasser, Benson, & Wisenbaker, 2002).

nspe tion

of the Residual Correlation Matrix

A s noted previously, as more factors are extracted th e entries in the

res idual correlation matrix

(R

v x

v)

approach zeroes.

If all

possible factors

are extracted, then the resid ual m atrix

wi l l

always consist only of zeroes.

DECISION

SEQUENCE

33



Logically, another approach to de te rmin ing the n u m b e r of notewor thy

factors involves examination

of the

res idual matr ix

as

successive factors

are

extracted. Compu ter packages provide the res idual matr ix on request . And

some packages count th e n u m h e r of entries in one triangle of the res idua l

correla t ion matr ix that

are

greater

than

|.05|

to

help inform this eva luation .

N o te

that

this criterion has nothing to do wi th p < .05 and that the cutoff

va lue is somew hat arbi trary. Researchers m ay invoke other cr i ter ia , and

m ight elect to use larger cutoffs when sample sizes are sm all and the standard

errors of the


a re

larger.

Parallel

nalysis

Horn (1965) proposed

a

strategy called parallel analysis that appears

to be among the best methods for deciding how many factors to extract or

reta in (Zwick

&

Velicer,

1986).

The

strategy involves taking

the

actual

scores on the m easured variab les and creating a random score m atrix of

exactly th e same rank as the actual data with scores of the same type

represented in the data set.

There

are various approaches to this s tra tegy.

The

focus

of the method is to create eigenv alues

that

take into account

th e sampling error that influences a g iven set of measured v ar iables . If scores are

random ly ordered , the related correlation m atrix wi l l approximate an id enti ty

matr i x , and the e igenva lues of that m atr ix

wi l l

f luc tua te around 1.0 as a

func-

tion

of sampling

error.

The range of fluctuations will tend to be narrower as

th e sample size is larger and the n u m b e r of measured var iables is sm aller.

If

one presumes

that

actual d ata are approximately normally dis tr ib uted ,

then the short

SPSS

syntax presented by Thompson and Daniel (1996)

m ay

be

used

to

conduc t

th e

analysis.

This

program creates randomly ordered

scores consisting of the same intege r valu es (e.g., 1, 2, 3, 4, or 5) represented

in some data sets such

as

Likert scale data.

Other approaches use regression equation s pu blish ed in prior sim ulatio n

studies to

pred ic t

th e

eigenvalues that would

be

produced

from th e

correla-

t ion m atr ices for random ly ordered d ata having a cer ta in rank. M ontanell i

and Humphreys (1976) produced one such equation that uses as inpu t only

the ac tual sample

size

and the nu m ber of measured var iables . Lautenschlager ,

Lance ,

and

F laher ty (1989) pub lished

an

al ternative formula that uses these

tw o

predictors

bu t

also uses

the

rat io

of

var iables-to-par t ic ipants

as another

predic tor var iable .

A more sophisticated approach exactly honors not only the types of

possible

values contained in the da ta set but also the exact values (and

their d escriptive statistics includ ing shape statistic s such as skewness) in

th e real data set. Table 3.2 presents the first 25 scores from A p p e n d i x A

on the

variables PERI, PER2, PER5,

and

PER6.

The data on these

variables

34 E XPL OR AT OR Y AND C O N F I R M A T O R Y FA C T O R ANALYSIS



TABLE 3 2

Variables Randomly Sorted for Parallel Analysis

Original data

ID

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

Mean

SD

PER1

8

5

6

5

5

7

8

7

1

9

9

4

6

7

9

9

8

8

9

8

9

7

7

8

8

7.08

1.93

PER2

7

7

5

5

5

7

8

7

3

9

9

3

6

7

9

9

8

5

9

7

9

7

8

9

8

7.04

1.84

PER5

3

4

5

4

5

7

6

5

1

5

7

6

7

2

7

5

7

4

6

5

9

7

5

5

6

5.32

1.75

PER6

2

5

3

4

4

8

7

3

1

7

9

5

7

3

5

5

6

8

7

6

9

6

6

3

6

5.40

2.14

Randomly sorted data

RAN1

9

8

7

5

5

5

9

9

8

8

8

7

1

8

7

9

9

7

6

4

8

7

6

8

9

7.08

1.93

RAN2

5

9

5

7

5

5

9

7

9

3

8

3

7

9

9

7

7

7

6

8

8

7

9

8

9

7.04

1.84

RAN5

3

7

5

4

4

5

5

6

2

5

1

7

7

5

7

9

6

4

7

6

7

5

6

5

5.32

1.75

RAN6

6

4

5

4

7

3

3

5

8

7

9

6

1

3

5

7

3

6

6

2

6

5

9

8

7

5.40

2.14

were independen t ly randomly ordered to create th e scores on va r iab les

RANI,

R A N 2 , RAN5, and RAN6.

N o te that the means and s tandard deviat ions on rela ted actual and

randomly ordered data match exactly (e.g., X P H R I = X R A N I

=

7.08; SDpgRj =

SD^AMi

=

1-93). Other d escriptive statistics, such as coefficients of skewness

and kur tos is ,

wou l d

also match fo r each ac tua l and randomly ordered vari-

able pair .

However, randomly ordered scores should have bivariate correlations

approaching zero with eigenvalues all f luc tua t ing around one. Table 3 .3

presents the correlation matrix for the actual data presented in Table 3.2,

and correlation

coefficients

for the same data after independent random

ordering of the scores on each var iable .

Note that

the correlation

coefficients

for

the

randomly ordered var iables

are

closer

to

zero

than the

correlations

among

the

actual scores.

DECISION

S E Q U E N C E

35



TABLE

3.3

Bivariate Correlations For Actual and Randomly Ordered Data

PER1/ PER2/ PER5/ PER6/

Variable

RAN1 RAN2 RAN5 RAN6

PER1/RAN1

.15

-.13

.41

PER2/RAN2 .87 .01

-.02

PER5/RAN5 .51 .46 -.55

PER 6/RAN6 .54 .43 .73

Note. Th e

coefficients above

t he

diagonal

are

correlations among

the

randomly ordered data.

The co-

efficients

below th e diagonal are correlations among the actual data in its original order as presented in

Table 3.2.

TABLE 3.4

Eigenvalues for Actual and Randomly Ordered Data

Number

1

2

3

4

Sum

Actual

data eigenvalue

2.776

0.828

0.276

0.119

4.000

Random

order eigenvalue

1.756

1.082

0.797

0.364

4.000

In

parallel analysis

the

eigenvalues

in

successive pairs from

the

actual

and

the randomly ordered data are compared. Factors are retained whenever

the

eigenvalue

for the

actual data

for a

given factor exceeds

the

eigenvalue

for

the

related factor

for the

randomly ordered scores.

These

eigenvalues

for

the

Table

3.2 data and the Table 3.3 correlation matrices are presented

in Table 3.4.

For this example, because 2.776 is greater than 1.756, the first factor

from

the

actual data would

be

extracted. Because 0.828

is

less than 1.082,

the second factor would not be extracted.

Brian

O'Connor

(2000)

has

written several

SPSS and SAS

syntax

files

to execute variations on

these

analyses. The programs can also be down-

loaded from http://flash.lakeheadu.ca/~boconno2/nfactors.html

FACTOR EXTRACTION METHOD

There

are

numerous statistical theories

that can be

used

to

compute

factor

pattern coefficients. Probably the most frequently used EFA extraction

method, perhaps because it is the default analysis in most statistical packages,

is called pr incipal components analysis (see Russell, 2002).

Principal components analysis assumes that the scores on measured

variables have perfect reliability.

Of

course, scores

are

never perfectly reliable

36 E X P L O R A T O R Y A N D CONFIRMATORY FACTOR

N LYSIS



(see Thompson,

2003).

Furthermore, th e analysis attem pts to reproduce th e

variance

or information in the sample data, rather than the population. Of

course, if the sample is reasonably representative of the population data,

th e sample factors should tend to match th e population factors.

Principal

components analysis, which actually

was the

analysis

d e-

scrihed in

chapter

2,

uses ones

on the

diagonal

of the

correlation

mat r ix .

This seems reasonable to the extent that scores o n a given m easured variable

should correlate perfectly with the same scores on the same variable.

However, if scores on a variable are not perfectly reliable, then cor-

relations

of

scores

on a

measured var iable with itself will

not be

perfect

(i.e., +1.0)

if

measurement error

is

taken into account. Logically,

it

might

be reasonable to alter the d iag on al entries of the c orrelation m atri x to take

measurement error

into

account.

It was

noted previously that co m m un ality coefficients

are

lower-bound

(or

conservative) estimates of score reliability. Another common factor

extraction method, principal axes factor analysis uses

the

communal i ty coeffi-

cients to replace th e ones on the diagonal of the correlation matrix.

Principal axes factor analysis often starts w ith

a

principal components

analysis.

Then the

c o m m un a l ity coefficients

from

this analysis

are

su bst i tu ted

respectively for the ones on the diagonal of the original correlation matrix.

It

is

im portant

to

emphasize that

no

off-diagonal data

are

altered

in

this

process. Then

a new set of

factors

and

their corresponding com m unali ty

coefficients are

estimated.

This

process

is

successively repeated un til

th e

communal i ty es t imates

stabilize. This process is called i terat ion. Iteration is a statistical process of

estimating a set of results, an d then

successively

tweaking results un t i l tw o

consecutive sets

of

estimates

are

su fficiently close

that the

iteration

is

deemed

to

have converged. Iteration

is

used

in

var ious ways throughout mult ivar ia te

statistics,

inc luding

in

other aspects

of

EFA, such

as

factor rotation, which

is discussed in the next section.

The principal axes communality iteration process does not always

converge. Estimates may sequentially bounce u p, bounce dow n, bounce up,

and so forth. If this happens, often the sample size is too small for the

num ber of m easured variables and the m odel being estimated.

Always carefully check whether i teration

has

converged. Some com-

puter programs print only innocuous-sounding warnings about failure to

converge, and then still

pr in t full results.

So the unwary

analyst

may not

notice

a

convergence problem. Programs also have b ui l t- in maxim um nu m-

bers of

iterations (e.g., 25) .

If a

solution does

not

converge, override

th e

default n um ber of i terations. If a solu tion does no t converge in 100 or 125

iterations, u sually the solution will not converge at any num be r of iterations.

Either

a different factor extract ion method must be selected or sample size

must

be

increased.




Principal

components

analysis

does

not

have

to be the f i r s t step of the

principal axes factor an alysis. An y reliab ility estimates that the researcher

deems reasonable can he used as initial estimates. For example, it the mea-

sured

va riab les are test scores, the relia bil ity coeffic ients reported either in

test ma nu als or in a prior "reliability generalization" (R G ) an alysis (V acha -

Haase, 1998) can be used as initial estimates placed on the diagonal of the

correlation matrix.

Another strategy for creating initial estimates of com munal i ty coeffi-

cients is available in most statistical packages. The multiple

R

2

of a given

measured variable with all the other measured variab les is used as the initia l

estimate

for that

variable's

h

2

(e.g.,

h

}

2

=

R

2

P E

R i

x

P E R 2 , P E R 5 , P E R 6 , ^ =

R

2

p E R 2 x

P E R I , P E R 5 , P E R 6 ) -

In theory, what one u ses as the ini tial com m un ality coefficients in this

iteration process

will

affect

how

many iterations must

be

performed (with

princip al components inv okin g ones on the diagonal of the correlation

matrix obviously taking

th e

most iterations)

but not

what

are the final

estimates.

If

this

is so,

given that

the

only downside

of

numerous iterations

is

wasting a little electricity, it doesn't matter too much which initial

estimation procedure

is

used.

Other factor extraction method s includ e alpha factor analysis, maxi-

m um

likelihood factor analysis, image factor analysis,

a nd

canonical factor

analysis.

Alpha factor analysis focuses on creating factors with maximum

reliability. Max imum likelihood

analysis focuses

on

c reating factors that repro-

duce

th e

correlation

or

covariance matrix

in the

population, versus

in the

sample. Image factor analysis focuses

on

creating factors

of the

latent variables

that exclude

or

m inim ize "unique factors" consisting

of

essentially only

on e

measured variable. Canonical

factor

analysis seeks to identify factors that are

maximally related to the measured variables.

FACTOR ROTATION

Factor rotation involves mov ing

th e

factor axes measuring

the

locations

of the measured variables in the factor space so

that

the nature of the

underlying constructs becomes more obvious to the researcher. Factor rota-

tion

can be

performed either graphically, guided only

by the

researcher's

visual jud gm ent, or em pirically using one of the d ozens of

different

algorithms

that have

been

developed over th e years. Analytical rotation has some

advantages over graphical rotation, because

th e

procedure

is

automated,

and

because

any two

researchers analyzing

the

same data will arrive

at the

same factor struc ture when using the same em pirical rotation method.

38

EXP LOR ATOR Y

A N D


F ACTOR

AN ALYSIS



TABLE 3.5

Correlation

and

Squared Correlation Coefficients

for 6

Variables

and

100 Participants

Variable

PER1

PER2

PER3

PER5

PER6

PER7

PER1

1.00

.85

.78

.40

.32

.48

PER2

72.3%

1.00

.72

.45

.32

.50

PER3

60.8%

51.8

1.00

.51

.40

.50

PER5

16.0%

20.3%

26.0%

1.00

.66

.80

PER6

10.2%

10.2%

16.0%

43.6%

1.00

.71

PER7

23.0%

25.0%

25.0%

64.0

50.4

1.00

Note.

Th e (unsquared)


are

presented

in the

bottom

triangle. T he squared

correla-

tion

coefficients are presented as

percentages

in the top

triangle;

values

above

43% are italicized.

Rotation

is not

possible when only

one

factor

is

extracted.

But in

vir tual ly

all cases involving two or m ore factors, rotation is u sually essential

to

interpretation.

T he first 100 scores on six LihQUAL+™ va r iab les (PERI , PER2,

PER3, PER5, PER6,

and

PER?) presented

in

Appendix

A can be

used

to

demonstrate

w hy

rotation

is so

essential

in

EFA. Table

3.5

presents

th e

bivar iate r and r

valu es associated w ith

these

data .

A n examination of the r va lues in Table 3.5 ( r e m e m b e r

that

r values

mus t

be squared before their magnitudes can be compared in an in terval

metric) suggests (a) the existence of two factors, consisting respectively of

PE RI , PE R2, and PER3, and of PER5, PER6, and PER 7, because a ll pairs

of

th e

variables

in these tw o

sets have

r

values greater

than

4 3 % ,

and (b)

the two

factors probably

are not

highly correlated, because

the r

va lues

between measured variables across the two sets are all less than 2 7 %. T he

unrotated pattern /s tructure coeff ic ients

and the

eigenvalues presented

in

Table

3.6 confirm th e expectat ion of two factors. The two factors reproduc e

84.2% (3.81 + 1 .23 = 5.05 / 6 = .842) of the var iance in the correlation

matrix involving the six measured variables.

However ,

th e

u nrotated factor pattern /s tructu re coefficients presented

in Table

3.6 do

n o t match

ou r

expectations regarding

th e

composition

of

th e factors. For example , as reported in the table, all six variables are

correlated

.70 or

more with Factor

I Yet the

F igure

3 .2

plot

of the

factor

space, based on the p attern/ struc ture coefficients reported in Table 3.6, d o

match ou r expectat ions regarding factor s tructure.

This

suggests

that

the

patterns in our unrotated results are exactly what w e expect, b u t that th e

patterns

cannot

be readily discerned

from

statistical results in the form of

unrotated results .

The idea of rotation is that th e locations of the axes in the factor

space are com pletely arbi trary. W e can move (i.e., rotate) these axes to any

DECISION S E Q U E N C E 39



TABLE 3.6

Unrotated Principal Components

Pattern/Structure Coe fficients

Unrelated factors

Squared

structure

Variable

PER1

PER2

PER3

PER5

PER6

PER7

X

I

0.81

0.81

0.82

0.80

0.70

0.83

II

-0.50

-0.46

-0.35

0.43

0.54

0.41

I

65.9%

65.6%

67.9%

63.6%

49.2%

69.3%

3.81

II

25.3%

20.9%

12.1%

18.8%

29.7%

16.4%

1.23

h

2

91 .2%

86.5%

80.0%

82.4%

78.9%

85.7%

Note. Pattern/structure coefficients

>

|.35|

are

italicized.

For the

principal components model

the

eigenval-

ue s

equal

the

sums

of the squared unrelated

pattern/structure coefficients

for a

given factor

(e.g.,

.659

+

.656 + .679 + .636 + .492 + .693 = 3.81).

O

1

0.5 -

n

1 -0.5

-0.5

1

0.5

A

Series

1

Factor

I

Figure 3.2. Table

3.6

variables presented

in

factor space before rotation.

location, and in any manner we wish. However, we may not move the

location

of any

variable

in the

factor space.

Persons first learning of rotation are often squeamish about the ethics

of this proced ure. Ce rtainly ethics have

an

important bearing

on the

exercise

of statistical choice. If w e have several mice in a d rug experiment, a nd Elwood

in the e xperim ental group is not responding w ell to the new intervention, i t

is unethical to take Elwood out for a li ttle walk a t 3:00 a.m. that results in

Elwood's "accidental" demise. That

is

unethical

scientific

conduct (and

cruelty to Elwood) . B ut factor rotation is not in any respect unethical (a s

long as we only move fac tor axes and not measu red va riable s).

Table 3.7 presents an exam ple of an em pirical rotation of the Table

3.6

pattern/s tructure

coefficient

matr ix .

Note

that (a) the communality

coefficients fo r each of the variables rem ain unaltere d by rotation and (b)

the total variance reproduced in the factors as a set (84.2% = [2.59 + 2 .46 =

5.05

/ 6 =

.842]) remains unaltered.

40

E XPL OR AT OR Y A N D C O N F I R M A T O R Y FACTOR ANALYSIS



TABLE

3.7

Rotated Principal Components Pattern/Structure Coe fficients

Variable

PER1

PER2

PER3

PER5

PER6

PER7

Trace

Rotated factors

I II

0.94 0.19

0.90 0.23

0.84 0.31

0.28 0.86

0.13 0.88

0.32 0.87

Squared structure

I II

87.5% 3.7%

81.4% 5.1%

70.1% 9.9%

7.9% 74.5%

1.8% 77.1%

10.6% 75.2%

2.59 2.46

/i

2

91 .2%

86.5%

80.0%

82.4%

78.9%

85.7%

Note. Pattern/structure coefficients

>

|.35|

are

italicized.

For the

principal components

model

th e

trace vari-

ance

reproduced by the

rotated factors equals

the

sums

of the

squared rotated pattern/structure coeffi-

cients for a

given

factor (e.g., .875 + .814 + .701 + .079 + .018 + .106 = 2.59).

However,

it is

true that rotation tends

to

redist r ibute

th e

location

of

variance within th e solut ion. In the unrotated solution the first factor

reproduces th e b u l k of the observed variance (3.81 / 6 = 63.5%). In the

rotated solution the two rotated factors reproduce 43.2% (2.59 / 6) and

41.0% (2.46 / 6) of the observed v ariance. The rotation has almost equalized

the location of the informat ion within the solution. This tends to happen

to the extent that rotations are performed in pursu i t o f "simple structure."

Thurstone (1935,

p.

156, 1947,

p.

335) proposed

the

basic concept

of

simple structure

and the

associated criteria

fo r

evaluating simple structure.

Think about what set of factor pattern coefficients would maximize our

abil i ty

to

interpret

th e

na ture

of the

latent construc ts und erlyin g scores

on

measured variables.

T he un rotated results presented in Table 3.6 w ere virtu ally impossible

to

interpret because

all of the

m easured v ariab les were very highly correlated

with

both

factors. A more interp retation -frien dly solution w ould strike a

balance between hav ing enoug h m easured variables correlated w ith

a

factor

that some measured variables reveal the underlying nature of a factor, but

not so many that th e factors cannot be d is t inguished from each other or

are so complex

that

their nature

cannot

easily he apprehended.

A s imple s t ructure would involve

from

a column perspective roughly

an equal number of measured variables being appreciably correlated with

each factor (e.g., 6/2 = 3), bu t from a row perspective m ost or all measured

variables being appreciably correlated with only one factor.

These

criteria

are met

very well

for the

rotated factor solution presented

in Table

3.7.

Empir ical (i .e., nongraphical) rotation methods perform

the

rotation

by applying an algorithm to derive a transformation matrix ( T p x p ) b y which

the unrotated pattern coefficient mat r ix ( P v x p ) is postm ul t ipl ied to yield

the rotated factors. There are dozens of formulas for deriv ing T p

x

p matrices,

DECISION SEQUENCE

41



several of w hich are availab le in various statistical packages. These c o m p u te r

programs wil l

pr in t the transformation matr ices on request. The more a

t ransformation matr ix deviates

from

be ing an identi ty matr ix, the more the

unrotated versus th e rotated factor pattern coefficients

wi l l

b e different .

Orthogonal Rotation

Figure

3.2 and Table 3.7 involve a case in which th e original 90-degree

angle between the two factors was m ain ta ined in the rotation process. This

means that the rotated factors wi l l remain orthogonal or uncorrela ted.

Factors are always orthogonal upon e xtraction, and rem ain unc orrelated if

orthogonal rotation is used.

The correlation

coefficient

ac tu ally is a fun ctio n of the cosine of the

angle between ei ther measured or composi te var iables .

When

two var iables

have been standardized to have a length or var iance of 1.0, as factors are,

the cosine of the angle b etw een tw o factors is the correlation o f the

factors

(see Gorsuch, 1983, chap. 4).

The

most common or thogonal rota t ion method,

and

indeed

the

most

common rotat ion of a n y ki nd , is the v a r imax rota t ion method developed by

Kaiser

(1958) .

This is the

default rotation

in

most statistical packages.

V a r i m a x tends to focus on m a x i m i z i n g the

differences

between the squared

pattern /s tructure coefficients on a factor ( i .e. , focuses on a column perspec-

tive ). In my experience, in about 85 % of exploratory factor analyses va r i max

will yield s imple s tructure.

Another

orthogonal rotation method

is

quar t imax

which tends

to

focus on s imple s t ruc tu re

pr imar i ly

from th e perspective of rows (i .e. , fo r

variables across

th e

factors) . Quart imax

is

most appropriate w hen

the first

factor is expected to be a large "G" or general factor that is saturate d by

disproportionately many of the variables.

E q u a m a x

is a compromise betwe en

var i max an d quar t imax methods .

Oblique Rotation

In some cases s imple s truc ture cannot be obtained us ing or thogonal

rotation. Typically this is indicated by var iables having pattern /s tructure

coefficients that are

large

in

absolute value

on two or

more factors (which

is sometimes cal led mult ivocal v s. un ivoca l ) . Such factors are difficult to

interpret. The researcher must then tu rn to oblique rota t ion in the pursu i t

of s imple s t ruc tu re .

However , as noted in chapter 2, w hen factors are rotated orthogonally,

the

pa t te rn coefficient

and the

s tructu re coeff ic ient m atr ices , which always

have the same ran k, also

wil l

contain exactly the same numbers . This occurs

because when factors are uncorrelated the interfactor correlation matrix

42 EXPLORATORY A N D C O N F I R M A T O R Y FACTOR

ANALYSIS



p )

is an identity matrix, and given that P y

x

p R F X F

=

S v x F i if R p x F

=

I p

x F

, then necessarily P

V x

F

=

S y

x

p .

But once

factors

are

rotated obliquely,

th e

factors

are

correlated,

and

P\/ x F Sy

x

F -

Correlations among the factors that deviate further

from

zero

lead

to greater

differences

between the pattern and the structure

coefficients.

A nd

once factors

are

rotated o bliqu ely,

th e

pattern coefficients

are no

longer

correlation

coefficients.

On ob lique ly rotated factors, any or even all of the

pattern coefficients can be outs ide th e range

—1

to +1.

When orthogonal rotation is performed in computer packages,

R F

X

F

will not be printed, because the interfactor correlation is by d efini t ion known

to be the identify matr ix . And because PV x F equals Sy

x

F. only one factor

matr ix

wi l l

be

printed, which

may be

somewhat ubiquitously labeled some-

thing like "varimax-rotated factor matrix." When

an

oblique rotation

is

performed, all

three matrices

(Py

x

F >

RF x

F )

and Sy

x

F)

will always

be

printed,

so

that

the

results

can be

correctly interpreted

by

consult ing

both

Py

x

p

a n d S

V x F

.

When

oblique rotation is necessary, p r om a x rotation (Hendrickson &

White, 1964) is almost always a good choice. Promax is actually conducted

as a series of rotations. The first rotatio n is an orthogonal rotation, such

as var imax . These pa t tern /s t ruc ture coefficients are then raised to some

exponential power,

k

which is sometimes called a pivot power. If an even-

numbered pivot power (e.g.,

k =

4) is used, the signs of the resulting

coefficients

are restored after the

exponential operation

is performed .

Raising

var imax pattern/s tructure coefficients

to an

exponential power

makes

all the

resulting coefficients closer

to

zero.

But the effect is

differential

across

origin al value s that

ar e

larger

in

magnitude versus smaller

in

magni-

tude. Fo r example, .9

4

is .656, whereas ,2

4

is .002. The resulting coefficients

tend to have simple structure.

N e xt , a "Procrustean" rotation is invok ed . Procru stean rotation can

also be useful as a rotation method on its own merits and not as part of

promax rotation,

as

when

a

researcher wants

to

rotate factors

to

best

fit a

theoretical structure (Thompson & Pitts, 1982), or when conducting an

EFA

meta-analysis (Thompson, 1989). Indeed , some test distrib utio ns have

been developed

fo r

s tatis tically

evaluating

the fit of

actual

and

theoret ical ly

expected structures,

or of

actual structures

from

d ifferent studies (Thomp-

son, 1992b).

Procrustean rotation derives its name from the Greek myth about an

innkee per w ith only one bed . If gu ests we re too short for the bed, he stretched

the guests on a rack. If guests were too tall for the bed , he cut off parts of

their legs.

Procrustean rotation finds the best fit of an actual matrix to some

target matrix (e.g., a hypothetically expected pattern matrix, the factor

solution for the same variables in a prior

s tudy) .

In the promax use of

DECISION SEQUENCE

43



Procrustean rotation,

th e

varim ax pattern/struc ture coefficients each raised

to the pivot power hecome th e target. T he resu lting Procrustean rotation

i s the promax solution.

The researcher can control the degree of correlation am ong th e factors

( R p

x

p )

hy

changing

th e

pivot power.

T he

higher

the

exponent,

the

greater

will be the

degree

of

correlation among

the

factors.

Another oblique rotation method

is oblimin.

Ob limin invokes

a

va lue

called delta

to

control

the

degree

ot

correlation among

th e

rotated factors.

Delta values

of

zero yield factors

that are

more highly correlated, whereas

large negative valu es produ ce factors that

are

more uncorrelated. However,

in cases where oblique solutions are required, promax is

usually

a very

good choice.

FACTOR SCORES

As

noted

in c hapter 1, sometim es factor analysis is used not to

identify

an d

interpret factors,

hut to

obtain

a

more parsimonious

set of

composite

scores (i.e., factor scores)

that are then

used

in

subsequent analyses (e.g.,

regression) instead

of the

measured variable scores. There

are

four methods

of obtaining factor scores (i.e., composite variable scores for each person

on

each

factor)

that

are briefly summarized here.

Chapter

4 includes more

detailed

comparisons of these methods.

The

most commonly used method

of

obtaining factor scores

(F^

x

y)

is called the regression

method.

As a first step, the measured variables are

each converted into z scores with means of zero and SDs of 1.0.

Then

the

following

algorithm

is

applied:

F N x F

=

Z N

x

v R V x v P V x

F -

The rightmost portion of the formula can be reexpressed as :

W v x F

R-VxV PvxF-

This is the "factor score matrix output" by

SPSS.

Second, Bartlett (1 937 ) proposed m ethods for estim ating factor scores

intended

to

minimize

the

influence

of the

un iqu e factors consisting

of

single

measured

variables not usu ally extracted in the analysis. Third, Anderson

and R ub in (1956) proposed methods sim ilar to Bartlett 's methods, b ut w ith

an added constraint that th e resu lting factor scores m ust be orthogonal.

All of

these factor score estimation procedures yield factor scores that

are themselves z scores,

if the

extraction method

is

principal components,

or

the

estimation method

is

Anderson—R ubin. This does

n ot

create

difficulties if

44

EXPLORATORY A N D CON FIRMATO RY FACTOR ANALYSIS



TABLE 3.9

z Scores and Standardized Noncentered Scores for the

Four

Measured Variables

z

scores

ID

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

Mean

SD

ZPER1

.48

-1.08

-.56

-1.08

-1.08

-.04

.48

-.04

-3.14

.99

.99

-1.59

-.56

-.04

.99

.99

.48

.48

.99

.48

.99

-.04

-.04

.48

.48

0.00

1.00

ZPER2

-.02

-.02

-1.11

-1.11

-1.11

-.02

.52

-.02

-2.20

1.07

1.07

-2.20

-.57

-.02

1.07

1.07

.52

-1.11

1.07

-.02

1.07

-.02

.52

1.07

.52

0.00

1.00

ZPER5

-1.33

-.75

-.18

-.75

-.18

.96

.39

-.18

-2.47

-.18

.96

.39

.96

-1.90

.96

-.18

.96

-.75

.39

-.18

2.10

.96

-.18

-.18

.39

0.00

1.00

ZPER6

-1.59

-.19

-1.12

-.65

-.65

1.21

.75

-1.12

-2.06

.75

1.68

-.19

.75

-1.12

-.19

-.19

.28

1.21

.75

.28

1.68

.28

.28

-1.12

.28

0.00

1.00

Standardized

CPER1

7.56

6.00

6.52

6.00

6.00

7.04

7.56

7.04

3.94

8.07

8.07

5.49

6.52

7.04

8.07

8.07

7.56

7.56

8.07

7.56

8.07

7.04

7.04

7.56

7.56

7.08

1.00

CPER2

7.02

7.02

5.93

5.93

5.93

7.02

7.56

7.02

4.84

8.11

8.11

4.84

6.47

7.02

8.11

8.11

7.56

5.93

8.11

7.02

8.11

7.02

7.56

8.11

7.56

7.04

1.00

noncentered

CPER5

3.99

4.57

5.14

4.57

5.14

6.28

5.71

5.14

2.85

5.14

6.28

5.71

6.28

3.42

6.28

5.14

6.28

4.57

5.71

5.14

7.42

6.28

5.14

5.14

5.71

5.32

1.00

CPER6

3.81

5.21

4.28

4.75

4.75

6.61

6.15

4.28

3.34

6.15

7.08

5.21

6.15

4.28

5.21

5.21

5.68

6.61

6.15

5.68

7.08

5.68

5.68

4.28

5.68

5.40

1.00

ote

Th e

means

of the

standardized noncentered

scores are the

same

as the

means

of

these

variables

in their

original

form, a s

reported

in Table

3 2

descriptives variables=cperl

to

cper6

.

compute btscrl= .55080 * cperl) + .62861 * cper2)

+

-.17846

* cperB) +

-.18121

*

cper6)

.

compute btscr2= -.11583 * cperl) +

-.22316

* cper2)

+

.60668 * cperS) + .61055 * cper6) .

print formats fscrl fscr2 btscrl btscr2 F8.3) .

Table

3.10

presents

both

th e regular

regression

factor scores and the

standardized

noncentered

factor

scores

for

these data.

Both sets of factor

scores

are

compared

here

only

fo r

heuristic purposes.

In

ac tua l research

th e

score es t imation method would be selected on the basis of the research

questions and the related preferences for

different

factor

score

properties.

EXPLORATORY

AND CONFJRMATORY FACTOR ANALYSIS



TABLE

3 10

Conventional Regression Factor Scores

F

2 5 X

2 )

and Standardized

Noncentered Factor

Scores for the EFA of the Four Measured Variables

Regression Standardized noncentered

ID

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

Mean

SO

FSCR1

.773

-.437

-.770

-1.037

-1.139

-.428

.386

.199

-2.300

1.115

.741

-2.295

-.970

.505

1.080

1.284

.368

-.522

1.013

.230

.537

-.259

.288

1.169

.470

0.000

1.000

FSCR2

-1 .824

-.442

-.483

-.485

-.138

1.334

.520

-.786

-1

.898

-.008

1.256

.797

1.230

-1 .826

.115

-.578

.582

.476

.339

.010

1.950

.763

-.052

-1

.089

.235

0.000

1.000

BTSCR1

7.170

5.960

5.627

5.360

5.258

5.969

6.783

6.596

4.097

7.512

7.138

4.102

5.427

6.903

7.477

7.681

6.765

5.875

7.410

6.627

6.934

6.138

6.685

7.566

6.867

6.397

1.000

BTSCR2

2.309

3.691

3.650

3.649

3.996

5.467

4.654

3.348

2.235

4.126

5.390

4.930

5.363

2.307

4.249

3.555

4.715

4.610

4.472

4.143

6.083

4.897

4.082

3.045

4.369

4.133

1.000

T he

standardized noncentered factor scores

are perfectly

correlated

w i t h their regular regression factor score counterparts (i.e., r^scRi x B T S C R I

=

1.0 and

f p s c R Z

x

R T S C R Z

=

1-0)i

anc

l

the

correlations across

the two

factor

scores also match across the two score types (i.e., T F S C R I x F S C R 2

=

r

B T S C R i x

R T S C R 2 *

which here

is

0.0).

For the

Tahle

3.10

results,

a

comparison

of the

means on the

s tandardized

noncentered

factor scores suggests

that the 25

participants on the

average rated their

l ibraries

higher

on

Factor

I

(6 .397)

than on

Factor

II

( 4 -133) .

M A J O R

CONCEPTS

The sequence of EFA involves num erous decisions (i.e., what associa-

tion

matrix

to

analyze,

h ow ma n y

factors

to

extract ,

factor

extraction method,

DECISION SEQUENCE

47



factor rotation method, factor score estimation method). For each of the

steps

of

analysis, there

are

myriad possihle decisions

or

ways

of

informing

judgment . Thus, across all the steps and all the choices fo r

each

step, th e

possihle number

of

com hinations

of

analytic

choices is

huge

It is

also important

not to

l imit

the

analysis

at any

stage

to a

single

analytic

choice.

For example, in deciding how many factors to extract,

researchers do well to consult several procedures (e.g., scree plot, parallel

analysis) and

then

evaluate

whether

various strategies corroborate

each

other. The most confidence can be vested in solutions fo r w hich factors are

stable

or

invariant over various analytic

choices.

The

purpose

of EFA is to

isolate factors that have simple structure

or ,

in

other words, are interpre table . Any thoug htful ana lytic choices that yield

clear factors ar e justif ied, including analyzing th e data in lots of different

ways

to "let th e data speak."

In practice, factor rotation is necessary. The only exception is when

a

single factor

is

extracted,

in

which case

no

rotation

is

possible.

Nor in

such a case is

rotation

needed, b ecause presum ably every variab le will saturate

the single factor, and the factor is w hatev er explains all the v ariable s bein g

highly

associated with

each other.

Usually when m u ltip le factors

are ex-

tracted, reasonable simple structure

is

realized with varimax rotation.

A nd

i f

these

factors

ar e

interpretable,

th e

correct rotation method

has

typically

been identif ied. However, even when

an

orthogonal rotation yield s simple

structure,

when

th e

purpose

of the

analysis

is to

obtain factor scores

for use

in subsequent statistical analy ses, if the researcher be liev es

tha t

the latent

constructs

have

notew orthy correlations,

an

oblique rotation (e.g., promax)

may be

sui table.

4 8 E XPL OR AT OR Y A N D C O N F I R M A T O R Y FACT OR A N A L Y S / S



4

COMPONENTS VERSUS

FACTORS CONTROVERSIES

In the previous chapter it was noted that there are myriad methods

wi th

which

to

extract factors, some

of

wh ich

are

avai lab le

in

commonly

used statistics packages.

It is

impor tant

to

understand

th e

extent

to

which

extract ion choices

m ay

affect factor interpretations, and, especially,

to

under-

stand why and when differences m ay arise across method s for a given data set.

For reasons that

wil l b e

explained mom entari ly ,

i t

m ight

be

argued that

wh e n

th e

n u m b e r

o f

var iables

is [at

least] moderately large,

fo r

example,

greater than

30, and the

analysis contains

virtually no

variab les expected

to have low com m unalit ie s (e .g. , .4) , practically any of the exploratory

[extraction] proced ures . . .

wil l

lead to the same interpretations.

(Gorsuch,

1983,

p.

123)

Similarly , Cliff (1987) suggested that "the choice of common factors or

components methods often makes

vir tual ly

no

difference

to the conclusions

of

a

s tudy"

(p.

349). These d y namics will

be

i l lustrated here with data

for

different variables

and

participants from

the

Appendix

A

scores.

Table 4.1 compares

three

of the more elegant extraction methods that,

unlike principal components analysis, each invokes iteration

as

par t

of the

extract ion process: alpha, m ax im um l ikel ihood, and image factor analysis.

As an

inspection

of the ta ble d results suggests, for these d ata the three

extraction methods yield reasonably comparable factor structures.

49



TABLE 4.1

Varimax-Rotated Pattern/Structure Coefficient Matrices

fo r

Three Factor

Extraction Methods

n =

100 Faculty)

Variable

PER1

PER2

PER3

PER4

PER5

PER6

PER?

PER8

PER9

PER10

PER11

I

.20

.12

.23

.16

.89

.87

.92

.72

.34

.21

.17

Alpha

II

.90

.82

.62

.86

.13

.14

.18

.31

.18

.23

.34

III

.18

.24

.22

.26

.22

.22

.20

.18

.44

.72

.56

I

.20

.11

.26

.19

.90

.87

.92

.72

.39

.20

.23

ML

II

.91

.86

.64

.84

.13

.15

.20

.32

.22

.23

.37

III

.16

.19

.12

.20

.18

.19

.09

.16

.32

.95

.40

I

.21

.12

.24

.18

.86

.84

.86

.71

.37

.25

.21

Image

II

.84

.82

.61

.80

.14

.16

.19

.32

.21

.31

.36

III

.19

.20

.19

.24

.20

.20

.18

.16

.35

.45

.42

Note.

Pattern/structure coefficients greater than |.4| are italicized. ML = maximum likelihood.

The largest difference occurs on the third factor, which involves only

2 or 3 of the 11

measured variables.

And the

results from

the

image factor

analysis differ somewhat more from the

first

tw o sets of results than these

tw o

sets

differ

from each other.

This is

partly because im age factor analysis

focuses on m inim izing the influence of factors involving single variables not

reflected

via

covariations

in the

"images"

of the

other

measured variables.

However,

the

d ifferences across extraction m ethods tend

to

decrease

as there are more measured variables. For example, regarding image analysis,

as

more measured variables are added

there

is an increasing likelihood

that

most measured variables will be correlated with another measured variable

or a combination of the other measured variables, so the influences of

"specific"

factors consisting

of a

single measured variable,

and not

represent-

ing true latent variables,

will

tend to become less as there are more m easured

variables.

Bear

in

mind that

th e

present small heuristic analysis involving

only

11

measured variables

is

unrea listic . Researchers would rarely

(i f

ever)

factor analyze

so few

measured variables.

MEASUREMENT ERROR

VERSUS

SAMPL ING ERROR

As

noted

in chapter 3, principal axes factor analysis is the same as

principal components analy sis, except that in principal axes factoring

only

the

diagonal

of the

correlation matrix

is

modified

by

removing

the

1.0's

originally alway s residing there and iteratively estimating the true commu-

50 EXPLORATORY A ND CONFIRMATORY

FACTOR

ANALYSIS



TABLE 4 2

Communality Coefficients If ) Through Four Iterations

n - 100 Graduate Students)

Iteration

a r i a b l e

PER1

PER2

PER3

PER4

PER5

PER6

PER7

PER8

83714

74596

67464

65751

75740

55652

73583

65091

89374

76327

69098

67103

79961

58434

77671

67977

2

92230

76468

69167

67066

81378

58885

78818

68493

3

93880

76277

68996

66857

81939

58870

79155

68521

4

94904

76061

68843

66692

82197

58798

79257

68479

nality coefficients. This is

done

to take into

consideration

the fact that all

scores have some measurement error (i.e., scores are never perfectly relia ble).

Table 4-2

illustrates

th e iteration

process

for a

principal axes factor

analysis involving th e first eight measured variables presented in Appendix

A, and

l imi t ing

the

analysis

to the 100

graduate students

(i.e., the first 100

scores in Appendix A). In this particular analysis, before iteration w as begun,

R

values were substituted on the diagonal of the correlation matrix.

These

results

involve

the R between a given measured variable and the seven

remaining measured variables in the analysis.

For

example,

as

reported

in

Table

4.2,

the

R

2

fo r

predicting

the 100

scores

on the

measured variable, PERI, with

the

scores

on the other

seven

measured variables in this analysis (i.e., PER2, P ER 3,. . . PER8) w as .83714.

The R for

predicting

the 100

scores

on the

measured variable, PER2, with

th e

scores

on the other

seven measured variables

in

this analysis (i.e., PERI,

PER3, . . . PER8) was .74596.

Using the R value s in this man ner is sensible, although other ini t ia l

estimates of communal i ty m ay also be plausible, because the reliabilities of

th e

scores

do impact th e

multiple regression effect size (Thompson, 2003).

If

a particular variable was perfectly unreliable (i.e., measured

nothing,

or

only

random

fluctuations

in

scores),

then

the R"

would theoretically

be zero.

Table 4.2 then

presents

the /T

est imates

fo r

each

of four

successive

steps

of

iteration. These

can be

obtained

in

SPSS

by

invoking

the

CRITERIA

subcommand:

factor variables=perl to per 2/

analysis=perl

to

per8/

criteria=factors(2)

iterate(1)

COMPONENTS

VERSUS

FACTORS CONTROVERSIES 51



PRINCIPAL

COMPONENTS VERSUS

PRINCIPAL AXES

FACTORS

By

far the two

most commonly used estimation methods

are

principal

components an d principal axes factors analyses. Principal

com ponents

analy-

sis

is the

default

method

in

commonly used statistical packages,

and is

used

with

considerable freq uenc y whe never exploratory factor analyses (EFA s)

are perform ed (Fabrig ar et a l., 1999; Russell, 2002). The

only

difference

between these tw o methods is that 1.0's are used on the d iagonal of the

correlation matrix

in

principal components analysis.

In

prin cipal axes factor-

ing,

th e

communali ty est imates

on the

diagonal

of the

correlation matrix

are iterativ ely estimated un til the iterations converge (i.e., rema in

fairly

stable

across a contiguous pair of estimates).

It would be a major understatement to say

that

"analysts differ quite

heatedly over the uti l i ty of princip al components as compared to comm on

or

principal factor analysis"

(Thompson

&

Daniel, 1996,

p.

201).

As

Cliff

(1987) noted,

Some author i t ies

insist

that componen t analysis is the only suitable

approach,

and that the c o m m o n

factors methods just superimpose

a lot

of ext raneous m umbo jum bo, dea l ing w i th fundamen ta l l y immeasurable

things, th e c o m m o n

factors. Feelings are,

if

anything,

even

stronger

on

th e o ther

side.

. . .

Some even insist that

th e

term "factor analysis" must

not

even

be

used

when a

components analysis

i s

performed,

(p .

349)

Indeed,

an

entire special

issue on

this controversy

was

published

in

the j ou rna l ,

M ult ivariate

Behavioral

Research

(Mulaik ,

1992). And Gorsuch

(2003) recently noted that "in following th e dialogue for the past 30 years,

th e only major change seems to be that th e heat of the debate has in-

creased" (p. 155).

A series of com parisons of princ ipal com ponents versus principal axes

factors for the App endix A d ata m akes this discussion concrete. Table 4.3

presents a principal com ponents

versus

a prin cip al axes factor analysis of

TABLE

4.3

Components

Versus

Principal

Axes

Factors for

our

Measured

Variables

n= 100 Graduate Students)

Varimax-rotated

components

Variable

PER1

PER2

PER5

PER6

.94

.94

.28

.13

II

.19

.22

.8 6

.92

If

92.3%

92.7%

82.1%

85.3%

Varimax-rotated factors

.8 8

.90

.27

.16

II

.22

.25

.7 9

.7 8

tf

83.0%

87.2%

70.4%

62.9%

Note.

Pattern/structure coefficients

greater

than |.4|

are

italicized. Given

tha

small number

of

variables,

the

convergence criterion fo r principal axes factor extraction was raised from the default of .001 to .003.

COMPONENTS VERSUS FACTORS CONTROVERSIES 53



TABLE

4 .4

Varimax-Rotated Components Versus Factors fo r Eight Measured

Variables With Factors Estimated with Ft

2

Starting Values

and 1.0 Starting

Values n =

100

Graduate Students)

Components

Principal-axes

factors

3

Principal-axes

factors

Variable

PER1

PER2

PER3

PER4

PER5

PER6

PER7

PER8

I

.94

.8 8

.8 2

.8 2

.27

.15

.33

.27

II

.19

.23

.32

.29

.8 8

.8 4

.8 5

.8 3

tf

91

.5%

82.8%

77.2%

76.1%

84.0%

73.0%

82.6%

76.7%

I

.96333

.82979

.75473

. 74967

.26440

.18993

.32802

.28089

II

.18636

.26108

.34148

.32049

.86862

.74214

.82784

. 77795

tf

96.3%

75.7%

68.6%

66.5%

82.4%

58.7%

79.3%

68.4%

I

.96362

.82971

.75466

.74959

.26447

.18985

.32802

.28089

II

.18625

.26112

.34151

.32054

.86841

. 74235

.82782

.77796

tf

96.3%

75.7%

68.6%

66.5%

82.4%

58.7%

79.3%

68.4%

Note. Pattern/structure coefficients greater than |.4|

are

italicized.

Communality

estimation

was

begun using

fl

2

starting values.

Com munality e stimation

was begun

using

1. 0

starting values.

four m easured variables. Altho ugh the p atterns of the results (two m easured

variables

saturate each

of the two

latent

variables) are

similar,

tw o differ-

ences appear.

First,

th e

pattern/structure coefficients tend

to be

smaller

for the

pr inci-

pal axes factors. Second,

th e

co mmu na l i t y

coefficients

tend

to be

smaller

for

the principal axes factors. These tw o outcomes are a function of changing

only

the diagonal of the correlation matrix from 1.0s to sm aller positive

numbers w i t h in the prin cipa l axes analyses.

Table 4-4 presents a p rincipal com ponents

analysis

versus tw o principal

axes factor analyses of eight m easured variables. W ith regard to the pr incipal

axes factor analyses,

the first

analysis again began with h est imation using

R

values of each measured variable with the rem aining seven m easured

variables, and estim ation continued until convergence was achieved in seven

iterations.

The second prin cipa l axes factor

analysis

began

h

est imation

using

1.0 values (i.e., the unaltered correlation matrix was

used

as a starting

poi nt), and estim ation continued un til convergence w as achieved in eight

iterations. Clearly

the two

sets

of

results

are

comparable, even

fo r

this

example w ith only eight m easured variables, which is why these results had

to be

reported

to

more than

tw o

decim al values.

This is the general pattern in regard to selecting starting points for

iteration. Theoretically, roughly

th e

same values

will be

realized regardless

of

the

initial estim ates. H owever, using

less

reasonable initial estim ates (i.e.,

1.0's vs. R or rel iabil i ty

coefficients) will

require m ore iterations. On a

modern microcomputer ,

th e

add it ional computat ions

from

more iteration

will

usually

go unnoticed.

54

EXPLORATORY AND

CONFIRMATORY FACTOR

ANALYSIS



The interpretations of the Ta ble 4-4 com ponents versus factors wou ld

be the same, even though the h values tend to be larger for the component

analysis.

And the two sets of results are more com parable for the Table 4-4

com parisons involv ing eight m easured variables than for the Table 4-3

comparisons involving

four

measured variables. Why?

Two factors affect the equivalence of the principal components an d

the princ ipal axes factors for a given d ata set. First, if the relia biliti es of the

scores being

analyzed

are quite high

(i.e.,

approach 1.0), then th e

h

values

will tend to each be roughly 1.0, which is the same

h

val ue used for each

variable in the principal components analysis. This m ight occur, for exam ple,

i

the researcher was analy zing m easured variables that were each scores on

tests each consisting of a large number of items.

Second, there is also a less obvious dyn am ic at work . The only difference

between princip al com ponents and princip al axes factor analyses involves

changing

the

diagonal

of the

correlation matrix when principal axes factor

analyses are conducted. But the impact of altering only the diagonal of the

correlation matrix

differs

depending on how many measured variables are

being analyzed

As

Snook

and

Gorsuch (1989,

p.

149) explained,

"As the

number

of

variables decreases, the ratio of diagonal to

off-diagonal

elements also de-

creases, and therefore the value of the communality has an increasing effect

on the analysis." For example, with four measured variables, there are 16

(4 X 4) entries in the correlation m atr ix, of which

four

are on the diagonal

(4 /16 = 25 .0%) . When there are eight m easured v ariables , there are 64

(8 X 8) entr ies in the c orre lation m atr ix, of wh ich eight are on the diagon al

( 8 / 6 4 = 12.5%).

When there are 100 measured variables, there are 10,000 entries in

the correlation matrix, of which 100 are on the diagonal (100 / 10,000 =

1.0%). And it is more common to factor analyze 10 0 than 4 measured

variables As there ar e m ore m easured variables, even it their reliabilities

are modest, principal components

and

principal axes factors tend

to be

increasingly similar (Ogasawara, 2000). And principal components have

some add itional desirable properties, as dem onstrated in chapter 5, that

makes

these

analyses attractive,

at

least

to

some researchers.

MAJOR CONCEPTS

There

are

numerous

EFA factor extraction methods

(e.g.,

alpha, canon-

ical,

image, m a xi m u m likelihood). However,

most f requent ly

seen in pub-

lished

research are principal components and principal axes methods. Princi-

pal

components

is the

default

method in

widely

used

statistical packages.

COMPONENTS VERSUS FACTORS CONTR OVERSIES 55



Principal axes m ethods acknowledge measurem ent error

by

iteratively

approximating

the

com m unali ty estimates

on the

diagonal

o f the

correlation

matrix.

When

score re liabilit y is high for the m easured variab les, there will

be

little difference between

th e

factors extracted using these

tw o

methods.

In any case, the differences in the two methods

will

also be smaller

as the number of factored variables increases.

This

is because iterative

estimation of comm onalities changes

only

the diagonal entries of the correla-

tion

matr ix ,

and the

diagonal

of the

matr ix

is

progressively

a

smaller con-

stituent of the correlation matrix as more measured variables are analyzed

(e.g.,

4 diagonal entrie s w ith 4 m easured variables is 4 out of 16 [25.0% ]

matrix entries;

8

diagonal entries with

8

measured variables

is 8 out of 64

[12.5%] ma trix entries). And in practice we usually factor

fairly

large num bers

of measured variables.

There is also a trade-off in considering measurement error variances.

Each sample data

se t

also contains sampling error variance.

The

more steps

we iterate to take into account measurement error, th e more statistical

estimates we are making, and the more opportunity we are creating for

sampling

error to influence our results.

56


FACTOR

ANALYSIS



5

COMPARISONS OF SOME

EXTRACTION, ROTATION,

AND SCORE METHODS

The exploratory factor analysis (EF A) statistics called structure coeffi-

cients were

first

introduced

in

chapter

2, and the

statistics called communal-

ity

coefficients were

first

introduced

in

chapter

3 .

Both

of

these statistics merit

further discussion.

This

chapter further elaborates

the

concepts und er ly ing

s tructure

and communali ty (h ) coefficients hy invoking factor score statis-

tics.

First,

however, some comparisons of factor score estimation methods

may be helpful.

FACTOR SCORES IN DIFFERENT EXTRACTION METHODS

As

noted

in

chapter

4, the two

most commonly used

EFA

approaches

are principal components and principal axes factor analyses. And, as noted

in

chapter

3,

there

are

four different methods with which

to

estimate

factor

scores.

Here three

factor score methods (regression, Bartlett,

and

Anderson-

Rubin)

are

compared

to

faci l i ta te deeper understanding

of the

factor score

estimation choices

and to

elaborate addit ional differences

in the two

most

common

extraction methods.

In real

research,

factor scores are typically only estimated

when

the

researcher elects to use these scores in further substantive analyses (e.g., a

57



mul t iva r i a t e analysis of var iance comparing m ean

differences

on three factors

across men and women) . In these applications th e researcher will use only

one

factor score estimation method. Here several methods

are

compared

s imultaneously only

for

heuristic purposes.

Factor scores

can he

derived

hy

invoking

SPSS

COMPUTE

sta tements

that apply the factor score coefficient matrix (Wy

x

F) to the measured

variahles in either z

score

or

standardized, noncentered form. This

has the

heuristic value

of

making this compu tation process concrete

and

explic i t .

However , in actual research it is

both

easier and more accurate to obtain

the factor scores by using the

SPSS

SAVE subcommand within the

FACTOR procedure.

For ex am ple , to obtain regression factor scores

from

a principal compo-

nents analysis involving

11

measured var iables (PERI through P ER U ),

th e

syntax would

be:

factor variables=perl to per 2/

analysis=perl

to

perll/criteria=factors 3)/

extraction=pc/rotation=varimax/

print=all/save=reg all

reg_pc)/ .

The execution of this commands yields three factor scores for each partici-

pant f rom a pr incipal

components

analysis that are labeled REG _PC1,

REG_PC2, and

REG_PC3.

Factor scores from a principal axes factor analysis estimated using th e

Bart le t t method would be produced with the syntax:

factor variables=perl

to per 2/

analysis=perl to perll/criteria=factors 3)/

extraction=paf/rotation=varimax/

print=all/save=bart all bar_paf)/ .

Factor scores were computed for all 100

faculty

using the first 11 measured

variables presented

in

Appendix

A fo r all six

combinations

of two

extraction

methods (principal components vs. principal axes factor analysis) and three

factor score estim ation m ethods (regression, Ba rtlett ,

an d

Anderson—Rubin) .

Table

5.1 presents th e

three

sets of principal components factor scores fo r

the first 5 and

last

5

participants

in the

data set. Notice that when principal

components methods a re used, all three factor score methods

yield

identical factor

scores

for a given participant (e.g., fo r participant 101, REG_PC1 =

BAR_PC1 = AR_PC1 = +.35) .

Table 5.2 presents the three sets of principal axes analysis factor scores

for the first 5 and last 5 participants in the data set. Notice that when

58 EXPLORATORY AND CONFIRMATORY FACTOR ANALYSIS



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COMPARISONS OF METHODS

59



princ ipal axes factor analyses method s

are

used,

all

three factor score methods

yield

different

fac tor scores

for a

given partic ipan t (e.g. ,

fo r

participan t 102,

RG_PAF1 = -.76; BR_ PAF1 = -.86; ARJPAF1 =

-.83).

Table 5.3 presents th e Pearson correlation coefficients among all pair-

wise

combinat ions of the 18 factor scores for the 100 participants in this

analysis.

Recal l

that all six

analyses invoked rotation

to the

varimax cri ter ion,

which

means that the three

factors

in all six

analyses were

perfectly

uncorre lated .

As the

correlation

coefficients in Table 5.3

indicate,

the

correlations

of th e

factor scores from

th e

principal components analyses exactly matched

th e

correlations (here

all

zero)

of the

factors themselves.

This

always (and

only always) happens

in

prin cipa l components analysis.

The

exact matching

of

factor score correlations with factor correlations is

another

appealing

feature

of principal components analyses.

STRUCTU RE COEFFICIENTS IN A COMPONENTS CONTEXT

As

noted

in chapter 2, correlations between measured variables and

composite variables

are

always called structure coefficients (r ) throughou t

the general l inear model (GLM; Courville Thompson, 2001). Table 5.4

presents

th e

Pearson

correlation

coefficients

for the 11

measured variables

with

the

three factor scores

from the

principal components analysis.

These

were

obtained

wi th

the

SPSS

syntax:

correlations variables=perl

to

peril with

reg_pcl to reg_pc3 .

Note that th e Table 5.4 structure coefficients

literally

computed as

bivar ia te correlation

coefficients

exact ly match

the

pat tern/s t ructure coeffi-

cients produced by the SPSS principal components analysis of the same 11

measured variables.

These

varima x-rota ted pat tern/s t ructure coefficients are

presented in Table 5.5.

COMMUNALITY

COEFFICIENTS AS

R

1

VALUES

In chapter 2 it was noted that communality coefficients indicate th e

proportion of variance of a measured variable that the factors as a set can

reproduce. Conversely, h

can be

viewed

as

characterizing

how

much

of the

variance of a measured variable was

useful

in delineating the extracted

factors.

COMPARISONS OF

METHODS 61



TABLE

5 .3

Factor

Score C or r e la tions

n = 100

Faculty)

Principal components

Variable

REG.PC1

REG-PC2

REG-PC3

BAR_PC1

BAR_PC2

B A R _ PC3

AR-PC1

AR_PC2

AR-PC3

RG-PAF1

RG-PAF2

RG-PAF3

BR-PAF1

BR_PAF2

BR_PAF3

AR_PAF1

AR_PAF2

AR-PAF3

REG_PC1

1 0000

.0000

.0000

1

0000

.0000

.0000

1 0000

.0000

.0000

.9903

.0054

.0485

.9904

.0061

-.0476

.9917

.0044

-.0023

Regression

REG_PC2

1 .0000

.0000

.0000

1 .0000

.0000

.0000

1 0000

.0000

-.0017

.9779

.1007

-.0064

.9748

-.0116

-.0050

.9793

.0383

REG-PC3

1 .0000

. o o o o l

.ooool

1 . O O O O l

. o o o o l

. o o o o l

1. OOOOl

.0596

.0543

.95711

-.03141

-.0577]

.95011

.0208)

.00351

.95351

BAR_PC1

1.0000

.0000

.0000

1 .0000

.0000

.0000

.9903

.0054

.0485

.9904

.0061

-.0476

.9917

.0044

-.0023

Bartlett

BAR-.PC2

1 .0000

.0000

.0000

1 .0000

.0000

-.0017

.9779

.1007

-.0064

.9748

-.0116

-.0050

.9793

.0383

Anderson-Rubin

BAR_PC3

1

.0000

. o o o o l

. o o o o l

1 .OOOOl

.0596

.0543

.95711

-.03141

-.05771

.95011

.02081

.00351

.95351

A R _ P C 1

1 .0000

.0000

.0000

.9903

.0054

.0485

.9904

.0061

-.0476

.9917

.0044

-.0023

AR_PC2

1

.0000

.0000

-.0017

.9779

.1007

-.0064

.9748

-.0116

-.0050

.9793

.0383

AR_PC3

1.0000

.0596

.0543

.95711

-.03141

-.05771

.95011

.02081

.00351

.95351

One

heur is t ic

to

mak e

these

concepts concrete is to

conduct

a

m u l t i p le

regression analysis in which the measured variables in tu rn are predicted

by the factor scores

from

the pr inc ipa l components analysis.

These

regression

analyses can be executed using the SPSS syntax:

regression variables=reg_pcl to reg_pc3

perl to perll/dependent=perl /

enter reg_pcl

to

reg_pc3

.

regression variables=reg_pcl to reg_pc3

perl

to

perll/dependent=per2

/

enter reg_pcl

to

reg_pc3

.

regression variables=reg_pcl

to

reg_pc3

perl

to

perll/dependent=perll

/

enter reg_pcl

to

reg_pc3

.

6

EXPLORATORY

AND C ONFIRM ATOR Y FACTOR ANALYSIS



Dependent Var iab le.

.

PER1

1

Wi ll in gne ss

to

help users

M ethod: Ente r REG _PC1 REG _PC2 REG_PC3

M ultiple

R

.93803

R Square .87990

Variables

in the

Equation

Variable B SE B Beta T Sig T

REG-PC1 .277091 .048319 .202837 5.735 .0000

R EG_ P C 2

1.224812 .048319 .896592 25.349 .0000

REG _PC 3 .255 119 .048319 .186753 5.280 .0000

(Con stan t) 7.650000 .048077 159.121 .0000

E n d

Block Number

1 Al l

r eques ted var iab les en t e r ed .

De pen de n t Variable. . PER2 1 G iving use rs in dividual att

M ethod: Ente r REG _PC1 REG_PC2 REG_PC3

M ultiple

R

.90666

R

Square .82204

Variables

in the

Equation

Variable

R E G PC 1

RE G

PC 2

REG_PC3

(Constant)

B

.167676

1 .353442

.333508

7.310000

SE B

.066672

.066672

.066672

.066338

Beta

.108283

.874029

.215374

T

2.515

20.300

5.002

110.194

S i g T

.0136

.0000

.0000

.0000

Figure 5.1.

M ultip le re gres s ion predictin g

pleasured

variables using

the

three factor

s co r es

n =

100 faculty).

Factor scores can also be used fo r heuristic purposes, to better grasp

the nature of both structure coefficients and communal i ty

coefficients.

Throughout

th e G L M , a structure coefficient, score-world statistic, can

be computed (l i teral ly) as the bivariate

product-moment

correlation of a

measured variable (e.g. , th e scores of n people on a factored variable) with

a

latent variable (e.g. ,

th e

principal components factor scores

of n

people).

And a

com muna lity coefficient,

an

area-world statistic,

in the

orthogonal

principal components case

can be

computed

as the R

2

between

th e

factor

scores

of n

people with

th e

scores

of n

people

on the

measured variables.

COMPARISONS OF METHODS 65



6

OBLIQUE ROTATIONS

AND

HIGHER-ORDER

FACTORS

In chapter

3 the

notion

of

factor rotation

was

introduced.

It was

noted

t h a t

ro tat ion is a lways poss ib le in exploratory factor analys is (EFA) whe never

there

is more

than

one factor , and

that rotat ion

is almost always necessary

in these cases

to

obtain s imple s t ructure .

O ne

class

of

methods involves

or thogonal

rotation

and

leaves

the

factors s til l perfectly uncorrelated.

A

second class o f me thods involves obl ique ro tat ion and allows t he unro ta ted

factors ( that a re

ini t ia l ly a lways uncorrelated)

to

become correlated. Ob lique

rotations present some interpretation challenges

tha t

mer i t fu r ther

discussion.

Table 5.5 (see chap. 5) presented the v ar im ax- ro ta ted p r inc ipa l compo-

nents

f rom

an

analysis

o f the first 11

measured var iab les presented

in Appen -

dix

A for

data provided

by the 100

facu l ty members .

This

solution

had

reasonably s imple s t ructure .

A few

measured var iab les were

no t

q u i t e

as

clear

in

regard

to

what factors they reflected.

For

example , PERU

had a

pa ttern /stru ctu re c oeffic ient on F actor III of .72, but on Factor II had a

pat tern/s tructure coeff ic ient

of

.34.

How ever , i t is im por t ant to reme mb er that a l l cor relat ion coeff ic ients ,

including structure coefficients , must be squared before their magnitudes

can be

compared. Thus, because

.72 is

51.8%,

and .34 is

11.6%,

the

larger

of these tw o s tructure coeff ic ients w as actual ly 4 .5 times larger than the

smaller value.

67



TABLE 6.1

Promax-Rotated Components for 11 Measured Variables n = 100 Faculty)

Pattern Structure

Variable

PER1

PER2

PER3

PER4

PER5

PER6

PER7

PER8

PER9

PER10

PER11

I

.04

-.07

.10

-.01

.95

.93

.94

.81

.20

-.03

-.10

II

.94

.92

.75

.90

-.06

-.05

.00

.20

-.12

.01

.17

III

-.03

.03

-.03

.05

.04

.04

.00

-.07

.70

.85

.75

I

.40

.31

.39

.37

.94

.93

.95

.85

.48

.38

.33

II

.94

.91

.78

.92

.34

.34

.38

.49

.31

.43

.51

III

.46

.46

.40

.50

.46

.45

.45

.41

.73

.84

.79

If

88.0%

82.2%

61 .5

84.7%

88.8%

86.9%

89.7%

75.5%

57.0%

71 .2%

64.7%

Note.

Pattern coefficients greater

than

|.4| are italicized.

Although the v a r ima x - r o t a t e d solution had s imp le s t r uc tu r e , for he u -

r is t ic purposes the solution was also rotated to the p r o ma x

criterion.

Because

S

V x

p equa ls

P

V X

H ( R F

X

H ) .

a n

d

R p x F

n o w

wi l l

no t equa l

I P

X

F .

the s t r u c t u r e

and the

pattern coef f ic ient matr ices wi l l

no longer be

equa l . Indeed , hecause

the

pa t te rn coef f ic ients

are no

longer cor re la t ion coef f ic ients , some, m a n y ,

or all of the pa t te rn coef f ic ients may be less than -1 or grea te r

than

+1.

Table 6.1 presents the pattern and the s t r uc tu r e ma t r i c e s f rom this

ana lys is .

Table

6.2

presents

the

fac tor

correlation

ma t r ix f r om

the

ana lys is .

In

chapter

2 the

notion

of the

c o mmu na l i t y c o e f f i c i e n t

was first

intro-

duced. In ac tua l i ty , the examp le invo lved a v a r ima x - r o t a t e d p r i n c i p a l c o m-

ponents so lu t ion. In an

orthogonal

ro ta t ion the

h

2

can be c o mp u t e d by

s u m m i n g the squared s t ruc ture coef f ic ients for a var ia b le ac ross the factors.

This

y ie lds

the total

var iance

of the

measu r ed va r i ab le common

to the

fac tors

as a set. The computation is

s implif ied

because the fac tor s are unco r r e -

la ted, and so

there

is zero over lap be tw een the common var iances of a g iven

measu r ed va r i ab le on any two fac tor s .

B ut

when factors are cor re la ted , the h is der ived by first computing

the

p roduc t s

of a

g iven measured var iab le ' s

pattern

coef f ic ient

on

each

TABLE 6.2

Factor Correlation Matrix

(R

3x 3

) for the

Table

6.1

Factor Pattern Matrix

Factor I II III

1.000

.398 1.000

III

.473 .502 1.000

68 E X P L O R A T O R Y AND CO N F I R M A TO R Y F A CTO R ANALYSIS



factor with the corresponding s truc ture coefficient on each factor , and then

s u mming

these

products .

For

example ,

for the

measu red var iah le PERI

in

the

Table

6.1 solution, h for this measured var iahle is computed as :

[0.040

x

0.399]

+

[0.936

x

0.937]

+

[-0.029

x

0.460]

=

0.016 + 0.877 + -0.013 = 87.96%.

It

should he

noted that

th e algorithm for a given measured var iab le ,

ZP

K

S« across the k factors , is a more general

formula

fo r h

2

. When factors

are uncorrelated, £P

K

SK across the k factors can be reexpressed as ZP

K

2

,

because

in

this case

the

pat tern

coefficients are

also s tructure

coeff icients .

In

o ther

words , fo r pr incipal components and pr incipal axes analyses th e

algorithm ZP^Sx across the fe factors

always

works , whether or not the

factors

are

correlated.

M A X I M I Z I NG S A M P LE

F IT VERSUS

RE PLI CA BI LITY

In th e present comparison the Tab le 5 .5 var im ax-ro tate d so lu t ion and

th e Table 6 .1

promax-ro tated so lu t ion would general ly lead

to the

same

interpretat ions regarding

the

m a k e u p

of the three

unde rlying constructs .

Where the so lu t ions d i f f e r is in the representat ions of the correlat ions among

the

latent constructs .

The first componen t in

Table

5.5

wou ld

be

interpreted

as

measur ing

th e

Library

as

Place.

This component w as

saturated w ith percept ions

of

l ibraries inc lud ing r espec t ive ly

the

i tems

7 . A

co n tem p la t i v e env i ro nmen t ,

5. A

haven

fo r

qu ie t

and

solitude,

6. A

meditat ive p lace,

and 8.

Space

tha t f ac i l i t a tes qu ie t s tudy .

The

second

com ponent: might

be

labeled Service Affe ct.

This compo-

n en t w as

saturated respect ive ly with

the

i tems

1.

W ill ingness

to help

users ,

2 . Giving users indiv idual a t ten t ion , 4 . Employees who are consistently

cour teous ,

and 3.

Employees

w ho

deal with users

in a

caring fashion.

T he third com ponent might be labeled Information

Access. This

component

w as

saturated respect ively with

the

item s 10.

Comple t e

runs

of

jour nal t i t les , 11. Interdiscip l inary l ibrary needs being addressed,

and

9.

C omprehens ive pr in t collections.

But

fo r

these data

th e

p rom ax- ro ta ted r esu l t s r epo rted

in

Table

6.1

would lead

to the

same factor labels

as those

produced

in the

var imax-

rotated solution. T he pat tern coeff ic ients in Table 6.1, like the

pat tern /

s tructure coeff ic ients presented

in

Table 5.5,

had

s imple s t ructure .

T he s tructure coeff ic ients in Table 6.1 ref lect a dynamic invo lv ing th e

components

being

fairly

highly correlated,

as

repor ted

in

Table 6.2. Most

of the s tructure c oeff ic ients in

Table

6 .1 are qui te notew or thy for a ll var iab le s

O BL I Q U E

ROTATIONS AND H I G H ER - O R D ER FACTORS 69



on a l l components . This is mere ly indica t ive of the fac tors having corre la-

t ions of .398, .473, and .502.

Of course , the com ponen ts a re sufficiently discre te tha t i t

st i l l

seems

reasonahle to recogn ize the ex i s t ence of three la ten t va r iab le s , because even

the two

mos t h igh ly cor re la ted com ponen ts on ly have 25 .2% com mo n

var iance

( . 502

2

=

. 2 5 2 ) .

But for a

case such

as

th is ,

how w ou l d one

se lec t

a

f a c t o r an a ly t i c so l u t i on ?

Three

re la ted

issues

seem

pa r t i cu la r ly

re levan t .

Analytic Purpose

In

som e fac to r ana ly t i c app l i ca t ions

the

researcher

is p r ima r i ly

in ter-

es ted in in terpre t ing the fac tors . In other appl ica t ions , fac tor scores are

genera ted fo r use in subs equ en t s ta t i s t i c a l analyses . As the present exam ple

suggests, when fac to r in te rp re ta t ion i s the focus, rota t ion choice may be

less

c r i t i c a l a s long as a g iven a na lys i s y ie lds a r e a son ab le s imp l e s t r u c t u r e .

B ut

w hen com pos i t e scores

are

going

to be

crea ted

for use in

sub sequen t

analyses ,

and w hen the ob l iqu e fac to rs a re h igh ly cor re la ted , some pre fe rence

migh t

be

given

to the

ob l ique so lu tion ,

so

th a t

the

cor re la t ions

of the

la tent

var iables wil l b e honored in

these subsequen t su bs tan t ive ana lyses . Ho wever ,

there is one o ther a ppro ach to es t im at ing fac tor scores , som et ime s ca l led

unit

weighting, tha t may finesse these concerns (see G orsu ch, 1983, pp.

268—276) .

As noted in chapter 4, eve ry sample con ta ins some id iosync ra t i c and

nonreproduc ib le va r iance

as a

func t ion

of

samp ling error.

The

more pa r ame -

ters w e e s t ima t e , the more po ten t i a l the re is for our re su l t s to cap i t a l i ze

excess ively on

u n i q u e

fea tures of the

s amp l e , w i th

the

consequence

that

resul ts wil l

genera l ize less

w e l l to new

samp le s .

One way to capi ta l ize

less

on id iosync ra t i c

fea tures

of the sam ple is to com pute fac tor score es t im ates

w i t h o u t

us ing

the fac tor

score we igh t ma t r ix ( W v

x

p ) -

One approach

s imply

ident i f ies those m easured va r iab le s tha t

def in i -

t ively

mea su re

a

single fac tor .

Then

composi te scores

on

each fac tor

are

com puted m ere ly by su m m ing the scores on the re leva n t measured va r iab le s ,

by av era ging these scores , by av eraging the

z

scores of only the re levant

var iab les ,

or by do ing wha teve r e l se the re sea rche r deems though t fu l and

reasonab le . For ex am ple , ac ross both the va r imax - ro t a t ed and the pro 'max-

rota ted resul ts

for the

p re s en t e xamp le ,

the

s ame m ea su red va r i ab le s wo u ld

be used

to

ob ta in un i t -we igh ted compos i t e scores

on the

three construc ts .

Parsimony

Centur ie s ago a ph i losopher named Wi l l i am of Occam a r t i c u l a t ed the

concept w e recognize today as the law of pars im ony. Thus, th is precep t i s

somet imes ca l led

Occam's

razor. Bas ically,

the law of

pars im ony says that

w h e n

tw o

exp l an a t i on s

fi t a set of

facts rough ly

equa l l y

we l l ,

a ll

things

70 EXPLORATORY

AND

CONFIRMATORY FACTOR ANALYSIS



equal, the

simpler explanation

is

more

l i ke l y to be

true.

We

tend

to p re f e r

parsimonious explanations because things that are true are also more l i ke l y

to replicate in

fu tu r e

research, and researchers

u s u a l l y

do not want to be

embarrassed by

making discoveries

that no

one else

can

replicate.

In

an EFA context, orthogonal solutions are more parsimonious

than

their oblique counterparts because f e w e r parameters are estimated when an

orthogonal rotation

is

implemented. This

is

because when

R

F x

F

equals

Ip

x F

,

then PV

x

F equals Sy

x

F-

In any s t a t i s t i c a l ana ly s i s ,

including EFA, what

we are

trying

to do is

fit a

model

to our

data.

The

more parameters

we

estimate,

the

greater

is

the likelihood that our model

w i l l

fit the sample data. For example, if n is

2, for any two measured

v a r i a b l e s ,

r

2

is

a l w a y s

+1. For a regression ana ly s i s ,

when

the

number

of

predictor variables equals

n — 1, the

degrees

of

freedom

error

is

zero,

and the

resulting

R is

always

+1,

regardless

of

what

the

predictor

variables

are.

The

same situation applies across

d i f f e r e n t

analyses.

For

example,

in

predictive

discriminant analysis (PDA), when

less

parsimonious equations

called quadratic versus linear rules

are

used,

the

prediction

hit

rate tends

to be higher in the original sample, but the prediction eff icacy of the same

prediction equation across other samples tends to be lower than in the

original sample. Thus, Huberty (1994, p. 64) noted a general preference for

the more parsimonious PDA equations, even though the fit of the model

to

sample data tends

to be

better

for the

less parsimonious solution. Elsewhere

he noted

that

the linear rule is suggested because of presumed greater

stability

[across samples]

than

that

for the quadrat ic rule, in general (p. 260).

Interpretation Difficulty

It must also be

sa id that

factor interpretation

i tself

is more complex

when

factors are correlated.

This

is because more parameter estimates must

he

simultaneously interpreted. Specifically, throughout

the

general linear

model (GLM), whenever weights

are not

structure coefficients,

both

stan-

da rd iz ed weights and structure

coe f f i c i en t s

must be examined to arrive at

correct interpretations.

This

is true with reference to analyses including mul tiple regression

(cf. Courville &

Thompson,

2001;

Thompson

& Borrello, 1985), canonical

correlation analysis (Meredith, 1964;

Thompson,

1984), and confirmatory

factor

analysis (cf. Graham

et

al.,

2003; Thompson,

1997).

This is

also true

in

EFA.

As

Gorsuch (1983) emphasized, proper

interpretation of a set of

factors

can probably only occur if at least S and P are both examined (p . 208).

A variable does not contribute to a factor only if the variable's pattern

and

structure

coe f f i c i en t s are both

zero.

If a

structure coefficient

is

near zero,

but the pattern coefficient is not, the measured variable contributes to the

OBLIQUE ROTATIONS

AND

HIGHER-ORDER

FACTORS

71



solut ion indi rec t ly

by m odify ing the

re la t ionship

of

other measured var iables

w ith the

factor. Across

the GL M

this

is

called

a

suppressor

e f f e t

(see Co urv il le

& Thompson, 2001; Horst, 1966).

For the Table 6.1 resul t s , the analysis does not present overwhelming

difficulties,

because the s t ruc tu re is so clear. The three factors

( Library

as

Place, Service Affect , and Inform ation Access ) w ere highly correlated

but

sti l l

sufficiently

discre te

to be

recognized

a s the

constructs described

by

the pattern coefficients. But the structure coefficients in this case clear ly

indica te tha t all 11 variable s are correlated w ith al l three c onstructs. Such

findings

suggest

the

presence

of

overarching superordinate const ruc ts ,

or

higher-order factors.

H I G H E R - O R D E R

FACTORS

When

variables are correlated (i .e . , R y

x

v ^ l v \ v ) >

then

factors can

be

computed

as

weighted aggregates

of the

measured va r i ab le s .

By the

same

token, when factors are correlated (i.e., R F

X

F IF

X

F ) >

then factors can be

extracted

from

this matr ix

as

wel l .

The factors extracted from interv ariab le correlat ions (o r other stat ist ics

measu r ing associat ions)

are

called

first-orde r

factors.

T he

factors

then ex-

t rac ted

from

the inter fac tor correlat ion s among the first-order factors are

called

second-order

factors.

If the

second-order factors

are

correlated,

then

third-order

factors

can be

ext rac ted. However ,

as

Kerl inger (1984) noted,

whi le ordinary fac tor

analysis is

probably

wel l

unders tood, second-order

factor

analysis,

a

vi ta l ly impor t an t pa r t

of the

analysis, seems

not to be

widely k n o w n

and

understood

(p .

x i v v ) .

It is argued here that these higher-order factors

should

be ext rac ted

whenever fac tors are corre la ted. As Gorsuch (1983) em phasized:

Rotating

ob l i que ly

in

factor analys is

impl ies

that

the

factors

do overlap

and that there are,

there fo re ,

broader areas of generalizabil i ty than j u s t

a

p r i m a r y fac to r . I m p l i c it

in all

oblique

rotations

are

h ig he r - o r de r factors .

It

is recommended that these [always] be

extracted

and examined so

that

the

inves t iga tor

may

ga in

the

ful les t poss ible

understanding of the

data.

(

P

. 2 5 5 )

Too few

researchers report ing correlated

first-order

factors conduct these

needed higher-order analyses.

The different levels of factor analysis give

different

perspec t ives on

the

da ta . Thompson (1990b)

offered th e

analogy

o f

hiking

in the

m oun t a in s

versus flying over them. When you are hiking you can see rocks and trees

and fish in

streams. When

you fly

over

the

mountains,

you can see the

pat terns made

by the

moun ta ins

as a

range.

The

perspect ives complement

72 EX P LO R A TO R Y A ND CONFIRMATORY

FACTOR

ANALYSIS



TABLE 6.3

Second-Order

Factor Pattern/Structure Coefficient Matrix

First-order factor

A

If

FACTJ

.773 59.8

FACTJI

.792 62.7

FACTJII

.832 69.2

each other, and each is needed to see patterns at a given level of specif ic i ty

ver sus

generality.

The literature includes various examples of higher-order factor analyses

(cf . Borrello & Thompson, 1990; Cook, Heath, & Thompson, 2001; Cook

&

Thompson,

2000; Thompson & Borrello, 1986). Here

both

the implemen-

tation

of a

higher-oider analysis using SPSS

and the

interpretation

of

such

an analysis are illustrated.

Example 1

For the Table 6 .2 interfactor correlation matrix (R ^

x

5) the second-

order factors can be eas i ly computed in

SPSS

by analyzing the correlations

among

the first-order

principal components scores.

The

syntax

is:

subtitle '5

n=100

Faculty

AAA

First-order

A A A A A A

.

execute .

temporary

.

select

if (ranktype eq 3) .

factor

variables=perl

to per 2/

analysis=perl to erll/criteria=factors(3)/

extraction=pc/rotation=promax/

print=all/save=reg(all

reg_pc}/ .

subtitle

6 Second-order $$$$$$$$$$$$$$$$$$$$$$$$' .

execute

.

factor

variables=reg_pcl to reg_pc3/

print=all .

This works only fo r principal components analysis, because only in

principal

components analysis

wi l l

the factor correlations and the factor

score correlations match exactly.

The

second-order factor derived

in

this

manner is presented in Table

6.3.

Higher-Order Interpretation Strategies

The interpretation of higher-order factors poses some special dif f icul -

ties. Factors

are

abstractions

of

measured variables. Second-order factors,

OBLIQUE

ROTATIONS AND

HIGHER-ORDER

FACTORS 73



then,

are abstractions of abstractions even more removed f rom the measured

va r iab le s .

Somehow

we

would like

to

interpret

the


in

terms of the measured variables, rather than as a manifestation of the factors

of

the

measured variables.

As Gorsuch (1983) suggested,

T o avoid basing interpretat ions upon interpretat ions of interpretat ions,

the relationships of the original variables to each level of the higher-

order factors are

de te rmined .

. . . Interpret ing from th e variables

should

improve

th e

theore t i ca l unders tanding

of the data and

produce

a

better

ident i f icat ion

of

each

higher-order

factor, (pp. 245-246)

There are three strategies for doing so.

Firs t ,

Gorsuch (1983) suggested

that the

second-order factors directly

ref lec t ing the measured variables could be evaluated by computing the

product matrix of the first-order

pattern

matrix times the second-order

pattern matrix ( i .e . , here

PH

x 5

P}

X

] = PH

x

i) . Second,

Thompson (1990b)

reasoned

that

an orthogonally rotated product matrix might be interpreted.

He

reasoned

that if

rotation

is

sensible

in

both

the first and the

second

order,

then

rotation of the product matrix to achieve simple structure might

also be

usefu l .

Of course, in the present example involving only one second-order

fac tor ,

rotation

is not possible. Rotation is only possible when there are two

or more factors.

Third, Schmid and Leimaii (1957) proposed an elegant method for

expressing

both the first-order and the


in

terms

of the

me a su r e d variables, but also res idua l iz ing (removing) all variance in the

first-order factors

that

is

also present

in the

second-order factors.

This

allows

the researcher to determine what, if any, variance is unique to a given level

o f analysis or perspective. Schmid and Leiman (1957) suggested that their

Schmid-Leiman solution not only preserves

the

desired interpretation char-

acteristics

of the

oblique solution,

but also

discloses

the

hierarchical structur-

in g of the variables (p. 53).

This analysis

can be

implemented using SPSS matrix commands.

The

input into the analysis includes (a) the obliquely rotated first-order pattern

matrix (here

P j i x ? )

an

d (b) the second-order pattern matrix (here

P ?

x

i )

augmented

by a

symmetrical (here 3 x 3 ) matrix containing zeroes off-

diagonal

and the

square roots

of

uniquenesses

on the

diagonal.

This augmented matrix, A, is of rank / by s + /, where / is the number

of f irst-order factors and s is the number of second-order factors. In the

current example, this augmented matrix

is

A3

x4

.

R e c a l l

that

a communality coefficient

(h

) indicates in an area-world

squared metric the percentage of variance in a measured variable

that

is

74

EXPLORATORY

A ND




TABLE

6.4

Augmented Matrix A

3 x 4

SECOND

0.773

0.792

0.832

U|

0.634

0.000

0.000

SORT of uniquenesses

Un

0.000

0.611

0.000

U

0.000

0.000

0.554

Note.

SORT (0.402) = 0.634.

containe d in the fa ctors as a set. The uniqueness ( i t ) i s an area-w orld

s tat i s t ic

ind ica t ing the percen tage o f var i ance in a measu red var i ab le

n o t

con ta ined in the fac tor s as a set (e.g., 100% - h

2

= u

2

, or 100% - 59.8% =

4 0 . 2 % ) .

Table

6.4 presents the m a t r i x , A^^ for this analysis .

The SPSS

sy n t ax

file for the

p resen t example

is:

MATRIX .

COMPUTE

PI =

{0.040, 0.936,

-0.029

0.917,

0 031

0.751, -0.026

0.900, 0.050

0.037

0.037

0 . 944

0.805

0.198

-0.070,

0.103,

-0.014,

0.947,

-0.061,

0.933, -0.049,

0 . 003, 0.003

0.202,

-0.070

-0.121,

0.700

;

-0.026,

0.010,

0 851

;

-0.096, 0.165,

0.753}

.

COMPUTE

P2 =

{0.773 ;

0.792

;

832}

COMPUTE

Pvh = PI * P2 .

COMPUTE

A =

{0.773, 0.634, 0.000, 0.000 ;

0.792, 0.000, 0.611, 0.000 ;

0.832, 0.000, 0.000, 0.554} .

COMPUTE SL = PI * A .

PRINT

PI /

FORMAT='F8.3' /

TITLED First-order Promax-rotated Pattern

Matrix

/

O B L I Q U E ROTATIONS

AND

H I G H E R - O R D E R

FACTORS 75



TABLE

6.5

Pattern Product Matrix (Pn

x 1

)

Variable SECOND

PER1 .748

PER2 .698

PER3 .653

PER4 .744

PER5 .715

PER6

.713

PER7 .735

PER8

.724

PER9 .640

PER10 .696

PER11 .683

TABLE 6.6

Schmid-Leiman Solution for Example 1

First-order

Variable

PER1

PER2

PER3

PER4

PER5

PER6

PER7

PER8

PER9

PER10

PER11

SECOND

.748

.698

.653

.744

.715

.713

.735

.724

.640

.696

.683

FACTJ

.025

-.044

.065

-.009

.600

.592

.598

.510

.126

-.016

-.061

FACTJ I

.572

.560

.459

.550

-.037

-.030

.002

.123

-.074

.006

.101

FACTJ

II

-.016

.017

-.014

.028

.020

.020

.002

-.039

.388

.471

.417

x mple 2

The second

example invo lves perceptions

of

l i b ra ry s e rv i ce qua l i t y

provided by

61,316

u n i v e r s i t y

students

and facul ty us ing the 25

LibQUAL+ i tems presen ted in

Table

6.7. Four first-order fac tors were

extracted. For heur is t ic purposes on ly , in this analysis two

second-order

fac tors

were extracted. The

Schmid-Leiman

so lu t ion is reported in Table 6.8.

The richness of higher-order fac tor analy s is can be communicated by

i n t e r p r e t i n g these results .

The two

second-order fac tors measure Lib rary

as

Place ( SEC_B )

and all

other perceptions

as

SEC_A.

H o w e v e r , the

nine

Service Af fec t and the six

Personal

Control i t ems pa r t i cu l a r ly

dominate the first second-order

fac tor .

O B L I Q U E

ROTATIONS

AND H I G H E R - O R D E R

FACTORS

77



TABLE 6.8

Schmid-Leiman

Solution

fo r Example 2

(n

61,316 Students

and

Faculty)

Second-order

First-order

Variable

PR1

PR4

PR11

PR14

PR15

PR17

PR18

PR20

PR24

PR2

PR10

PR13

PR21

PR23

PR 3

PR8

PR9

PR19

PR22

PR5

PR6

PR7

PR12

PR16

PR25

Sum of

squares

SEC_A

.655

.624

.690

.635

.660

.704

.713

.610

.689

.200

.206

.262

.328

.260

.526

.566

.559

.416

.512

.628

.685

.708

.736

.761

.625

8.531

SECJ3

.153

.201

.260

.265

.256

.234

.216

.294

.310

.754

.820

.771

.705

.824

.281

.169

.274

.324

.406

.106

.239

.129

.202

.231

.344

4.312

FACTJ

.422

.450

.291

.403

.469

.366

.430

.382

.378

-.025

-.028

.028

.035

.005

-.060

.048

.033

.037

-.052

-.041

.000

-.039

.061

.103

.079

1.494

FACTJ

I

-.011

-.001

.003

.009

.005

-.006

-.008

.006

.004

.124

.132

.117

.095

.120

-.009

-.018

-.005

.014

.009

-.002

.012

-.001

.003

.003

.010

0.071

FACTJII

.024

-.012

.072

.005

-.022

.020

.004

-.057

-.029

.045

.025

.001

-.023

-.043

.010

.072

.027

-.021

-.044

.364

.306

.399

.306

.260

.068

0.576

FACTJV

-.045

-.062

.041

-.038

-.053

.047

.007

.054

.077

-.064

-.042

-.014

.086

.067

.419

.254

.308

.247

.449

-.019

.033

-.016

.012

.044

.228

0.694

set printback=listing .

DATA LIST

records=3 /

ROWTYPE_ (A8) FACTOR— (F2.0)

Prl Pr4 Prll Prl4 Prl5 Prl7 Prl8 Pr20 Pr24

Pr2 PrlO Prl3 Pr21 Pr23

Pr3 Pr8 Pr9

Prl9 Pr22

Pr5 Pr6 Pr7 Prl2 Prl6 Pr25

(1X,9F5.3/11F5.3/5F5.3).

BEGIN

DATA

FACTOR 1 .655 .624 .690 .635 .660 .704 .713 .610 .689

.200 .206 .262 .328 .260 .526 .566 .559 .416 .512 .628

.685 .708 .736 .761 .625

OBLIQUE ROTATIONS

AND HIGHER-ORDER FACTORS

79



TABLE 6.9

Equamax-Rotated Product Matrix

n =

61,316 Students and Faculty)

Variable

PR1

PR4

PR11

PR14

PR15

PR17

PR18

PR20

PR24

PR2

PR10

PR13

PR21

PR23

PR3

PR8

PR9

PR19

PR22

PR5

PR6

PR7

PR12

PR16

PR25

A

.653

.622

.688

.632

.658

.702

.711

.607

.686

.193

.198

.255

.321

.252

.523

.564

.556

.413

.508

.627

.683

.707

.734

.759

.622

Second-order factor

B

.159

.207

.266

.271

.262

.241

.223

.300

.316

.756

.822

.773

.708

.826

.286

.174

.279

.328

.411

.112

.245

.136

.209

.238

.350

FACTOR 2

.153 .201 .260 .265 .256 .234 .216 .294 .310

.754 .820 .771 .705 .824 .281 .169 .274 .324 .406 .106

.239

.129 .202 .231 .344

END DATA.

FACTOR

MATRIX=IN(FAC=*)

/

ANALYSIS=prl to pr25 /

PRINT=ROTATION /

CRITERIA=ITERATE(75) /

ROTATION=equamax

.

T he

equam ax- ro ta t ed pa t t e rn / s t ruc tu re coe fficients

for the

analysis

are

reported in Table 6.9. In the present example fac tor in terpre ta t ions would

be l i t t l e a l te red by this rotat ion.

O f course , th e produc t m a t r ix would neve r be obliquely rotated. That

would

imply the exis tence of higher-order fac tors . In conduc t ing ob l iqu e

80 E X P L O R A T O R Y AND C O N F I R M A T O R Y FACTOR ANALYSIS



rotations, always extract

the

progressively higher-order factors imp l ied

hy

nonzero inte rfacto r correla t ions, unt i l (a) there is only a single higher-orde r

factor o r (b) the factors at the highest level of analysis have s imp le stru ctu re

when ro ta ted or thogonal ly .

M A J O R

CONCEPTS

If

s imple s t ruc ture cannot be obtained with orthogonal rotation, re-

flected in

m any var iab les having p a t t e rn / s t ruc ture coef fi c ien t s tha t

are

large

in

magni tude

on two or

more factors, oblique rotation

may be

needed

to

derive interpretable factors . However, more parameters must

be

est imated

in

obl ique rotat ion. As a consequence, obl iq ue resul ts not o nly (a) tend to

fit sample data better but also (b)

tend

to yield factors that replicate less

well .

As a consequence, unless obl iqu e resul ts are substa nt ia l ly

be t te r

than

orthogonal resul ts , or thogonal resul t may be pre fe r red .

Whenever obl ique

ro ta t ion is

perform ed, higher-order factors

are im-

plied

and should be derived and interpreted . The Schmid-Leiman so lu t ion

is

part icular ly useful in such cases. T he

SPSS

syntax p resented in the chapter

invokes

SPSS

MATRIX p rocedure s and can be used as a m odel to perform

these analyses using this widely ava i l ab le software.

OBLIQUE ROTATIONS AND HIGH ER-O RDER FACTORS 81



7

SIX TWO-MODE TECHNIQUES

Psychologists of ten think

in

terms

of

different types

of

people, such

as wo rkaholics or Type A personalities. Yet the factor analyses discussed in

previous chapters d o

n ot

address question s regard ing types of people Instead,

all the previous discussion

implicitly

involved investig ations of the structures

under lying variables.

The

previous analyses involved

a raw

data matrix

in

which people

were represented as data rows and the variables were organized as colum n

fields wi th in the data matr ix . When da ta are analyzed in this manner ,

intervariable relationships and factors of the measured variables are the

focus of the analysis.

But

data matrices

do not

h av e

to be

organized

in

this manner.

For

example, data could be organized such that variables define the rows and

people define the columns. When da ta are organized in this ma nner, interper-

son relationships are the focus o f the analy sis, and people factors are the

result. Indeed, there

are

other data structures

that are

possible

as

well.

The

mathemat ics

of the

analysis remain

the

same across these varia-

tions.

The

computer package need

not

know what data features

are

being

correlated and analyzed

CATTELL'S "DATA BOX "

Raymond Cattell (1966a) sum ma rized six prim ary exploratory factor

analysis (EFA) choices in the form of his "data box." Cattell suggested that

83



data could involve (a) either one or more than one

variable,

(h) either one

or

more than

one person, and (c)

either

one or

more than

one occasion of

measurement.

These features

of the data are termed

modes.

Some data sets might involve several variables, several people,

and

several

occasions

of

measurement.

It is

theoretically possible

to do a

factor

analysis that simultaneously considers all three modes. Tucker (1966) pro-

vided some guidance on performing these analyses.

However, three-mode analyses

are

extraordinarily complex,

and

com-

monly used

software

does not support these methods. Thus, only two-mode

analyses are routinely seen in the li terature. Within each two-mode combina-

tion (e.g., variables and people, measured at only one occasion), there are

two

possible organizations

of the

data matrix (e.g., people

define the rows

and

variables

define the

columns,

or

variables

def ine th e

rows

and

people

define

the columns). Thus, there are six two-mode techniques. Each two-

mode technique

was

labeled

by

Cattell using

a

d i fferent

capital letter (e.g.,

R Q).

Six Two-Mode Techniques

R-Technique

When

people

define the

rows

of the

data matrix,

and

columns

are

used to represent different variables, the analysis examines the structure

underlying variables. This analysis is termed R-technique factor analysis. All

of

the

previous examples

in

this book were R-technique examples. This

is

the most commonly used two-mode EFA technique, and is so common

that people invoking this analysis often

do not

even label

the

analysis

R-technique.

Q-Technique

When

variables

define

the rows of the data matrix, and columns are

used to

represent

different

people,

the

analysis

identif ies

people factors

(rather than factors of variables).

This

analysis is termed Q technique factor

analysis. Q-technique is the

second

most

widely

used EFA method.

P-

Technique

When

variables

are

factored,

and

occasions constitute

the

rows

of the

data matrix,

th e

analysis

is

called

P-technique

factor analysis. P-technique

m ay involve a single-person study. However, the analysis might also involve

means, or medians, summarizing for each variable and each occasion all the

scores as a single number.

84 EXPLORATORY

AND




of categories with each category co ntain ing exact ly a p rede termined num ber

of objects. The sorted o bjec ts can be anyth ing (e.g., ind ex cards conta ining

statements, photographs

of

ar t ) .

The

sorting criteria

can vary

(e.g.,

how

much participants agree or disagree with the statements, aesthetic valuing

of

the

ar t) .

Typically

the fixed single distributio n is developed to be sym me trical

and

quasi-normal

in

shape.

An

exam ple might involve sorting index cards

containing statements into categories

1

(m ost agree) through

7 most disagree)

using the

following

dis t r ibut ion:

Category 1 2 3 4 5 6 7

n

o f

cards

2 4 6 9 6 4 2

In th is example

every

column of data (i .e. ,

every

person's data) would

involve exactly two 1's, exactly

four

2's, and so forth. The mean of

every

column would be 4-00

SD =

1.58) with a coefficient of skewness of 0.00

and a

coefficient

of

kurtosis

of

— 0.50 .

Ranking

In a ranking strategy par ticipa nts are instructed to ra nk order the

objects

(e.g., index cards with statements) using some criterion (e.g.,

from

most agreem ent to most di sag reem ent). The appe al of this strategy is that

the method yields grea ter score va rian ce. Because more in form atio n is being

collected, this,

in

tu rn , theore t ica l ly

will

yield more sensi t ive

or

stable

interperson correlations,

and

thus better person factors.

The

downs ide

of

the strategy is that participants may find rank ordering 60 or 100 objects

to be

tedious

and

painful.

Mediated Ranking

Thompson (1980a) suggested

an

alternative two-stage strategy

fo r

col-

lecting ranked data that makes the ranking process more manageable for

participants. First, participants are asked to perform a conventional Q sort.

Second,

the

part icipants

are

then asked

to

rank order

the

objects with in

each

category. This means th at fewer objects must

be

simultaneously considered

by

the part icipants when performing a given ranking task.

euristi

Example

The

hypothetical data presented

in Table 7.2 can be

used

to

make

concrete the discussion of Q-technique interpretation issues. The example

presumes that five students and four faculty rank ordered 20

features

of

l ibrary

service quality

from 1

(most important)

to 20 least

important).

Table

88 EXPLORATORY AND CONFIRMATORY FACTOR

ANALYSIS



TABLE

7.2

Q Technique

Heuristic

Data

S t u d e n t

F a c u l t y

I t e m

1 2 3 4 5 1 2 3 4 I t e m c o r e

1 1 1 1 6 1 3 1 1 1 1 2 0 1 0

2 0 S A E m p l o y e e s w h o i n s t i l l c o n f i d e n c e i n u s e r s

2 1 6 1 3 1 6 1 6 1 6 7 7 9 1 I A C o n v e n i e n t b u s i n e s s hours

3

1 3 1 1 1 1 1 3 8 1 1 1 7 I P S p a ce t h a t f a c i l i t a t e s q u i e t s tu d y

4 8 8 7 8 1 3 9 9 2 0 9 S A E m p lo y e e s d e a l u s e rs i n a c ar in g f a s h i o n

5 1 2 1 2 1 1 2 1 2 1 0 2 0 1 0 1 0 S A E m p l o y e e s u n d e r s t a n d n e e d s o f t h e i r u s e rs

6 7 7 8 1 7 8 8 1 6 8 I A

C o m p l e t e r u ns

o f

j o u r n al t i t l e s

7 1 1 1 2 9 1 1 6 1 6 8 1 7 S A G i v i n g u s e r s i n d i v i d u a l a t t e n t i on

8 1 5 1 7 1 5 1 5 9 3 3 3 1 6 S A

E m p l o y e e s

w h o a r e

c o n s i st e n t l y c o u r te o u s

9 9 9 9 7 1 5 1 7 1 3 1 7 3 I A

C o m p r e h e n s i v e p r i n t c o l l e c t i o n s

1 0 4 4 4 4 4 1 2 1 2 1 3 1 2 S A

E m p l o y e e s k n o w l e d g e

t o

an s w e r u s e r q u e s t i o n s

1 1

1 7 1 5 1 7 1 7 6 1 3 1 7 1 2 1 3 P C li b we b

s i t e e n a b l i n g

m e

l o c a t e i n f o

o n m y o w n

12 6 6 3 6 17 4 4 6 4 LP A

contemplative environment

1 3

3 3 6 3 3 6 6 4 6

L P A

p l a c e

f o r r e f l e c t i o n a n d

c r e a t i v i t y

14 14 10 14 19 14 15 15 11 14 1A Timely document delivery interlibrary loan

1 5 1 0 1 4 1 0 1 8 1 0 1 4 1 1 1 4 1 5 S A R e a d i n e s s t o r e s p o n d t o users q u e s t i o n s

1 6

2 0 2 0 18 10 1 9 1 1 1 4 1 5 1 1 P C

C o n v e n i e n t a c c e s s

t o

l i b r ar y c o l l e c t i o n s

1 7

1 9 1 9 2 0 1 4 2 0 5 5 2 5 L P A h a v e n f o r q u i e t a n d s o l i t u d e

1 8 2 5 2 2 5 2 2 5 2

L P A

c o m f o r t a b l e

a n d

i n v i t i n g l o c a t i o n

1 9

5 2 5 5 2 1 9 1 8 1 9 1 8 S A W i l l i n g n e ss t o h e l p u s e rs

2 0

1 8 1 8 19 2 0 18 1 8 1 9 18 1 9 I A

I n t e r d i s c l i b r a r y n e e d s b e i n g a d d r e s s e d

7.3 presents

th e

v arima x-rotated principal components pattern/structure

coefficients from

this

analysis.

The f i r s t question addressed by the analysis is "How many types of

people

are

there?" Here

th e

answer

was

two. Presumahly

th e

decision

to

extract two factors was guided hy various evalua tion criter ia (e.g., eigenvalues

greater than 1.0, scree test, parallel

analysis)

as it would he in any

other

EFA.

TABLE 7.3

Varimax Rotated Pattern/Structure Coefficients

Person

STUD1

STUD2

STUD3

STUD4

STUDS

FAC1

FAC2

FAC3

FAC4

Factor

I

96

94

85

8

75

03

07

.09

13

II

02

.01

12

23

.14

93

9

74

76

ote

Values greater than |.40|

are

presented

in

italics.

SI X TWO-MODE TECHNIQUES 89



TABLE 7.4

Salient Factor Scores

for

Factor

I

Sorted

by

Descending Levels

of Agreement

Factor score

Item

I II

SPSS variable label

19

18

13

10

7

16

20

17

-1.37

-1.35

-1.27

-1.27

-1.12

1.32

1.48

1.60

1.66

-1.51

-.90

39

96

26

1.52

-1.33

SA

Willingness to help users

LP A

comfortable

and

inviting location

LP A

place

for

reflection

and

creativity

SA Employees knowledge to answer user

questions

SA

Giving users individual attention

PC

Convenient access to library collections

IA Interdisc library needs being addressed

LP A haven for quiet and solitude

The

second

question addressed is

Which

people belong to the different

types?"

Here the five students saturate Factor I, and the four faculty saturate

Factor II. The pattern/structure matrix has simple structure. Would that

real results were always so

def in i t ive )

The third question addressed is Which variables were the basis for

delineating

the

different person

factors?" To

address this question, factor

scores are computed for each variable on both the person factors. These

factor scores are in ?-score

form,

as are all factor scores except standardized,

noncentered

factor scores.

The

variables with factor scores less

than

-1.0

or

greater

than

+1.0

are more than one standard deviation in distance from the factor score

means

of

zero.

These are the

items

of

greatest

importance or

least importance

primari ly to the persons defining the two factors. Table 7.4 presents all the

factor scores for Factor I, the student person factor, for

which

factor scores

were greater than |1.0|. Table 7.5 presents all the factor scores for Factor

I I , the

faculty

person factor, for which factor scores were greater than

|1.0|.

TABLE 7 .5

Salient Factor

Scores for

Factor

II

Sorted

by

Descending Levels

of Agreement

Factor score

Item I II SPSS variable label

3

18

17

12

20

19

26

-1.35

1.60

-.55

1.48

-1.37

-1.55

-1.51

-1.33

-1.28

1.52

1.66

LP Space that facilitates quiet study

LP A comfortable and inviting location

LP A haven for quiet and solitude

LP A contemplative environment

IA Interdisc

library needs being

addressed

SA Willingness to help users

90

EXPLORATORY

AND

CONFIRMATORY FACTOR

ANALYSIS



In the present exam ple, lower rankings (e.g., 1, 2) indicated items

ranked most important.

And all the

noteworthy pattern/structure coefficients

in

Table

7.3 are

positive. This means that factor scores

in

Tables

7.4 and

7.5 that

are

smallest (i .e. , biggest nega tive value s) were items deemed most

important, whereas factor scores

that

are

largest were items deemed least

important.

The

factor scores

for

Factor

I

presented

in Table 7.4

suggest that

the

s tudents were most concerned that library staff are service oriented and tha t

the

library

is a comfortable place for reflection. At the other extreme,

the

students were least concerned about

the

library being quiet

and

about

library collections.

The

factor scores

fo r

Factor

I I

presented

in

Table

7.5

suggest

that th e

faculty

were most concerned about

the

library being quiet

and

least con-

cerned about

the

helpfulness

of

library

staff and the

interdisciplinary needs

of users being ad dressed.

These

hypothe t ica l

faculty

only care about w hether

the library coverage in their own d isciplines is ad equate

Variables with large factor scores and different signs across

factors

reflect

w hat v iews

part icularly differentiate

the two person

factors.

For exam-

ple,

s tudents want l ibrary

staff to be

helpful

(F[ = -1.37),

whereas this

is

less

important

to

faculty

(Fu =

+1.66). However,

both

s tudents

and faculty

w an t

th e

library

to be

com fortable

(Fj =

-1.35

and FH = — 1.51),

even though

they disagree wh ethe r this in volv es quie t

(Fj =

+1.60

and FU = -1.33).

MAJOR

CONCEPTS

In

many research situations, factoring variables

via

R-technique

ad-

dresses im port ant issues, such

as

bet ter unders tand ing

th e

nature

of

intelli-

gence or

self-concept.

But both

people

and

t ime

can

also

be

factored.

Factoring people

is

often

of

interest

in

psychology, because psychological

theory often is abou t type s of people (e.g., Type A perso nalitie s).

Q-technique factor analysis is the second-most comm only used EFA

method , after R-technique.

In

Q-technique, d eciding

how

many people

factors

to

extract addresses

the

question "How many types

of

people

are

there?" across

th e

variables used

in a

given analy sis. Persons' pa ttern

or

s t ructure

coefficients large

in

m a g n i t u d e

(plus or

min u s )

on a

given facto r

can be

examined

to

de te rmine

Which

factored people belong

to a

given

type?" And the

factor scores that

are

large

in

magnitude

can be

examined

to

de te rmine

Which

variables were

the

basis

for

del ineat ing

the different

person

factors?"

SIX

TWO-MODE TECHNIQUES

91



8

CONFIRMATORY ROTATION A N D

FACTOR INTER PRETA TION ISSUES

There are

var ious s i tua t ions

in

which

the

researcher

m ay

w a n t

to

evaluate

how

we l l

tw o

di f feren t

factor structures

f it

each other.

J u s t

as

unrelated factors c a n n o t be readi ly in terpre ted, as reflected in the Table

3.6

results (see chap.

3),

factors across samples often can n o t

be

accu ra te ly

compared unless the two structures are compared analytically using Procrus-

tean rota t ion methods.

These m eth o ds rotate one st ruc ture to a best-fit posi t ion w ith a

second target structure.

If the two

factor matrices

do n o t

match under these

condi t ions, they can n o t

f it

u n d e r

an y

other condit ions,

and the

s t ruc tures

s imply

are

incompat ib le .

Some

example applications

of

these methods

in -

c l u d e

empir ica l ly evaluat ing

1. w h e th e r the factor pattern /struc ture matrices for a given set of

variables across di f feren t samples

are

invariant across samples;

2 .

whether there

are

substantively based differences

in

s t ruc ture

fo r a set of

i tems across groups (e.g. , graduate students

vs.

facul ty) ;

3. whether the st ruc ture is replicable using in ternal replicabil i ty

( T h o m p s o n , 1994, 1996) cross-validation methods (i .e . ,

r andomly sp l i t t i ng

the

sample

to

in form judgment s abou t

replicabil i ty); and

93



4 - the factor adequ acy of an obtained pa ttern /stru ctu re m atrix

by

eva lua t ing

its fit to a

theoretical ly expected target stru ctu re

consist ing of

ones

(o r negative ones) and zeroes (Thompson

&

Pit ts, 1982).

Thompson (19 92 b) provid ed a part ial stat ist ical significan ce test d istr ibu tion

for evaluating these factor correlations across samples.

BEST-FIT

ROTATION

E X A M P L E

For heur is t ic purposes,

th e

invariance

of the

pr inc ipal components

var imax-rota ted pa t tern /s t ruc ture coeff ic ients involv ing percept ions of li-

brary

service qual i ty w i th regard to 11 var iables (PE RI through P E R U )

wil l

be evaluated. T he comparison involves th e st ruc ture for 100

facul ty

reported in T ab le 5.5 (see chap. 5) being compared wi th the corresponding

s t ruc tu re

of 100 graduate s tudents (see Append ix A). The gradu a te s tuden t s'

s t ruc tu re wi l l be

used

as the

target matrix

for

this analysis.

Ve ldman (1967 , pp .

2 38 — 2 4 4)

expla ined the mathemat ica l theory

under ly ing

the method . H e

noted that

th e cosines among the factors across

the two

solut ions equal

the

factor correlat ions across samples.

H e

also no ted

that this matrix of factor cosines can be used to rotate a second mat r ix to

its best-fi t posi t ion with a target matr ix .

The cosine m atri x is comp uted using the m atrix algebra fo rmu la:

C j

x

i = ( A ' i

x

K B

K x

j ) ' j

x

|

V j

x

j

E

A x

j ~ • V ' j

x

| ,

where I = the n u mb e r of fac tors in the A target pat tern matr ix; J = t he

num ber of fac tors in the B pat tern matr ix; and K in the nu mber of var iables

in the two analyses. This algori thm requires first the computa t ion of

Q i x i =

A ' |

x K

B

K x

j ( A '

l x K

B K

X

J ) J

X

I .

The matr ix V |

x

j is the mat r ix of pat tern coefficients extracted from Q [

x

j The

matr ix E

A x

j has

zeroes

off the

d i a g o n a l

and on the

diagonal

th e

e igenvalues

of

Q i x i .

This analysis can be conducted in SPSS using th e syntax presented

in

Ap p e n d i x

B. By the

way, this syntax invokes

th e

MATRIX procedures

avai lable in SPSS, i f one learns how to type syntax rather than execut ing

SPSS via point and c l ick. In App end ix B, a l l the syntax between the

MAT R I X c o mma n d a nd t he EN D M AT RIX command execu tes ma t r ix

algebra. T he adventurous reader can access an SPSS ma n u a l to discover

how to command

SPSS

to execute any matr ix procedure (e.g., transpose,

inversion, mul t ip l ica t ion)

o r

matr ix formula .

9 E X P L O R A T O R Y A N D CONFIRMATORY FACTOR ANALYSIS



TABLE

8 1

Factor Correlations (Cosines)

Student

target

F CTJ

F CTJI

F CTJII

F CTJ

-.007

1.000

.022

Faculty matrix

F CTJI

1.000

.006

.024

F CTJII

-.024

-.022

.999

Table 8.1

presents

the

factor correlat ions across

the two

samples .

The

resu l t s indicate that the factors were h ighly congruent or invariant across

samples (e.g. , r^

x

u g = 1.00), al though the order of the factors was

d i f f e r en t

in the two

samples (e.g. , Factor

I in the

s tuden t sample

w as

Fac to r

II in

th e facu l ty

s a m p l e ) .

Table 8 .2

presents

th e

fac to r pa t t e rn / s t ruc tu re m at r ix

of the 100 f acu l ty

after

rotation to

best

fit

w i t h

the

s t ruc tu re

of the 100

gradu ate s tudents .

The s t ruc tu res are very similar across the two groups in th is data set .

The p rogram also produces cosines ( i .e . , here correlations) betw een

the

var iables

in the

target pat tern /s t ructure matr ix

and the

var iables

in the

best-f i t ro ta t ed pa t t e rn / s t ruc tu re c oe f f i c i en t ma t r ix . These cosines among

variables should all be reasonably large to ind ica te

that

the two solutions

fit reasonably wel l at the level of each of the var iables , and not jus t at the

factor level . If some var i ab les ' cosines are smaller , these are the var i ab les

that

could

not be

related wel l

in the

best- f i t

rotation.

For

th e

Table

8 .2 d a t a , th e lowest of the 11 va r iables ' cosines was .94

for variable PER9, and the 10 remaining

cosines

were all .98 or larger.

These

resul t s con f i rm fac to r f it even across the 11 measu red var i ab les .

TABLE

8 2

Pattern/Structure Matrix

for 100

Faculty Rotated

to

Best-Fit Position With

th e

Pattern/Structure Matrix

for 100

Graduate Students

Factor

Variable

PER1

PER2

PER3

PER4

PER5

PER6

PER7

PER8

PER9

PER10

PER11

FACT_I

.890

.868

.726

.866

.115

.124

.167

.315

.087

.209

.321

FACT_II

.204

.109

.231

.164

.9 04

.892

.905

.792

.323

.166

.105

F CTJII

.213

.239

.185

.266

.240

.239

.223

.170

.677

.801

.730

C O N F I R M A T O R Y R O T A T IO N

A N D

F A C T O R I N T E R P R E T A T I O N

95



F A C T O R

I NTER P R ETATI ON CONSIDERATIONS

These

results afford

an

opportunity

to

state some general principles

of

resul t

interpretation. The statement of these principles requires speci fy ing

which

aspects

of

resul t s

are and are not

relevant

to the

interpretation process.

Factor Order

Both

hefore and

af ter

rotation, factors are ordered by the amount of

information they

contain.

Prior to rotation, the eigenvalues characterize

the amount of information

contained

in the unrot ted factors. For example,

if the first eigenvalue associated with the first principal

component

was 3.0,

and 11 variables were included in the R-technique analysis, the result

indicates that 27.2% (3.0 / 11) of the variance in the variables can be

reproduced

by the first

component.

Factor rotation redistributes the variance across the factor in service

of pursuing simple structure. The eigenvalues are not relevant to evaluating

the

information contained

in

particular rotated factors. Computer packages

print sums

of

squares

to

address

the

issue

of the

information

contained

within specif ic rotated factors.

In

general,

the

order

of the

factors

is not of

interest

in

evaluating

the

solution.

For

example,

here the

analyst expected

three

factors

in both the

student and the f acu l ty samples: Service A f f ec t , Information

Access,

and Library as

Place.

These dimensions did emerge in

both

samples. No

particular factor ordering

was

theoretically expected

in

this analysis.

It

does

not

matter that

the

factors emerged

in a d i f f e r en t

sequence

in

these analyses.

In

some analyses

there

might

be an

expectation

that one

factor w i l l

dominate

the

factor space (i.e., contain

a

hugely disproportionate amount

of

variance).

Only

in such cases would a theoretical expectation regarding

f ac t o r

ordering

be relevant.

Reflection of Factors

Some factors involve pattern

an d

structure coefficients with

d i f f e r e n t

signs. Because most people

find it

easier

to think in

positive terms,

if the

larger

pattern

and

structure coefficients have negative signs

on a

given

fac tor , it is completely appropriate for the analyst to reflect the factor by

reve rs i ng the signs of the coefficients on any given factor (Gorsuch, 1983,

p. 181).

This

is

appropriate because psychological variables typically have

no

intrinsic scaling

direction. For

example,

an

achievement test could

be

scored

by

counting either

th e

number

o f

right answers

or the

number

o f

wrong

answers.

96


AND CONFIRMATORY FACTOR ANALYSIS



How ever , w hen a g iven fac to r i s r e f l ec t ed , al l the signs of al l the patte rn

and the s t r u c t u r e coef f ic ien ts on the fac to r mus t be rever sed . I t i s not

necessary to a c k n o w l e d g e the reflect ion process in descr ibing resu l t s . And

in a g iven so lu t ion , any c o m b i n a t i o n o f d i f f e r en t fac tors m ay b e r e f l e c t e d , '

i nc lud ing , o f course,

none.

Salience of Variables

It is common to see researchers invok ing some c r i ter ion to de te rmine

w hat va r iables are sal ien t to the in terp retat ion o f a g ive n facto r . V alues o f

pat tern /s t ructure coeff ic ien ts

of

|.3|, |-35|,

and

|.4|

are

used wi th some f r e -

q u e n c y . Such v a l u e s

m a y h e useful in

se t t i ng

th e

m i d p o i n t

o f

gaps b e tween

smal ler an d larger coeff ic ien ts o n a g iven fac to r .

Researchers

m ay

u n d e r li n e

o r

ita lic ize

th e

ent r ies

in

pa t t e rn / s t ruc tu re

matr ices that meet thei r cr i ter ia , to help readers see pat terns . Or researchers

m ay sort th e rows of the mat r i ces by the va lues of the biggest abso lu te

coef f ic ien ts on

d i f f e r en t

factors .

Some researchers blank ou t nonsa l i en t coe f f ic i en ts to make s imp le

st ructure more apparent . Unl ike i ta l ic iz ing

coef f ic ien ts

o r sor t ing factor rows,

omi t t i ng nonsa l i en t coe f f i c i en t s is n ot good pract ice . Doing so dep r ives

other

researchers of the oppor tun i ty to conduct secondary analyses , such as

i nvok ing d i f f e re n t ro ta t ion methods ,

o r

c o n d u c t in g best- f i t ro tat ion

o f

their

resul t s

w i t h yours .

It

is impor tan t not to ge t too rigid in determining sal ience cr i ter ia .

Use a sal ience cr i ter ion in a g i v e n s t u d y that does a good job for the g iven

resul t s in h igh l igh t ing s imp le s t ruc tu re .

Naming

actors

An impor tan t part of the interpretation process is n a m i n g the factors .

This

is

where

the art of the

fac tor analy t ic process

is

exercised . Factor names

are usual ly one- o r two-word phrases .

Factors should

never

be named af ter mea sured var iables . A fac tor i s

inherent ly an aggregat ion o f several measured factors . A factor ref lected in

only a s ing le measured var iable ( i .e . , a speci f ic factor ) is inheren t ly n o t

a t rue factor . Thus, a ll real fac tors invo lve mu l t ip l e var i ab les , and to th is

extent m u s t

be

n a m e d

in a

man ner r e f l ec t ing

th e

o veral l pat tern

o f

c o n t r i b u -

tion

o f di f fe ren t

variables

to the

factor ' s def in i t ion

(o r

conversely ref lect ing

th e unde r ly ing cons t ruc t 's ma n i fes t a t ion v ia the measu red var i ab les ) .

Factor naming also must be in fo rmed by the

d i f f e r en t i a l

sa tu ra t ion o f

a

g iven factor

by

g iven factored en t i t ies .

F or

example ,

in an

orthogonal

R -

technique analys is , i f one var iab le has a pat te rn /s t ru ctu re coeff ic ien t o f +.7

o n a

factor ;

and another

var i ab le

has a

pa t t e rn / s t ruc tu re coe f f i c i en t

o f

+ .5 ,

C O N F I R M A T O R Y R O T A T I O N A N D F A C T O R IN T E R P R E T A T I O N 97



do not forget tha t the first va r iab le has 49% of its va r iance in c o m m o n w i t h

the f ac to r , whereas the second measured var iab le has on ly 25% c o m m o n

variance . Thus, the

first

var iab le should have rough ly doub le in f luence on

selection of the factor label.

Fac to r names mus t be

suff iciently

broad to cap ture all of the p r i m a r y

relationships suggested

by the

pa t te rn

and

s t ruc ture coef f ic ients .

But there

is

ano the r cavea t

that

m u s t

be

m e n t i o n e d .

When

conduct ing fac tor analyses

and

com paring resul ts w i th those

in

pr ior re la ted s tud ies ,

it is

impor tan t

to not

p lace undue re l i ance

on

construct names assigned

by

previous researchers .

Resea rche rs may have nam ed the i r f ac to r s in ways tha t d o not seem ins igh t fu l

or even reasonable. If the fa ctor labels from pr ior s tud ies are accepted w i tho ut

scru t iny ,

inva l id compar i sons wi th re la ted p r io r f i n d i n g s m a y re su l t .

M A J O R C O N C E P T S

In

va r ious s i tua t ion s re sea rche rs may want to d e te rmine how we l l two

d i f f e ren t

f ac to r s t ruc tu res

for a

g iven

set of

fac tored ent i t ies (e .g ., var iab les

if

R - t ec h n i q u e

is

u s e d )

f it

each other. Empi r ica l me thods usua l ly m u s t

be

used for

this purpose , because ap parent

d i f ferences may be

method ar t i fac ts ,

and emp i r ica l me thods p rov id e a quan t i f i ca t ion o f degree o f s t ruc tu re cor re -

spondence.

The

A p p e n d i x

B

SPSS syntax

can be

used

as a

mode l

to

conduct

such best-f i t rota t ions .

In ev aluat ing correspondence of fac tors across samples or subsam ples ,

factor

order

is usual ly not a

cons idera t ion. Sim ilar ly ,

a

given fac tor across

samples

may be

or ien ted

in

oppos i te d i rec t ions , such that

the

coef f ic ients

for

a

fac to r

in one

sample mus t

be

ref lec ted

by

changing

a ll th e

signs

o f

a ll coefficients

on the fac tor . Factor or ienta t ions are comple te ly arbi t ra ry ,

but we

usually prefer

the

predominance

o f

l a r g e m a g n i t u d e

coefficients to

be

posit ive, because most people

find it

eas ier

to

th ink pos i t ive ly .

When

present ing fac tor s t ruc tures , researchers of ten highl ight

coeffi-

cients that they deem sa l ien t . Such salience criteria

are

usua l ly

set by

looking

for gaps be tween near-zero and no nzero co ef f ic ients (e .g . , absolute .4) in a

given solut ion. Sal ience cr i ter ia should not be used wi th excess ive

r ig id i ty .

Factors are synthet ic var iab les re f lec t ing un de r lying la tent cons tructs ,

and therefore should never be named af ter measu red var iab les . Eq ual ly

impor tan t , when read ing p r io r re sea rch ,

do not

accep t

the

factor labels

of

others at

face

va lue . When conduc t ing either sub jec t ive or empir ica l fac tor

meta-analyses (Thompson, 1989), focus on the inva r iance of the f ac to r

meanings across s tud ies , notwiths tand ing

th e

labels assigned

by

d i f f e r en t

researchers.

98 E X P L O R A T O R Y AND C O N F I R M A T O R Y F A C TO R

A N A L Y SIS



9

I N T E R N A L REPLICABILITY ANALYSES

In

quantitative research,

all

statist ical methods, including factor analy-

sis,

are susceptible to the influences of

three

kinds of errors: sampling error,

measurement error,

and

model specification error.

As

noted

in

chapter

4,

sampling error

is

that portion

of

variance

in

sample data

that is

unique

to

a given sample, and in

effect

reflects the personality of a given sample.

B y

definition, the sampling error in a given sample does not exist in the

population.

e surement

error is the

variance

in

data that

is due to

inherent

imperfec t ions in measurement (Thompson, 2003). We estimate these inf l u -

ences by computing ratios of reliable variance to the total observed score

variance; these ratios

are

called reliability

coefficients. In

every study

it is

important to estimate the reliability of the data in

hand

in the study and

not

merely cite reliability coefficients

from

the

manual

or

prior studies

(Vacha-Haase, Kogan, &

Thompson,

2000). As Wilkinson and the Task

Force on Statistical Inference (1999) emphasized:

It

is important to remember that a test is not

reliable

or unrel iable. . . .

Thus ,

authors should provide

re l iabi l i ty

coefficients

of the scores fo r

th e data being analyzed even when th e focus of their research is not

psychometric , (p.

596)

Model specification error

is the

failure

to use all and

only

the correct

variables,

or the f a i l u r e to use the correct statistical analysis, or

both,

thus obtaining

99



incorrect resul ts . Of course, as Ped haz ur (1982 ) has noted, The rub , how -

ever,

is

that

the

t rue model

is

seldom,

if

ever, known

(p.

229 ) .

And as

Du ncan (19 75 ) has noted, Indeed it wou ld req u ire no elaborate sophistry

to show that we

wi l l

never have the 'r ight ' model in any absolute sense

(p.

101).

Sampling error is par t icular ly pernic ious in mu l t ivar iate analyses , be-

cause more complex analyses afford more opportunit ies fo r these influences

to be

m anife s ted . With respect

to

exploratory factor analys is (EFA ), Gorsuch

(1983) noted, In factor analysis , one has nu merou s poss ibi l i t ies f or

capital iz-

in g on chance. Most extraction procedu res, inclu ding principal factor

solu t ions ,

reach the ir criterion by such capital iz at ion. The same is tru e

of rotat ional procedures, including those that rotate for simple structure

(p. 330) .

There

are two

major ways

of

empir ical ly address ing resul t repl icabi l i ty

quest ions: external repl icabi l i ty analyses an d internal repl icabi l i ty analyses

( H u b e r ty , 1994;

Thompson,

1996). Ex ternal repl icab i l i ty s tudies involve

collect ion of independent data from different par t ic ipants . These are true

replicat ion studies and provide the best evidence of resul t repl icabi l i ty by

direc t ly

and exp l ic i t ly address ing repl icabi l i ty concerns .

However, many researchers

find

themselves unable

to

a lways conduc t

external

repl icabi l i ty s tudies . In these situat ions researchers may use some

combination of three major logics for internal repl ica bi l ity analyses: cross-

val idat ion, th e jackkn i fe , and the bootstrap (Thompson, 1993b, 1994).

These

three methods combine the

subjects

in hand in

different ways

to

dete rmine wh ether resul ts are stable across sample variat ions, that is, across

th e idiosyncracies of in d iv id u a l s that make general izat ion in social science

so challeng ing (Thomps on, 1996,

p.

29 ) .

Here both EFA cross-validation and bootstrap methods wil l be il lus-

trated. These are, respectively, the simplest and the most complex internal

replicabil i ty methods. Jac kk nife applicat ions have been discussed elsewhere

( H u b e r t y ,

1994).

Across the three major internal re plic ab il ity analys is choices it is

important to remember that none of the methods is as useful as true replica-

t ion

s tudies .

A s

Thompson

(1993b) noted, because

all three strategies are typical ly based on a single sample of subjec ts in

w h i c h th e sub jec t s u s u a l l y

h a v e

m u c h in common (e .g. , point in t ime

of

measu rement , geographic or igin) re lat ive

to

what they wou ld have

in common

w i t h

a separate

sample ,

the three

methods

a l l

y ie ld somewhat

inf lated

es t imates of repl ica bi l i ty . Because inf lated es t imates of repl ica-

b i l i t y

provide a bet ter es t imate of rep l icab i l i t y than no estimate at all

( i .e . , s t a t i s t i ca l s ign i ficance t e s t ing) ,

these

procedures can st il l b e usefu l

in

focusing on the sine

qua non

of science, (pp. 368-369)

00 EXPLORATORY A ND

C O N F I R M A T O R Y F A C T O R ANALYSIS



Cross-validation and bootstrap applicat ions wil l be il lustrated here using

for heu r is t ic purposes the first 30 cases of data and var iab le s PE RI through

PER8 from Appendix A .

EFA

CROSS-VALIDATION

Cross-validation methods executed

as

internal repl icabi l i ty analyses

randomly split the data into tw o subgroups. Analyse s are conducted sepa-

rately

w ithin each sub group, and then an empirical analys is must be execu ted

to quant i fy the degree of fit across the two analyses. These methods have

been used across the general l inear model ,

from

reg ression to canonical

correlation analyses (cf. Grossman, 1996;

Thompson,

1994). The methods

are

logical ly straightforward

and do not

requ ire specialized sof tware .

Table

9.1

presents

th e

cross-validation su bgrou p assignments ( 1

or

2 )

for the 30

cases produced

by

flipping

a

coin.

For

logistical purposes

as

is typical ly done, becau se keeping t rack

of the

subgroups

is

thereby faci l i ta ted,

here the 30 cases were split 16/14 rather

than

15/15. Table 9.2 presents th e

pat tern/s t ruc ture matr ices

from

principal components analyses rotated to

the varimax criterion for both subgroups.

T he best-f i t rotat ion procedure must then be invoked to al ign th e

factor s t ruc tures

to

opt imal

fit.

E ither so lut ion

can be

used

as the

target

f or

this analysis, al though some preference migh t be afforded to using the larger

sample as the target .

Table

9.2 also presents th e

second

solut ion rotated to

best-f i t

posi t ion w ith

th e

so lu t ion

for the f irs t

subgroup .

In

this

smal l

heur is t ic

example

th e

solut ions

in the two

subgroups

are

reasonably congru ent.

This is

reflected in the

cosines

of the two

solu t ions

vir tual ly

being

an

ident i ty matr ix.

The biggest disparity across the subgroups involves the variable PER8,

which in the second subgroup had pat tern/s t ruc ture coeff ic ients on Factor

of

.667

and on

Factor

II of

.596.

In the first

sub group, PER8

had a

t r iv ial

pat tern/s t ruc ture coefficient

on

Factor

I

( .158

2

=

2 . 5 % )

but a

huge va lue

on

Factor

II

(.840

=

70 .6%) .

This

dispari ty

is

also reflected

in the

cosines

for

the eight variable pairs

each

being greater than .94, except for PER8,

which had a cosine of .808.

Cross-validation can g ive some insight regarding potential resul t

repli -

cabi l i ty .

If

solu t ions wil l

n ot

replicate across sub groups

in an

inte rnal replica-

bi l i ty analys is involv ing a single sample,

then

replicat ion with a

t ru ly

inde-

pendent second sample seems part icularly un like ly.

And i t

should

be

emphasized that

the

fu l l

sample resul t s are a lways the basis for our interpre tat ions . T he

subsample resul t s are

only

employed as a vehic le for

exploring

th e

repl icabi l i ty

of the fu l l

sample results .

T he f u l l

sample results

are

used

INTERNAL

REPLICABILITY ANALYSES

101



TABLE

9.1

Cross-Validation

Random

Subgroup Assignments and Bootstrap

Resampling Assignments

C r o s s v a l i d a t i o n

Case

3

4

5

6

7

8

9

2

3

4

5

6

7

8

9

2

2

22

23

24

25

26

7

28

29

3

S u b g r o u p

Draw

3

4

5

6

7

8

9

2

3

4

5

6

7

8

9

2

2

22

23

24

25

26

7

28

29

3

29

8

3

24

3

7

5

6

22

2

26

3

4

5

2

22

5

26

4

5

4

8

24

22

25

R e s a m p l e

2

3

4

2

7

3

25

7

7

28

6

4

9

3

5

7

2

24

6

28

5

2

5

28

25

23

3

2

27

26

2

7

4

23

8

29

6

3

3

5

5

28

4

9

3

24

26

9

9

4

6

9

8

4

25

9

7

25

4

4

23

8

2

3

8

7

7

3

8

22

5

9

3

7

28

5

fo r interpretation, because we tend to vest more confidence in resu l t s

as sample

size

increases, and the fu l l s ampl e

analysis

involves

more

subjec ts

than do the s u bgr ou p ana l ys es . (Thompson,

1994,

p.

1 7 2 )

The subgroups are only used a s a vehic le to evaluate the possible replicability

of th e factor solution for the complete sample.

EFA

BOOTSTRAP

The bootstrap is an elegant computer-intensive internal replicab il ity

analysis conceptualized fo r several analyses by Efron and his col leagues .

Diaconis

an d

Efron (1983) provide

an

accessible art icle su mm ariz ing this

1 02 E X P L O R AT O R Y

A N D

C O N F f R M A T O R Y FA CTOR ANALYSIS



TABLE 9.2

Varimax-Rotated Factor Pattern/Structure Matrix for Tw o Subgroups

(n , = 16; n

2

= 14)

Subgroup 1

(target)

Variable

PER1

PER2

PER3

PER4

PER5

PER6

PER7

PER8

I

.896

.954

.784

.877

.376

.326

.261

.158

II

.340

.110

.474

.354

.792

.845

.906

.840

Subgroup

2

.929

.942

.809

.596

.345

.356

.228

.667

II

.214

.250

.496

.487

.898

.890

.926

.596

Subgroup 2

after

best-fit

I

.923

.935

.796

.583

.322

.333

.205

.657

II

.238

.274

.576

.502

.906

.899

.932

.613

Note. Pattern/structure coefficients greater than |.5| are presented in italics.

logic

in layperson's terms. Lunneborg's (2000) book provides a more techni-

cal and comprehensive view.

The

bootstrap

can be

conceptualized

as a

method

of

using

the

sample

data

in

such

a

manner that

the

sample data

are

themselves used

to

estimate

the distribution of a large number of sample results from the population

(i.e., the sampling distribution).

This

process of the estimation pulling itself

up by its own

bootstraps explains

why the

approach

is

named

the

bootstrap.

One way of

explaining

the

technique, although somewhat

s impl i f ied ,

begins

by describing the creat ion of a mega-file created by copying the sample

data set of size n

onto

itself ( i .e . , concatenating) a ve ry large number of times.

Then repeated resamples

of

cases,

each of

exactly

size

n

are

drawn

from the

mega- f i l e ,

and in each resample the results are analyzed independently.

The

number

of

resamples

is u su a l ly at

least

1,000, although 2,000 to

5,000 resamples

are

recommended.

In a

given resample,

a

given case might

be

drawn

not at

all, once,

or

several times.

And

this case might

be

drawn

not at all, once, or several times in the next resample. After all the resamples

are drawn, mean (or median) parameter estimates (e.g., R

2

, factor pattern/

structure coef f i c ients ) are computed across the resamples.

The

standard deviations

of the

estimates

are

also computed. These

are

empirical (as

against theoretically based) estimates

of the

standard errors

o f

each of the

parameter estimates.

There

is

only

one diff icul ty in

applying

the

bootstrap

in

multivariate

statistics such as EFA: When more than one set of weights (e.g., factors)

can be

computed,

the

weights across

the

resamples must

be

rotated

to

best

fit

w i t h

a common target matrix prior to computing averages and standard

errors. This is because in the first resample a construct might emerge as

Factor I, but in the next resample might emerge as Factor II. Factors must

I N T E R N A L

KEPLJCABILfTY

ANALYSES

103



TABLE

9.3

First Four Eigenvalues From Four Resamples n =

30)

Factor

Resample

1

2

3

4

Mean

SE/SD

I

3.867

4.798

5.357

5.190

4.803

0.577

II

2.523

1.776

1.312

1.553

1.791

0.454

III

0.600

0.675

0.548

0.605

0.607

0.045

be rotated to

bes t - f i t

position so that averages of apples and apples are being

computed across

the

resamples.

Of

course, these mechanics

are not d i f f i cu l t

when they

are

automated,

and bootstrap applications across various mult ivariate analyses have been

reported

and recommended (Thompson, 1988, 1992a, 1995). In EFA the

bootstrap can be used to

inform

the decision of how many factors to extract,

to help evaluate the replicahility of the structure, or both.

Factor Extraction Bootstrap Procedures

The

bootstrap

can be

used

as

another procedure

to

decide

how

many

factors to extract, in addition to the methods described in chapter 3. In this

heuristic example only four resamples are drawn. This makes the s u m m a r y

manageable, but of course is not a realistic description of an actual boot-

strap analysis.

As reported in Table 9.1, in the first resample, the first case drawn

w as

Case #1. In each draw all the data for a given case are extracted into

the

resample.

Case #1 was

also drawn again

in

Draw

21.

In

each

resample the eigenvalues are computed.

Then

means of the

eigenvalues

and the standard deviations can be derived. Because the resam-

ples are being used to estimate sampling distributions, these standard devia-

tions are actually empirically estimated standard errors (i.e., SDs of the

parameter estimates

in the

sampling distribution).

Table

9.3 presents the first three of the eight eigenvalues

from

principal

components analyses

of the

data from

the four

resamples.

The

table also

presents

the

mean eigenvalues across resamples

and the

standard errors

of

these estimates.

For

example,

the

mean eigenvalue across

four

resamples

for

the second factor without rotation was 1.791 (SEj. = 0.454). The mean

eigenvalue across four resamples

for the

third factor

was

0.607 (SE^

=

0.045).

F or these results the extraction of two factors is suggested. If the third

eigenvalue mean

had

been 0.999,

and the

standard error

had

been small

104

E XP L ORA T ORY

A N D


FACTOR

ANALYSIS



bu t

nontr iv ial , the extraction of the third factor might have been warranted.

In other cases, even

if an

e igenvalue mean

is

slightly greater than 1.0,

a

relat ively

large standard error would

suggest

that not much fai th can be

vested

in

this eigenvalue consistently being greater than 1.0;

here

some

thought might he g iven to not extrac t ing the factor associated with this

re la t ively

smal l and unstable e igenvalue .

Pattern Structure Coefficients and Standard Errors

Bootstrap estimation of factor pattern/stru ctu re coefficients and their

stand ard errors requ ires the spe cifi catio n of a common factor space or targe t

to wh ich all resampled so lut ions can be rotated to a position of best fit.

The

s t r u c t u r e

in the

actual sample

can be

used

as the

target

for

this purpose.

Another

choice is a

theoretically expected factor structure.

In the

present example,

the

target matrix reflected

an

expectation that variables

P E R I

throug h PER 4 wou ld have +1.0 coefficients on Factor I , and zeroes

on Factor II. The target matrix also reflected an expectation that var iables

PER5 through PER8 would have +1.0 coefficients on Factor II, and zeroes

on

Factor

I.

Table 9 .4 presents the v ar imax-rotated pat tern/s t ruc tu re matr ices from

principal components analyses

in the

four

resamples. Note that

the first

factor

inc lu des var iables PE RI through PER4

in

Resamples 2

and #3 but

not

#1 and #4.

Table 9 .5 presents the four s t ruc tures each rotated to best-f i t posit ion

w i t h the target matrix. Table 9.6 summarizes the means of the paramete r

est imates

and

their standard errors.

Table 9 .6

also reports

th e

mean parameter

est imates d iv ided

by

their respective standard errors.

The ratio of a paramete r estimate to i ts standard error (SE ) is very

important in statist ics, and so this ratio is given m any names. These names

include critical ratio, Wald statistic, an d t. W e usual ly expect these t val ue s

to be

greater

than

roughly |2.0|

for

parameters that

w e

wish

to

interpret

as

being nonzero.

Overal l ,

the

heurist ic results,

if

they

had

been real , would have sup-

ported

a

conclusion that

the

fac tor s t ruc tu re

w as

reasonably replicable .

This

is reflected

in the fact that the em pirica lly estimated standard errors for

large

va lues

in

Table

9.6 are

quite small (e.g., .039, .009).

These

standard

errors indicate

that th e

noteworthy parameters

in the

model

do not

vary

mu ch across various configu rations of the people considered in the bootstrap

resamples. If we mix up our

sample cases

in

many

different

configurat ions

and

always

sti l l get the same basic factor structure, this would give some

hope that the

s t ruc ture

may

also replic ate across

the

configurat ion

of

par t ic i -

pants in a new sample .

I N T E RN A L RE P L I C A B I L J T Y A N A L YSE S 1 05



TABLE

9.4

Varimax-Rotated Pattern/Structure Matrix From Four Resamples n

K

= 30)

Factor

Variable

~i ll~~

Resample 1

PER1 -.066 .921

PER2 -.067 .958

PER3 .347 .836

PE R4 .251 .808

PER5 .847 .219

PER6 .904 .031

PER7 .917 -.021

PER8 .797 .135

Resample 2

PER1 .966 .136

PER2 .964 .130

PER3 .927 .266

PER4 .759 .431

PER5 .128 .927

PER6 .344 .784

PER7 .182 .845

PER8 .159 .775

Resample 3

PER1 .869 .297

PER2

.947 .127

PER3 .810 .493

PER4 .859 .292

PER5 .555 .670

PER6 .226 .899

PER7 .251 .882

PER8

.228 .825

Resample 4

PER1 .067 .947

PER2

.215 .933

PE R3 .485 .836

PER4 .541 .632

PER5

.922 .203

PER6 .911 .228

PER7 .878 .148

PER8 .788 .329

MAJOR

CONCEPTS

A ll statis tical methods, inclu ding factor analysis , are susceptible to the

influences of three kinds of errors: sampling error, measurement error, and

model specification error.

To

address

th e

influences

of

sampling error,

re -

searchers replicate resu lts wi th independ ent samples (i .e. , external replica-

/ 06 EXPLORATORY A ND CONFIRMATORY FACTOR ANALYSIS



TABLE 9.5

Four

Resample Solutions Rotated to Best Fit With Target

Factor

Variable

I II

Resample

1

PER1 .920 -.079

PER2 .957 -.080

PER3 .841 .335

PER4 .812 .240

PER 5 .231 .844

PER 6 .044 .904

PER7 -.008 .917

PER8 .146 .795

Resample

2

PER1 .968 .125

PER2 .966 .119

PER 3 .930 .255

PER4 .764 .422

PER5 .139 .926

PER 6 .354 .779

PER7 .192 .843

PER8

.168 .773

Resample

3

PER1 .864 .312

PER2 .944 .143

PER3 .802 .507

PER 4 .854 .306

PER5 .544 .679

PER 6 .211 .903

PER7 .236 .886

PER8 .214 .828

Resample

4

PER1 .950 .012

PER2 .944 .159

PER3 .863 .435

PER4

.663 .503

PER5

.256 .909

PER6 .282 .896

PER7 .199 .868

PER8

.375 .767

tion). When external

replication

is

impossible

or

impract ica l , internal rep l i -

cab i l i ty methods , such as cross-validation or the bootstrap, may be usefu l .

Exploratory factor analysis cross-validation methods can be conducted

by randomly spli t t ing the sample and using SPSS to factor analyze both

data sets, and

then

us ing the Appendix B bes t - f i t rotation to q u a n t i f y degree

of factor

fit

across

the two

subsamples . However ,

the

basis

for

actua l fac tor

interpretation is a lways the s t r u c tu r e from the total sample. The subsamples

INTERNAL REPUCABiL/TY ANALYSES

10 7



TABLE 9.6

Bootstrapped

Estimates of Pattern/Structure Coefficients and Their

Standard Errors

Variable

PER1

PER2

PER3

PER4

PER5

PER6

PER7

PER8

Mean

.926

.953

.859

.773

.293

.223

.155

.226

Factor

I

SD/SE

.039

.009

.046

.071

.152

.115

.095

.090

23.47

102.33

18.49

10.86

1.93

1.94

1.62

2.52

Mean

.093

.085

.383

.368

.840

.871

.879

.791

Factor

II

SD/SE

.146

.096

.096

.102

.098

.053

.027

.024

0.63

0.88

3.99

3.62

8.60

16.45

32.58

33.09

Note. Pattern/structure coefficients greater than |.4|

are

italicized.

are created only

to

gain insight regarding

the

character

of the

total data

set and not for factor interpretation.

The EFA bootstrap (Thompson, 1988) is a particularly powerful inter-

nal replicability method. The bootstrap is so p o w er f u l because it resamples

the data in so many

d i f fe rent

combinations. However, as a practical matter,

specia l i zed software must be used to conduct these analyses.

10 8 EXPLORATORY AND CO NFIRMATO RY FACTOR

ANALYSIS



10

CONFIRMATORY FACTOR AN ALYSIS

DECISION SEQUENCE

Although the basic concepts of factor analysis are roughly a century

old (see Spearman, 1904), more recent extensions of the exploratory factor

analysis (EFA)

concepts

have created

the

basic methods

for

confirmatory

factor analysis (CFA;

see

Joreskog, 1969).

Both EFA and CFA are

part

of

the

general linear model (GLM).

Thus,

many

of the

concepts introduced

previously

with regard

to EFA

apply

in CFA as

well. Indeed,

an

important

emphasis

in

this book

is

elaborating

the

similarities between

EFA and CFA

as

part

of a

single underlying GLM, while also highlighting

th e

differences

in the two sets of methods.

Of course, the basic difference in CFA is that one or more underlying

models (i.e., how many factors, which measured variables are

thought

to

reflect

w hich la tent v ariab les, whe ther the factors are correlated )

must

be

specified

even

to run the

analysis.

There

are

also statistical analyses

that

are possible

in CFA but

impossible

in EFA

(e.g., allowing error variances

to be

correlated).

And some procedures that are routine in EFA, such as factor rotation,

are irrelevant in CFA. This is because the a priori mode ls thems elves typic ally

specify simple structure,

by

constraining certain factor pattern

coefficients

to be

zero while

freeing

other pattern

coefficients to be

estimated.

Confirmatory

factor analysis

is a

very important component within

a

broader class

of

methods called

structural equation modeling

( S EM) ,

or

some-

109



t imes

covariance structure analysis.

CFA specifies th e

"measurement models"

delineating how measured variahles reflect certain latent variables. Once

these measurement models a re deemed satisfactory, then th e researcher can

explore path models (called structural models) that link th e latent variables.

Structural

equation modeling is beyond the scope of the present vol-

ume.

Excellent book-length treatm ents of SEM are availab le elsewhere (see

Byrne, 1994, 1998, 20 01 ). Cha pter-length expositions

are

also available

(see Thompson, 2000c). Schumacker

and

Marcoulides (1998)

and

Marcou-

lides and Schumacker (2001) describe recent extensions of

these

methods.

Suffice it to say that even w hen the researcher is app lyin g the wid er

range of analyses possible with SEM , CFA mod els are often a critical start ing

place for such analyses (Thompson, 2000c). It makes little sense to relate

constructs w ith in an SEM m odel if the factors

specified

as part of the model

are not worthy of further attention.

CONFIRMATORY FACTOR ANALY SIS VERSUS EXPLORATORY

FACTOR ANALYSIS

In EF A, all parameters im plicit in a factor model m ust be estimated .

For example, if there are 9 measured variables, and 3 princip al components

are

extracted and then rotated to the promax cr i ter ion, then 27 ( 9 x 3 )

pattern coeff icients , 27 ( 9 x 3 ) s tructure coeff icients , 9 comm unali ty

coeffi-

cients

h ; or

their corresponding uniq uenes ses estimated

as u = 1

—

h" ) ,

and 3 factor correlation

coefficients ( r ]

X

n , r\x\\\, i i i x i n )

are estimated.

Of course, because Pg

x

3 R^ x 3

=

S

y x

3 given P, if either the factor

correlation matrix (R ^

x

3) or the factor structure m atrix (8 9

x

5) is m a themati -

cally

determined, so is either R or S, and thus in actua lity only 30 parameters

(2 7

+ 3) are

estimated. N ote

that th e

uniquenesses

are

also mathematically

determined once

th e

pat tern

and

structure matrices

are

estimated.

In CFA, th e researcher can "constrain" or "fix" certain param eters to

mathematically "permissible" values (e.g., a variance m ay be constrained to

equal any positive number; a correlation, r, may be constrained to equal

-1, +1, or any

number

in

between), while

at the

same time "freeing"

the

use of the input data to derive estimates of other model parameters (e.g.,

factor p attern coefficien ts, factor varian ces) . For example, the researcher

might constrain variables VI, V2, and V3 to reflect only Factor I, variables

V4, V5, and V6 to reflect

only Factor

II, and

variables

V7, V8, and V9 to

reflect only Factor II I, w ith the th ree facto rs allowed to be correlated.

In EFA, th e researcher m ay expect certain coefficients, but these expec-

tations cannot be incorporated into the analysis. In CFA a researcher must

110 EXPLORATORY AND

CONFIRMATORY

FACTOR ANALYSIS



declare as input into the analysis one or more

specific

models, each containing

some "fixed" and some "freed" parameters.

The original computer software for CFA (and S EM ), such as LISRE L

(i.e., analysis of Linear Stru ctura l RELationships; Joreskog & Sorbom, 1984),

required

th e researcher to input these model expectations using ma trix

algebra. A thorough knowledge of both th e uppercase and the lowercase

Greek alphabet was also required

Modern software, such as AMOS (Arbuckle & Wothke, 1999) an d

EQS (Bentler, 1995), instead uses a graphics-oriented user interface. The

researcher's expec tations are declared using the software to dr aw a diag ram

representing

th e

input model.

The

results

are

then reported

by the software

in text form (i.e., sets of coefficients printed fo r different matrices) , but also

as

an

output diagram with estimates printed

on the

input model diagram.

By

tradition, in these v arious compu ter programs measured variables

are

represented

by

squares

or

rectangles. Latent variables

are

represented

as

circles or ovals. Regression paths or, in the factor analysis case, pattern

coefficients, are

represented

by

one-headed arrows. These

are

drawn

from

factors

to measured variab les to reflect the premise that the latent variables

are

in

fact

an

underlying influence

on the

manifestation

of the

factors

in

the form of scores on the measured variables. Correlations (and cova riances)

are represented as two-headed arrows draw n to connect either m easured or

latent variables in pairs for which correlations or covariances are freed to

be nonzero and are estimated in the analysis.

Figure

10.1 presents the inp ut graphic model declaration implied by

an EFA involving nine measured variables and three correlated factors.

Table 10.1 presents

a

sequential count

of the 30

param eters that

a re

estimated

as

well as a sequential count of the parameters mathematically determined

once th e estimates are in hand.

Figure 10.2 presents th e related CFA inpu t model presum ing that each

of the

nine measured variables

is

associated with only

one

factor, three

measured

variab les per factor. The absence of paths between selected pa irs

of

measured and latent variables (e.g., between V I and Factor II, between

V2 and Factor II) implies that these factor pattern coefficients have been

constrained

to be

zero.

This

model declares simple structure

on its face,

because

no

measured variable

is

allowed

to

function

as an

indicator

fo r

more

than on e

factor. Thus,

if

model

fit is

adequate ,

no

rotation

will be

necessary.

Table 10.2 presents

a

sequential count

of

parameters estimated,

no t

estimated, and implied by this model. However, one difference between

EFA

and CFA arises in this presentation. Note from Table 10.2 that th e

nine

uniquenesses

u

) are counted as estimated in the CFA analysis. In

the EFA

analyses these were taken

as

implied, because once

th e

pat tern

an d

structure

coefficients are in

hand,

the

communal i ty coefficients h

) are

CONFIRMATORY FACTOR

ANALYSIS

DECISION SEQUENCE

I 11



factor \Y

Figure

10 1

Exploratory factor analysis factor model with nine measured variables

an d three correlated factors

computed by simple multiplications and additions along a given row of the

factor matrix, an d

then

uniquenesses ar e computed as u

2

= 1 —

h

2

.

Figure

10.3 presents yet another factor model for this research problem.

This

figure

illustrates

some

of the

model elements that

are not

possible

in

EFA.

First, the uniquenesses or error variances involving el and e2 have

been allowed

to

covary

or be

correlated

in

this model.

No

error variances

can be correlated in an EFA model, but in CFA the correlations of various

112

EXPLORATORY

AND




TABLE

10.1

Counts of Parameters Estimated

and

Implied) by the Figure

10.1

Factor Model

Pattern Structure

Variable

V1

V 2

V 3

V4

V5

V 6

V7

V8

V9

Factor r

I

II

III

1

2

3

4

5

6

7

8

9

—

28

29

II

10

11

12

13

14

15

16

17

18

—

30

M l

19

20

21

22

23

24

25

26

27

—

I

31)

32)

33)

34)

35)

36)

37)

38)

39)

II

40)

41)

42)

43)

44)

45)

46)

47)

48)

III

49)

50 )

51 )

52)

53)

54)

55)

56)

57)

u

2

58 )

59 )

60)

61 )

62)

63)

64)

65)

66)

Note. Coefficients mathematically

determined

and thus

implied

once estimates are in

hand

are

noted

in

parentheses.

pairs

of

error variances

can be

estimated,

as

deemed necessary

by the

researcher.

Second,

only tw o factors are allowed to be correlated, and this correla-

tion

is

estimated

as part of the

model.

In EFA, factors must be either all

correlated or all uncorrelated.

Third,

an "equality" constraint, designated by the same letter (here

a)

b eing placed on the paths for the factor pat ter n coeff icients for three

measured variables on Factor III, h as been imposed.

This

means

that

the

input data will be consulted to estimate

these three

pa ttern coefficients, but

th e estimate has been constrained such

that

th e same single number must

be used for all three of these pa ttern coef ficients. Eq ual ity constraints are

not possible within EFA.

Equali ty constraints have several possible uses

in

EFA,

but one use is

to

test

a

theoretical exp ectation

that all

measured variables

on a

given factor

measure

that

factor equa lly well.

Fo r

exam ple, when scoring

on

test

subscales

is done

by summing item scores on a given subscale, the implication is that

th e items on the subscale measure th e underlying

construct

equally well.

This premise can be directly tested in CFA.

In CFA one may also (and should ) test rival models. Equality con-

straints might

be

imposed

in one

model; however,

in a

riv al model

all freed

pattern coefficients might be estimated with no equality constraints.

This

rival model might reflect a belief that the test items do reflect the intended


DECISION SEQUENC E

113



TABLE

10.2

Counts

of Parameters Estimated, Not Estimated, (and Implied) by the

Figure 10.2

Factor

Model

Variable

V1

V2

V3

V4

V5

V6

V7

V8

V9

Factor r

I

II

III

1

2

3

—

—

—

—

—

—

—

10

11

Pattern

II

III I

(22)

(23)

(24)

4 (25)

5

(26)

6

(27)

7

(28)

— 8 (29)

— 9 (30)

—

12 —

Structure

II

31 )

32 )

(33)

(34)

(35)

(36)

(37)

38 )

(39)

III

40 )

41 )

42 )

(43)

(44)

45 )

46 )

47)

48 )

u

2

13

14

15

16

17

18

19

20

21

Note. Coefficients mathematically determined and thus implied once estimates are in hand are noted in pa-

rentheses. Fixed parameters are designated by dashes ( — ). Here 18 factor pattern coefficients are con-

strained to be zeroes. And the three factor variances are fixed to equal 1

.Os,

thus making the factor covari-

ances equal factor correlation coefficients, because for example r . „ = COV,

x

„/ (1.0 x 1.0).

ival Models

Like SEM, CFA involves testing the fit of models to data. When

models fit well, as reflected in various statistical fit indices, some researchers

erroneously infer that

the

correct model

has

been specified,

and

that

in

some sense

the

model

is

proven.

In fact, it is conceivable that several models may fit a given data set.

As Thompson (ZOOOc) noted, "the fit of a single tested model may always

be an

artifact

of

having tested

too few

models"

(p.

278).

Thus it is

desirable

to test the fit of several r iva l models when conducting a CFA.

The fit of a preferred model is more impressive when that fit occurs

in the context of testing several r ival models, especially when some of the

rival models are theoretically plausible and are not articulated so as to create

straw person arguments. The fit of a model is also more impressive when

the

tested model

is

strongly

disconfirmable.

Disconfirmability

is partially a function of the degrees of freedom fo r

the model test. Degrees of freedom indicate, given the presence of one or

more sample statistics (e.g., the mean of the sample data, the r between

two measured variables), how many pieces of information in the analyses

remain

free to

vary

after the

estimate

is in hand. For

example,

in the

presence

of knowledge that the sample mean is 3, given any two scores in the data

set

(1, 3, 5), the researcher can always accurately determine the third score.

CONFIRMATORY FACTOR

ANALYSIS

DECISION

SEQUENCE

115



Figure 10.3. Confirmatory factor analysis CFA) factor model with three model

features unique

to

CFA.

For

this problem

th e

degrees

of

freedom

is a

function

of the

sample

size, n,

and

equals

n — 1.

Tahle 10.3 presents the c om puta tions for stat istic al significan ce testing

in any r or

m ult ipl e regression

(R )

analysis involving

p

pre dictor variables

(e.g., for the

bivar iate correlation,

p = 1). There are two

related situations

in which

^ C A L C U L A T E D

(and the corresponding

^ C A L C U L A T E D

also) is unde-

116

EX PLORAT ORY

AN D CONFIRMATORY FACTOR

ANALYSIS



TABLE

10.3

Statistical

Significance Calculations

for r

2

and R

2

Problems

Source

Model

Unexplained

Total

Sum-of-

squares

A

B

C

Degrees of

freedom

D

E

F

Mean

square ^ C L C U L T E D

G J

H

I

r

2

/

ft

2

K

Note. D = p (i.e., the number of

predictor

variables). F = n -1 .E = / ? - 1 -p. G = A / D . H = B / E . I = C/

F. J = G/ H. K = A / C .

fined

and a statistical significance test cannot be conducted. First, regardless

of

sample

size, if the sum-of-squaresuN^xpLAiNEO '

s

zero

(and correspondingly

the r

2

or R

2

is

100%),

the F

cannot

be

calculated, because

the

mean

s q u a r e u N E X P L A i N E D

w

'"

De zero

>

an

J division by zero is mathematically imper-

missible.

Second,

whenever

the

degrees

of freedomMODEL

equals

the

available

degrees of freedom (i.e., fo r these analyses

n —

1), the

sum-of-squaresuNEX-

P L A I N E D

W

'1I always

be

zero.

F or

example,

if X and Y are

both

variables (i.e.,

SD

X

> 0 and SD

Y

> 0), and n = 2, r

2

must be 100%, and the sum-of-

s q u a r e s u N E x p L A i N E D

must

^Y

definition both

be

zero. Similarly,

in a

regression

analysis

involving an y tw o predictors, the R

2

wil l always be 100% whenever

n = 3.

Clearly, an R" of 100% is entirely unimpressive when 10 predictor

variables are used and n is 11 However, an R of 100% would be very

impressive (even though the result still could not be tested for statistical

significance ) if n = 1,000, because this regression model would be highly

disconfirmable given

that

th e residual degrees of freedom

UNEX

pLAiNEn

equaled 989.

The

residual degrees

of

freedom

are an

index

of the

potential discon-

firmability of any model. The good fit of models is more impressive when

degrees of freedom

UN E

xpLAiNED

ar

e

barge,

because as a function of the residual

degrees

of

freedom

the

model

is

parsimonious

and yet has

good explana-

tory ability.

In CFA (and SEM), however,

total

degrees of freedom for tests of

model fit are not a function of the sample

size,

n. Instead, the total degrees

of

freedom

in CFA are a

function

of the

number

of the

unique entries

in

the covariance matrix, which is a function of the number of measured

variables (v) in the analysis. The total degrees of freedom for

these

tests

equals [ < v v + 1)] / 2). For example, in a study involving i> = 9 measured

variables,

th e

available degrees

of

freedom equal

([9 (9 + 1)] / 2) =

([9 (10)]

/ 2) = 45. This

corresponds

to

evaluating

9

variances

and 36

covariances (i.e., there are 9 + 36 = 45 unique entries in the associated

variance/covariance matrix).

CONFIRMATORY

FACTOR ANA LYSIS DECISION

SEQUENCE 117



In CFA (and SEM), each parameter e.g., factor pat tern coefficient,

model error variance,

factor

co rrelation) that is estimated

"costs"

one degree

of

freedom.

This

means that each a nd every model involving nine measured

variables and

est imation

of

an y

4 5

parameters will always perfectly reproduce

or

fit the

data.

The disconfirmable and/ or plausible models for a CFA are context-

specific, and

some such models

can

only

be

developed

by

consulting theory

and related em piric al research. How ever, several rival CF A models may

often be useful

across

a fairly

wide array

of

problems:

1 . Independence model. This

model

specifies

that

the

measured

variables

are all perfectly uncorrelated with each other, and

thus

tha t

no

factors

are

present.

The

model estimates only

the mo del specifica tion varianc es of each of the meas ured

variables.

This model

is

d isconf irmable

but usually not plausi-

ble.

H owever, the stat ist ics qua ntifying the fi t of this model

are

very useful

in

serving

a s a

baseline

fo r

evaluat ing

the fits

of plausible rival models.

2. One-factor model.

This model

is

disconfirmable,

but usually

not p lausib le for most research situations. Aga in, how ever,

even

if

this m odel

is not

theoretically plausible,

the fit

stat ist ics

from this model test can be

useful

in cha racte rizing the degree

of

sup eriori ty

of a rival

mul t i fac tor model .

And it is

desirable

to

rule

out the fit of this model even when m ultifa ctor models

are

preferred,

so

that support

for the

mul t i factor model

is

then

stronger.

3.

Uncorrelated

factors model. In EFA, as noted in chapter 3,

orthogonal rotat ions

are the

default

an d usually

provide simple

structure.

However, in CFA, correlated factors are usually

expected

and

almost

always

provide

a

better

fit to the

data.

However , it is

helpful

to quantify th e degree of superiority of

model

fit for a

correlated versus uncorrelated factors model.

Estimating

the factor correlations for /fac tors costs (/ [f - 1]) /

2 degrees of freedom, for exam ple, (3 [3 - 1] ) / 2 = (3 [2] ) /

2 = 3). Uncorrelated factor models ar e more parsimonious

when there are two or more

factors.

The quest ion is whether

the sacrifice of

pars imony (and

of

related

disconfirmability) is

worthwhile in service of securing a better model fit .

Model Identification

A

model

is said to be

"identified" when,

for a

given research problem

and data set,

sufficient

constraints are imposed such that there is a single

118

EXPLORATORY

AN D

CON FIRMATORY

FACTOR ANALYSIS



apples to apples as to how well each m easured varia ble reflects th e underly-

in g

construct.

The

selection

of

scaling

for the

latent variables

will not affect th e

model fit statistics. The scaling decisions are basically arbi trary and can

reasonably be made as a function of the researcher's s tyli stic preferences.

These are not the only conditions necessary to achieve model identifi-

cation.

When a single measured varia ble is posited to reflect multip le un de rly-

in g

factors, evaluating model identification becomes technically

difficult.

Bollen (1989)

and

Davis ( 1993 ) provided more details.

Fortunately,

modern computer software

is

relatively sophisticated

at

dete cting model und erid ent ific atio n. Most packages (e.g., AM OS) provide

diagnostics such

as

suggestions

of how

m any more param eters need

to be

"fixed" or constrained , and some guidan ce as to wh ich para me ters in particu-

lar

are good candidates for achieving model identification.

Matrix of Association Coefficients to Analyze

As noted in chapter 3, in EFA there are num ero us choices that can

be made regarding the matr ix of association coefficients to be analyzed.

Popular EFA

choices include

th e

Pearson

produc t— moment

r matr ix ,

th e

Spearman's

rho

matr ix ,

and the

variance/covariance matrix.

A s

noted

in

chapter 3, the

default

choice in statistical packages is the Pearson r matr ix ,

and

th is

is the

selection used

in the

vast preponderance

of

published

EFA

studies.

The

statistical

software

for CFA

supports

an

even larger,

dizzying

array

of

choices.

These

include coefficients with which many readers

may be

unfamiliar, such a s the po lyserial or the polychoric cor rela tion coefficients .

But

th e default choice in most CFA software is the covariance m atrix .

This

matr ix

has

covariances

off the

main diagonal. Remember that

T X Y

=

COVxv / [(SD

X

) (SD

Y

)], and that therefore COV

XY

= r

X Y

(SD

X

) (SD

Y

). The

correlation

is a

"standardized" covariance from which

the two

standard

deviations have been removed

by

div is ion .

The variance/covariance matrix has variances on the main diagonal.

Because the covariance of a var iable with itself equals the var iance of the

variable, it is

more common

to

hear researchers refer

to

th is matrix more

simply as the covariance matrix.

The computation of

both

th e Pearson r and the covariance matrices

presumes that th e data are inte rval ly scaled. This means that the researcher

believes that

th e

underlying measurement tool yielding

the

scores

has

equal

units

throughout. For example, on a ruler every inch (e.g.,

from

1" to 2",

from

11" to

12")

is the

same amount

of

distance

(or

would

be if the

ruler

were

perfect). Note tha t the a ssumption is that th e ruler has equa l intervals,

and not that the scores of different people are all eq uid ist an t.

120

EXPLORATORY

A N D

CONFIRMATORY FACTOR

ANALYSIS



In psychology sometimes the interval scaling of the measurement is

more debatable when scores on abstract constructs are being estimated. For

example,

if

Likert scales

are

used

to

measure attitudes,

and 1 = never,

2 =

often, 3 = very

often,

and 4 = always, most researchers would doubt that

th e

data were intervally (continuously) scaled. However,

if 1 =

strongly

disagree, 2 = disagree, .3 = agree, and 4 = strongly agree, many (not all)

researchers would deem the scores intervally scaled, or at least approximately

intervally scaled.

Of course, whenever

there

is some doubt regarding the scaling of data,

or regarding the

selection

of matrix of associations to analyze, it is thoughtful

practice

to use

several reasonable choices reflecting

different

premises.

When

factors are invariant across analytic decisions, the researcher can vest greater

confidence in a view

that

results are not methodological artifacts.

In CFA, tor some complex statistical reasons, it is almost always prefera-

ble

to use the

coyariance matrix, rather

than the

Pearson

r

matrix,

as the

matr ix to be

analyzed. There

are

some conditions

in

which

the

Pearson

p r o du c t— m o m e n t matrix may he correctly used (see Cudeck, 1989). How-

ever, the covariance matrix will always yield correct results, as long as the

data are appropriately scaled and distributed.

Multivariate

Normality

As

noted

in

chapter

7, the

shapes

of the

data distributions being

correlated

affect the

matrices

of

association actually analyzed

in

EFA. Tradi-

tionally, considerable

attention

is paid to

these

issues in

Q-technique

EFA

applications, while less

attention

is focused on these

concerns

in R-technique

EFA applications, even when the Pearson product-moment r matrix is the

basis for analysis.

However, many CFA results are particularly affected by data distribu-

tion shapes. Certain statistical estimation theories presume multivar iate

normality, especially with regard to the standard errors for the estimates of

model parameters.

And

some

CFA

(and SEM) model

fit

statistics also

presume normality.

Even univariate normality is not as straightforward a s many researchers

erroneously believe. Misconceptions regarding univariate normality occur,

in part, because many textbooks only present a single picture of a univariate

normal distribution:

z

scores

that

are normally distributed (i.e., the "standard

normal" distribution).

There is only one univariate normal distribution for a given mean and

a given standard deviation. But for every different

mean

or every different

standard deviation, the appearance of the normal distr ibution changes.

Some

normal distributions when plotted in a histogram have the classic bell shape

that many associate with the normal or Gaussian curve.

C O N F I R M A T O R Y FACTOR

ANALYSIS




But bells come

in

infinitely many shapes. Some bells appear fairly flat

and wide, whereas others appear tall and narrow. Of course, one universally

true statement about normal dis tribution s with different appearances is that

they all are unimodal an d symmetrical (i.e., coefficient of skewness = 0)

and

have

a

coefficient

of

kurtosis

of

zero.

Bivariate

normality is

even m ore complica ted. Because there

are now

scores on two va riables for each person, a three-dimen sional space is requ ired

to represent th e distribution, an d three-dimensional objects must be located

in this space to represent the scores of different people. The objects used

might

be

little

BBs or

ballbearings,

one for

each person.

An X-axis and a perpendicular Y-axis might be drawn on a piece of

paper. A

vertical axis extending through

the X,Y

intercept might

be

used

to measure the frequencies at different score pairs. For example, if

C olleen

and Deborah both had scores of 1,1, their BBs would be stacked on top of

each

other

at

this particular score pair coordinate.

If

both

variables

are

z scores

and

normally distributed,

the BB s in

bivari-

ate data w ould create a literal three-dim ensio nal b ell shape. If we traced around

th e outer edges of the BB s on the paper, a circle wou ld be drawn on the paper.

If

we made a slice through the BBs anywhere up the frequenc y axis, parallel

to but moving away

from

the piece of paper, concentric and n arrow ing circles

would be

defined un til

we

reached

the top of the

bell.

If we

sliced th rou gh

the

BBs with a

knife,

in any location only with the restriction that the knife cut

al l

the way

down through

th e

vertical frequenc y axis, every slice would

yield

a univariate standard normal distribution.

If

the Y scores were more spread out than the X scores, the signa ture

or foot of the bell defined by trac ing on the paper a round the BB s wou ld

be an ellipse rather than a circle. If we sliced upward on the vertical axis,

but par allel to the paper, every slice would yield a concentric ellipse until

we reached the top of the bell. If we sliced throug h the BB s w ith a kn ife,

in any

location only with

t he

restriction

that th e

knife

cut all the w ay

down

through

th e

vertical frequency axis, every slice would yield

a

univariate

normal distr ibu tion , albeit potentially slices that were narrower or wider

than each other.

Univariate normality of both variables is a necessary but not sufficient

condition

fo r

b ivariate normality. B ivariate normality

of all

possible pairs

of

variables

is a

necessary

but not

sufficient condition

fo r

mult ivariate normali ty.

Of course, multivariate normality requires that more than three dimensions

are conceptualized. These are not easy dynam ics to picture or to understand.

Henson (1999) provided a very

useful

discussion of these concepts.

There a re primari ly tw o ways

that

researchers can test th e multivariate

normality of their data. First, if more than 25 or so measured variables are

involved, the grap hical techniqu e (not involving a statistical significance

test) described

by Thompson

(1990a)

can be

used.

J

2 2

EXPLORATORY

AND




Second,

a statistical test of the mu ltiva riate c oefficient of kurto sis,

using the null hypothesis that kurtosis equals

zero,

is available in com monly

used CFA/SEM computer programs. These computer programs also provide

a "critical ratio" t

or

Wald statistic)

of

this p aram eter estimate

to its

standard

error.

Researchers

hope that

th e mu ltivaria te kurtosis

coefficient

is close to

zero, and that the null hypothesis is not r ejec ted.

What can he

done

if the

assumptions

of

m ultivaria te normality cannot

he met? One choice is to use a parameter estimation theory, described in

a subsequent section of this chapter, that does not require this assumption.

H owever, "distribution-free" estimation theory does require even larger sam -

ple

sizes than

do

more commonly used theories.

A second alter nat ive is to create item "parcels" by bund ling together

measured variables.

For

example,

if

eight measured variables

are

presumed

to measure Factor I, the data can be reexpressed as four measured variables.

For example,

each

person's score on the measured variable

that

is most

skewed left is

added

to the

person's score

on the

measured variable that

is

most skewed right,

the

score

on the

measured variable that

is

second-most

skewed left is

added

to the

score

on the

measured variable

that is

second-

most skewed right, and so forth. However, this strategy is subject to the

general limitations

of

parc eling methods (cf. Bandalos

&

Finney, 2001;

Nasser

&

Wisenbaker, 2003).

Outlier Screening

It is well-known that

outliers

can

grossly

influence statistical results for

a data set, even if the number of outliers is small and the sample size is

large.

Outliers can influence CFA results (see Bollen, 1987)

just

as they

can other analyses.

Yet misconceptions regarding outliers abound. Some researchers incor-

rectly believe that outliers have anomalous scores on all possible variables

with respect

to all

p ossible statistics.

One can

almost visualize evil, aberrant

people whose foreheads are (or should be) branded with large capital Os.

Outliers are persons with aberrant or anomalous scores on one or more

variables

with respect

to one or

more statistics.

But

most outliers

are not

outliers

on all variables o r w ith respect to all statistics. Consider th e scores

of

Tom, Dick,

and

Harry

on the

variables

X and

Y :

Person

X Y

Tom 1 2

Dick

2 4

Harry

499 998

CONFIRMATORY FACTOR ANALYSIS DECISION

SEQUENCE 1 23



TABLE

10.5

Covariance

and Correlation Matrices for Eight

Measured Variables

n= 100

Faculty)

Measured variable

Variable

PER1

PER2

PER3

PER4

PER5

PER6

PER7

PER8

PER1

1.847

1.768

1.340

1.681

0.921

0.962

0.986

1.344

PER2

0.844

2.374

1.278

1.787

0.816

0.797

0.835

1.468

PER3

0.653

0.549

2.280

1.488

0.992

1.026

1.132

1.176

PER4

0.825

0.774

0.657

2.248

1.043

1.019

0.967

1.338

PER5

0.328

0.256

0.318

0.336

4.274

3.543

3.375

3.322

PER6

0.349

0.255

0.335

0.335

0.844

4.122

3.231

3.150

PER7

0.388

0.289

0.400

0.344

0.872

0.850

3.506

2.935

PER8

0.461

0.444

0.363

0.416

0.749

0.723

0.730

4.605

Note. Covariances are not italicized; Pearson

rs

i.e.,

standardized covariances) are italicized. Pearson

r

C O V x v

/

SDx

x SCM

e .g. , 0.844

=

1.768

/

1.847

5

x

2.374

5

)). COV

XY

=

r

m

x SD

*

x S

f <

e

9 >

1

-

768

=

0.844

x

1.847

s

x

2.374

s

).

distances

can

also

he

obtained within SPSS

by

performing

a

regression

listing the measured variables as the predictor variables in a mult iple regres-

sion analysis and using any other variable as the dependent variable. For

this

example,

if

participant

ID

number

was

used

as the

pseudo-dependent

variable, the SPSS

syntax would

be:

regression

variables=id perl

to

per8/

descriptive=mean stddev corr/

dependent = id/eiiter perl to per8/

save=mahal mahal)

sort cases by mahal D) .

print formats mahal F8.4)

list variables=id perl to per8 mahal/cases=999

However, even when we know that a given person is an o u t l i e r on a

given set of variables in regard to

p a r t i c u l a r

s ta t is t ic s , it is not

a lways

clear

what should be

done

with the person's scores. If the researcher interviews

the outlier and determines that b izar re scores were intentionally generated,

then

deleting

the

person's scores

from

the

data

set

seems only reasonable.

But when

we

cannot

determine

the

reason

for the

score aberrance,

it

seems

less reasonable

to

delete scores only because

we are

unhappy about their

atypicality

with respect to deviations

f rom

the means.

One resolution refuses to represent all analytic

choices

as being either /

or. The analysis might be conducted both with the outlier candidates in the

data and with

them

omitted. It interpretations are robust across var ious

decisions,

at

least

the

researcher

can be

confident

that the resu l ts are not

artifacts

of

methodological choices.

C:ONFIRMATORY

FACTOR

ANALYSIS DECISION SEQUENCE

125



It is

also critica l

to

take

into account th e

analytic context when making

these decisions. The context is very different if one is doing research in a

previously

unexplored area versus an arena in which n ume rous studies have

been conducted. When

other

studies

have

been

done

th e researcher can

feel

more comfortable

that

outliers are not c omp romising gene ralizab ility,

i f results replicate across stud ies. By the same

token,

researchers may be

somew hat less concerned about outliers when they

are

also conducting

interna l replicability analyses.

A s Cohen, Cohen, Wes t , and Aiken (2003) argued,

The

extent

of

sc ru t iny

of these

statistics

depends on the

nature

of the

study. If the s tudy is one of a series of replications, inspection of graphical

displays for any

obvious

outl iers

is

normally

sufficient. Consistency of

the findings across replications provides assurance that the presence of an

outlier is not responsible for the results

[italics added]. On the

other hand,

i f

the

data

set is

un ique

and

un l ike ly

to he

replica ted (e.g.,

a

s tudy

of

40 ind iv idua ls wi th a rare

medical

disorder), very careful scrutiny of

th e

data

is in

order, (pp. 410-411)

Parameter stimation

Theory

Conventional graduate courses in applied statistics teach ana lyses such

as

t

tests, analysis

of

variance, multiple regression, descriptive discriminant

analysis,

and cano nical correlation analysis. All of these an alyses use sample

data to compute sample statistics (e.g., sample means, correlation coeffi-

cients) that

are

then taken

as

estimates

or

corresponding population param e-

ters

(e.g.,

p., a, p).

All of

these analyses invoke

a

single statistical theory

fo r

parameter

estimation called

ordinary

least squares

(OLS).

These estimates seek to max-

imize

the

sum-of-squaresMonEL,

or its

mu ltivariate equivalent,

and the

corres-

ponding r

l

effect size analog (e.g., R

2

, r\

2

) in the sample data.

However,

there

are (many) other parameter estimation theories that

can also be used. Indeed, it can be argued

that

classical statistical analyses

are not

very

useful, and may

even routinely lead

to

incorrect research

conclusions, because classic al methods are so hea vily influe nced by outlie rs

(see

Wikox, 1998).

Two alternative parameter estimation theories are

fairly

commonly

used

in CFA (and SEM). Hayduk (1987) and Joreskog and Sorbom

(1993)

provided more detail on these and other

choices.

When sample size is large,

and if the model is correc tly specified, the va rious estimatio n theories

tend

to

yield consistent

and

reasonably accurate estimates

of

population parame-

ters (e.g., population factor pattern coefficients).

First, maximum likelihood theory is the de fau lt parameter estimation

theory in most statistical packages. This estimation theory attempts to

/ 2 6

EXPLORATORY

AND CONFIRMATORY FACTOR

ANALYSIS



estimate th e

true population parameters,

and not to

reproduce

th e

sample

data or minimize the sample

s u m - o f - s q u a r e s u N E x r L A i N E n

while maxim izing

th e

related sample

effect

size (e.g.,

R ',

Tp).

This is

accomplished

by

applying

maximum

likelihood

theory

to the

sample

covariance

matrix

to estimate

the population covariance ma trix, 2

V

x v, and then derivin g factors to repro-

duce that matrix rather than the sample covariance matrix.

Obviously, estimating population parameters with a statistical theory

that optim izes this estimation (rather than features

of the

sample data)

is

very

appealing. Better estimation of population parameters should yield

more replicab le results. This e stima tion theory also has the appeal

that

most

of

th e

statistics

that

evaluate model

fit are

appropriate

to use

when this

theory has

been

invoked. H owev er, m ax im um likelihood estimation theory

does require that

th e

data have

at

least

an

approximately mu ltivaria te

normal distribution.

Second, asymptotically

distribution-free

(AD F) estimation theory may

be invoked. As the name of this theory imp lies, these estimates tend to he

accurate

p aram eter estima tes regardless of distribution shapes as the sam ple

size is increasingly large. This is an appealing featu re of the theory. H owe ver,

very

large sample sizes

of

more th an 1,000 cases

are

required

to

invoke this

theory for some analytic problems.

Of course, even CFA m ax im um likelihood estima tion often requires

larger sample

sizes than

would EFA modeling for the same data set (MacC al-

l u m ,

Browne, (StSugawara,

1996). W est,

Finch,

and

Curran

(1995)

reviewed

some of the relevant issues and choices regarding th e distributional assump-

tions of variou s para me ter estimation theories.

Jackson (2001) found

that

sample

size per se, and not the

number

of

parameters being estimated,

is the

critical ingredient

in

deriving accurate

CFA estimates. And sample size is less of a

concern

when the average

pattern

coefficient is

|.80|

or

greater.

POSTANALYS1S DECISIONS

Once

the analysis is in hand, various postanalysis decisions arise. First,

the goodness of fit tor

each

rival model must be evaluated.

Second, specific

errors

in

model specification must

be

considered.

Model Fit Statistics

A unique feature of CFA is the capacity to quantify the degree of

model fit to the data. However, there are dozens of fit

statistics,

and the

properties of the fit

statistics

are not

com plete ly understood.

As

Arbuckle

and W othke

(1999) observed, "Model evalu ation

is one of the

most unsettled

CONFIRMATORY FACTOR

ANALYSIS DECISION SEQUENCE

\ 27



and

difficult

issues connected with s tru ctu ral modeling [and CFA]. . . .

Dozens of statistics . . . have heen proposed as measures of the mer it of a

model" (p. 395).

Classically,

Monte

Carlo

or simulation studies are used to explore th e

behaviors of statistics under variou s condition s (e.g., differe nt sample

sizes,

various

degrees of violations of distributio nal assumptions). In Monte Carlo

research, populations with known parameters

are

created (e.g., with

an

exactly known degree

of

deviation from normal i ty) ,

and

numerous random

samples (e.g.,

1,000,

5,000) from

these

populations

are

drawn

to

evaluate

how the statistics perform.

M any of the pre vious sim ulatio n studies of

CFA/SEM

model fit statistic s

specified

populations associated with known models,

and then

drew samples

and tested the fit of the known models. Such simulations studies are not

realistic

as regards real research situa tions , because in real research the

investigator does not usually know the true model. Indeed, if we knew the

true model,

we

would

not be

doing

th e

modeling study

Better simulation studies of fit statistics have created populations fo r

certain m odels, and

then

intentionally evaluated the behavior of

fit

statistics

fo r va rious conditions and v ariou s model m isspecifications. Recent s imula-

tion stud ies in this venue have pro vided im portant insights re garding how

the fit statistics perform (cf. H u & Rentier, 1999; Fan,

Thompson,

& Wang,

1999; Fan,

Wang,

& Thompson, 1997), but there is still a lot to learn

(Rentier,

1994).

Four

fi t

statistics

are

most commonly used today.

In

general,

it is

best

to consult several fi t statistics when evaluating model fit and to evaluate

whether th e results corroborate each other.

First, the

Jj statistical

significance

test

has

been

applied since the origins

of CFA. As noted in chapter 2, in EFA the reproduced correlation matrix

( R v x V

+

)

c a n

°e computed by mul t ip lying P V X F by Ppxv ' - T he residual

correlation matrix (Ry

x

v )

l s

computed by subtract ing R y

x

v

+

from R y x X -

The

same basic idea

can be

applied

in

CFA.

One can

estimate

how

much of a covariance matrix is reproduced by the parameter estimates, and

how m uch is not. In CFA, the n ul l hypothesis t ha t is tested is essentially

whether th e residua l covariance estim ate equals a matrix consisting only

of

zeroes.

Un like most sub stantive applications of null hypothesis statistical sig-

nificance

testing,

in CFA one

does

not

w an t

to

reject this null hypothesis

(i.e., does not wa nt statistical sign ifica nce ), at least for the preferred model.

The degrees of freedom for this test statistic are a function of the number

of measured variables and of the number of estimated param eters. For exam-

ple, if there are

eight m easur ed variables,

the

d/Toj^i.

»

re

36 ([8 / (8 + 1)] /

2). If eight factor pat ter n coeffic ients, eight error variances, and on e factor

1 28 EXPLORATORY A N D CONFIRMATORY FACTOR ANALYSIS



covariance are estimated, the degrees-of-freedorriMODEi equal 19 (i.e., 36

17).

How ever, as with conventional statistical significa nce tests, though

the sample size is not used in computing the degrees of freedom for the test,

the sample size is used in comput ing the %

2

. Even for the same input

covariance matrix and the same degree of model fi t , Jj is larger for a larger

sample size. This makes this

fi t

statistic

not

very useful when sample size

is

large, as it

usually should

he in

CFA, because with large sample sizes

al l

models will tend to be rejected as not fitting. A s B entler and

B onnett

(1980)

observed, "in ve ry large samples vir tua lly all models that one might consider

would

have to be rejected as untenable. . . .

This

procedure

cannot

generally

be justified,

since

the

chi-square value

. . . can be

made small

by

simply

reducing sample size" (p. 591).

The % test is not very useful in evaluat ing the fit of a single model.

But the test is very

useful

in evaluating the comparative f i t s of nested models.

Models are nested when a second model is the same as the first model,

except that one or more additional parameters (e.g., factor covariances) are

estimated or freed in the second model.

The % "s and

their degrees

of

freedom

are

addi t ive.

For

example,

if for

the first

model positing

tw o

uncorrelated factors

the %

2

is

49.01 df = 2 0 ) ,

and for the second model estimating th e same parameters except that th e

factor

covariances

are freed to be

estimated

the yj is

31.36 df

=

19),

then

the differential fit of the two

models

can be

statistically tested using

%

2

=

17.65 (i.e., 49.01 - 31.36) with df = 1 (20 - 19). The ^ C A L C U L A T E D of this

difference

can be

determined using

th e

Excel spreadsheet function

= c h i d i s t ( 1 7 . 6 5 , 1 ) ,

which here yields f) = .000027.

When

ma ximu m likelihood estimation theory is used, and the mu ltiva r-

iate normality assumption

is not

perfectly met,

the J j

statistic will

be

biased.

H owe ver, Satorra and B entler (1994) developed a correction for this bias

that may be

invoked

in

such cases.

Second,

th e

normedfit index (NFI; Bentler

&

B onn ett, 1980) compares

the / for the tested model against the % for the baseline model presuming

that the measured variables are completely independent. For exam ple, if

the %

2

for the tested C FA model was 31.36, and th e %

2

for the baseline

independence model pr esum ing the measured variables were perfectly uncor-

related

was

724.51,

the NFI

would

be

(724.51

-

31,37)

/

724-51

=

693.15

/

724.51

= 0.96. The NFI can be as

high

as

1.0.

Generally, models with NFIs

of

.95 or more may be deemed to fit reasonably well.

CONFIRM ATORY FACTOR ANALY SIS DECISION SEQUENC E 1 29



When

maxim um likelihood estimation theory is used, and the multivar-

iate normality assumption is not

perfectly

met, the standard errors for the

parameter estimates

will

also

be

inaccurate. Satorra

and

Rentier (1994)

proposed methods fo r estimating th e correct standard errors, an d conse-

quently correct critical ratios, using

these

standard errors as denominators,

by taking into account data distribution shapes. Fouladi (2000) found that

these

corrections work reasonably we ll if sample

sizes

are 250 or more. Most

modern software provides these corrections as they are needed.

Parameters Erroneously

Fixed

For

each

parameter

fixed to not be

estimated that could

be

estimated,

th e CFA/SEM

software

can

estimate

th e

decrease (improvement)

in the

model fit

jj

resulting from

freeing

a given previously fixed parameter instead

to be estimated. In some packages (e.g., A MO S )

these modification

indices

are "lower-bound" or conservative estimates, and if a fixed pa ram eter is

freed the ~ f c reduction predicted by the software m ay be better than predicted.

Obviously, fixed parameters with the largest m odific ation indices are among

th e

leading candidates

fo r

model respecification.

Model

Specification

Searches

Confirmatory factor analysis is intended to be, as the name implies,

confirmatory. Respecifying

a CFA

model based

on

consultation

of

crit ical

ratio and modification index statistics is a dicey business, i f the same sample

is

being used

to

generate

these

statistics

an d

then

to

test

the fit of the

respecified model.

Using

th e

same sample

to

respecify

th e

model,

and then

test

th e

respecified model, increases capitalization on sampling error variance an d

decreases the replicability of results. And using the same sample in this

manner also turns

th e

analysis

into an

explanatory one, albeit using

CFA

algorithms.

As

Thompson (2 00 0c) emphasized, model respec ification should

never

"be based on b lind, dust-bow l empiricism . Models should only be respecified

in

those

cases in which th e researcher can articulate a persuasive, rationa le

as

to why the

modification

is

theoretically

and

practically defensible"

(p .

272) . Model respecification

is

reasonable when

each change can be

justified

theoretically, and when the fit of the respecified model can be evaluated

in a

holdout

or

independent sample.

MAJOR CONCEPTS

Exploratory factor analysis and CFA are part of the same general linear

model

(Bagozzi

et al., 1981). However, CFA models can estimate some

CONFIRMATORY

FACTOR ANALYSIS DECISION SEQUENCE

13 \



parameters that

cannot

he evaluated within EFA. For exam ple, only in CFA

can the covariances of error variances be nonzero and estimated. In EFA

factors can he

either correlated

or

uncorrelated,

hut in CFA

addit ionally

some factors may be allowed to he correlated whereas

other

pairs of factors

in

th e

same model

are

constrained

to he

uncorrelated.

O r,

uniquely

in

CFA,

factor correlations can he

freed

hut constrained to be equal. And in CFA,

unlike

EFA, all the pat tern coeff icients for a factor can he freed to be

estimated

but

constrained

to be

equal.

Confirmatory factor

analysis

m odels req uire the specification of one

or more hypothesized factor structures, and the analyses directly test and

quantify

the

degrees

of fit of rival

models. Because more

than on e

model

may

fit in a given

study,

it is important

to

test the fit of multiple plausible

models, so

that

any case for fit of a given model is more persuasive.

Only

"identified" factor models

can be

tested

in

CFA.

In

CFA, degrees

of freedom are a function of number of measured variables and not of sample

size. A

necessary

hut not sufficient

condition

fo r

model identification

is

that

th e

number

of

estimated parameters (e.g., pattern coefficients, factor

correlations, error varia nce s) cannot exceed th e ava ilable degrees of freedom.

Ideally,

considerably fewer parameters

will be

estimated

than

there

are

degrees

of

freedom,

so

that

the

model

is

more

disconfirmable. Fit

when more

degrees of freedom are unspent is inherently more impressive.

Some

factor

pattern coefficients or factor variances must also he constrained to obtain

model identification.

Like

EFA,

CFA

requires various decisions, such

as

what matr ix

of

associations (e.g., covariance m atr ix)

to

analyze,

an d

what estimation theory

(e.g., maximum likelihood) to use.

Some

estimation theories require that

data are m ultivariate normal. And screening data fo r outliers can be part icu-

larly

important when v ery elegant methods, such

a s

CFA,

are

brought

to

bear.

Confirmatory factor analysis model tests yield various statistics that

quantify degree of model fit. It is generally best practice to consult an array

of

f it

statistics (Fan

et

al., 1999;

H u &

Rentier, 1999)

in the

hope that they

will

confirm

each other.

The

analysis also yields criti cal ratios

that

evaluate whether freed

parameters were indeed necessary, and modification indices that indicate

whether fixed parameters instead should have been estimated. Respe cifying

models based

on

such results

is a

somewhat dicey proposition, unless theory

can be

cited

to j u s t i f y

model revisions.

Such

"specification searches" are,

however, completely reasonable and appropriate when th e respecified model

is

tested with

a

new, independent sample.

132 EXPLORATORY A N D CONFIRMATORY FACTOR ANALYSIS



1 1

SOME CONFIRMATORY

FACTOR

ANALYS IS

INTERPRETATION PRINCIPLES

Many princ iples of conf i rmatory fac tor analys is (CFA) mirror

those

in explora tory fac tor analys is (EFA ; e .g ., the impo rtance of con sul t ing both

pa t t e rn and s t ru c tu r e coe f f i c ien t s ) , g ive n the re la t ionsh ips o f these tw o

analyses wi thin

th e

genera l l inear model (GLM). However ,

there are

also

aspects

of CFA t h a t are

u n i q u e

to CFA

model tes ts .

For

e x a m p l e ,

in EFA we

a lmos t a lways

use

on ly s t anda rd ized f ac tor

pa t te rn

coef f ic ien ts . How ever , in CFA both uns tanda rd ized pa t t e rn coeffi-

c ien t s (ana logo us to regress ion uns tan da rd ized b we igh t s ) and s t anda rd ized

pat tern coef f ic ients (analogous to regress ion s tand ardized (3 we ights ) a re

c o m p u t e d .

The s ta t is t ic a l s ignif ican ce tests of

i n d i v i d u a l

C F A p a t t e r n

coeff ic ients

are conducted us ing the uns tandardized coef f ic ients . These coeff icients a re

c o m p u t e d

b y

d i v i d i n g

a

g iven pa rame te r

by i ts

re spec t ive st anda rd e r ror

to

compute a t (or Wald or c r i t ica l ra t io) s ta t is t ic . Noteworthy uns tandardized

coefficients us ing

these

tests also indicate the noteworthiness of the corres-

pond ing s t anda rd ized pa rame te rs , a s in mul t ip le reg res s ion .

H o w e v e r , w h e n p a r a m e t e r s a re compared w ith each

o ther ,

usua l ly th e

s t andard ized

coef f ic ients a re reported, again as in m ul t ip le reg res sion . T he

standa rdization allows apples- to-apples

comparisons of related parameters

with in a single model.

133



Five aspects of CFA p rac t ice are elaborated here . These inc lude (a)

a l t e r n a t i v e

wa y s

to

create

ident if ied

models ,

(b )

some es t imat ion theory

choices,

(c) the

impor tance

of

consu l t ing

b o th

p a t t e r n

and

s t ruc tu re coeffi-

cients when C FA factors are cor rela ted , (d) com par ing

f i t s

of nes ted models ,

and (e) ex tract ion o t h igher -order factors .

In

these i l lus tra t iv e analyses , th e Append ix A da ta p rov ided by the

100 facu l ty m e m b e r s for the f i r s t eight i tems are used . Readers may f ind i t

useful

to conduct paral le l analyses us ing the append ed data provided by the

100 g radua te s tuden ts .

MODEL IDENTIFICATION

CHOICES

A s

explained

in

chap ter

1 0 ,

parameters cannot

b e

est imated unless

th e

m o d e l

is

" iden t i f ied . "

A

m o d e l

is

u n id e n t i f i e d

o r

under iden t i f i ed

if for

t h a t model an inf in i te num ber o f es t imates fo r the paramete r s a r e equa l ly

p laus ib l e .

A necessary bu t insuff ic ient condit ion for ide nt if ic at io n is that the

number o f paramete r s be ing es t imated mus t be equa l to ( j u s t ident i f ied) or

less t han

(ove r iden t i f ied ) the ava i lab le deg rees o f f r eedom. In th i s exam ple

the

n u m b e r

of

m e a s u r e d v a r i a b l e s

is

eight . T herefore ,

the

ava i lab le deg rees

of f reedom

are 36 ([8 x (8 + 1)] / 2).

T h e

models tes ted here presum e that i tems PE R I through PER4

reflect

only

an

under ly ing f ac to r named "Service Af fec t"

and

tha t i t em s PER5

through PER8 reflect only an u nde r ly in g factor named "Lib rary as Place."

Thus ,

a total of up to

eight factor pat tern coefficients

are

es t imated . Also,

th e er rors var iances for

each

of the e igh t measu red var iab les are es t imated .

I t is assumed here tha t the factors are unc orrela ted .

H o w e v e r ,

for

th is model

w e

c a n n o t es t im ate these

16 (8 + 8)

paramete r s

because th e scales of the two la ten t factors have no t

been

indicated . These

scales

of

m e a s u r e m e n t

of

la ten t constructs

are

a r b i t r a r y . T h a t

is,

there

is

no

in tr ins ic reason

to

th ink tha t "L ib ra ry

as

Place" scores

in the

p o p u l a t i o n

should have

a

p ar t icu la r va r iance

or

s tandard dev ia t ion .

Bu t we

m us t dec la re

some scale for these late nt va riab les, or w ith

each

of

infini tely

m a n y

di f ferent

possible var iances there w ould

b e

inf in i te ly m any corresponding sets

of

o ther

parameters being es t imated .

There are two

bas ic ways

to

iden t i fy

C FA

models.

First , for a

g iven

fac tor we can "f ix" or constrain

any

g iven un stand ard ize d factor pa t tern

coefficient

to be

a ny m a th e m a t i c a ll y p l a u s i b l e n u m b e r . H o w e v e r , usua l ly

the number 1 .0 is used for th is purpose. In doing so , we are scal ing the

var iance of the la ten t construct to be measured as some func t ion of the

measured

v a r i a b l e

fo r

w h ic h

we fix the

pat tern coeff ic ien t .

Some

researchers

134 E X P LOR A TOR Y A ND CONFIRMATO RY FACTOR ANALYSIS



prefer to f ix the

pa t t e rn

coef f i c i en t for the measu red var i ab l e be l i eved

to be

mos t p sy chom et r i ca l ly r e l i ab l e . Ho wev er ,

the

se l ec t ion

is

bas i ca l ly

arb i t r a ry .

Second, w e c a n

es t imate

all the

pa t t e rn coef f i c i en t s

and f ix the

fac tor

var iances to any m athe m at i ca l ly p l aus ib l e o r "admiss i b l e" num b er .

Because v ar i ance s canno t be nega t ive , adm iss i b l e num be rs fo r var i an ce

es t imates

m ay no t be

n e g a t i v e .

And the

factor var iance

c a n n o t b e

zero ,

o r we wou ld be eva lua t ing the

inf luences

of a var ia b le tha t was no longer

a

v a r i a b l e .

When

factor var iances

are f ixed, usual ly th e

var i ance paramete r s

a re

f ixed to be

1 .0 .

This

approach

has a

s ingu la r advan tage :

When

we a re

es t imat ing fac to r covar i ances ,

th e

covar iances

of the

fac to r s wi th

each

o t h e r now

become fac to r co r re l a t ions , because g iven tha t

r\

x

n =

COV[

x

n /

(SDj

x SD j i ) , when the fac to r var i ances a re both 1.0, w e o b t ai n C O V ]

x l

j /

( 1 x 1 ) .

This

makes in t e rp re t a t ion more s t r a igh t fo rward .

In add i t ion ,

t

s ta t i s t ics a re o n l y o b t a i n e d fo r es t imated fac to r pa t t e rn

coefficients .

So i t i s help fu l to kno w the

s t a t i s t i c a l

s ign i f i cance , o r W ald

rat ios,

fo r

each factor

p a t t e r n

coef f i c i en t ,

a n d

these

w o u l d

no t be

a v a i l a b l e

i f th e

model were

ident i f ied b y

cons t r a in ing some fac to r pa t t e rn coef f i c ien t s .

T h e

approach also

has the

a d v a n t a g e t h a t

a l l the

pa t t e rn

coeff ic ients

fo r a

g iven factor

can be

com pared w i th each o ther , b ecause they were

a ll

freed. In

genera l ,

w e m a y

expec t

t h a t

pa t t e rn

coeff ic ients on a

g iven fac to r

wil l b e

s i m i l a r

in

m a g n i tu d e s .

Table

11.1 compares three analyses

o f

these

da t a wi th th ree

d if fe ren t

model i den t i f i ca t ion s t r a t eg ies hav ing been invoked . In Model 1

a rb i t r a r i l y

the

uns t and ard ized pa t t e rn

coeff ic ient for

v a r i a b l e P E R I

on

Factor

I

("Service

Affect") was f ixed to

equal 1.0,

and the

uns t andard ized pa t t e rn coef f i c i en t

tor var iab le PER8 on Factor I I ("L ib rary as Place") was f ixed to equ al 1 .0 .

T he

fac to r var i ances were

b o t h

"freed"

to be

es t imated .

In

Model

2

a rb i t r a r i ly

the

uns t andard ized pa t t e rn

coeff ic ient fo r

v a r i a b l e

P E R 2

on

Factor

I

( "Serv ice A f fec t " )

w as f ixed to

equal 1.0,

and the

u n s t a n -

dardized

pa t t e rn coef f i c i en t

fo r

v a r i a b l e P E R 5

on

Fac to r

II ( " L ib ra ry as

Place")

was f ixed to

equal 1.0.

T he

fac to r var i ances were

b o t h

"freed"

to

b e

es t imated .

In

M o d e l

3 ,

po r t r ayed g raph ica l ly

in

Figure 11.1 ,

a ll

e igh t pa t t e rn

coefficients

were es t imated . How ever ,

to identify th e

model ,

b o t h

fac to r

var iances were const rained

to be

1.0.

N o t e

f rom

T a b l e

11.1 that

th e

s t andard ized pa t t e rn coef f i c ien t s

and the

fac tor

s t ru ctu re coeff ic ien ts are ide nt ica l across the three m odels, reg ard less of

how the

model s were

iden t i f i ed . And the f i t

s t a t i s t i cs

a re

i d e n t i c a l .

S o m y

preferen ce for ident i fy ing CFA models by f ix ing factor var ianc es i s arb i t rary

bu t perm iss i b l e as a mat t e r o f s ty l i s t i c

choice .

SOME INTERPRETATION PRINCIPLES 135



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J 3 6 E X P L O R A T O R Y ' AND CONFIRMATORY FACTOR A N A L Y S I S



e1

e 2 )

(

e 3 j ( e 4

Perl iPer2J P e r 3

Per4i

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tt

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Per?

r

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V T

A A

( e7 ) ( e8

Figure

11.1. C on f irmatory fac to r ana lys is fac to r model wi th e ight meas ured variables

and two

co r re la ted fac to rs wi th fac to r var iances cons t ra ined

to

equal 1.0. ServAf f

=

"S erv ice

Af f ec t " ; LibPlace = "Library as P lace. "

SOM E

I N T E R P R E T A T I O N

P R I N C I P L E S

137



D I F F E R E N T

ES TIM A TIO N TH EO R IES

As noted in chapter 10, classical s tatis t ic s (e.g. , analysis of variance,

regress ion, canonica l corre la t ion analys is ) invoke

a

s ta t is t ica l es t imat ion

theory called ordinary

(east

squares

(OLS).

However , the re are n u m e r o u s

o ther s ta t is t ica l es t imat ion theories

t h a t

c a n b e invoked to e s t ima te mode l

pa rame te rs . T he theory most commonly used in CFA (and in s t r u c t u r a l

equa t ion m odel ing [SEM]) is m axim um l ike l ihood theory. Most of the f it

s tatis t ics

have

heen developed for this theory.

How ever , max im um like l ihood e s t ima t ion does p re sume that data a re

mul t ivar i a t e n o r m a l l y d i s t r i b u t e d .

A n o t he r

es t imat ion theory, asymptot i -

cally d i s t r i b u t ion - f ree (AD F) theory , can b e invoked i f the d i s t r i bu t iona l

assumptions

o f

m a x i m u m li k e li h o o d

c a n n o t b e

me t . However ,

A D F

es t ima-

t ion

does presume large sample s ize.

For com para t ive purposes ,

Ta b l e

1 1.2 presents

the

s tandardized pat tern

and the s t ruc tu re coeff ic ients for the Mode l 3 f it to the facul ty da ta us ing

three different es t imat ion theories .

N o t e t ha t th e

pa t t e rn

an d

s t ruc tu re

coefficients es t imated in a l l three models are reasonably comparable .

These analyses were compared pr im ari ly for heuris t ic purposes . How -

ever, i t is not entirely a bad th ing to inv oke d if fe ren t analytic s trategies for

the

same da ta , j u s t

to

conf i rm t ha t results

are not

ar t i fac ts

o f

ana lyt ic choices .

In this ins tance , probably only the maximum l ike l ihood e s t ima tes would

b e

reported,

but the

researcher would sleep easier knowing

t ha t

s imilar

factors

emerged for the other analyses.

CONFIRMATORY

FACTOR

ANALYSIS PATTERN

A N D

S T R U C T U R E C O EF F IC IEN TS

In EFA, or thogonal solut ion s a lmost a lways provide s im ple s t ruc ture ,

are the defaul t models in s ta t is t ica l packages, and d o m i n a t e the l i t e ra tu re .

How ever , in CFA, i t i s more usua l to test either correlated factors models

or b o t h

uncorre la ted

and

correlated factors models .

That is , usua l ly b o t h

C FA models are reasonably plaus ib le , and we w a n t to tes t p laus ib le

r iva l

models

t o

have

the

strongest possible supp ort

for our

preferred model , presum -

in g that

this model u l t imate ly

has

best

fit .

When factors a re u ncor re la ted , the CFA s tandardized fac tor pa t te rn

and the s t ruc tu re coeff ic ients for g iven v ar iab les on given fac tors exact ly

equa l each other , as ref lected in the i l lus t ra t ive resul ts reported in

Ta b l e

11.2. Bu t wh en CFA factors are correlated , exa ctly as is the case in EFA ,

factor pa t t e rn and fac tor s t ruc ture coef f ic ients are no

longer equal.

For mul t ip le regress ion (Courvi l le &

T h o m p s o n ,

2001; Th o mps o n

& B orre llo , 1985) , desc r ipt ive disc r im inan t analys is (H ub erty , 1 994) , and

/

3 8


A N D


FACTOR

A N A L Y S I S



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SOME

INTERPRETATION

PRINCIPLES

139



canonical co r r e l a t io n ana lys i s ( c f . Cohen

& Cohen,

1983,

p.

4 5 6 ; L e v i n e ,

1977 , p . 2 0 ; Mere d i th , 196 4 ; Thompson, Z O O O a ) , w h e n w e i g h ts an d s t r u c t u r e

c oe f f i c i e n t s

a r e

u n e q u a l , s e n s i b l e i n t e r p r e t a t i o n u s u a l ly r e q u i r e s e x a m i n a t i o n

o f bo th se t s o f r e su l t s . S t ruc tu re c o e f f i c i en t s a r e equa l ly im p o r t an t i n C FA

(cf .

G r a h a m

et

a l . , 2 003 ;

Thompson,

1997) .

Thompson, Cook, and

H e a t h

( 2 0 0 3 ) p r e s e n t e d

i l l u s t r a t i ve

s t r a t eg i e s fo r r e p o r t i n g r e s u l t s fo r co r r e l a t ed

CFA f ac to r s .

P u t d i f f e r e n t l y , a s B e n t l e r a n d Y u a n ( 2 0 0 0 ) e m p h a s i z e d , i n C F A w h e n

fac tors a re

corre la ted , "even if

a

f ac to r do es

n o t

i n f l u e n c e

a

v a r i a b l e , that

is , its p a t t e r n c o e f f i c i e n t o r w e i g h t is zero , th e c o r r e s p o n d i n g s t r u c t u r e

c oe f f i c i e n t , r ep resen t ing the co r r e l a t io n or c o v a r i a n c e of the v a r i a b l e w i t h

that

f ac to r , gene ra l ly w i l l

not be

zero"

(p.

3 2 7 ) . There

are

th ree po ss i b l e

s i t ua t io ns th a t a r i s e w i th co r r e l a t ed f ac to r s , no t c o un t ing co m bina t io ns o f

t h e s e s i t u a t i o n s .

nivoc l

Models

O n e

po ss i b i l i t y

is

t h a t

each

m e a s u r e d v a r i a b l e

re f lec ts th e

i n f l u e n c e s

of only a s ing le one of the co r r e l a t e d f ac to r s . Mo d e l 4 p resen ted in Table

11 .3 invo lves such a case.

F o r such "un ivo ca l " m o de l s

in

w h i c h each m e a s u r e d v a r i a b l e s p e a k s

o n

beh a l f

o f a

si n g le l a t e n t v a r i a b l e ,

th e

freed s t a n d a r d i z e d p a r a m e t e r s

an d

th e f ac to r s t ruc tu re co e f f i c i en t s a lw a ys exac t ly match. F o r e x a m p l e , P E R I

had a pa t ter n c oe ff ic ie nt o f Facto r 1 in Mod el 4 o f .950; the corresp ond ing

s t r u c t u r e

c oe f f i c i e n t also w a s exa c t ly .950 .

H o w e v e r , note that none of the s t r u c t u r e c oe f f i c i e n t s a s so c i a t ed w i th

p a t t e r n c o e f f i c ie n t s fixed to equa l zeroes were themselves zeroes . For e x a m p l e ,

P E R I

h ad a pa t t e rn co e f f i c i en t o n Factor I I i n Mo de l 4 f i xed to equa l ze ro ,

b u t the

c o r re s p o n d i n g s t r u c t u r e c o e f f i c ie n t

was

.398.

In other w o rds , w h en fa c to r s a r e co r r e l a t ed , a l l m easured v a r i ab l e s a r e

co r re l a t ed w i th a l l f ac to r s . This dyn am ic m us t h e co ns ide red a s pa r t o f r e su l t

i n t e r p r e t a t i o n . The h i g h e r the f a c t o r c o r r e l a t i o n s are in a b s o l u t e v a l u e s ,

t h e m o r e a l l v a r i a b l e s

w i l l

be co r r e l a t ed w i th a l l t h e f ac to r s .

Correlated Error Variances

A unique f e a t u r e of CFA v e r s u s EFA is the p o s s i b i l i t y of estimating

covariances

or

correlations

of m e a s u r e m e n t e r ro r variances. If the model is

correctly

spec i f ied , then

the

e r r o r

variances of the

m e a s u r e d v a r i a b l e s r e f l e c t

score u n r e l i a b i l i t y ( s e e Thompson, 2 0 0 3 ) .

For

e x a m p l e ,

the var iance of the

m e a s u r e d v a r i a b l e P E R I

was 1.847.

In Model 4 the estimated error variance for this v a r i a b l e was .178. This

means that,

if the

model

is correctly spec i f ied , the u n r e l i a b i l i t y of t h i s

140

EXPLORATORY

A N D


FACTOR ANALYSIS



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SOM INTERPRET TION PRIN IPLES



measured var iab le could

he

e s t ima ted

as

.178

/

1.847

=

.096

(o r

9 .6%).

The

corresponding re l i a b i l i ty of this measured var iab le could be es t imated as

(1.847 - .178) / 1.847 = 1.669 / 1.847 = .904 (or

90 .4% ;

or 100% - 9 .6%) .

Measurement error variances in classical s tatis t ics are usua l ly taken as

be ing random

and

therefore independent . However ,

in

some cases elements

of measurement error may be sys temat ic and overlapping. For example , in

a s tudy involv ing pe rcep t ions of gender roles , if two i t ems in a given s tu d y

invoked s t rong hom ophob ic emo tional reac t ions , the error var ianc es for

these

tw o

i t ems migh t

b e

corre la ted even t h o u g h t h i s d y n a m i c

is

ex t raneous

to the p r im ary f ea tu re s of the model .

Model

5

allows

the two

error var iances

for the

Factor

I

measured

var iab les ,

"Giving users in d iv id u a l a t tent ion" and "Employees who deal

with users in a caring fashion," to c o v a r y . T h i s add i t iona l s ing le pa rame te r

es t imate cos ts one addi t ional degree of f reedom, as reported in

Ta b l e

11.3.

B ut

only a l lowing error covariances to be nonzero does no t

change

the

s t ruc tu re coeff icients associated w ith freed pat tern coef f ic ients, and the

freed

pat tern coef f ic ients

and

re la ted s t ruc ture coef f ic ients

wi l l

remain equal .

Multivocal Models

Model

6 is

presented graph ica l ly

in

Figure 11 .2 .

In

this model there

is o ne

measured va r iab le (PER8) th rough which b o t h factors speak. Note

t h a t

fo r such var iab les the freed pat tern coef f ic ients no longer equal the

associated structure coeff icients (i .e . , .206

^

.489,

an d

.711

^

.793) .

C O M P A R I S O N S

OF MODEL FITS

In any CFA (o r SEM) m odel tes t ing, mu l t ip le models ma y f it the same

da ta . T h u s , i t is not best practice to test only a single, preferred model

(Th o mps o n , 2 0 0 0 c ) .

Ins tead, i t i s des i rab le to tes t m ul t ip le plau s ib le r iv a l

models .

Then,

if the p referr ed mod el is the only m odel w ith reasona ble f i t ,

the case for t h a t mo del i s correspondingly s t ronger .

In a fac tor analyt ic context ,

r iva l

models would of ten inc lude a model

speci fying independence

of the

measured v ar iab les ( i .e .,

no

fac tors ) ,

a

mode l

posit ing a s ingle factor, and m odels posit in g

b o t h

unco rrelated an d correlated

factors . T a b l e 11.4 presents

fit

s tatis t ics

for

these models

fo r

these data.

The s ta t is t ica l s ignif icance tes t for the b ase l ine independence model

pos i t ing no factors results in re jec t ion of the nu l l hypothes is t ha t the mode l

fits.

N o t e

t h a t th e ^ C A L C U L A T E D v a l u e for this test is reported in scientif ic

nota tion as "1.45E-134." This means t h a t af ter the decimal , the p is 133

zeroes fol lowed by

"145."

142

EX PL O RA T O RY

A N D

CO N FJ RMA T O RY FA CT O R ANALYSIS



e1 ) e2

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1 1 1

Per t

t

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e5 i e6 e7 e8

Figure

11.2.

C on f irmatory fac to r ana lys is M odel #6 (Fac to rs I an d II mu l t ivoca l throug h

P E R 8 ) . ServAf f

=

"Service Affect" ; LibPlace

=

"Library

as

P lace. "

SOM E

I N T E R P R E T A T I O N

P R I N C I P L E S 1 4 3



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144

E X P LO R A TO R Y A N D CO NF I R M A TO R Y F A CTO R A NA LYS IS



The

p

is not

e q u a l

to

zero, regard less

o f

w h a t

the

s ta t i s t i ca l packa ge

m ay

p r in t to r e la t ive ly f ew dec im al p laces . A common e r ro r in pub l i shed

l i t e ra tu re is er roneously repor t ing that p = .000, wh ich means th a t the

researcher impossibly ob ta ined an impossible

resul t

(wh ich is imposs ib le ) .

Such

sm all values can b e acc ura tely repor ted as p < .001, o r exac t va lues

can be

derived using Excel spreadsheet functions such

as:

= c h i d i s t ( 7 2 4 . 5 1 , 2 8 ) ,

which here y ie ld s p = "1 .45E-134 ."

For

these tes ts , Model 4 provides reasonable fit. Th is is tr u e n o tw i th -

s tand ing

th e p

value ( .03685) associated with

th e

tes t .

A s

noted

in

chap te r

10, the p values fo r m o d e l fit tests are inflated by the large sam ple sizes

usually used

in

CF A ,

as are

s ta t is t ical s ignif icance tes ts

in o ther

contexts

( T homp son ,

1996) .

The s ignif icance tes t is more

useful

in look ing at comparative fit of

nested models . In th i s example , Model 4 posi ted correla ted factors, e s t im a te d

17 paramete r s (8 pat tern coeff ic ien ts , 8 er ror var iances , and 1 covar iance) ,

and reta ined 19 (36 - 17) degrees o f fr eedom. Model 3 posi ted uncorrela ted

factors , es t imated 1 6 paramete r s (8 pat tern coeff ic ien ts , 8 e r ro r var iances ) ,

and

reta ined 20 (36 - 16) degrees of f re e d o m .

T he

models

are

nes ted because Model

3 is

Model

4

w i th

one

p a r a m e te r

( a covar iance) r emoved . Th i s m e a n s t h a t

th e

differences

in the

m o d e l

fi ts

c a n b e q u a n t i f i e d b y s u b t r a c ti n g the %

:

va lu es (48 .27 - 31.36 = 16.91) an d

the

corresponding degrees

o f

f reedom

( 2 0 - 19 = 1 ) . Th e

i m p r o v e m e n t

in

Model 4 over Model 3 is

s ta t i s t ical ly

s ignif icant (p =

.00004) .

H I G H E R- ORD E R M OD E L S

A s

exp la ined

in

chap te r

6, in EFA

h ig her -o rder fac to r s

can b e

ex t r ac ted

where lower -order factors

a re

correla ted . G i v e n

the

G L M ,

the

same pr inci -

ples

a p p ly

i n

CFA.

One di f ference in the

analyses , however ,

is th a t

h igher -

order factors

in CFA a re

e x t r a c t e d w i th

th e

factor covar iances /correla t ions

f ixed to b e

zeroes.

In CFA

h igher -order analys is ,

t he

factor covariances

are

posi ted

to he

accounted

for by the

h igher -order factors .

Three First-Order Factors

Figure 11.3 presents a h igher -order m odel invo lv ing 12 measured var i -

ables and 3 f i r s t -order factors for the data provided by the 100 faculty. In

C FA

h igher -o rder ana lyses , usual ly there

are

th ree

or

more

first-orde r

factors

so tha t

th e

h igher -o rder paramete r s wi l l become ident if ied (see Byrne, 1998,

p p . 1 7 0 - 1 7 3 ) .

SOME

INTERPRETATION PRINCIPLES

145



N /

e2 ) ( e3 ( e4

Per5

Per6

Per?

_J

i

]1

J L

.

Per8

ti

i

^_L.^

ti

e5 ( e6 e7 e8 )

Figure 11.3.

Second-order model wi th three f i rs t-order factors. ServAff

=

"Service

A ffec t " ;

LibPlace = "Library as Place";

InfoAcc

= "Information

A cces s . "

The m odel g raph ica l ly po r t r aye d in F igu re 11 .3 has 78 deg rees o f

f reedom

( [12 x (12 + 1) ] / 2) . The model expends 27 of the 78 degrees of

f reedom to es t im ate 12 er ror var ianc es of the me asured var iab les , 9 f i rs t -

o rder f ac to r pa t te rn coefficients , 3 var i anc es of the f i r st -order factors, and

3

second -o rder f ac to r pa t te rn coef f ic ien ts

(12 + 9 + 3 + 3 =

2 7 ) . Three

first-order pattern coefficients were fixed to equal Is , and the second-order

fac tor ' s v a r i a n c e was f ixed to equa l 1 .

146

E X P L O R A T O R Y A ND CO NF I R M A TO R Y F A C T O R A N A L Y S I S



Table 11.5 presents

th e

p a r a m e t e t e s t i m a t e s

and f i t

s ta t i s t ics

fo r

th is

analys i s . As

b e f o r e,

note

that pa t t e rn coef f i c i en t s

fixed to

equa l ze ro

do not

i m p l y

that co r respond ing s t ruc tu re

o r

o ther co r re l a t ions

a re

also zero .

Note that these r esu l t s sugges t

a

r easonab le model

fit.

These three

latent

v a r i a b l e s

d o

m e a s u r e

a

second -o rder f ac to r .

But the

" In fo rmat ion

Access"

first-order

fac to r

has the

h ighes t co r re l a t ion wi th

the

second -o rder

f ac t o r

(r = .990). The f i rs t -order factor , "Library as Place," has the lowes t

of

th e three

co r re l a t ions wi th

th e

second-order factor

(r =

. 6 0 2 ) .

This

a n a ly s i s reflects

the

r ichness

o f

s e c o n d - o r d e r f a c t o r a n a l y s i s

and i t s

capac i ty

t o m o d e l c om p l e x h i e r a r c h i c a l d y n a m i c s .

Two First Order Factors

A

second-order

CFA

c o r r e s p o n d i n g

to

M o d e l

#4 can be

execu ted ,

if

addi t ional const rain ts

are

imposed

so

that

the

model becomes

iden t i f ied ,

because there

are

on ly

two first-order

factors .

The

constraint i l lus t ra ted

in

Figure 11.4 is impos ing a res t r ic t ion that the two second -o rder p a t t e r n

coef f ic ien ts for the f i rs t -order fac to r s a re to be es t imated , b u t that these

es t imates m us t be equa l . In SEM/CFA s o f t w a r e , s u c h r e s t r i c t i o n s a r e c o m m u -

n ica t ed b y u s ing a s ing le l e t t e r fo r two o r more p aram ete r s cons t r a ined to

b e e q u a l .

Table

11.6 presents

the

paramete r es t imates

and the fit

s ta t i s t ics associ-

a t ed wi th th i s mode l . No te that the f i t s ta t i s t ics repor ted in Table 11.6 fo r

t h i s second -o rder m ode l

are

iden t i ca l

to the fit

s ta t i s t ics repor ted

in Table

11.3 for the correlated f i rs t -order fac to r mode l , Model #4 .

In

CFA, h igher -o rd er f ac to r ana ly s i s r eexp resses

the

correlat ions amo ng

the f i r s t - o rde r

fac to r s

b y

m o d e l i n g

these

d y n a m i c s

in an

a l t e rna t ive model .

H o w e v e r , the fits of these h igh er -o rder mode l s and the co r respond ing first-

order m odels are the s a m e , w h e n t h e r e a re f ew er than four f i r s t - o rde r fac to r s

fo r

a g iven second -o rder f ac to r . N ever the l ess , t he h igh er -o rd er ana lys i s does

p rov ide add i t iona l i n s igh t , because on ly in th i s mode l a re the co r re l a t ions

of

m e a s u r e d v a r i a b l e s

and f i r s t -order

f a c t o r s w i t h h i g h e r - o r d e r c o n s t r u c ts

q u a n t i f i e d .

M A J O R

CON CEPTS

The decis ion of how to

iden t i fy

a fac to r mode l i s a rb i t r a ry and does

no t impa c t f i t s t a t i s ti cs . Ho w eve r , s ty l i s t i ca l ly , mode l

id e n t i f i c a t ion

v i a f ix ing

f a c to r var i ances

to 1 has

s o m e a d v a n t a g e s , s u c h

a s

m a k i ng f ac to r c o v a r i a n c e s

a l so equ a l f ac to r co r re l a t ions

an d

a l l o w i n g c o m p a r i s o n s

o f

e s t i m a t e d p a t t e r n

coef f ic ien ts

f o r a l l t h e m e a s u r e d v a r i a b l e s o n a g i v e n f a c t o r .

SOME

IN T E R P R E T A T IO N PRINCIPLES 147



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EXPL ORAT ORY AND CONFIRMAT ORY F A C T O R

ANALYSIS



© 1 ) © 2 © 3 04

ServAffHH v1

Pe

— i

> r 5

t

1

Per6

Pe

i

\\1

1

i

|

Per8

e5 e6

)

e7 e8

gure

n.4. Second-order model (Sec-O rder) with two first-order

factors. S e r v A f f :

Service Affect ; LibPlace

=

Library

a s

Place.

S O M E INTERPRETATION P R I N C I P L E S 149



12

TESTING MODEL INVA RIAN CE

The

sine qua non

of

research

is f inding effects

that repl icate under

stated conditions. When factor analytic results replicate across samples or

analytic decisions, the results are said to be invariant. Confirmatory factor

analysis (CFA) is ideally suited for testing invariance, because CF A allows

constraining different types of result equalit ies (e.g., invariance of pattern

coefficients,

invariance

of

factor correlations)

in

different models

and

quant i -

fies the

degrees

of

mo del invarian ce across these variation s.

Here three sets o f invariance analyses a re reported using t he Appendix

A da ta . The examples involve all 12 measured variables and the graduate

s tudent

(HI =

100)

and

faculty

(n^ =

100) data. These analyses illustrate

testing factor invariance across different respondent groups. Al ternat ives

are equally plausible, such as investigating invariance over random ly created

subgroups,

across data

collected

from

the

same people over t ime,

or

across

the same role group but different geographic origins (e.g., s tudents from

dif feren t

regions of the country) .

SAME MODEL, DIFFERENT

ESTIMATES

The

least restrictive test

of

factor invariance presumes only that

the

same model

fits in

different groups

b ut

that

the

parameters estimated

in the

different groups are independently estimated in each group. Figure 12.1

specifies

an input model for such an analysis.

153



e1

e2 e3 e4

e9 e10

6 1 2

1

r

Perl

1

Per2

V

\

Pe

t

1

r

1

I

> r3

1

r

Per4

Pe

r

;r9

,—

--.

1

t i

1

r

i

Perl 6

Peril

] 1 _ _

^er1S

(LibPlacd

1

x\

e5

e6 e7 e8

F/gure

12.1.

Confirmatory factor analysis factor model with

1 2

measured variables

and 3

correlated factors with factor variances constrained

to

equal 1.0. ServAff

=

"Service

Affect";

InfoAcc = "Information Access"; and LibPlace = "Library as Place."

In

this analysis

in both

groups

there are 78

degrees

of

freedom,

because

there

are 12

measured variables ([1 2

x (12 + 1 )] / 2). The specified

model

posits

that

each of three correlated factors, with

factor

variances con-

s t rained

to equal Is, are estim ated . Parameters freed to be est imated in both

groups are 12

factor pattern coefficients,

1 2

error variance s

for the

measured

variables,

and 3

factor covariances/correlations

(12 + 12 + 3 =

2 7 ) .

154

EXPLORATORY

AN D

CONFIRM ATOR Y FACTOR ANALYSIS



Thus,

there

are in both groups 51 (78 - 27) unspent degrees of f r e e d om ,

for

a

t o t a l

of 102 (2 X 51)

avai lable degrees

of

freedom after paramete r

es t imat ion .

Table

12 .1 presen ts

the

s tandard ized fac tor pa t te rn

coefficients , the

factor s t ruc tu re

coefficients ,

the factor covariances (here also correlations,

because

the

fac tor var iances

are all

fixed

to

equal

1), and fi t

statistics

for

the analysis. The fit s ta t is t ics suggest that the m od e l is a plaus ib le fit to the

data provided by both groups.

There are some differences in the paramete rs es t imated ind ependen t ly

across the two groups. For example , the i tem PER3 ("Employees who d e a l

with users in a caring fashion") is more correlated with the "Service Affect"

factor for

graduate s tudents (r

s

2

=

.835

2

=

69.7%)

than fo r

faculty (r

s

2

=

.692 = 47.9% ). The

d i f ference

in the squared s tructure

coefficients ,

wh i ch

can be

directly compared because squaring

has

been invoked,

is

no tewor thy .

S tu d e n t s def ine

affect of

service

in

l ibraries more tha n faculty

in

t e rms

of

being deal t wi th

in a

caring fashion

by library staff.

Con versely , the i tem PER6 ("A m edi t a t ive place") is more correla ted

wi th the "Library as Place" factor for faculty (r

s

2

= .908

Z

= 82 .4%)

than

for

s tuden ts (r

2

=

.753

2

=

56.7%).

In

this case, faculty percept ions

of the

library

are more correla ted wi th a quie t and contemplat ive a tmosphere.

The factor correla t ions are reasonably similar. The biggest differences

occur on the correla tion of "Service Af fect" with "Library as Place" for

gradua te

s tuden ts

(r |

x

u =

.569

=

32 .4%) versus facul ty

(r j

x

\\ =

.418

2

=

1 7 . 5 % ) .

Nevertheless ,

the

single model

fit

reasonably well

in both

groups,

and

the parameter es t imates were reasonably s imilar across groups. Furthe rmo re,

the

detected differences seem reasonably

to

reflect n a tu ra l

differences in

needs and

interests

of the two

g roups

o f

l ibrary users.

SAME

MODEL,

EQUAL ESTIMATES

Figure 1 2.2 presents a more con strained test of factor inv arianc e across

the two groups. In this case the three un s tan dard ized factor pat tern

coeffi-

cien ts

are

con strained

to

equal

Is. The

remain ing nine pa t te rn

coefficients

are

freed

to be

es t imated ,

but

w i t h

the

constrain t that

the set of

nine under-

standardized estimates mus t be equal across the two groups, as reflected by the

use of

le t ters wi t hin

the

d rawing

of the

mode l .

Also

es t imated independen t ly

in the two g roups are 12 error variances for the measured variables , the

variances

of the 3

factors,

and the 3

factor covariances .

Thus, from

the

avai lable

total of 156

degrees

of

freedom

(78 X 2), a

total of 45 (9 + 2 [12 + 3 + 3] = 9 + 2

[18]

= 9 + 36)

degrees

of

f reedom

are spent on parameter e s t imation , leaving 1 11 degrees of freedom un spen t .

TESTING MODEL INVARIANCE 1 5 5



TABLE 12.1

Factor

Pattern/Structure

Coefficients and Fit

Statistics

for

First-Order

Factor

Analysis With 12

Measured Variables

(n, = 100 Graduate Students; n

2

= 100 Faculty)

Group/

variable/

statistic

Graduate

PER1

PER2

PER3

PER4

PER5

PER6

PER7

PER8

PER9

PER10

PER11

PER12

O x I I

r

\ x III

1 XIII

Faculty

PER1

PER2

PER3

PER4

PER5

PER6

PER?

PER8

PER9

PER10

PER11

PER12

Ixll

XIII

1 XIII

f

df

f/df

NFI

CFI

RMSEA

Pattern

II

students

0.945 —

0.886 —

0.835 —

0.822 —

— 0.907

— 0.753

— 0.890

— 0.834

— —

— —

— —

— —

0.569

0.642

0.567

0.944 —

0.882 —

0.692 —

0.882 —

— 0.933

— 0.908

— 0.932

— 0.798

— —

— —

— —

—

—

0.418

0.689

0.596

135.03

102

1.32

0.924

0.980

0.040

III I

— 0.945

— 0.886

— 0.835

— 0.822

—

0.516

— 0.429

— 0.507

— 0.475

0.827 0.531

0.739 0.474

0.813 0.522

0.613 0.393

— 0.944

— 0.882

— 0.692

— 0.882

—

0.390

— 0.380

— 0.390

— 0.334

0.576 0.397

0.716

0.493

0.663 0.456

0.480 0.331

Structure

II

0.538

0.504

0.475

0.468

0.907

0.753

0.890

0.834

0.469

0.419

0.461

0.347

0.395

0.369

0.290

0.369

0.933

0.908

0.932

0.798

0.343

0.426

0.395

0.286

III

0.607

0.569

0.536

0.528

0.514

0.427

0.505

0.473

0.827

0.739

0.813

0.613

0.650

0.607

0.477

0.608

0.556

0.541

0.555

0.475

0.576

0.716

0.663

0.480

Note. Parameter estimates "fixed"

to be

zeroes

are

reported

as dashes

("—").

All

critical ratios

> |2.0|.

NFI

= normed fit index; CFI = comparative fit index; RMSEA = root-mean-square error of approximation.

1 5 6


FACTOR ANALYSIS



e-\J ( e2 ) ( e3 ) ( e4 )

1

Perl

V ~ : y

1 n

e9)

(e1o)

ellj

1612,,

Per2 Per3

1

J_

1 1

Per4

Per9| Perl6

Peril

^er1

i I I

Per6

Per? Per8

Figure 12 2 Confirmatory factor analysis factor model with 12 measured variables

and 3

correlated factors with equality constraints.

ServAff

=

"Service

Affect";

InfoAcc

=

"Information Access"; and LibPlace = "Library as Place."

In

other

words the equality constraint means

that

the Figure 12.2 model

is

more

parsimonious than the Figure 12.1 model by a factor of 9 degrees

of

freedom

(111-102).

Table 12.2 presents the standardized factor pattern coefficients the

factor structure coefficients

the

factor

correlations, and fit

statistics

for the

TESTING MODEL

1NVARIANCE

7



TABLE

12.2

Factor Pattern/Structure Coefficients and Fit Statistics for First-Order

Factor Analysis With Equality Constraints

on

Unstandardized Factor

Pattern Coefficients n - i = 100 Graduate Students; n

2

= 100

Faculty)

Group/

variable/

statistic

Pattern

II

III

I

Structure

II

III

Graduate students

PER1

PER2

PER3

PER4

PER5

PER6

PER7

PER8

PER9

PER10

PER11

PER12

Ixll

X

III

r \\

x H I

Faculty

PER1

PER2

PER3

PER4

PER5

PER6

PER7

PER8

PER9

PER10

PER11

PER12

O x I I

r

\ x

III

X

III

f

df

I

z

df

NFI

CFI

R MS EA

(0.944)

—

0.893 —

0.812

—

0.834 —

— (0.907)

—

0.789

— 0.884

— 0.816

— —

— —

— —

— —

0.568

0.641

0.565

(0.947) —

0.870 —

0.735 —

0.874

—

—

(0.932)

—

0.898

— 0.935

—

0.814

— —

— —

— —

— —

0.422

0.699

0.603

144.47

111

1.30

0.919

0.980

0.039

—

0.944

— 0.893

—

0.812

— 0.834

— 0.515

—

0.448

—

0.502

— 0.463

0.825 0.529

(0.763)

0.489

0.798 0.511

0.607 0.389

— 0.947

— 0.870

— 0.735

—

0.874

—

0.393

— 0.379

— 0.395

— 0.344

0.589 0.412

(0.658) 0.460

0.690 0.482

0.488

0.341

0.536

0.507

0.461

0.474

0.907

0.789

0.884

0.816

0.466

0.431

0.451

0.343

0.400

0.367

0.310

0.369

0.932

0.898

0.935

0.814

0.355

0.397

0.416

0.294

0.605

0.572

0.520

0.535

0.513

0.446

0.500

0.461

0.825

0.763

0.798

0.607

0.662

0.608

0.514

0.611

0.562

0.542

0.564

0.491

0.589

0.658

0.690

0.488

Note.

Parameter estimates

"fixed"

to be zeroes are reported as dashes ( — ). Parameter estimates

"fixed"

to equal

1.0 for the

unstandardized model

are

reported

in

parentheses.

All

critical ratios

>

|2.0|.

NFI =

normed

fit index; CFI = comparative it index; RMSEA = root-mean-square error of approximation.

158

E X P L O R A T O R Y AND CONFIRMATORY

FACTOR ANALYSIS



analysis .

The fi t s tatistics suggest

that

the mod el is again a plau sible fi t to

the

da ta p rov ided

by

both groups.

The fit is not as

good

as the fit of the

less con strained m odel ,

but

never the less remains plaus ib le . Further res tr ic t ions on

equal i t ies

might be

imposed (e.g. , constraining

factor

covariances

to be

equal ) ,

bu t

such addi-

t ional con strain ts w ould necessi tate some reasons

as to why each set of

add i t iona l paramete rs might be expected to be i n v a r i an t .

Al te rna t ive ly ,

a

sequence

of

equal i ty cons t ra in t s m ight

be

imposed

to

detect

w here, i f a t a l l , invariance breaks down across groups (Byrn e, 2001;

Joreskog

&.

Sorbom, 1993).

The

sequence

of

invariance tes ts wi th equal i ty

cons t ra in t s might

proceed as follows: (a )

e q u a l i t y

of

uns tand ard ized pa t te rn

coefficients; (b)

equal i ty

of

u n s t an d a rd i ze d p a t t e rn

coefficients an d

factor

covariances; (c) equal i ty of uns ta nd ardized pat ter n

coefficients,

factor covari-

ances,

and

factor variances;

and (d)

equal i ty

of

u n s t an d a rd i z e d p a t t e rn

coefficients,

factor covariances , factor varianc es ,

an d

e r ror var iances

of

mea-

sured

variables.

SAME MODEL, TEST INVARIANCE

OF

PARAMETERS

The ul t ima te tes t of mod el in varian ce is us ing some or all of the specific

parameter es t imates

from on e

sample

in a

model tes t wi th

an

i n d e p e n d e n t

sample. Here

the

uns tand ard ized pa t te rn

coeff icients for the

model es t imated

wi th g rad ua te s tuden ts were

f i t to the

da ta p rov ided

by the

facul ty .

Figure

12.3

presen ts

the

inpu t mode l .

Table 12 .3 presen ts the s tandardized factor pat tern coefficients, the

factor

s t ruc tu re

coefficients,

the factor correlations, and fi t s tatistics for the

analysis .

The f i t s ta t is t ics

suggest

tha t the model is near the bou nd ary of

plausible

f i t to the data provided by the

facul ty .

Ne ver th e less , the deg ree

of fit is

s t r ik ing g iven

that

selected

specific

parameters from

the

g radua te

s tudents were used

in the

tes ted model .

In

pract ice , researchers usually hope t ha t

the

same cons t ruc ts

are

measured

across group s but usually do not expect factor pat tern

coeff icients

to be

completely invarian t . Nevertheless , such tes ts i l lus tra te

the

power

of

CFA to find the

boundar ies

of

resul t invariance.

MAJOR CONCEPTS

Science i s abou t i so la t ing and und ers tan d ing la ten t cons t ruc ts. We

seek con structs that repl icate unders ta ted condi t ions and that are useful for

stated purposes.

In

other words ,

w e

seek factors that

are

invar ian t over

samples and analyses .

TESTING

MODEL INVARIANCE

159



e 1 ) e 2 ) 6 3 e 4

1

e9j

(elQ) ( e11J e12

Per9 RerlCJ eril

v

(LibPlacq

1

Per5

Per6 Per?

65 ) 66 67 68

Figure 12.3. Confirmatory factor analysis factor model

with 12

measured variables

and 3 correlated factors

with

1 2 unstandardized pattern coefficients from graduate

students constrained as fixed. ServAff = Service Affect";

InfoAcc

= Information

Access";

and

LibPlace

=

"Library

as

Place.

For exam ple, do w e obtain the same or simila r factors across ind epen -

dent samples of people? Do we obtain similar factors when we analyze two

similar but different

sets

of

measured

variables? Do we

obtain similar factors

when we invoke different estimation theories?

CFA is ideally suite d to tes tin g invarian ce propositions, because of its

flexibility

and

capacity

to quantify

degrees

of

model fi t .

When

rival models

for

a given da ta set are nested, we can quant i fy the degree to which more

160 EXPLORATORY A ND CONFIRMAT ORY

FACTOR

ANALYSIS



TABLE 12.3

Factor

Pattern/Structure Coefficients and Fit Statistics for First-Order

Factor Analysis for Faculty Data n = 100) Fitting Graduate Students'

Unstandardized Factor Pattern Coefficients

Variable/

statistic

PER1

PER2

PER3

PER4

PER5

PER6

PER7

PER8

PER9

PER10

PER11

PER12

1.x II

XIII

1l x III

x

2

df

f/df

NFI

CFI

RMSEA

Pattern

II

[0.966] —

[0.902] —

[0.816]

—

[0.901] —

— [0.917]

— [0.837]

— [0.929]

— [0.801]

— —

— —

— —

— —

0.460

0.783

0.575

106.71

63

1.69

0.879

0.946

0.084

III

I

— 0.966

— 0.902

—

0.816

— 0.901

— 0.422

—

0.385

— 0.428

— 0.369

[0.708] 0.554

[0.753] 0.590

[0.808] 0.632

[0.613] 0.480

Structure

II

0.445

0.415

0.375

0.415

0.917

0.837

0.929

0.801

0.407

0.433

0.464

0.352

III

0.756

0.706

0.638

0.705

0.527

0.481

0.534

0.461

0.708

0.753

0.808

0.613

Note. Parameter estimates fixed to be zeroes are reported as dashes ( — ). Parameter estimates fixed

to

equal

unstandardized pattern coefficients for the graduate students are reported in brackets. A ll critical

ratios > |2.0|. NFI = normed fit index; CFI = comparative fit index; RMSEA = root-mean-square error of

approximation. 78-(12

+ 13)

freed estimates

= 36.

parsimonious models estimating

fewer

parameters

fit

less wel l ,

as noted in

chapter

11. Remember that less parsimonious models (i.e., models estimating

more parameters) fit as well as or better than more parsimonious models,

until at the extreme w e estimate all possible parameters (i.e., df = 0),

at

which

point

every

j u s t - i den t i f i ed

model

with

zero degrees

of

freedom

fits perfectly.

Because

% s for nested models are additive, we can subtract the %

2

for

th e

less parsimonious model (e.g.,

%

2

=

31.36, df

= 19) from the %

2

for the

more parsimonious model (e.g., Jj = 48.27,

df

= 20) to quantify h ow

much

better

the

less parsimonious model

fits

(e.g.,

2

48.27

-

31.36

=

16.91).

Because degrees of freedom

from

nested models are additive, we can also

subtract the degrees of freedom (e.g., 20 - 19 = 1). This allows calculation

of

th e p

value (e.g.,

p =

.00004

fo r f =

16.91 with

df = 1) for the

null

hypothesis that

the

nested models

fit

equally

wel l .

TESTING MODEL

INVARIANCE

1 6 1



In CFA, we can fi t s imi la r but d i f ferent models to d i f feren t types o f

samples or to the same sample over time. We can fit the same models to

different samples but free the parameters to be est imated independent ly in

each

data set . Or we can fit the same model and also use some or all of the

est imates

in one

sample with

the

da ta

from

an

independent sample (see

Thompson

et

a l . , 2003) . Analyses

of the

last kind

are the

u l t ima te

in

tests

of invar iance ,

and the

u l t im a t e

in

parsimony, because

at the

ex t reme

we

are es t ima t ing no pa rameter s wi th new da ta when w e fi t only the parameters

from

a

prior study.

] 62

EXPLORATORY

AND

CONFIRMAT ORY FACTOR ANALYSIS



A P P E N D I X

A

LibQUAL+™ Da t a

0 0 1 2 8 7 5 5 3 2 3 3 6 5 5 7

002 2 5 7 5 5 4 5 5 6 3 4 3 6

0 0 3

2 6 5 5 6 5 3 5 3 3 5 4 7

0 0 4 2 5 5 4 6 4 4 4 4 2 1 5 4

0 0 5

2 5 5 5 5 5 4 6 2 4 4 5 7

006 2 7 7 7 8 7 8 7 6 6 8 7 5

007

2 8 8 7 7 6 7 6 7 6 5 6 5

008 2 7 7 7 7 5 3 3 6 5 4 5 8

9 2 1 3 1 1 1 1 1 1 1 1 1 1

0 1 0

2 9 9 9 8 5 7 5 2 7 6 7 7

1 1 2 9 9 9 7 7 9 9 7 4 6 5 9

1 2 2 4 3 3 5 6 5 4 5 2 5 1 4

013

2 6 6 5 7 7 7 7 7 6 5 6 7

0 1 4

2 7 7 2 7 2 3 2 3 6 6 5 9

0 1 5 2 9 9 9 9 7 5 4 5 4 1 1 4

016

2 9 9 9 9 5 5 5 7 6 7 6 6

017

28899766868

9 9

0 1 8

2 8 5 8 5 4 8

5

8 4 3 4 7

019 2 9 9 9 9 6 7 6 6 7 9 9 9

0 2 0 2 8 7 7 9 5 6 6 4 5 6 2 9

021 2 9 9 8 9 9 9 9 8 8 6 6 7

022

2 7 7 7 8 7 6 7 6 7 8 8 7

023

2 7 8 7 6 5 6 5 6 6 5 7 7

0 2 4 2 8 9 7 8 5 3 4 5 9 9 7 7

025 2 8 8 8 8 6 6 6 7 9 6 9 9

026

2 8 9 9 9 6 6 6 6 8 8 9 8

027

2 8 8 8 8 8 7 8 8 8 9 8 7

028

2 9 9 9 7 6 7 7 6 9 9 6 9

0 2 9

2 6 5 6 5 5 6 6 5 4 5 4 5

0 30 2 8 8 7 8 7 8 8 7 8 7 6 7

031 2 7 7 7 8 8 7 8 7 7 6 8 8

3 2 2 1 1 2 1 1 1 1 1 6 6 5 9

0 33 2 8 8 8 8 8 7 7 8 6 1 6 7

0 34

2 8 8 6 7 5 6 5 6 8 9 8 7

035

2 8 8 8 5 7 4 8 5 6 6 6 7

0 36 2 8 8 8 8 5 5 6 5 7 7 7 8

0 3 7 2 6 4 5 6 7 7 7 8 5 5 4 4

1 6 3



0 3 8

2 5 4 5 3 3 4 3 4 6 7 3 3

0 3 9

2 5 7 6 7 7 5 6 7 6 6 5 5

0 40 7 7 7 8 6 7 7 7 7 6 8 8

0 41 2 6 8 7 7 8 7 7 7 7 9 7 7

0 42

2 7 8 8 5 6 6 6 6 6 5 5 6

0 43 2 8 8 7 7 5 6 5 6 6 7 6 7

0 4 4 2 7 7 5

7 4 5 4 4 8 7 6 8

045 2 9 9 9 9 8 9 8 8 9 7 7 9

046 2 7 7 7 7 7 7 7 7 7 7 7 1

047 2 7 5 7 6 4 7 5 4 7 9 6 7

0 4 8

2 1 3 3 1 4 5 4 4 9 6 7 7

0 49 2 7 8 6 7 8 8 8 8 7 6 7 7

0 5 0 2 8 8 7 8 8 7 7 8 7 8 7 7

051

2988864686767

052

2 9 9 8 9 8 7 8 7 8 7 9 9

053 2 6 6 5 6 8 4 6 7 7 7 6 7

0 5 4

2 8 8 8 4 5 4 5 8 7 9 6 7

0 55

2 8 8 7 9 2 2 2 1 5 2 6 8

0 5 6

2 8 8 6 8 3 3 5 6 6 5 4 7

057

2 9 9 9 9 2 2 2 2 8 9 9 9

058 2 8 8 8 8 7 7 8 8 8 8 8 8

059

2 6 5 6 7 6 7 6 7 5 6 6 7

0 6 0

2 8 9 7 7 6 5 6 6 7 7 6 7

0 6 1

2 7 7 7 7 7 7 7 8 7 7 7 7

0 6 2 2 6 7 7 6 4 4 5 7 6 3 7 3

063

2 8 8 7 8 8 6 7 9 7 7 7 8

064

2 8 7 8 8 4 4 4 3 8 2 6 4

0 65 2 8 9 8 7 7 7 7 7 6 6 7 7

066 2 8 8 9 8 8 8 8 8 7 9 6 7

0 6 7 2 7 7 7 7 7 7 6 8 7 6 7 8

0 6 8 2 6 3 7 9 4 3 3 3 9 4 5 5

069

2 9 8 8 9 8 8 7 8 6 7 6 9

0 7 0 2 6 6 6 5 4 4 4 4 5 256

0 71

2 8 8 9 8 7 7 7 7 8 6 7 8

0 7 2 2 8 8 8 8 7 6 8 7 7 7 7 8

0 73 2 9 9 7 8 4 5 6 7 6 7 6 7

0 7 4 2 8 8 8 8 6

6

6 6 8 8 8 8

075

2 8 8 8 7 8 7 8 7 6 8 8 9

076 2 8 8 7 8 4 5 6 8 8 5 6 7

0 77

2 6 6 6 6 7 7 5 6 6 5 3 7

0 7 8

2 7 5 5 4 3 5 5 6 2 2 6 8

079

2 4 4 5 8 7 8 8 8 6 4 6 9

0 8 0

2 9 9 8 7 8 1

5

99699

64 APPENDIX A



0 8 1

2 8 5 8 7 3 1 6 1 7 7 6 9

0 82 2 9 7 7 9 9 8 7 9 8 8 8 7

0 83 2 8 8 8 8 6 4 7 7 7 9 6 6

0 8 4

2 7 7 7 7 6 6 6 7 6 7 6 8

8 5 2 4 2 3 4 1 4 1 1 1 1 1 1

0 86

2 6 7 5 6 5 6 6 6 5 6 6 7

0 87

2 6 6 7 7 8 5 6 9 6 7 6 7

0 8 8

2 8 8 7 8 6 5 7 7 8 5 6 9

089 2 9 9 9 9 9 6 9 9 8 9 9 7

090 2 5 5 5 5 3 3 44.7 5 5 7

091 2 8 8 8 8 7 7 7 7 6 9 7 9

092 2 5 5 4 5 7 7 7 7 7 6 7 5

093

2 8 8 8 8 6 6 6 6 4 4 6 7

094

2 8 8 9 8 8 6 8 8 9 5 7 9

0 95

2 8 7 7 7 8 6 7 6 7 7 7 8

0 96 2 8 8 8 8 7 7 7 6 7 5 7 8

0 97 2 7 8 7 8 7 5 8 5 7 8 7 7

0 9 8 2 4 5 2 1 5 1 2 2 1 2 6 2

0 99

2 8 8 1 8 3 5 8 3 4 2 1 7

100

2 9 8 8 9 7 4 9 8 9 8 9 9

1 1 3 7 7 7 7 6 7 6 6 7 6 6 8

1 2 3

8 8 6 5 4 4 4 7 6 7 5 8

1 3

3 5 5 5 5 5 7 5 6 4 5 5 5

104 3 9 9 9 9 6 6 6 7 5 6 9 7

1 5

3 9 9 8 8 5 3 5 6 8 6 6 8

106 3 6 7 5 7 6 5 5 6 6 7 8 4

1 07 3 8 6 9 7 6 8 8 7 5 5 6 7

108 3 8 7 8 8 6 4 6 7 6 5 6 7

1 0 9 3 6 6 6 5 5 5 5 4 6 7 5 8

1 1 0 3 8 7 7 8 6 8 4 6 6 8 7 7

3 9 8 9 9 9 9 9 9 8 5 8 9

1 1 2

3 7 7 8 7 6 8 7 5 6 6 7 5

1 1 3 3 7 7 7 6 6 6 6 6 4 6 7 6

1 1 4

3 7 7 7 8 1 2 2 1 7 7 8 3

1 1 5 3 7 7 8 7 7 7 7 7 7 5 64

1 1 6 3 8 9 8 8 6 5 6 9 4 3 5 7

1 1 7 3 5 4 5 5 7 7 5 5 1 3 5 3

1 1 8

3 9 9 8 9 7 7 7 9 7 6 8 9

1 1 9 3 8 8 8 8 7 7 7 8 6 7 8 8

1 2

3 7 6 8 8 7 6 7 6 7 8 7 5

1 2 1 3 8 8 8 9 8 8 8 8 4 7 6 7

1 2 2 3 8 7 8 8 5 4 6 5 6 7 6 7

1 2 3 3 5 3 4 5 2 2 2 2 2 2 2 7

APPE NDIX A 165



1 2 4 3 9 9 9 9 8 8 8 8 6 7 6 9

1 2 5 3 88 7 93 3 3 5 4 5 68

1263 999998896 7 8 8

1 2 7

3 9 9 8 9 8 8 8 8 8 7 8 9

1 2 8

3 9 9 7 9 7 7 7 6 8 7 8 7

1 2 9 3 3 3 7 3 7 5 6 3 7 3 6 6

1 30

3 9 9 8 9 6 7 7 7 7 7 8 8

1 3 1 3 8 8 6 7 3 5 4 5 8 6 8 8

1 3 2

5 5 5 5 2 3 3 2 6 6 5 5

1 3 3

3 7 6 8 8 5 5 5 6 5 4 5 8

1 3 4 3 8 8 9 8 6 6 6 7 8 7 8 8

1 3 5 3 8 8 9 8 8 9 8 8 7 3 7 7

1 3 6 3 7 6 8 8 4 4 4 6 8 6 7 8

1 3 7 3 8 8 8 8 2 2 2 2 1

1 6 4

1 3 8

3 8 7 8 9 6 7 5 7 6 8 6 6

139 3 7 8 8 7 6 7 6 7 4 8 8 9

140 3 9 9 7 8 7 8 7 8 5 6 8 8

1 4 1

3 8 7 7 8 6 7 8 7 7 6 6 7

1 4 2

3 8 7 6 7 6 6 6 6 7 2 5 8

1 4 3

3 7 7 7 5 3 2 5 3 9 3 7 4

144 3 8 8 8 8 9 8 8 8 8 8 8 7

1 4 5 3 8 7 8 7 3 5 5 5 6 8 5 6

1 46 3 7 7 6 7 6 5 6 7 7 5 7 6

1 4 7

3 8 8 7 8 5 6 5 6 6 3 5 3

1 48

3 5 7 5 7 6 6 6 6 5 5 6 7

149 3 9 9 9 9 7 7 7 7 9 9 9 7

1 5 3 8 8 7 8 5 5 5 7 1 5 6 8

1 5 1 3 8 5 8 8 8 8 8 8 7 2 6 6

1 5 2 3 9 9 9 9 5 6 5 9 9 8 8 5

1 5 3 3 9 9 8 9 8 8 8 9 8 9 8 8

1 5 4 8 8 8 8 7 7 7 8 8 8 7 7

1 5 5 3 6 5 5 5 3 2 3 2 1 2 5 3

1 5 6

3 5 5 3 5 5 5 5 7 5 3 7 7

1 5 7

3 7 6 7

7 3 3 4 4 3 6

7 4

1 5 8 3 7 6 6 7 5 5 7 6 7 4 8 6

1 5 9 3 9 8 9 9 6 6 6 7 7 9 8 9

160

3 7 7 7 7 6 6 6 2 8 8 6 7

1 6 1 3 8 8 3 7 5 5 5 4 7 5 6 7

162

3888888897 3 72

1 6 3 3 9 9 9 9 1 1 1 1 6 5 6 9

164

3 9 9 9 9 6 6 7 6 6 3 8 7

1 6 5 3 7 7 6 7 6 7 6 6 6 7 5 6

1 6 6 3 9 5 9 8 9 9 9 3 9 8 9 7

66 APPENDIX A



A P P E N D I X

B

SPSS Syntax

to

Execute

a

Best-Fit Factor Rotation

SET PRINTBACK=LISTING .

COMMENT Type in the pattern matrix that is the target as

matrix A ; Then type in the pattern matrix to be

rotated to

'best-fit'

position with the target as

matrix B ; Type in a matrix N_A consisting of all

zeroes with the same number or rows as A and B ;

Type in a square matrix of all zeroes with the number

of

rows and columns equal to the number of orthogonal

factors

in A and B .

MATRIX

.

COMMENT 100 Grad Students,

COMPUTE A =

{ .92134, .18690,

.21945,

.28211,

.28360,

.84662,

.85348,

.82875,

.80378,

.13580,

Varimax Pattern Matrix

.84248,

.75606,

.80837,

.22292,

.17110,

.29969,

.22454,

.30781,

.15795,

.27222,

.18737

.25625

.35335

.19958

.26935

.03794

.22423

.26075

.81976

.23077,

.80478

.20840, .80207 } .

COMMENT 100 Faculty, Varimax Pattern Matrix

COMPUTE B =

.18675

;

.21537

.16198

.24099

.21722

.21639

.19870

.14477

.66769

.79192

.71967

}

{ .20284

.10828,

.23051,

.16361,

.90807,

.89654,

.90832,

.79337,

.33664,

.18185,

.11897,

COMPUTE

N_A =

{ -0 ;

. 0 ;

.0

;

.0 ;

.0 ;

.89659,

.87403,

.73176,

.87285,

.12646,

.13470,

.17788,

.32388,

.10530,

.22943,

.33942,

169



.0

;

-0

.0 ;

0 ;

0 ;

.0

} .

COMPUTE DIAG_M

=

{ . 0 , . 0 , . 0 ;

0 , . 0 , . 0 ;

.0,

.0, .0 } .

COMPUTE N_B=N_A .

PRINT A /

FORMAT='F8.2 ' /

TITLE=

1

First Pattern Matrix (Target)' /

SPACE=4

/

RLABELS=Perl, Per2, Per3, Per4, PerS, Per6,

Per7, Per8, Per9, PerlO, Peril

/

CLABELS=Fact_I, Fact_II, Fact-Ill

/ .

COMPUTE A_N=A

.

-LOOP #1=1

TO

NROW(A)

.

+

LOOP #J = 1 TO NCOL(A) .

COMPUTE A_N(#I,#J)=A(#I,#J) ** 2 .

+

END

LOOP

.

-END LOOP .

PRINT

A_N /

FORMAT='F8.4' /

TITLE='First Pattern Matrix (Target) Squared' /

SPACE=4 /

RLABELS=Perl, Per2, Per3, Per4, Per5, Per6,

PerV, Per8, Per9, PerlO, Peril

/

CLABELS=Fact_I, Fact_II, Fact_III

/ .

-LOOP #J=1 TO NCOL(A) .

+

LOOP #1=1 TO NROW(A) .

COMPUTE N_A(#I)=A_N(#I,#J) + N_A(#I) .

+

END LOOP .

-END LOOP

.

PRINT

N_A /

FORMAT='F8.3' /

TITLE=

>

Row Sum of Squares for First Pattern Matrix' /

SPACE=4

/


PerV,

PerS, Per9, PerlO, Peril / .

LOOP

#1=1

TO

NROW(A)

.

COMPUTE N_A(#I) = 1.0 / (N_A(#I) ** .5) .

END LOOP .

PRINT

N_A /

FORMAT='F8.3'

/

TITLE='Normalization Factor

for

Rows'

/

SPACE=4

/

RLABELS=Perl, Per2, PerS, Per4, PerS, Per6,

170

APPENDIX

B



Per7, Per8, Per9, PerlO, Peril / .

-LOOP #J=1 TO NCOL(A) .

+ LOOP #1=1

TO

NROW(A)

.

COMPUTE A_N(#I,#J)=A(#I,#J)

*

N_A(#I)

.

+ END LOOP .

-END LOOP

.

PRINT A_N /

FORMAT='F8.4' /

TITLE='First Pattern Matrix (Target) Normalized' /

SPACE=4 /


PerV, Per8, Per9,

PerlO,

Peril

/

CLABELS=Fact_I, Fact_II,

Fact__III

/ .

PRINT B /

FORMAT='F8.2'

/

TITLE='Second Pattern Matrix'

/

SPACE=4

/


PerV,

PerS, Per9,

PerlO,

Peril /


/ .

COMPUTE B_N=B .

-LOOP #1=1 TO NROW(B) .

+ LOOP #J=1

TO

NCOL(B)

.

COMPUTE B_N(#I,#J)=B(#I,#J)

** 2 .

+ END LOOP .

-END LOOP .

PRINT

B_N /

FORMAT='F8.4' /

TITLE='Second

Pattern Matrix Squared' /

SPACE=4 /

RLABELS=Perl, Per2, Per3, Per4,

Per5,

Per6,

PerV, PerS, Per9, PerlO, Peril /

CLABELS=Fact_I, Fact_II, Fact_III / .

-LOOP #J=1 TO NCOL(B) .

+

LOOP #1=1 TO NROW(B) .

COMPUTE N_B(#I)=B_N(#I,#J)

+

N_B(#I)

.

+ END LOOP .

-END LOOP

.

PRINT N_B /

FORMAT='F8.3'

/

TITLE='Row Sum of Squares for Second Pattern Matrix' /

SPACE=4 /

RLABELS=Perl,

Per2,

Per3,

Per4, Per5, Per6,

PerV, PerS, Per9, PerlO, Peril / .


COMPUTE N_B(#I)

= 1.0 /

(N_B(#I)

** .5) .

END

LOOP .

PRINT

N_B /

FORMAT='F8.3' /

TITLE='Normalization Factor for Rows' /

A P P E N D I X

B 171



SPACE=4

/


Per7, PerS, Per9, PerlO, Peril / .

-LOOP #J=1 TO NCOL(B) .

+


COMPUTE B_N(#I,#J)=B(#I,#J)

*

N_B(#I)

.

+ END LOOP .

-END LOOP .

PRINT B_N /

FORMAT='F8.4'

/

TITLE='Second Pattern Matrix Normalized'

/

SPACE=4

/

RLABELS=Perl, Per2,

Per3,

Per4, PerS, Per6,

Per?, PerS, Per9,

PerlO,

Peril /

CLABELS=Fact_I, Fact_II, Fact-Ill / .

COMPUTE

A_T=TRANSPOS

(A_N)

.

PRINT

A_T /

FORMAT=

'F8.2' /

TITLE='A_N Transpose'

/

SPACE=4

/

RLABELS=Fact_I, Fact_II, Fact-Ill /

CLABELS=Perl, Per2, Per3, Per4, PerS, Per6,

Per7, PerS, Per9, PerlO, Peril

/ .

COMPUTE B_T=TRANSPOS(B_N) .

PRINT B_T /

FORMAT='F8.2'

/

TITLE='B_N

Transpose' /

SPACE=4

/

RLABELS=Fact_I,

Fact_II, Fact-Ill /

CLABELS=Perl, Per2, Per3, Per4, PerS, Per6,

PerV,

PerS, Per9, PerlO, Peril / .

COMPUTE RI=A_T * B_N .

PRINT

RI /

FORMAT= > F8.3 ' /

TITLE='A_N Transpose times B_N' /

SPACE=4 /


CLABELS=Fact_I,

Fact_II, Fact-Ill / .

COMPUTE RI_T=TRANSPOS(RI) .

PRINT

RI_T /

FORMAT='F8.3' /

TITLE='Transpose of (A_N Transpose times B_N)' /

SPACE=4 /


CLABELS=Fact_I, Fact_II, Fact-Ill

/ .

COMPUTE QUAD=RI * RI_T .

PRINT QUAD

/

FORMAT='F8.3' /

TITLE='A_N Trans * B_N * Trans of (A_N Trans * B_N)' /

SPACE=4 /

] 72

APPENDIX

B



TITLE=

>

D= trans (trans A times B) times Eigenvectors' /

SPACE=4 /

RLABELS=Fact_I,

Fact_II, Fact_III /


-LOOP J=l TO NCOL(A) .

COMPUTE EE=EIG(J) .

+ LOOP

1=1 TO

NCOL(A)

.

COMPUTE D(I,J)=D(I,J) * EE .

+ END LOOP .

-END LOOP .

PRINT D /

FORMAT=

>

F9.3' /

TITLE='D

= D

times Eigenvalues

**

-1.5'

/

SPACE=4 /

RLABELS=Fact_I,

Fact_II, Fact_III /

CLABELS=Fact_I,

Fact_II, Fact_III / .

COMPUTE D_T=TRANSPOS(D) .

PRINT D_T /

FORMAT='F9.3' /

TITLE='D transposed' /

SPACE=4

/

RLABELS=Fact_I,

Fact_II, Fact_III

/

CLABELS=Fact_I,

Fact_II, Fact_III / .

COMPUTE C=EIGVEC

* D_T .

PRINT

C I

FORMAT='F9.3' /

TITLE='Factor Correlations (Cosines)'

/

SPACE=4

/

RLABELS=Fact_Ia, Fact_IIa, Fact_IIIa /

CLABELS=Fact_Ib,

Fact_IIb, Fact_IIIb

/ .

COMPUTE C=D * VEC_T .

COMPUTE B_ROT=B * C .

PRINT

B-ROT /

FORMAT='F8.3'

/

TITLE='B rotated to Best-Fit with A' /

SPACE=4

/

RLABELS=Perl,

Per2, Per3, Per4, Per5, Per6,

PerV, Per8, Per9, PerlO, Peril /


COMPUTE BROT_N=B_ROT

.

-LOOP

#1=1 TO NROW(A) .

+ LOOP #J=1 TO NCOL(A) .

COMPUTE BROT_N(#I,#J)=B_ROT(#I,#J)

** 2 .

+

END

LOOP

.

COMPUTE N_A(#I)= .0 .

-END LOOP .

PRINT BROT_N /

FORMAT='F8.4' /

TITLE=

>

Best

Fit Pattern

Matrix

(Target)

Squared'

/

SPACE=4 /

174

APPENDIX

B




PerV, Per8, Per9, PerlO, Peril

/

CLABELS=Fact_I, Fact_II,

Fact__III

/ .

-LOOP

#J=1 TO NCOL(A) .

+

LOOP

#1=1 TO

NROW(A)

.

COMPUTE N_A(#I)=BROT_N(#I,#J) + N_A(#I) .

+

END

LOOP

.

-END LOOP .

PRINT N_A /

FORMAT='F8.3' /

TITLE=

1

Row

Sum of Squares for

Best

Fit

Matrix'

/

SPACE=4 /


PerV,

Per8, Per9, PerlO, Peril / .


COMPUTE N_A(#I)

= 1.0 /

(N_A(#I)

** .5 } .

END

LOOP

.

PRINT N_A /

FORMAT='F8.3'

/

TITLE='Normalization Factor

for

Rows'

/

SPACE=4

/


PerV, Per8, Per9,

PerlO,

Peril

/ .

-LOOP

#J=1

TO

NCOL(A)

.

+


COMPUTE BROT_N(#I,#J)=B_ROT(#I,#J)

*

N_A(#I)

.

+ END

LOOP

.

-END LOOP .

PRINT BROT_N /

FORMAT='F8.4' /

TITLE=

>

Best Fit Pattern Matrix (Target) Normalized' /

SPACE=4 /


PerV, Per8, Per9,

PerlO,

Peril

/


/ .

COMPUTE BROTN_T=TRANSPOS(BROT_N)

.

COMPUTE T_M=A_N * BROTN_T .

COMPUTE TEST=DIAG(TJVI) .

PRINT

TEST /

FORMAT='F8.3'

/

TITLE='Test Vector Cosines for Variables' /

SPACE=4 /


PerV, Per8 , Per9, PerlO, Peril / .

END

MATRIX

.

APPENDIX B 175



NOTATION

C v x v

T he

symm etr ic va r iance/cova r iance

(or

simply covar iance) ma tr ix wi th

var iances on the diagonal and covar iances of measured va r iables wi th each

other

off the

diagonal.

D

2

,

The Ma ha lano bi s distan ce of the i th person's set of scores from the set of

means

on the

measured va r i ab les .

F

N x V

The scores on the latent factors.

I P X F

The

ident i ty ma tr ix which, when postmult ipl ied times

an

original conform-

able matr ix , y ie lds

as an

answer

the

or iginal matr ix ( l ike

the

n u mb e r

1

in algebra) .

P V X F The f irs t-order

factor pat tern coeff ic ient matr ix .

P F X H T he

second-order fa ctor pa ttern coefficient m a tri x.

P V X H The

product

(first-order b y

second-order) factor pat tern coefficie nt m atrix .

R F X F The

symmetr ic ma t r ix

of

bi v ar i a t e correla t ions (e .g. , Pearson

r's)

among

the f factors.

R V X V The

symm etr ic ma t r ix

of

biv ar ia te correla t ions (e .g. , Pearson

r's)

among

the v


R V X V The

reproduced

correlation

matrix (i .e.,

the

portion

of the

intervar iable

correla t ion matr ix that the factors [i.e., the pattern coefficients]

can

re-

produce).

R V X V The

residual correla t ion ma tr ix ( i .e . ,

the

por t ion

of the

i n t e r v a r i ab le

correla t ion matr ix

that

the factors [i.e., the pattern coefficients]

cannot

re-

produce) .

R V x v ' The

inver ted intervar iable

correlation

matrix, which satisfies

th e

equat ion:

RyxV

RvxV

=

IvxV'

S \ x F

The f i r s t -order

factor s t ructure

coefficient

m a t r i x .

W

V

x F

The factor score coefficient m at rix.

Z

N

x

v The measured va r iables in z score form, such

that

the columns have means

of

zero

and

s tandard devia t ions

(and

v ar iances)

of one.

177



GLOSSARY

Communality coefficient h

2

). A statistic in a squared metric indicating how much

of the variance in a measured variable the factors as a set can reproduce, or

conversely, how much of the variance of a given measured variable w as useful

in delineating

the

factors

as a

set.

Component.

A set of pattern

coefficients

for the measured variables that can be

used

to estimate factor scores, derived without iterative estimation of communality

coefficients. (Some researchers consider factor a synonymous term, but others

feel that factor is a term that should be reserved fo r weights derived when

iteration is invoked.)

Critical ratio or

or Wald statistic).

A parameter estimate divided by its standard

error,

which is expected to be greater

than

roughly |2.0| for parameters

that

one

expects

to he

nonzero.

Eigenvalues

A . ) .

Area-world, variance-accounted-for statistics that characterize the

amount of information present in a given factor or function. (Eigenvalues a re

also sometimes synonymously called characteristic roots. )

Factor rotation. Graphic v isua l

or

mathematical movement

of the

axes measuring

the factor space used in exploratory factor

analysis

so that the factors can be

more readily interpreted.

Factor

scores. The latent (synthetic or composite) variables computed on each

factor (like the Y scores in multiple regression, or the discriminant function

scores

in

descriptive discriminant

analys is)

that

ma y

then

b e

used

in

fur ther

statistical analyses (e.g., t tests, analyses

of

variance)

in place of the

mea-

sured variables.

First-order factors. The pattern or pattern/structure coefficients for the factored

entities (measured variables in the commonly used R-technique analysis).

General linear model GLM).

The

notion

that all statistical analyses are correla-

tional and can yield

effect

sizes analogous to r

2

, and

apply

weights to measured

var i ab les to obtain scores on the composite variables that are actually the

focus of all these analyses.

Identification.

A model is said to be identified when, for a given research problem

and

data set,

sufficient

constraints

a re

imposed such

that there

is a

single

s et

of parameter estimates yielded

by the

analysis.

Inadmissible. A statistical estimate that is mathematically implausible (e.g., a

negative variance, or an r

that

is outside th e range of

— 1

to +1) or a solution

that contains

one or

more estimates that

are

implausible.

Invariance. The

degree

to

which

a

factor

or a

solution

can be

reproduced across

samples, across

different

analytic methods, or

both.

Inversion. The

matrix algebra process

of

solving

to find a

matrix that when post-

mult ipl ied by the original matrix yields the identity matrix; inverted matrices

are

used

in

matrix algebra

to

accomplish division.

17 9



Iteration. A

sta t i s t ica l

process of est i ma t in g a set of

results,

and then successively

tweaking them u n t i l two consecutive sets of es t imates are sufficiently close

that the i terat ion is deemed to have converged.

Loadings.

A n amb iguous and con fusing slang term used by some au thors for pat tern

coefficients,

and b y some au thors for structure coefficients, and by some auth ors

for both even w i th in the same ar t icle even when the coefficients are not equal .

Measurement error. The

var iance

in

scores that

is

deemed unrelia ble; this area-

world

proport ion can be est ima ted a s 1.0 minus a rel ia bi l i t y coefficient.

Modification

index.

The est imated decrease ( improvement) in the model fit chi-

square resulting from freeing

a

given prev ious ly

fixed

parameter instead

to

be est imated.

Oblique. An angle of more or less than 90° between some rotated factors, which

means that the factors (or components) and factor scores are correlated.

Orthogonal. The 90° angle

of

al l unrotated and some rotated factors , which s tat is t i -

cally

means the factors (or components) are perfectly uncorrelated.

Outliers.

Persons with aberrant

or

anomalous scores

on one or

more var iables with

regard

to one or

more statistics.

Pattern coefficients. The weights appl ied to the measured var iables to obtain scores

on the

factor analysis latent va r iables (cal led

factor

scores). These weights

a re

analogous

to the (3

weights

in

mu lt ip le regression,

the

s tandardized d iscriminant

func t ion coefficients

in

descr ipt ive discr iminant analys is ,

and the

s tandardized

canonical funct ion coefficients

in

canonical correlat ion ana lysis .

Pattern/structure coefficients.

The

term

for the

entries

in the

single exploratory

factor

analysis

coefficient

ma tr ix produced

when

factors are first extracted or

after rotat ion when or thogonal rotat ion

has been

invoked,

and in CFA

when

factors are

constrained

to be

uncorrelated.

Procrustean rotation. A method that rotates a pattern (or pat te rn/s t ruc ture) coeffi-

cient mat r ix

to

best-f i t posi t ion wi th

a

theoretical

or

actual pat tern coeffi-

cient ma t r ix .

Reflection. The process of chan ging a ll the signs of pat tern and s tru ctu re coefficients

on a

given factor

so

tha t

all or

most

of the

largest values

on the

factor

are

pos i t i ve in sign.

Sampling error. The uni que , id iosyncra t ic var iance in a given sample that does

not exist (a) in the popula t io n , which has no idiosyncrat ic var iance due to a

sampling process, or (b) in other samples, which do have sam pling error, but

sampling error var iances that are at least somewhat different.

Second-order factors.

The pat tern or pat tern/s tructure coefficients for the second-

order factors defined

by the first-order

factors.

Simple structure. A theoret ical pat tern of the dispersion of near-zero and large

nonzero pattern coefficients tha t opt imizes interp reta bi l i t y of the factors by

there being (a) enough v ar iab les ref lect ing a construct and (b) most v ar iab les

180 GLOSSARY



reflect ing

only

a

single factor such

that the

factors

can be both

labeled

and

dis t inguished from each other.

Structure

coefficient.

The

Pearson

product— moment

cor re la t ion

of a

factored ent i ty

(measured variables

in the commonly

used

R-technique

analysis) with

a la-

tent factor.

GLOSSARY 181



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assessment.

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factors extracted

from the

correlation

matrix

versus

th e

covariance

matrix: A n

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th e Love

Relationships

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How

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190 R E F E R E N C E S



INDEX

ADF estimation theory. See A symptot i -

cally

distribution-free est ima tion

theory

Alpha factor extract ion, 38

AMOS,

111, 120, 131

Analysis of

var i ance (ANOVA) ,

12

Ana ly t i c purpose, 70

A nderson-R ubin est imation method,

4 4

Area world statistics, 11, 12

Asymptotically distribution-free (ADF)

estimation theory, 127, 13 8

Bartlett method,

58

Bell curves, 121-122

Beta (]3) weights, 16

Binet , Alfred, 3

Bivar ia te

normali ty,

122

Bootstrap

methods,

102-108

in CFA/SEM, 129-130

in EFA, 128-129

Canonical factor analysis, 38

CFA. See C onfirmatory factor analysis

C F1 (comparat ive fit index), 130

Characteristic values. Se e Eigenvalues

Communality coefficients, 19-21

in i t ia l estimates, 38, 51

as rel iabil i t ies , 99

as

R

2

v alues, 61-64

Comparat ive fi t index (CFI), 130

Composite variables (latent variables,

synthetic va riables), 9, 10

Confirmatory factor analysis (CFA),

5-7, 30,

133-151

alternative estimation theories in ,

138, 139

and comparisons of model fits, 142 ,

144-145

decision

sequence in, 109-132

EFA

vs., 110-114

and

higher-order models, 1 4 5 — 1 5 0

matrix of

association coefficients

in ,

120-121

maximum likelihood theory

in, 138

model identifica t ion in,

118-120,

134-137

mult ivar ia te

normali ty in,

121-123

outl ier screening in , 123—126

and parameter est imation theory,

126-127

pattern and

structure

coefficients in ,

138,

140-143

postanalysis decisions in , 127—131

preanalysis

decisions

in ,

114—127

Confirmatory

rotat ion,

93-95

Conformabili ty, 13-14, 25

Constructs,

5

Convergence, 37

C orrelated error v ari anc es, 140,

14 2

Correlat ion coefficient (r), 10-11

Correlat ions

Interpret by squar ing, 19

Covariance matrix,

30—31

C ovariance structure analysis.

Se e

Struc-

tural

equation

modeling (SEM)

Crit ical rat io (t test), 105, 123

Cross-validat ion, EFA, 101-103

Data

intervally

scaled,

29, 120

Data box (C at te l l) , 83-86

Delta

values,

44

Diagonal

of a

matr ix ,

14

Disconfirmability, 115-118

Division of matrices, 15, 25

EFA. See Exploratory factor analysis

Eigenvalue

greater than 1.0 rule (Kl

rule), 32

Eigenvalues

(character is tic v alues),

21-22

Elbow (scree plots ),

33

191



Equality constraints, 113, 14 7

Error(s)

correlated error variances, 140, 14 2

measurement, 50-52, 99

model specification, 99-100

root-mean-square error of approxima-

t ion ( R M S E A ) , 130

sampling,

50-52,

1 00

Est imates

admissible, 135

Estimation theory(-ies)

asymptotically distribution-free, 127,

138

maximum likelihood,

126—127

ordinary least squares (OLS), 138

Exploratory

factor analysis (E FA ),

5-7,9

bootstrap, EFA, 102-108

C FA vs., 110-114

cross-validat ion, EFA, 101-103

factor

extraction method in , 36—38

factor rotat ion in ,

38-44

factor scores

in , 44—47

linear sequence of decisions in,

27-28,

47-48

matrix of

association coefficients

in ,

28-31

number

of

factors

in , 31—36

orthogonal vs. oblique solutions in,

71

sample

size in , 2 4 — 2 5

Ex terna l replicab ility analyses, 100

Factor adequacy, 94

Factor analysis

development

of, 3

and

nature

of

constructs,

5

an d

parsimony,

5

purposes

of, 4-5

two major classes of, 5— 7

and val id i ty , 4 -5

Factor extraction

alpha,

38

image,

38

maximum likelihood, 38

Factor interpretation

naming of factors,

97—98

salience,

97

Factor rotation, 38-44. Se e

also

Oblique

rotations

Factors

naming of ,

97-98

number of (in EFA), 31-36

possible number

of, 17—18

reflection

of , 96-97

Factor scores,

16 ,

44-47

Anderson— R ubin , 4 4

Bart let t ,

58

in different extraction methods,

57-61

First-order

factors, 72

Fi t statistics, 127-130

General

linear

model (GLM), 7, 109

interpretat ions in , 7 1 — 7 2

weights in , 15—16

G

theory

of

intelligence,

3

Guilford, Joy, 4 -5

Heywood cases,

20

Higher-order factors

CFA, 145-150

EFA, 72-81

Identif ication,

model,

118-120,

134-137

Ident i ty matr ix , 14-15

111 conditioned

matrices,

15

Image factor extraction, 38

Inadmissabil i ty, 20

Independence model,

11 8

Intell igence,

3

Internal replicabili ty

methods,

99-108

bootstrap, EFA, 102-108

cross-validation, EFA, 101-103

external vs.,

100

and sampling error, 99-100

Invariance, test ing fo r model,

153-162

with same model, different esti-

mates, 153-156

with same model, equal estimates,

155,

157-159

with

same model, test invar iance

of

parameters,

159—161

Inverse,

matr ix ,

15

IQ tests, 3

Iterat ion,

37

192 I N D E X



Kl rule (eigenv alue greater than 1.0

rule) ,

32

Laten t var i ables . S ee

Composite v a r i a b l e s

L i b Q U A L + ™ s t u d y ,

28, 39

LISREL, 111

Loadings, 19

Mahalanobis dis tance, 124

M a t r i x

algebra, 9,

12-15,

25

Matrix(-ces) , 13-15

of association coeffi cients ,

28—31

comformable ,

13-14

covar iance ,

30—31

div i s ion

of, 15, 25

ident i ty , 14-15

ill

conditioned,

15

inverse of, 15

mul t ip l ica t ion

of ,

13—14,

25

symmetr ic , 14

transpose of, 13

M a x i m u m

likelihood factor extract ion,

38

Maximum likelihood theory, 126-127

M ean ,

11

Measured (ob served) var i ables , 9

Measurement error

defined,

99

sampl ing

error vs.,

5 0 — 5 2

M edian , 11

Mediated ranking, 88

Model

proof

of, 115

Model fit statistics,

127-130

Model specification error, 99-100

Model specif ica t ion searches, 131

Models

nested r iva l , 12 9

plausible, 115

Modes, 84

Modif ica t ion indices ,

131

Monte Car lo s imula t ions , 24, 128

Mult ip l ica t ion of matrices , 13-14, 25

Multivariate

Behav ioral Research, 53

M ul t i v a r i a t e

no rma l i t y ,

121-123

Mult ivoca l models , 142

N a m i n g of factors , 97-98

N F1

(normed fit index), 129

N orma l i t y

b i v a d a t e , 1 2 2

m ul t i v a r i a t e , 1 2 1 — 1 2 3

univar i a te ,

121-122

Normed fi t index (NF1), 129

O bl imin ,

44

Oblique rotat ions , 42-44, 67-81

and higher-orde r factors, 72-81

a nd m ax im iz in g sample f i t vs.

replicabi l i ty,

69-72

Observed variables . S ee Measured

va r i ab les

Occam's razor, 70-71

OLS.

Se e

Ordinary leas t squares

One-factor model,

118

Ordina ry leas t squares (OLS), 126, 138

Orthogonal

ro ta t ion , 42, 67

O-technique

factor analysis , 85

Outlier screening, 123-126

Para l le l

analysis ,

34-36

Parameter es t imat ion theory,

126—127

Parcels

I t em, 123

Parsimony, 5, 70-71

Pars imony-adjusted s ta t is t ics ,

130

Pa t tern coefficients

in

CFA, 138, 140-143

in GSM , 16-17

Pa t tern /s t ruc ture

coefficients, 19

PD A (predic t ive d i sc iminant ana lys i s ) , 71

Pearson correlation coefficients, 10, 61

Pearson K

factors affecting,

30, 124

Pearson product-moment biv aria te

corre la t ion mat r ix , 29

Pearson

r, 2 9 — 3 1

Pencil test, 33

Pos tmul t ip l ica t ion , 15

Predic t ive d i sc iminant a na lys i s (PD A ) ,

71

Principal-axes factor analysis ,

37

i terat ion process for ,

51

principal-components analysis vs . ,

53-55

Pr inc ipa l components ana lys i s , 36-38

pr inc ipa l axes analysis vs . , 5 3 — 5 5

Procrustean rotat ion,

43-44

I N D E X 193



Promax rotat ion, 43-44

P-technique factor analysis,

Q

sorts,

87-88

Q-technique

factor analysis, 84 ,

data collection

in,

87-88

uses of, 86

6-91

R .

See

Correlation coefficient

Rank/ranking,

13, 88

R eflection

of

factors,

96-97

Regression method,

44

Rel iab i l i ty

coefficients,

12, 99

Rel iab i l i ty generalizat ion (RG) analysis,

38

Repl icab i l i ty

analysis

external,

100

internal,

99-108

R eproduced correlat ion ma trix,

16-17,

22-23

Resamples,

10 3

Residua l

correlation matrix,

23-24

inspection

of , 33-34

R G

analysis ,

38

R oot-mean-square error

of

approximation

( R M S E A ) , 130

R otat ion, factor.

See

Factor

rotation

R -techruque factor analysis,

84

Salience

of variables, 97

Sampling error, 100

measurement

error

vs., 50-52

Schmid-Leiman solution, 74

Scores,

factor.

See

Factor scores

Score

world statistics,

11

Scree

test,

32-33

SD (standard deviat ion), 11

Second-order factors,

72

SEM.

Se e

Structural equation modeling

Sequent ia l

factor extraction,

22-24

Simple structure, 41

Spearman's rho, 29, 30

Specific factors, 3

SPSS,

22, 36, 44, 45, 51, 58, 61-63, 73 ,

74-81, 85

Standard deviat ion

(SD), 11

Standard errors.

Se e

Error(s)

Standardized regression coefficients, 1 6

Statist ical significance tests, 31-32

S-technique factor ana lysis, 85

Structural equat ion modeling (SEM),

109-110

Structural

models, 110

Structure coefficients

in

CFA, 138, 140-143

in EFA, 61

in GSM, 18-19

Superscripts, 11

Suppressor effect,

72

Symmetric matrix,

14

Synthetic variables.

Se e

C omposi te

variables

Third-order factors,

72

Three-mode analysis, 84

Transapose(s), 13

T-technique factor analysis, 85

t test. See

C ri t ical rat io

Two-mode analysis,

83-91

O-technique, 85

P-technique,

84

Q-technique, 84, 86-91

R -technique ,

84

S-technique,

85

T-technique, 85

Uncorrelated factors model,

118

Uniqueness,

74

Unit weight ing, 70

Univar ia te normal i ty , 1 2 1 — 1 2 2

Univocal models, 140, 14 1

Val idi ty ,

4-5

Variables

composite (latent, synthetic), 9, 10

measured

(observed),

9

number

of, 10, 49-50

salience

of, 97

Variance, 11

Visual

scree test,

33

W ald s ta t is t i c ,

105,

123

W eights/weight ing,

9, 15-16

b e t a ( W , 16

uni t ,

70

194

7NDEX



ABOUT

THE

AUTHOR

Bruce

Thompson

is a professor and distinguished research scholar at the

Department of Educational Psychology at Texas A&M University, ad junc t

bruce thompson-exploratory and confirmatory factor analysis_ understanding concepts and...

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