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A Primer on Structural Equation Models:

Part 1. Confirmatory Factor Analysis.

Michael A. Babyak, PhD1 and Samuel B. Green, PhD2

1 Department of Psychiatry and Behavioral Sciences, Duke University Medical Center,

Durham, NC

2 Division of Psychology in Education, Arizona State University, Tempe, Arizona

Correspondence to: Dr. Michael Babyak, Box 3119, DUMC, Durham, NC 27710;

email: [email protected]; fax: (919) 684-8629

Word Count: 6,098

Tables: 1

Figures: 3

A Primer on Structural Equation Models:

Part 1. Confirmatory Factor Analysis.

Michael A. Babyak and Samuel B. Green

Confirmatory Factor Analysis 1

Abstract

In the first of a two-part didactic series on structural equation modeling, we present an

introduction to the basic concepts underlying confirmatory factor analysis. We use

examples with simplified fictitious data to demonstrate the underlying mathematical

model and the conventions for nomenclature and graphical representation of the model.

We then show how parameter estimates are generated for the model with the maximum

likelihood function. Finally, we discuss several ways in which model fit is evaluated and

also briefly introduce the concept of model identification. Sample code in the EQS and

Mplus software language is provided in an online appendix. A list of resources for

further study also is included.


Structural equation modeling (SEM) is a general data-analytic method for the

assessment of models that specify relationships among variables. SEM involves

investigating two primary models: the measurement model that links measures to factors

and the structural model that links factors to each other. In the first installment of this

two-part series, we will discuss confirmatory factor analysis (CFA), which is the method

for specifying, estimating, and assessing measurement models. The second installment

will be published at a later date and will focus on the full structural equation model,

which includes both measurement and structural components. SEM is highly flexible–it

can be used to carry out a large variety of analytic procedures. With flexibility, of

course, comes complexity, and even two overly long papers would not provide sufficient

space to address the many facets of modern SEM. Nevertheless, we hope to provide at

least a glimpse and intuitive understanding of the basic concepts of SEM. In the

following pages, we will offer an introduction to confirmatory factor analysis, including

the purpose of CFA, the specification of models, computation of estimates of the model

parameters, and assessment of model fit.

SEM in psychosomatic and medical research

Although SEM is used quite frequently in some fields, such as psychology,

education, sociology, and genetics, research using SEM appears comparatively

infrequently in psychosomatic and medical journals. There is at least a small irony to the

relative scarcity of SEM in medical and psychosomatic research in that the technique

actually has its direct roots in biology. In the 1920s, geneticist Sewall Wright first

developed an important component of SEM, path analysis, in an attempt to better


understand the complex relations among variables that might determine the birth weight

of guinea pig offspring (1). In the late 1970s by Joreskog and Goldberger (2) developed

and championed the full SEM model, successfully integrating factor analytic technology

with path analysis. SEM has begun to appear more often in psychosomatic research, and

even has made begun to make an occasional foray into high profile medical journals. For

example, in a recent paper published in the New England Journal of Medicine, Calis et al.

(3) used the path modeling portion of SEM to estimate a set of complex associations

among malaria, HIV, and various nutritional deficiencies. In a recent commentary on

post-traumatic stress syndrome (PTSD) that appeared in JAMA, Bell and Orcutt {Bell,

2009 #158} explicitly point out the potential utility of SEM in their area of study:

“Structural equation modeling is particularly well suited for examining complex

associations between multiple constructs; such constructs are often represented as latent

constructs and are assumed to be free of measurement error.” Nevertheless, very few

papers using SEM to address any topic have graced the pages of JAMA. Of course, we

tend to agree with Bell and Orcutt. SEM is particularly useful as an aid in understanding

of how variables might interrelate in a system. It is especially useful when those

variables are comprised of several facets that overlap to some extent. In the

psychosomatic medicine domain, for example, Rosen et al. (4) use SEM to estimate the

association of global subjective health with psychological distress, social support and

physical function. Rosen et al. operationalize these four study variables as latent

variables, that is, factors instantiated by a set of observed measures that were thought to

reflect the respective factors.


Our fictionalized example

In order to demonstrate the CFA part of SEM, we will focus on a fictionalized

example based on a real question in psychosomatic research. Our example bears some

similarity to the model studied by Rosen et al., although ours is, of necessity, much

simpler. We draw on the literature has accumulated in our field suggesting that a variety

of psychosocial constructs, including trait hostility, anger, anxiety, and depressive

symptoms, appear to be risk factors for the development of coronary artery disease

(CAD) (for a review, see Suls and Bunde (5)). Despite the large number of papers

published about the topic, however, a number of interesting fundamental questions

remain. For example, do depression, hostility, anger, and anxiety each uniquely pose a

risk for heart disease? Or, because these variables tend to overlap, are they really just

manifestations of a broader trait underlying “negative affect? And is it really that

underlying trait which confers the risk?

In this first installment, we will focus on how SEM can be used to study the

question of whether these variables might be manifestations of one or more broader

dimensions. Understanding the measurement properties of the variables under study is a

critical first step in carrying out any empirical research study—we have to know what we

are measuring and how well we are doing it before drawing any robust conclusions about

findings that concern those variables. In the case of our example, in this preliminary

‘measurement model’ phase we can address questions about the measurement properties

of the variables under study, such as: are hostility, depression, and anxiety really distinct

constructs? Or are they really just slightly different manifestations of the same negative

affectivity phenomenon? Understanding the measurement properties of the negative


affect variables will later help illuminate the “specific versus general” risk question. Of

course, in the world outside the confines of this tutorial, we also would draw from

substantive theory to make such arguments work. Since our focus here is within the more

narrow borders of the SEM technique itself, we will not delve very deeply into the

substantive aspects of the analyses we will present. Our aim here is to simply get your

feet wet in understanding the technique might be applied to such questions.

Purpose of Confirmatory Factor Analysis

CFA, as well as exploratory factor analysis (EFA), defines factors that account for

covariability or shared variance among measured variables and ignore the variance that is

unique to each of the measures. Broadly speaking, either can be a useful technique for

(a) understanding the structure underlying a set of measures, (b) reducing redundancy

among a set of measured variables by representing them with a fewer number of factors,

and (c) exploiting that redundancy and hence improving the reliability and validity of

measures. However, the purposes of EFA and CFA, and accordingly the methods

associated with them, are different. The goal of EFA is to discover a set of as-yet-

unknown or unverified factors based on the data, although a priori hypotheses based on

the literature may help guide some decisions in the EFA process. In other words, we start

with correlations among some set of variables, and although we may have some reason to

believe they will cluster following a certain pattern, we generally submit the correlations

to the software and allow the algorithm to tell us how many factors there may be and

what particular variables belong to those factors. In contrast, in CFA, we have to start

with one or more explicit hypotheses about the number of factors and how the variables


are related to those factors. CFA accomplishes this by assessing constraints imposed on

factor models based on the a priori hypotheses about measures. If these constraints are

inconsistent with the pattern of relationships among measured variables, the model with

its imposed constraints is rejected. Given the focus of CFA is on hypothesized models,

let’s first describe how these models are specified before considering how the parameters

of models are estimated and how the fit of models to the data are assessed.

Model Specification

With CFA, we hypothesize a model that specifies the relationship between

measured variables and presumed underlying factors.1 The model includes parameters we

want to estimate based on the data (i.e., freely estimated parameters) and parameters we

constrain to particular values based on our understanding of our data and the literature

(i.e., constrained or fixed parameters). It is the constraints on model parameters that

produce lack of fit.

In this section we will consider three prototypical CFA models, each with a

different substantive interpretation. We will present each prototypical model and discuss

it in the context of our negative affect example. The example for the first two prototypes

involves postulating a factor structure underlying four measures: a trait hostility measure,

an anger measure, an anxiety measure, and a depressive symptoms score. Each of the

measures is derived from summing the items on a self-report instrument designed to

measure that construct. For the third prototypical model, we extend this example by

breaking the depressive symptoms measures out into three domains of symptoms,

affective, somatic, and cognitive. 1 We use the terms factor and latent variable interchangeably throughout this paper.


Single Factor Model

The single factor model is the simplest CFA model. Nevertheless, we devote

considerable attention to it in order to introduce the basic concept of CFA and

conventional SEM terminology.

A single factor model specifies a unidimensional model, hypothesizing that a

single dimension underlies a set of measures. Unidimensionality is often seen as a

desirable characteristic of a set of measures, in particular because multiple measures can

be reduced to a single measure, thus improving parsimony, and also because the single

dimension is typically more reliable than a given individual component. As with any

structural equation model, a single factor model can be presented pictorially as a path

diagram or in equation form. Figure 1 is a graphical representation of a model with a

single factor (F1) underlying four measures, X1 (hostility), X2 (anger), X3 (anxiety), and

X4 (depressive symptoms). By convention, the factor is depicted as a circle, which

represents a latent variable, while the observed measures are squares, which represent

observable or indicator variables. A single-headed arrow between two variables indicates

the direction of the effect of the one variable on the other. Within the context of our

example, we are postulating that a factor called negative affect (F1) underlies or

determines the observed scores on the hostility, anger, anxiety, and depressive symptom

measures. Statistically, we believe these four measures are correlated because they have a

common underlying factor, negative affect. In other words, the model reflects the belief

that changes in the unobserved latent variable, negative affect, are presumed to result in

changes in the four variables that we have actually measured.


Continuing with Figure 1, a variable with arrows pointing only away from it is

called exogenous. A variable with one or more arrows pointing to it, even if one or more

arrows are pointing away from it, is called endogenous. One equation is associated with

each endogeneous variable. Accordingly, the model in Figure 1 involves four endogenous

variables and therefore four equations:

1 11 1 1

2 21 1 2

3 31 1 3

4 41 1 4

X = λ F + EX = λ F + EX = λ F + EX = λ F + E

.

The lambdas ( )λ are factor weights or loadings, which can be interpreted essentially like

regression coefficients. For example, for every one unit increase in the negative affect

factor, F1, the expected change in hostility, X1, will be λ11.

Observed measures are not likely to be pure indicators of a factor, but almost

certainly contain unique components, frequently referred to as residuals or errors (E). A

unique component for a measure includes two kinds of variability--reliable information

that is specific to that measure but not related to the factor, and unreliable information,

otherwise known as measurement error. Because errors are not directly observable, they

are also latent variables and are represented in our path diagram as circles. For the

hostility measure in our example, the unique component might include the specific

component of agitation as well as measurement error due to inattentiveness of

respondents and ambiguity of the items on this measure.

Finally, our path diagram also includes double-headed curved arrows. If an arrow

begins and returns to the same exogenous variable, it represents the variance of that

variable. A double-headed arrow could also be drawn between any two errors to represent


a covariance between them, but we chose not to include error covariances in our model to

avoid unnecessary complexity.

The model parameters, or unknowns, which we seek to estimate or constrain

based on our understanding of a study, are associated with the single-headed and double-

headed arrows in our diagrams and, by convention, are shown as Greek letters. In

addition to the lambdas, the parameters for the model in Figure 1 are the variance of the

factor ( )1

2Fσ and the variances of the errors ( )1 4

2 2E Eσ - σ . As shown at the bottom of the

figure, the model parameters can also be presented in three matrices: the phi matrix ( )Φ

containing the variances and covariances among factors, the lambda matrix ( )Λ that

includes all factor weights, and the theta matrix ( )Θ that includes the variances and

covariances among the errors.

We now turn to a concept that concerns all SEM, including our present CFA

models, the idea of “free” or “fixed” (also referred to as constrained) parameters. When

we conduct a conventional multiple regression, we typically deal directly only with free

parameters, those we wish to estimate.2 In a multiple regression model, the free

parameters are the regression coefficients. We tell the software or algorithm ahead of

time to calculate those values. In SEM, however, we specify not only which parameters

are to be estimated, but most critically, we also specify which parameters are constrained

or fixed, that is, which parameters are not to be estimated. In SEM, parameters can be

constrained in a number of ways, including fixing them to a specific value or to be equal

2 There are in fact a number of constraints even in a conventional multiple regression, but these are typically just part of the underlying assumption of the simple for of the model. For example, unless we explicitly specify the model differently, the relations between the predictors and response variable are all assumed to be linear, which is in a broad sense, a constraint. There are a number of other such constraints in most statistical models.


to each other. For example, in our negative affect CFA model, we could constraint the

loading (l) for the anxiety variable to be some value that we had obtained in a prior study.

Or, we could constrain the loading for anxiety to equal to loading of anger. These

constraints are generally used when we have a very specific theoretical question about

those parameters. A constraint that is far more frequently used is one in which the

parameter is constrained to a value of zero. For example, our single factor model includes

no covariances among errors (i.e., all zeros in the off-diagonal positions of the theta

matrix). Substantively, these constraints on the error covariances reflect our belief that

there are no other factors other than the negative affect factor that systematically

influence the variability in the measured variables. Also reflecting that there are no

additional factors underlying the measures, we don’t specify any additional parameters in

the phi and lambda matrices. This is a less obvious constraint in that they are made by

simply omitting any reference to additional factors; the net result, however, is that of

constraining the variances and weights of additional factors to zero. This latter point will

become clearer when we present our second example below. If one or more of these

constraints are incorrect, the model is likely to fit poorly and be rejected.

Another important class of constraints are those used to define an arbitrary metric

for a factor. The metric constraint is often a bit mysterious to SEM novices, and while

the precise mathematical details are not critical to our purposes here, we will describe

how this constraint is applied and what it accomplishes. Factor metrics are arbitrary

because they are latent variables—they are unobservable variables and hence do not have

an inherent metric. We can assign a metric for a factor by either fixing its variance to 1,

as we did, or fixing one of its weights to 1. The choice should have no effect on the fit


for the relatively simple models considered in this article. Fixing the factor variance to 1

in effect standardizes the factor, and the resultant loadings can be interpreted as

standardized. In more complex models, however, it may be necessary to fix one of the

factor weights (lambdas) to 1 rather than to fix the variance of the factor to 1.

Researchers often select the weight associated with the measured variable that is believed

to have the strongest relationship with the underlying factor. In our negative affect

example, we might choose to fix the weight of the best developed depressive symptom

measure to 1. The result of fixing a factor weight to 1 puts all the other loadings in a

scale that is relative to the depressive symptom scale. Very broadly speaking,

constraining one of the loadings to equal 1 is a distant cousin to selecting a reference

category for a set of dummy variables in a linear model, in that it provides a point of

comparison for the other effects. In addition to setting the metric of a latent variable, the

constraint also helps the algorithm estimate the remaining free parameters by making it

more likely that the model is identified. We will discuss the concept of identification in

more detail later in this tutorial, but the central idea of identification is that we cannot

have fewer data points than unknowns (free parameters) in the model. Finally, as we

noted above, the constraints to define a factor’s metric do not influence model fit. All

other model constraints, however, have potential effects on the fit of the model to the

data.

All of the free parameters in our model (i.e., those not constrained to 1 or 0) are

estimated based on the data. If the model fits, we interpret these estimated parameters to

evaluate, for example, which measure is the best indicator of the factor. As in the case of

regression analysis, interpretation is best performed by examining the standardized


weights (i.e., when the factor and measures are transformed to z-scores). If the model

fails to fit, we should not interpret the estimated parameters because their values may

have been adversely affected by the potential misspecification of the model. If our model

1 provides adequate fit (which we will define later) and each of the factor loadings is

substantial, we would conclude that there is evidence to support the idea that there may

be a single latent variable underlying the four observed measures. However, this result

does not mean that the one factor model is the only structure that might produce good fit.

The good fit only means that it is one of the possible models that fits well. Apart from

the obvious theoretical implications of a well-fitting one factor model with high factor

loadings, the result also suggests that we might feel fairly safe using just the single

negative affect latent variable as a predictor (or as an outcome or mediator) in a more

extensive structural model. Of course, if the model does not fit, we will have to test

alternative models in order to understand whether there is a structure that might fit the

observed data better.

Correlated Factors Model

Our second model is a correlated factors model, which specifies that two or more

factors underlie a set of measured variables and also that the factors are correlated. For

simplicity, we will consider a two-factor model, but our discussion is relevant to models

with more than two factors.

In Figure 2 we present a model for our four measures but now with two correlated

factors. As with our path diagram for a single factor model, we have circles for latent

variables (i.e., factors and errors), squares for measured variables, single-headed arrows


for effects of one variable on another, double-headed curved arrows for variances of

exogenous variables, and a double-headed arrow for the covariance between the two

factors. Within the context of our negative affect example, we might speculate that the

hostility and anxiety measures are related to one another by the shared characteristic of

agitation and also, to some degree, distinct from the two depressive symptom measures.

In other words, the model should include a factor (F1) affecting the hostility and anger

measures (X1 and X2), and another factor (F2) affecting the anxiety and depressive

symptom measures (X3 and X4).

Model parameters are associated with all single-headed and double-headed arrows

and are presented in matrix form at the bottom of Figure 2. Constraints can be imposed

on the model parameters. As previously presented, we can define the metric for factors by

constraining their variances to 1 or one of their weights to 1. In this instance, we

arbitrarily chose to set the factor variance to 1 ( )1 2

2 2F Fi.e., σ = 1andσ = 1 .

All constraints besides those to determine the metric of factors can produce lack

of fit and are evaluated in assessing the quality of a model. For example, the effects of

factors on measures, as shown by arrows between factors and measures in the path

diagram, can be represented as equations,

1 11 1 2 1

2 21 1 2 2

3 1 32 2 3

4 1 42 2 4

X = λ F + 0 F + EX = λ F + 0 F + EX = 0 F + λ F + EX = 0 F + λ F + E

.

As shown, the equations indicate that a number of factor loadings are constrained to zero

such that each measured variable is associated with one and only one factor. The

specified structure is consistent with the idea of simple structure, an objective frequently


felt to be desirable with EFA. In addition, a measure is less likely to be misinterpreted if

it is a function of only one factor. Given the advantages of this structure, researchers

frequently begin with specifying models that constrain factor loadings for a measure to be

associated with one and only one factor. In other words, each measure has one weight

that is freely estimated, and all other weights (potential crossloadings) between that

measure and other factors are constrained to 0.

Other parameters in our model that may be freely estimated or constrained are the

covariance between the factors and the variances and covariances among errors. (a) With

CFA, we typically allow the factors to be correlated by freely estimating the covariances

between factors. If we constrained all factor covariances to be equal to zero (i.e.,

orthogonal factors) and also constrained many of the factor loadings to be equal to zero

(e.g., each measure being associated with only one factor), we would be hypothesizing a

model that does not allow for correlations among measures associated with different

factors. This model is likely to conflict with reality and be rejected empirically. In

addition, this model would be inconsistent with many psychological theories that suggest

underlying correlated dimensions. The decision to allow for correlated factors is in stark

contrast with practice in EFA, where researchers routinely choose varimax rotation

resulting in orthogonal factors. However, in EFA, we can still obtain good fit to data in

that all factor loadings are freely estimated (i.e., all measured variables are a function of

all factors), permitting correlations among all measured variables. (b) We usually think of

our measured variables as being unreliable to some degree and thus must freely estimate

the error variances. In most CFA models, we begin by constraining all covariance


between errors to be 0. By imposing these constraints, we are implying that the

correlations among measures are purely a function of the specified factors.

If this model fits our data, we again have a structure that is consistent with the

data, but still cannot rule out other specifications that also might fit. A good fit for the

two factor model suggests that although all four measures may share variability, anger

and hostility may represent an underlying construct that is relatively distinct from the

construct that underlies the anxiety and depression measure. We might interpret the

anger and hostility factor as something like ‘opposition’ or perhaps ‘aggression,’ while

the depression and anxiety might be interpreted as something like ‘withdrawal.’ Again,

this result would have implications regarding whether we would be better off using these

two separate latent variables rather than the single negative affect variable.

In practice, we would typically compare the unidimensional model with the two

correlated factor model. We can do this formally in SEM by comparing the difference

between the fit of the models. We pointed out earlier that a well-fitting model does not

guarantee that it is the correct model. For this reason, SEM procedures such as CFA are

at their scientific best when there are several theoretically plausible models available to

compare. We will discuss fit a bit later, and model comparison in the second installment.

For now, we turn to one more type of model structure, just to further illustrate the kinds

of models that can be represented.

Bifactor Model

A bifactor model may include a general factor associated with all measures and

one or more group factors associated with a limited number of measures.(6, 7) In Figure


3 we present a model for six measures with one general factor and one group factor.

Keeping with our negative affect example, we replace the single depressive symptom

measures with three separate scores of affective, cognitive, and somatic symptoms of

depression. Let’s say that X1, X2, and X3 are the hostility, anger, and anxiety measures,

and the X4, X5, and X6 are the affective, cognitive, and somatic depressive symptom

measures. Due to space limitations, we will only briefly describe the specification of this

model.

As typically applied, we are unlikely to obtain a bifactor model with EFA in that

an objective of this method (with rotation) is to obtain simple structure, which is

generally intolerant to a general factor. In contrast, in CFA, we choose which parameters

to estimate freely and which to constrain to 0. Thus, we can allow for a general factor as

well as group factors. Most frequently bifactor models have been suggested as

appropriate for item measures associated with psychological scales (See (8)). Although

interesting measures are likely to assess a general trait or factor, they are also likely to

include more specific aspects of that trait, that is, group factors. In contrast with the

previous model, factors for a bifactor model are typically specified to be uncorrelated

(i.e., the factor covariances are constrained to 0). In our example, this model suggests that

the three depressive symptom measures are to some extent distinct from the other three

measures, but that a broader general factor, which might be called negative affect, also

underlies all six measures. 3

3 See Reise, Morizot, and Hays (2007) for a discussion of bifactor models. They suggest, for example, that items on an appropriately developed scale of depression would assess not only the general factor of depression, but also subsets of items would assess group factors representing such aspects as somatization and feelings of hopelessness.


Estimation of Free Parameters

Next we will consider how free parameters are estimated. We will discuss the

estimation using the model presented in Figure 1 with four measures and a single factor.

Hats (^) are placed on top of model parameters in recognition that we are now working

with sample data as opposed to model parameters at the population level.

SEM software typically allows a variety of input data formats, including raw

case-level data, the observed covariances among the study measures, or the correlations

and standard deviations of the measures. Regardless of the form of the data that you enter

into the software, the standard maximum likelihood estimation algorithm ultimately uses

the variances and covariances among the measured variables. If you input data as a

covariance matrix, the software will use this matrix directly; if you input data as raw

cases or correlations and standard deviations among measures, the software will convert

them to a covariance matrix before conducting the SEM analyses. These variances and

covariances are elements in the sample covariance matrix, S . The specified model with

its freely estimated parameters tries to reproduce this covariance matrix. The reproduced

matrix based on the model (also called the model-implied covariance matrix) is ModelΣ̂ and

the equation linking it to the model parameters is


1

2

13

4

Model

211 E

2E21 2

2F 11 21 31 41E31

2E

41

ˆ ˆ ˆˆ ˆ Σ = Λ Φ Λ + Θ

λ̂ σ̂ 0 0 0ˆ ˆ0 σ 0 0λ ˆ ˆ ˆ ˆσ̂ λ λ λ λ + ˆ0 0 σ 0λ̂

ˆ0 0 0 σλ̂

′

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎡ ⎤⎡ ⎤= ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦

(1)

The details of the equation are not important, but rather it is crucial to understand that the

values of the model parameters dictate the quantities in the reproduced covariance matrix

among the measured variables. The objective of the estimation procedure is to have the

variances and covariances based on the model parameters (i.e., the values in ModelΣ̂ ) to be

as close to the variances and covariances among measures in our sample data (i.e., the

values in S ). Stepping back from the technical details for a minute, we are assuming that

some knowable process in the population exists that has generated the set of variances

and covariances that we have observed among our study variables. In SEM, we use our

substantive knowledge of the field to hypothesize what that process might have been by

specifying what we think the model looks like. The model structure we specify

ultimately corresponds to a model-implied variance-covariance matrix. To the extent that

we have specified something like a plausible model, the values in that model-implied

matrix ought to be similar to the observed matrix. In practice, the constraints imposed on

the model do not permit perfect reproduction of S ; that is, Modelˆ≠S Σ . To summarize the

steps of estimation so far, we postulate a model that specifies how we believe the latent

and measured variables are related, and more importantly, how they are not related (the

constraints). The algorithm generates estimates of the parameter values, which in turn


determine the implied matrix among measures. The implied matrix should be similar to

the sample covariance matrix if the model is to be considered a good fit. Below we

consider how the parameter estimates are actually generated and then how the implied

matrix is used to evaluate the fit of the model.

In contrast to regression analysis and many other statistical methods, equations

are not available for directly computing the freely estimated parameters. The estimates

are instead computed by an iterative process, initially making arbitrary guesses about the

values of the model parameters and then repeatedly modifying these values in an attempt

to have the values between S and ModelΣ̂ be as similar as possible. The process stops

when a criterion is met that suggests that the differences between S and ModelΣ̂ cannot be

smaller.

A very simple example might be helpful at this point, plugging in some fictional

values into our one factor model example. (The code and input data for this example can

be found at http://www.duke.edu/web/behavioralmed). The reported results are based on

an SEM analysis using the software program EQS. Let’s say that the variances for the

hostility, anger, anxiety, and depressive symptom measures are 1 and all covariances

among measures are .36. A variance-covariance matrix of the 4 observed measures would

look like this, with the values in the main diagonal being the variances of each of our

variables, and the off-diagonal elements the covariances between any two given

variables:

1 .36 .36 .36.36 1 .36 .36

S.36 .36 1 .36.36 .36 .36 1

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦


(This set of values is highly improbable in the real world, but for convenience we have

created a covariance matrix S as a correlation matrix.) In addition, in specifying the

model, let’s say we fix the variance of our underlying factor to 1 to define its metric. The

SEM software package begins with very rough estimates of 1 for all factor loadings and

all error variances. For these estimated parameters, the reproduced covariance matrix

among the measured variables based on Equation 1 is

[ ][ ]Model

1 1 0 0 0 2 1 1 11 0 1 0 0 1 2 1 1ˆ Σ 1 1 1 1 1 +1 0 0 1 0 1 1 2 11 0 0 0 1 1 1 1 2

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

.

Clearly, the reproduced and sample variances and covariances (on the right with 2s along

the main diagonal and 1s in the off-diagonal positions) are not very similar to the 1s and

.36s in S. The software then takes another guess, revising its estimates so that the factor

loadings are all .68 and the error variances are .64.

[ ][ ]Model

.68 .64 0 0 0 1.102 .462 .462 .462

.68 0 .64 0 0 .462 1.102 .462 .462ˆ Σ 1 .68 .68 .68 .68 +

.68 0 0 .64 0 .462 .462 1.102 .462

.68 0 0 0 .64 .462 .462 .462 1.102

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

Now the values in ModelΣ̂ are more similar to the values in S, but still not exactly the

same. In the next iteration, all factor loadings are estimated to be .605, while the error

variances are estimated to be .640. With two additional iterations, the final estimates are

.600 for all factor loadings and .64 for all error variances.

[ ][ ]Model

.60 .64 0 0 0 1 .36 .36 .36

.60 0 .64 0 0 .36 1 .36 .36ˆ Σ 1 .60 .60 .60 .60 +

.60 0 0 .64 0 .36 .36 1 .36

.60 0 0 0 .64 .36 .36 .36 1.

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦


For this artificial example, the model parameters reproduce perfectly the sample

covariance matrix among the measures. In other words, the fit of the model to the data is

perfect—a highly unlikely result in practice. We also note here that we interpret the

loadings in the conventional way for a statistical equation with estimated weights. In this

case, for every one unit increase in the underlying factor, the score of the observed

measure is expected to increase .64 units.

How does the algorithm know when S and ModelΣ̂ are similar enough to stop

iterating? Mathematically, it is necessary to specify a function to define the similarity.

The most popular estimation approach is maximum likelihood (ML), and, with this

approach, the iterative estimation procedure is designed to minimize the following

function:

1ML Model

ˆ ˆF = log - log S + trace(S ) - pModel−Σ Σ , (2)

where p is the number of measured variables. It is not crucial to understand the details of

the equation. What is important to know is that each iteration (set of parameter guesses)

produces a value for FML and that FML is a mathematical reflection of the difference

between S and ModelΣ̂ for a given set of parameters. When FML is at its smallest value, S

and ModelΣ̂ are as similar as they can be, given the data and the hypothesized model. The

values of the parameter estimates at this point in the iterative process are the maximum

likelihood estimates for the CFA model.

For our example above, FML becomes smaller with each iteration, as shown in

Table 1. At step 5, it recognizes that there was no change from step 4 to step 5 and that


there were no changes in the parameter estimates, so the process stops and the estimates

at the last step are the maximum likelihood estimates.

-----------------------------------

Table 1 About Here

-----------------------------------

Although researchers most frequently minimize FML to obtain estimates in SEM,

it is sometimes preferable to choose other functions to minimize. For example, a different

function—the full information maximum likelihood (FIML) function—is preferable if

some data on measures are missing. When modeling item-level data (such as Likert-type

items) a weighted least squares (WLS) function is generally preferred for estimating

model parameters.

To summarize our steps so far, we specify a model, the algorithm then generates a

series of guesses for the parameters, trying to find parameters that imply a covariance

matrix ( ModelΣ̂ ) that is as similar as possible to the observed covariance matrix (S). When

the changes in the parameters can no longer make the implied matrix any more similar to

the observed matrix, the algorithm stops, and the final parameter estimates are reported.

We now turn to how fit is formally evaluated.

Assessment of Global Fit

We must assess the quality of a model by examining the output from SEM

software to determine if the model and its estimated parameters are interpretable. We first

scan the output for warning messages and rerun analyses when appropriate. Second, we

assess local fit. Examples include evaluation of individual estimated parameters to ensure


they are within mathematical bounds (e.g., no negative variances or correlations about

1.0), are within interpretational bounds (i.e., no parameter estimates with values that defy

interpretation), and are significantly different from zero based on hypothesis tests. Third,

we examine global fit to determine if the constrained parameters of a model allow for

overall good fit to the data. We will concentrate our attention on global judgments of fit

here.

As previously described, the fit function is used to assess whether the estimated

model parameters are optimal with respect to fit to the data. Given it is deemed useful for

assessing global fit in the estimation of model parameters, it is not surprising that the fit

function is also a central component of all global fit indices, as we describe next.

Testing the Hypothesis of Perfect Fit in Population

We can assess the hypothesis that the researcher’s model is correct in the

population. More specifically, we can ask whether the reproduced covariance matrix

based on the model ( ModelΣ ) is equal to the population covariance matrix among the

measures (Σ ). We can state this question in the more familiar form below: the null

hypothesis of equality between the model-implied and population covariance matrix is

given below as H0, while the alternative hypothesis that the two matrices are different is

given as HA.

0 Model

A Model

H : 0H : 0

Σ −Σ =Σ −Σ ≠

(3)

Two comments are worth noting about how this question is posed in SEM. First, in most

non-SEM applications of hypothesis testing, rejection of the null hypothesis implies

support for the researcher’s hypothesis. In contrast, in SEM rejection of the null


hypothesis indicates that the researcher’s hypothesized model does not hold in the

population—the model-implied and population matrices are “significantly different.”

Second, no model is likely to fit perfectly in the population, and thus we know, a priori,

that the null hypothesis concerning the researcher’s hypothesis is false.

The test of the null hypothesis is straightforward. The test statistic, T, is a simple

function of sample size (N) and the fit function:

( ) MLT = N -1 F (4)

(or ML=T N F , as computed in some SEM software packages). In large samples and

assuming the p measured variables are normally distributed in the population, T is

distributed approximately as a chi square. The degrees of freedom for the chi square are

equal to the number of unique variances and covariances in the covariance matrix among

measured variables (i.e., ( )1 2p p +⎡ ⎤⎣ ⎦ ) minus the number of freely estimated model

parameters (q), that is,

p (p 1)df q2+

= − . (5)

In most applications with some degree of model complexity, a sample size of 200 or

greater is recommended for T to be distributed approximately as a chi square. However, a

greater sample size may be required to have sufficient power to reject hypotheses of

interest, including hypotheses about particular parameters or set of parameters.

Unfortunately, this test of global fit suffers from the same problems that a

conventional hypothesis does. If the null is not rejected, it may be due to insufficient

sample size, that is, a lack of power. In addition, non-rejection does not imply that the

researcher’s model is correct—it is incorrect to “accept the null hypothesis.” In fact, it is


likely that a number of alternative models produce similar T values. If the hypothesis is

rejected, we can only conclude what we knew initially: the model is imperfect. In

addition, note that the formula for T is highly dependent on sample size. If the sample

size is large, the T value will necessarily be large, and even small and possibly

unimportant discrepancies between the model-implied and observed covariance matrix

will yield significance. It is our observation that tests of models are routinely

significant—meaning that we conclude our model does not fit—when sample size

exceeds 200.

Fit Indices: Assessing Degree of Fit

Because the chi-square fit test is affected by sample size, a wide variety of other

measures of fit have been proposed. Two indices that are used frequently are Bentler’s

comparative fit index (CFI) and the root mean square error of approximation (RMSEA).

CFI. The CFI is a comparison of the fit of the researcher’s model to the fit of a

null model. The null model is highly constrained and unrealistic. More specifically, the

model parameters are constrained such that all covariances among measured variables are

equal to zero (implying all correlations are equal to zero). Accordingly, we expect a

researcher’s model to fit much better than a null model.

In the population, CFI is defined as

null model researcher's modelpop

null model

λ - λCFI =λ

. (7)

λ is a non-centrality parameter that is an index of lack of fit of a model to a population

covariance matrix. λ is zero if a model is correct and becomes larger to the degree that

the model is misspecified. We would expect the null model to be a badly misspecified


model in most applications of SEM; therefore, null modelλ would be large. In comparison,

researcher's modelλ should be much smaller. Accordingly, we expect to obtain high CFI values

to the extent that the researcher’s model is superior to a null model that implies

uncorrelated measured variables.

In the formula for the CFIpop, we can substitute T - df for λ to obtain a sample

estimate of CFI:

( ) ( )( )

null model null model researcher's model researcher's model

null model null model

T df - T dfCFI =

T df− −

− (8)

According to Hu and Bentler(9), a value of .95 or higher indicates good fit. This cutoff is

consistent with the belief that a researcher’s model should fit much better than the

unrealistic null model. It must be noted that cutoffs for fit indices are problematic and a

preferable approach is to use these indices to compare fits for various alternative models.

RMSEA. The RMSEA is a fit index that assesses lack of fit, but does not use the

unrealistic comparison of a null model. The sample estimate is also a function of T and

df:

( )( )

researcher's model

researcher's model

T N -1 1RMSEA = -df N -1

. (9)

To the extent that the model fits [i.e., small ( )researcher's modelT N -1 ] and the model involves

estimating few model parameters (large researcher's modeldf ), RMSEA should approach zero.

RMSEAs of less than .06 indicate good fit according to Hu and Bentler(9), but again this

cutoff should be treated as a rough-and-ready rule of thumb.


Underidentification and Other Problems in Estimation

After presenting the example, let’s return to discuss one technical issue. A

requirement for estimating the parameters of a model is that the model must be identified.

Identification means simply that the information about your sample must equal or exceed

the needs defined by the estimation of your model parameters. Information about your

sample is captured in the variances and the covariances among the measured variables.

This information is used to estimate the unconstrained model parameters—potentially,

the factor loadings, the variances and the covariance among the factors, and variances

and covariances among the errors. The t-rule states that the number of freely estimated

parameters (q) must be less than or equal to the number of unique variances and

covariances among the measured variables, which is equal to p (p 1) 2+ . Another way to

express the t-rule is that the degrees of freedom for the chi square test cannot be negative

(see Equation 5).

The bad news is that even if your model passes the t-rule, the model may still be

underidentified (i.e., not identified). This occurs if the number of parameter for a portion

of the model exceeds the available sample information. For example, a model might

include one factor with freely estimated loadings on the first 2 measures and a second

factor with freely estimated loadings on the remaining 2 measures. All other loadings are

constrained to 0; the factor covariance and error covariances are constrained to 0; and the

factor variances are fixed to 1 to set their metric. In this example, the variances and

covariances for each pair of measures are available to estimate only the model parameters

for these measures. Because each pair of measured variables are linked to one and only


one factor, it is as if two CFAs are being conducted—one for the first pair of measures

and another for the second pair. The consequence is that the model cannot be estimated

because the number of freely estimated parameters for any pair of measures (2 loadings

and 2 error variances) exceeds the amount of sample information (2 variances and 1

covariance between these measures).

Additional identification rules are available. The 3-indicator rule may be applied

for the example just described. If each measure has only one estimated factor loading

(others constrained to 0), the covariances among factors are constrained to 0, and the

covariances among the errors are constrained to 0, then a model is identified if each

factor has estimated loadings on at least 3 measures (as opposed to 2 measures as

described in our previous example). In most SEM applications, factors are allowed to be

correlated, and then a model is identified if each factor has estimated loadings on at least

2 measures. For the 2-indicator rule, the same conditions must hold as with the 3-

indicator rule except the covariances among factors are freely estimated.

There is both bad and good news about the use of the 2- and 3-indicator rules. The

bad news is that they are not applicable for many CFA models. For example, they are not

helpful in determining if a bifactor model with both group and general factors is

identified. The good news is that available software is likely to give warning messages if

the model is underidentified. More bad news is that it is not always obvious what the

warning messages mean and what, if anything, should be done to remedy the problem.

In fact, the messages might suggest other estimation problems such as empirical

underidentification or bad start values. With empirical underidentification, the model is

identified mathematically, but nevertheless the parameters of the model cannot be


estimated because of the data. For example, a CFA model with two factors might meet all

the requirements of the 2-indicator rule, but may still not be able to be estimated if the

freely estimated covariance between factors is 0 (or close to 0). In this case, because the

factors are uncorrelated, 3 measures are required per factor. Alternatively, for the same

example, if the estimated factor loading for a measure is 0, it cannot be counted as one of

the indicators for a factor.

The other estimation problem is bad start values. As described earlier, the

estimation process in CFA is iterative and requires start values that are created by the

SEM software. With more complex models, the start values created by the program may

be bad in that they do not produce adequate estimates. In this instance, the researcher

may ask the program to conduct more iterations to get a good solution or may be forced

to supply their own start values for parameter estimation. Researchers might use

estimates from exploratory factor analysis or other CFA models to supply start values.

In conducting CFA, no researcher wants to see warning messages about parameter

estimates. Our piece of advice is not to deny the presence of warning messages, but rather

to acknowledge their presence and work through them with someone you trust can help

you (i.e., your local SEM expert).

Conclusions

In many applications, researchers who apply exploratory factor analysis could use

confirmatory factor analysis. To the extent that researchers have some knowledge about

the measures that they are analyzing, they should be conducting CFA. There are real

benefits to specifying rigorously one’s beliefs about measures, assessing those beliefs


with indices that allow for disconfirmation of these beliefs, and at the end being able to

specify which alternative model produces the best fit. It may require more thoughtfulness

upfront than EFA, but the outcome is likely to be more informative if the methods of

CFA are applied skillfully.

As we noted at the beginning of this piece, SEM is capable of carrying out a fairly

staggering variety of types of analytic procedures. We have presented one of the two

procedures that are fundamental to the full structural equation model. In our next

installment we will present the second fundamental procedure, path analysis, and also

how CFA and path analysis combine to form the full structural model. In the coming

installment, we also will present several important concepts that we had to omit in the

present paper. These concepts include how to compare competing models, approaches to

modifying models, indirect effects and mediation, and more general considerations, such

as sample size and power.


References

1. Wright S. Correlation and causation. Journal of Agricultural Research

1921;20:557-85.

2. Joreskog KG, Goldberger AS. Estimation of a model with multiple indicators and

multiple causes of a single latent variable. Journal of the American Statistical

Association 1975;70:631-9.

3. Calis JCJ, Phiri KS, Faragher EB, Brabin BJ, Bates I, Cuevas LE, de Haan RJ,

Phiri AI, Malange P, Khoka M, Hulshof PJM, van Lieshout L, Beld MGHM, Teo

YY, Rockett KA, Richardson A, Kwiatkowski DP, Molyneux ME, van

Hensbroek MB. Severe Anemia in Malawian Children. N Engl J Med

2008;358:888-99.

4. Rosen R, Contrada R, Gorkin L, Kostis J. Determinants of perceived health in

patients with left ventricular dysfunction: a structural modeling analysis.

Psychosom Med 1997;59:193-200.

5. Suls J, Bunde J, Suls J, Bunde J. Anger, anxiety, and depression as risk factors for

cardiovascular disease: the problems and implications of overlapping affective

dispositions. Psychological Bulletin 2005;131:260-300.

6. Rindskopf D, Rose T. Some theory and applications of confirmatory second-order

factor analysis. Multivariate Behavioral Research 1988;1988:51-67.

7. Yung YF, Thissen D, McLeod LD. On the relationship between the higher-order

factor model and the hierarchical factor model. Psychometrika 1999;64:113-28.

8. Reise SP, Waller NG, Comrey AL. Factor analysis and scale revision.

Psychological Assessment 2000;12:187-297.


9. Hu L, Bentler PM. Cutoff criteria for fit indexes in covariance structure analysis:

Conventional criteria versus new alternatives. Structural Equation Modeling

1999;6:1-55.


Suggested Readings

Introductory Reading:

Brown TA. Confirmatory factor analysis for applied research. New York: Guilford

Press; 2006.

Green SB, Thompson MS. Structural equation modeling in clinical research. In: Roberts

MC, Illardi SS, editors, Methods of Research in Clinical Psychology: A Handbook.

London: Blackwell; 2003. p 138-175.

Kline RB. Principles and practice of structural equation modeling (2nd ed). New York:

Guilford Press; 2005.

Glaser D. Structural Equation Modeling Texts: A primer for the beginner. Journal of

Clinical Child Psychology 2002; 31: 573-578.

More Advanced Reading:

Bollen KA. Structural Equations with Latent Variables. New York: Wiley; 1989.

Edwards JR, Bagozzi RP. On the nature and direction of relationships between

constructs and measures. Psychol Methods 2000; 5:155-174.

MacCallum RC, Roznowski M, Necowitz LB. Model modifications in covariance

structure analysis: The problem of capitalization on chance. Psychol Bull 1992; 111:

490-504.

McDonald R, Ho M-HR. Principles and practice in reporting structural equation analyses.

Psychol Methods 2002; 7: 64-82.

Wirth RJ, Edwards MC. Item factor analysis: Current approaches and future directions.

Psychol Methods 1989; 12:58-79.


Table 1. Estimation of Model Parameters by Minimizing of FML for Our Example

Steps in iterative

process

Estimates of factor

loadings at each step

Estimates of error

variance at each step FML

1 1 1 .55194

2 .680 .640 .01523

3 .605 .640 .00006

4 .600 .640 .00000

5 .600 .640 .00000


Figure Captions

Figure 1. A Single Factor Model

Figure 2. A Correlated Factors Model

Figure 3. A Bifactor Model


F1

1

2Fσ 1=

21λ 31λ11λ 41λ

E1

1

2Eσ

E2

2

2Eσ

E3

3

2Eσ

E4

4

2Eσ

X1 X2 X3 X4

11

21

31

41

λλ

Λ =λλ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

1

2

3

4

2E

2E

2E

2E

σ 0 0 00 σ 0 0

Θ = 0 0 σ 0

0 0 0 σ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

1

2FΦ = σ 1⎡ ⎤=⎣ ⎦

Path Diagram:

Matrices:

F1

1

2Fσ 1=

21λ 31λ11λ 41λ

E1

1

2Eσ

E2

2

2Eσ

E3

3

2Eσ

E4

4

2Eσ

X1 X2 X3 X4

F1

1

2Fσ 1=

21λ 31λ11λ 41λ

E1

1

2Eσ

E2

2

2Eσ

E3

3

2Eσ

E4

4

2Eσ

X1 X2 X3 X4

11

21

31

41

λλ

Λ =λλ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

1

2

3

4

2E

2E

2E

2E

σ 0 0 00 σ 0 0

Θ = 0 0 σ 0

0 0 0 σ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

1

2FΦ = σ 1⎡ ⎤=⎣ ⎦

Path Diagram:

Matrices:


Path Diagram:

Matrices:

F1

21λ32λ11λ

42λ

E1

1

2Eσ

E2

2

2Eσ

E3

3

2Eσ

E4

4

2Eσ

X1 X2 X3 X4

F2

1 2F Fσ

1

2Fσ 1=

2

2Fσ 1=

1

2

3

4

2E

2E

2E

2E

σ 0 0 00 σ 0 0

Θ =0 0 σ 0

0 0 0 σ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

1 1 2

1 2 2

2F F F

2F F F

σΦ =

σσ

σ⎡ ⎤⎢ ⎥⎣ ⎦

11

21

32

42

λ 0λ 0

Λ =0 λ0 λ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

Path Diagram:

Matrices:

F1

21λ32λ11λ

42λ

E1

1

2Eσ

E2

2

2Eσ

E3

3

2Eσ

E4

4

2Eσ

X1 X2 X3 X4

F2

1 2F Fσ

1

2Fσ 1=

2

2Fσ 1=F1

21λ32λ11λ

42λ

E1

1

2Eσ

E2

2

2Eσ

E3

3

2Eσ

E4

4

2Eσ

X1 X2 X3 X4

F2

1 2F Fσ

1

2Fσ 1=

2

2Fσ 1=

1

2

3

4

2E

2E

2E

2E

σ 0 0 00 σ 0 0

Θ =0 0 σ 0

0 0 0 σ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

1 1 2

1 2 2

2F F F

2F F F

σΦ =

σσ

σ⎡ ⎤⎢ ⎥⎣ ⎦

11

21

32

42

λ 0λ 0

Λ =0 λ0 λ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦


Path Diagram:

Matrices:

F1

F2

1

2Fσ 1=

2

2Fσ 1=

E6

6

2Eσ

X6

21λ11λ 41λ31λ

51λ 61λ

X5 E5

5

2Eσ

E4

4

2Eσ

X4X3 E3

3

2Eσ

E2

2

2Eσ

X2X1 E1

1

2Eσ

42λ 52λ 62λ

11

21

31

41 42

51 52

61 62

λ 0λ 0λ 0

Λ =λ λλ λλ λ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

1

2

3

4

5

6

2E

2E

2E

2E

2E

2E

σ 0 0 0 0 00 σ 0 0 0 00 0 σ 0 0 0

Θ =0 0 0 σ 0 00 0 0 0 σ 0

0 0 0 0 0 σ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

1

2

2F

2F

σ 0Φ =

0 σ⎡ ⎤⎢ ⎥⎣ ⎦

Path Diagram:

Matrices:

F1

F2

1

2Fσ 1=

2

2Fσ 1=

E6

6

2Eσ

X6

21λ11λ 41λ31λ

51λ 61λ

X5 E5

5

2Eσ

E4

4

2Eσ

X4X3 E3

3

2Eσ

E2

2

2Eσ

X2X1 E1

1

2Eσ

42λ 52λ 62λ

F1

F2

1

2Fσ 1=

2

2Fσ 1=

E6

6

2Eσ

X6

21λ11λ 41λ31λ

51λ 61λ

X5 E5

5

2Eσ

E4

4

2Eσ

X4X3 E3

3

2Eσ

E3

3

2Eσ

E2

2

2Eσ

X2X1 E1

1

2Eσ

E1

1

2Eσ

42λ 52λ 62λ

11

21

31

41 42

51 52

61 62

λ 0λ 0λ 0

Λ =λ λλ λλ λ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

1

2

3

4

5

6

2E

2E

2E

2E

2E

2E

σ 0 0 0 0 00 σ 0 0 0 00 0 σ 0 0 0

Θ =0 0 0 σ 0 00 0 0 0 σ 0

0 0 0 0 0 σ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

1

2

2F

2F

σ 0Φ =

0 σ⎡ ⎤⎢ ⎥⎣ ⎦

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