Chapter 13Repeated-Measures andTwo-Factor Analysis of Variance

PowerPoint Lecture Slides

Essentials of Statistics for the Behavioral Sciences Eighth Edition

by Frederick J. Gravetter and Larry B. Wallnau

Chapter 13 Learning Outcomes

•Understand logic of repeated-measures ANOVA study

1

•Compute repeated-measures ANOVA to evaluate mean differences for single-factor repeated-measures study

2

•Measure effect size, perform post hoc tests and evaluate assumptions required for single-factor repeated-measures ANOVA

3

• Measure effect size, interpret results and articulate assumptions for two-factor ANOVA

Ch 13 Learning Outcomes (continued)

• Understand logic of two-factor study and matrix of group means4

• Describe main effects and interactions from pattern of group means in two-factor ANOVA5

• Compute two-factor ANOVA to evaluate means for two-factor independent-measures study6

7

Tools You Will Need

• Independent-Measures Analysis of Variance(Chapter 12)

• Repeated-Measures Designs (Chapter 11)• Individual Differences

13.1 Overview

• Analysis of Variance– Evaluated mean differences for two or more

groups– Limited to one independent variable (IV)

• Complex Analysis of Variance– Samples are related; not independent

(Repeated-measures ANOVA)– Two independent variables are manipulated

(Factorial ANOVA; only Two-Factor in this text)

13.2 Repeated-Measures ANOVA

• Independent-measures ANOVA uses multiple participant samples to test the treatments

• Participant samples may not be identical• If groups are different, what was responsible?

– Treatment differences?– Participant group differences?

• Repeated-measures solves this problem by testing all treatments using one sample of participants

Repeated-Measures ANOVA

• Repeated-Measures ANOVA used to evaluate mean differences in two general situations– In an experiment, compare two or more

manipulated treatment conditions using the same participants in all conditions

– In a nonexperimental study, compare a group of participants at two or more different times

• Before therapy; After therapy; 6-month follow-up• Compare vocabulary at age 3, 4 and 5

Repeated-Measures ANOVA Hypotheses

• Null hypothesis: in the population there are no mean differences among the treatment groups

• Alternate hypothesis: there is one (or more) mean differences among the treatment groups

...: 3210 H

H1: At least one treatment mean μ differs from another

General structure of the ANOVA F-Ratio

• F ratio based on variances– Numerator measures treatment mean differences– Denominator measures treatment mean

differences when there is no treatment effect

– Large F-ratio greater treatment differences than would be expected with no treatment effects

effect treatment no withexpected es)(differenc variance

treatments between es)(differenc varianceF

Individual differences

• Participant characteristics may vary considerably from one person to another

• Participant characteristics can influence measurements (Dependent Variable)

• Repeated measures design allows control of the effects of participant characteristics– Eliminated from the numerator by the research

design– Must be removed from the denominator

statistically

Structure of the F-Ratio for Repeated-Measures ANOVA

ally)mathematic removed sdifference l(individua

effect treatmentno with expected es)(differenc variance

s)difference individual(without

eatmentsbetween tr es)(differenc variance

F

The biggest change between independent-measures ANOVA and repeated-measures ANOVA is the addition of a process to mathematically remove the individual differences variance component from the denominator of the F-ratio

Repeated-Measures ANOVA Logic

• Numerator of the F ratio includes– Systematic differences caused by treatments– Unsystematic differences caused by random

factors are reduced because the same individuals are in all treatments

• Denominator estimates variance reasonable to expect from unsystematic factors– Effect of individual differences is removed– Residual (error) variance remains

Figure 13.1 Structure of the Repeated-Measures ANOVA

Repeated-Measures ANOVA Stage One Equations

N

GXSStotal

22

treatment each insidetreatmentswithin SSSS

N

G

n

TSS treatmentsbetween

22

Two Stages of the Repeated-Measures ANOVA

• First stage– Identical to independent samples ANOVA– Compute SStotal, SSbetween treatments and

SSwithin treatments

• Second stage– Done to remove the individual differences from

the denominator– Compute SSbetween subjects and subtract it from SSwithin

treatments to find SSerror (also called residual)

Repeated-Measures ANOVAStage Two Equations

N

G

k

PSS subjectsbetween

22

_

bjectsbetween_suatmentswithin tre SSSSSSerror

Degrees of freedom for Repeated-Measures ANOVA

dftotal = N – 1

dfwithin treatments = Σdfinside each treatment

dfbetween treatments = k – 1

dfbetween subjects = n – 1

dferror = dfwithin treatments – dfbetween subjects

Mean squares and F-ratio for Repeated-Measures ANOVA

error

errorerror df

SSMS

treatmentsbetween

treatmentsbetweentreatmentsbetween df

SSMS

_

__

error

mentstreat between

MS

MSF

F-Ratio General Structure for Repeated-Measures ANOVA

)(

)(

sdifferenceindividualwithout

sdifferenceicunsystemat

sdifferenceindividualwithout

sdifferenceicunsystemateffectstreatment

F

Effect size for the Repeated-Measures ANOVA

• Percentage of variance explained by the treatment differences

• Partial η2 is percentage of variability that has not already been explained by other factors

or

subjectsbetween total

eatmentsbetween tr2

SS SS

SS

errorSSSS

SS

eatmentsbetween tr

eatmentsbetween tr2

In the Literature

• Report a summary of descriptive statistics (at least means and standard deviations)

• Report a concise statement of the ANOVA results– E.g., F (3, 18) = 16.72, p<.01, η2 = .859

Repeated Measures ANOVApost hoc tests (posttests)

• Significant F indicates that H0 (“all populations means are equal”) is wrong in some way

• Use post hoc test to determine exactly where significant differences exist among more than two treatment means– Tukey’s HSD and Scheffé can be used– Substitute SSerror and dferror in the formulas

Repeated-Measures ANOVA Assumptions

• The observations within each treatment condition must be independent

• The population distribution within each treatment must be normal

• The variances of the population distribution for each treatment should be equivalent

Learning Check• A researcher obtains an F-ratio with df = 2, 12

in a repeated-measures study ANOVA. How many subjects participated in the study?

•15

A

•14

B

•13

C

•7D

Learning Check - Answer• A researcher obtains an F-ratio with df = 2, 12

in a repeated-measures study ANOVA. How many subjects participated in the study?

•15

A

•14

B

•13

C

•7D

Learning Check

• Decide if each of the following statements is True or False

•For the repeated-measures ANOVA, degrees of freedom for SSer

ror could be written as [(N–k) – (n–1)]

T/F

•The first stage of the repeated-measures ANOVA is the same as the independent-measures ANOVA

T/F

Learning Check - Answer

•dferr

or = dfw/

i

treat

ments – dfbe

twn

subjec

ts•Wit

hin treatments df = N-k; between subjects df = n-1

True

•After the first stage analysis, the second stage analysis adjusts for individual differences

True

Repeated-Measures ANOVA Advantages and Disadvantages

• Advantages of repeated-measures designs– Individual differences among participants do not

influence outcomes– Smaller number of participants needed to test all

the treatments• Disadvantages of repeated-measures designs

– Some (unknown) factor other than the treatment may cause participant’s scores to change

– Practice or experience may affect scores independently of the actual treatment effect

13.3 Two-Factor ANOVA

• Both independent variables and quasi-independent variables may be employed as factors in Two-Factor ANOVA

• An independent variable (factor) is manipulated in an experiment

• A quasi-independent variable (factor) is not manipulated but defines the groups of scores in a nonexperimental study

13.3 Two-Factor ANOVA

• Factorial designs– Consider more than one factor

• We will study two-factor designs only• Also limited to situations with equal n’s in each group

– Joint impact of factors is considered• Three hypotheses tested by three F-ratios

– Each tested with same basic F-ratio structure

effect treatment no withexpected es)(differenc variance

treatments between es)(differenc varianceF

Main Effects

• Mean differences among levels of one factor– Differences are tested for statistical significance– Each factor is evaluated independently of the

other factor(s) in the study

21

21

:

:

1

0

AA

AA

H

H

21

21

:

:

1

0

BB

BB

H

H

Interactions Between Factors

• The mean differences between individuals treatment conditions, or cells, are different from what would be predicted from the overall main effects of the factors

• H0: There is no interaction between Factors A and B

• H1: There is an interaction between Factors A and B

Interpreting Interactions

• Dependence of factors– The effect of one factor depends on the level or

value of the other– Sometimes called “non-additive” effects because

the main effects do not “add” together predictably• Non-parallel lines (cross, converge or diverge)

in a graph indicate interaction is occurring• Typically called the A x B interaction

Figure 13.2 Group Means Graphed without (a) and with (b) Interaction

Structure of the Two-Factor Analysis of Variance

• Three distinct tests– Main effect of Factor A– Main effect of Factor B– Interaction of A and B

• A separate F test is conducted for each• Results of one are independent of the others

effecttreatmentnoisthereifexpectedsdifferencemeanvariance

treatmentsbetweensdifferencemeanvarianceF

)(

)(

Two Stages of the Two-Factor Analysis of Variance

• First stage– Identical to independent samples ANOVA– Compute SStotal, SSbetween treatments and

SSwithin treatments

• Second stage– Partition the SSbetween treatments into three separate

components: differences attributable to Factor A; to Factor B; and to the AxB interaction

Figure 13.3 Structure of the Two-Factor Analysis of Variance

Stage One of the Two-Factor Analysis of Variance

N

GXSStotal

22

menteach treat insideSSSS treatmentswithin

N

G

n

TSS treatmentsbetween

22

Stage Two of the Two Factor Analysis of Variance

• This stage determines the numerators for the three F-ratios by partitioning SSbetween treatments

N

G

n

TSS

row

rowA

22

N

G

n

TSS

col

colB

22

BAtreatments betweenAxB SSSSSSSS

Degrees of freedom for Two-Factor ANOVA

dftotal = N – 1

dfwithin treatments = Σdfinside each treatment

dfbetween treatments = k – 1

dfA = (number of rows) – 1

dfB = (number of columns)– 1

dferror = dfwithin treatments – dfbetween subjects

Mean squares and F-ratios for the Two-Factor ANOVA

reatmentst within

reatmentst withinreatmentst within df

SSMS

AxB

AxBAxB

B

BB

A

AA df

SSMS

df

SSMS

df

SSMS

within

AxBAxB

within

BB

within

AA MS

MSF

MS

MSF

MS

MSF

Two-Factor ANOVA Summary Table Example

Source SS df MS F

Between treatments 200 3

Factor A 40 1 40 4

Factor B 60 1 60 *6

A x B 100 1 100 **10

Within Treatments 300 20 10

Total 500 23

F.05 (1, 20) = 4.35*F.01 (1, 20) = 8.10**

(N = 24; n = 6)

Two-Factor ANOVA Effect Size

• η2, is computed to show the percentage of variability not explained by other factors

treatments withinA

A

AxBBtotal

AA SSSS

SS

SSSSSS

SS

2

treatmentswithinB

B

AxBAtotal

BB SSSS

SS

SSSSSS

SS

_

2

treatments withinAxB

AxB

BAtotal

AxBAxB SSSS

SS

SSSSSS

SS

2

In the Literature

• Report mean and standard deviations (usually in a table or graph due to the complexity of the design)

• Report results of hypothesis test for all three terms (A & B main effects; A x B interaction)

• For each term include F, df, p-value & η2 • E.g., F (1, 20) = 6.33, p<.05, η2 = .478

Interpreting the Results

• Focus on the overall pattern of results• Significant interactions require particular

attention because even if you understand the main effects, interactions go beyond what main effects alone can explain.

• Extensive practice is typically required to be able to clearly articulate results which include a significant interaction

Figure 13.4Sample means for Example 13.4

Two-Factor ANOVA Assumptions

• The validity of the ANOVA presented in this chapter depends on three assumptions common to other hypothesis tests– The observations within each sample must be

independent of each other– The populations from which the samples are

selected must be normally distributed– The populations from which the samples are

selected must have equal variances (homogeneity of variance)

Learning Check

• If a two-factor analysis of variance produces a statistically significant interaction, then you can conclude that _____

•either the main effect for factor A or the main effect for factor B is also significant

A

•neither the main effect for factor A nor the main effect for factor B is significant

B

•both the man effect for factor A and the main effect for factor B are significant

C

•the significance of the main effects is not related to the significance of the interaction

D

Learning Check - Answer

• If a two-factor analysis of variance produces a statistically significant interaction, then you can conclude that _____

•either the main effect for factor A or the main effect for factor B is also significant

A

•neither the main effect for factor A nor the main effect for factor B is significant

B

•both the man effect for factor A and the main effect for factor B are significant

C

•the significance of the main effects is not related

to the significance of the interaction

D

Learning Check

• Decide if each of the following statements is True or False

•Two separate single-factor ANOVAs provide exactly the same information that is obtained from a two-factor analysis of variance

T/F

•A disadvantage of combining 2 factors in an experiment is that you cannot determine how either factor would affect participants’ scores if it were examined in an experiment by itself

T/F

Learning Check - Answers

•Main effects in Two-Factor ANOVA are identical to results of two One-Way ANOVAs; but Two-Factor ANOVA provides Interaction results too!

False

•The two-factor ANOVA allows you to determine the effect of one variable controlling for the effect of the other

False

Figure 13.5 Independent-Measures Two-Factor Formulas

Figure 13.6 Example 13.1 SPSS Output for Repeated-Measures

Figure 13.7 Example 13.4 SPSS Output for Two-Factor ANOVA

AnyQuestions

?

Concepts?

Equations?