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STRUCTURAL EQUATION MODELLING

MODERN METHODS

(Chapters 14, pages 681704, 714-735, 755 - 785)

Often applied psychologists are part of a research and development team

trying to design interventions to help people improve their quality of life.

Ideally, these interventions are based upon a theory that models aspects of

human behaviour within particular social contexts. The more precise the

theory and the more it is based upon empirical evidence, the more specific

components of an intervention can be designed based upon sound scientific

principles as well as upon the experience and intuition of dedicated

practitioners. Indeed, theory is useful both because it suggests ways to probe

the effectiveness of conventional practices and because it suggests innovative

practices that have not been considered in the past.

It bears repeating (stifle your yawns!), that modern theories must be complexin nature if they are to have relevance to individuals designing social

programs. As well, practitioners can only manipulate some of the pertinent

variables that influence program participants. Testing theory in this situation

is not addressed adequately by conventional hypothesis testing research

methods, even though these methods provide strong support for central parts

of the theory. More and more, researchers wish to test the plausibility of their

theoretical models as a whole; that is they wish to examine the interaction of

an entire set of variables in a field setting that includes important social

outcomes. The growing interest in structural equation modelling (SEM)

reflects this desire.

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Whatever the SEM program selected by the researcher, the underlying

statistical procedure uses matrix algebra to solve a set of simultaneous

equations that are specified by a particular theory. These equations are

directly derived from a path diagram that shows the relationships among the

theoretical constructs (the structural model) and the relationships among the

theoretical constructs and the measures of these constructs (the measurement

model). Using these equations and initial start-up values, SEM programs

follow an iterative algorithm which converges on an optimal solution (it is the

different iterative procedures and their criteria for convergence that

distinguishes the various SEM programs). The extent to which this solution

can reproduce the variancecovariance matrix among the variables in the

data set becomes a test of the fit of the theoretical model (similar to the

factor analysis criterion that the factor solution should reproduce the

correlation matrix among the variables entered into the analysis).

Thus the results of a variety of goodness-of-fit indices are an important

outcome of any SEM program and it is possible to compare different theories

in terms of their relative goodness-of-fit. As well, the results of the analysis

can indicate whether adding certain paths will significantly improve the fit

between the model and the data. This is a valuable, albeit, post hoc procedure

that suggests specific modifications to the theory that warrant investigation

and replication in a subsequent study.

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The degree to which a model fits the data is only one of the outcomes of

interest, however. Of equal interest are the values of path coefficients that

estimate the strength and direction of both direct and mediated relationships

among the variables in the model. Indeed, SEM can model mediating

psychological processes in a much more direct way than traditional

experimental designs.

Combining the results of experiments that establish a causal link from an

independent variable to a dependent variable with the results of SEM that

place this causal relationship within a theoretically specified network of

relationships provides far more compelling evidence in support of a theory

than either kind of evidence alone. The path coefficients are unbiased

estimates of population parameters and they can be tested to see if they are

significantly different from zero. As well, the relative value of standardized

path coefficients (range -1 through 0 to +1) indicate their importance in

predicting specific outcomes.

Another name for structural equation modelling is confirmatory factor

analysis. Researchers using this name are specifically concerned with

generating evidence that a psychological construct takes the form specified by

theory (often the construct is multidimensional; e.g., sexism has both a hostile

and a benevolent aspect which are related). This type of analysis is usually

used when the researcher is developing tests which measure the construct

adequately (with reliability, content validity, and construct validity).

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To repeat, psychologists use structural equation modelling techniques to test

the adequacy of a theoretical model and to estimating the strength of causal

paths using path coefficients. This modelling technique also allows the

reliability of the measuring instruments to be estimated by specifying the

measurement model and a structural model and testing the viability of both

types of models simultaneously; a process that involves using a complex

iterative computer program to analyse a large data set.

In this course only a basic introduction to SEM as it is used to conduct a

confirmatory factor analysis or to assess the plausibility of a theoretical model

with recursive paths will be covered. As well, the exercises and assignment

will give you practice with one of the most widely used windows program,

EQS (installed on the Arts Lab computers in room 31 Arts). The Department

has a site licence for EQS so your supervisor can install it in her/his lab.

Note that if only one measured variable (manifest variable) is used to index

each underlying construct in a theoretical model, the SEM procedure conducts

a path analysisan analysis that only includes measured variables.

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An Impor tant Note:In the past, researchers often assumed that theoretical

models are recursive. This is limiting now and will likely be even more

limiting in the future. People react to events with a sense of agency rather

than as pawns controlled by fate, and their actions count. The ability of

theory to model complex feedback loops and the ability of statistics to test the

plausibility of such theories is becoming increasingly important.

To use a clinical example, victims of violence often do not label themselves in

this way. Rather active coping attempts by these individuals allow them to

avoid applying the label of victim to themselves and to successfully adapt.

Understanding how some individuals are able to do this while others succumb

to this extremely stressful experience is essential if professional psychologists

are to help effectively.

Similarly, in social psychology, stereotyping is only of concern because this

invidious process is perpetuated through selective perceptions guided by

unreasonable expectations. Again, understanding the feedback loop that

results in group members confirming the negative expectations implied by

their groups stereotype is a complex problem that involves reciprocal causal

influences between stereotyper and stereotypee in an interpersonal interaction

that extends across time.

More positively, consider the development of friendship which, at its heart,

involves increasingly intimate and reciprocal interactions as the lives of two

people become inextricably bound together. As all these examples show,

reciprocal causality lies at the heart of many important psychological

processes and the requirements for statistical techniques to help model these

processes are in increasing demand.

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CONVENTIONS USED IN SEM DIAGRAMS:

RECTANGLES = MEASURED OR MANIFEST VARIABLES

OVALS = CONSTRUCTS OR LATENT VARIABLES OR FACTORS

ARROW = CAUSAL PATH; DOUBLE ARROW = CORRELATION

EXOGENOUS VARIABLES = INDEPENDENT VARIABLES

ENDOGENOUS VARIABLES = DEPENDENT VARIABLES WHICH ARE

SOMETIMES INDEPENDENT VARIABLES AS WELL.

Canadian

Identity

Perceived

Discrimination

Cultural

Identity

Illegitimate

X Unstable

Intentions

(Protest)

Emotions

Em2

Em3

Em3

IDC3

IDC2

IDC1

PD3

PD2

PD1

ID1

ID2

ID3

exogenous

latent

variable

endogenous

latent

variable

E = Error

D = Disturbance

exogenous

manifest

variable

endogenous

manifest

variable

endogenous

manifest

variable

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Assumptions Underlying Structural Equation Modelling

1. All variables must have linear relationships with each other.

2. Outliers must be identified and dealt with prior to the main analysis.

3. The variables in the analysis should be normally distribution(multivariate normality). This assumption is more crucial for modern

forms of SEM and so a preliminary analysis must involve examining each

variable for skewness and kurtosis. Transformations that create a normal

distribution for these variables are then applied. If a transformation does

not achieve a normal distribution, then an iterative estimation procedure

that is robust to violation of this assumption must be used.

4. Absence of multicollinearity is necessary as the computer executes matrixinversions in each iteration. Most SEM programs give the determinant of

the variancecovariance matrix as part of the output so that this

assumption can be examined.

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5. Sometimes SEM programs have difficulty analysing a data set that containsvariables measured on scales that vary considerably in range and mean

value, a situation which results in covariances that are tremendously

different in size. Rescale some of the variables before running the analysis.

6. The possibility of a specification error haunts any researcher usingstructural equation modelling. The solution is to examine the residual

variancecovariance matrix. The residuals should be small and centred

around zero. Non-symmetrical residuals (some small and some large)

suggest that the model estimates some parameters well and others poorly.

One reason for this is that a causal path between variables in the model has

been mistakenly set to zero (the theory is wrong). If this is true, then post

hoc procedures can be used which suggest how the model can be

improved by adding paths. Then replication using another sample is

required. The other reason why residuals are large and non-symmetrical is

that the model is misspecified. There is no easy solution to this problem

but at least the analysis pushes the researcher to examine the theory more

critically.

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7. Large sample sizes are required in order to run modern structuralequation modelling programs. Generally, the minimum sample size for all

SEM programs can be estimated by multiplying the number of

parameters that the program is estimating by ten. This means that usually

EQS requires a sample size of at least 200 research respondents and other

programs require more. However, experienced applied researchers with

messy data say that even that number may not be enough for the program

to convergence on a final solution. That is, with smaller sample sizes and,

therefore, more unstable estimates, the program simply may not be able to

find an optimal solution. Part of this problem can be caused by the default

start values used by the SEM program being very different from the actual

values of the parameters. Therefore, if estimates of these parameters can

be obtained from past research, they should be specified as the initial start

values in the analysis.

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8. Structural Equation Modelling is based upon a mathematical procedurethat tests the ability of a theoretically derived model to reproduce the

variancecovariance matrix among measured variables in a data set.

The use of the variancecovariance matrix preserves the scale of the

original variables. Rescaling these variables by, for example, adding or

subtracting a constant does not change the results of the analysis.

However, rescaling variables through the use of sample statistics is more

problematic as it alters the value of the 2 statistic that is the basis fortesting the goodness-of-fit of the model. When variables are standardized,

the rescaling involves sample statistics because deviations from the meanare divided by the sample's standard deviation. Hence, the developer of

EQS, Peter Bentler, warns researchers not to use correlations whenever

possible. In some circumstances, a researcher has no choice because he or

she is doing a secondary analysis on a correlation matrix from a published

article. In this instance, the EQS program alerts users of the program to

the fact that the analysis may not be correct because a correlation matrix

has been used.

Of course standardized path coefficients are very useful because they

reflect the relative strength of different paths in the model. Thus, after the

analysis on the variancecovariance matrix has been done, the computer

calculates these standardized path coefficients from the unstandardized

path coefficients and their standard errors.

Perhaps in the future this limitation will be overcome, but right now it is

important for researchers to know that they should analyse the variance

covariance matrix, NOT the correlation matrix, whenever possible (the

default option in EQS).

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Model Specification Using the Bentler-Weeks Estimation Method

The Bentler-Weeks model takes a regression approach to structural equation

modelling. However, the matrix equation specifies both measured variables

(also called manifest variables) and the latent variables (constructs or factors)

that are presumed to underlie responses to the measuring instruments. In this

model, both types of variables can be exogenous or endogenous.

Remember from chapter 5 that the matrix algebra equation for multiple

regression is:

y = x . b + e

In this equation there are k regression coefficients (in a k x 1 vector) which

need to be estimated and there are always enough equations to provide an

estimate of these parameters (as long as N > k). The reason why it is always

possible to estimate the regression coefficients is because it is assumed that 1)

the independent variables are measured without error, 2) the independent

variables have a direct causal influence on the dependent variable and no

other variables, and 3) the residuals associated with the dependent variable

are not correlated with the independent variables (no specification error). In

other words, these assumptions allow the research analyst to fix the values of

many parameters that could, theoretically, vary and it is the specification of

these parameters that allows the computer to estimate unique values for the

regression coefficients (the estimated parameters) using a least squares

solution.

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In the same way, a more complex regression equation can be written

describing the relationships among the endogenous (dependent) and

exogenous (independent) variables specified by the structural (theoretical)

model and the measurement model. However, unlike multiple regression,

SEM asks the researcher to choose which parameters he or she will fix and

which to estimate. However, the researcher can not allow all possible

parameters to vary freely because this would always result in an under-

identified model (not enough degrees of freedom available to estimate the

parameters in the model). Thus the researcher must define both the

measurement model and the structural model in such a way that enough

parameters are fixed in value so that there are degrees of freedom available to

test the plausibility of the model in its entirety. That is, the model must be

over-identified.

Bearing this in mind, the fundamental Bentler-Weeks regression equation can

be expressed as:

= B . + . (q x 1) (q x q) (q x 1) (q x r) (r x 1)

where (eta) is a q x 1 vector of the q endogenous (dependent) variables; B(beta) is a q x q square matrix of path (regression) coefficients which are

estimates of the relationships among the endogenous variables; (gamma) isa q x r matrix of path coefficients which are estimates of the relationships

between the endogenous variables and the exogenous (independent) variables;

and (xi) is a r x 1 vector of the r exogenous variables.

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Notice that this method involves solving q equations. That is, there is an

equation for each of the q endogenous variables and there are no equations

for the exogenous variables because their variability is explained by variables

outside the model. However, the r exogenous variables have variances and

covariances that need to be estimated. These variances and covariances are in

an r x r variancecovariance matrix called (phi).

Altogether, then, the parameters that need to be estimated are in the B, , and matrices and the path diagram is used to set some of these parameters tofixed values (usually 0 or 1) so that there are enough degrees of freedom

available to test the goodness of fit of the model to the data. Start values for

the parameters are then entered into the matrices. These start values can be

set by the computer or they can be estimated and entered by the researcher.

The computer then estimates the variancecovariance matrix among all the

measured variables using the criterion for convergence specified by the

researcher and compares it to the actual variancecovariance matrix. New

parameter estimates are calculated and entered as start values in the next

iteration. The computer stops the iterations when the estimate of the variance

covariance matrix cannot be improved.

Notice in this form of structural equation modelling 1) the B, , and matrices contain parameter estimates for both the measured and the latent

variables (factors) and 2) the and vectors are not estimated but deriveddirectly from the data set.

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Using the Path Diagram to Derive Equations

Representing the Theory

Consider the path diagram shown below for the Skiing Satisfaction example

in TF (Fig 14.4, p. 692).

F2 = skisatF1 = loveski 1.0

V1 = yrsski

V2 = daysski

V5 = senseek*

V4 = foodsat

V3 = snowsat

*

E1* 1.0

*

1.0

*

D2*

1.0

*

1.0

E3*1.0

*

E4*1.0

*

1.0

*

1.0

*

1.0

*

1.0

1.0

*

1.0

In this diagram the stars (*) indicate a path coefficient or a variance that

needs to be estimated. The number of stars equals the total number of

parameters that need to be estimated (see later diagrams).

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Consider the elements in this path diagram. Remember, measured variables

are represented by squares and latent variables (factors) by ovals.

Endogenous variables have causal paths leading to them, while exogenous

variables do not. Residuals of the endogenous measured variables are

included in the model as exogenous variables labelled E (for error), while

residuals of the endogenous latent variables (factors) are included in the

model as exogenous variables labelled D (for disturbancesthe error in

prediction). Notice that endogenous and exogenous variables can be either

manifest or latent. Also note that usually the path from the errors and

disturbances are fixed at 1, but their variance is estimated. This reflects the

preference of most researchers to estimate the residual error and to not be

concerned with estimating the path coefficient from unknown variables

outside the model.

By convention, this model fixes the variance of the exogenous latent variable,

love of skiing, at 1. This is the researcher's decision. Alternatively, he or she

could have set one of the path coefficients from this latent variable to one of its

indicators to 1, as is done for the path from ski trip satisfaction to snow

satisfaction. Essentially this latter convention gives the latent variable the

same scale as the chosen indicator and is advocated strongly by statisticians,

including Bentler. Allowing all the paths and the variance to vary freely is not

an option, however, as this will usually cause the model to become under-

identified.

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How then are equations derived from the path diagram? First, it is important

to realize that the number of equations that you need to write is equal to the

number of endogenous variables. In this example, four measured variables

(number of years skied, number of days skied, snow satisfaction, and food

satisfaction) are endogenous and one latent variable, ski trip satisfaction, is

endogenous. Therefore, there are five endogenous variables represented in

five equations. Four of these equations represent the measurement model as

they link the latent factors to the measured variables. The fifth equation

represents the structural model specified by theory; in this case that the love

of skiing and sensation seeking are the causal determinants of ski trip

satisfaction.

Next you need to write the five equations in the same way as the matrix

equation:

= B . + .

The matrix equation for this example is shown below (see TF page 694), where

q = 5, r = 7, and * = parameter needs estimating:

V1 0 0 0 0 0 V1 0 * 1 0 0 0 0 V5

V2 0 0 0 0 0 V2 0 * 0 1 0 0 0 F1

V3 = 0 0 0 0 1 V3 + 0 0 0 0 1 0 0 E1

V4 0 0 0 0 * V4 0 0 0 0 0 1 0 E2F2 0 0 0 0 0 F2 * * 0 0 0 0 1 E3

E4

D2

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Consider the first endogenous variable, Number of years skied (V1). In the

path diagram this variable is predicted by the latent factor, love of skiing, (F1)

and variables outside the model (E1). Thus, the equation for this variable is:

V1 = * F1 + 1 . E1

In its full form, however, it would be written:

V1 = (0 . V1 + 0 . V2 + 0 . V3 + 0 . V4 + 0 . F2) + (0 . V5 + * . F1 + 1 . E1 + 0 . E2 + 0 . E3 + 0 . E4 + 0 . D2)

Remember when two matrices are multiplied each element of the first row in

the first matrix is multiplied by the corresponding element in the first column

of the second matrix and then added, and so on. Thus, the full form of this

equation is equivalent to the first line of the matrix equation shown above.

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In the same way, the other equations for the endogenous variables are

equivalent to the corresponding lines of the matrix equation specified by the

Bentler-Weeks SEM model. This means that writing these equations is

equivalent to writing the matrix equation for this SEM procedure and writing

these equations is easily done using the path diagram. The five equations

implied by this particular path diagram are:

V1 = * F1 + E1

V2 = * F1 + E2

V3 = 1 F2 + E3

V4 = * F2 + E4

F2 = * F1 + * V5 + D2

Note that in these equations the path coefficients from E1, E2, E3, E4, and D2

are set to 1.

In addition, the model specifies the variances and covariances among the

exogenous variables that need to be estimated This means that the variances

for V5, E1, E2, E3, E4, and D2 need to be estimated, that the variance of F1

is set to 1, and that all the covariances are set to 0, specifying the variance

covariance matrix, (see page 695 which shows this diagonal matrix).

Together this information is sufficient for the computer to execute the

iterative estimation procedure (section 14.4.4 shows how the computer goes

through one of these iterations). Specifically, start values replace the stars (*)

in these matrices. Then the SEM program estimates the variance

covariance matrix among the measured variables.

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Methods of Achieving Convergence

So far we have discussed how, in general, the computer converges on a best

estimate of the variance covariance matrix for the measured variables.

However, there are several different criterion for achieving convergence. The

most common of these is the Maximum Likelihood (ML) method followed by

the Generalized Least Squares (GLS) method. Essentially, the Maximum

Likelihood method converges on an estimated variancecovariance matrix

which maximizes the probability that the difference between the estimated

and the samples variance - covariance matrices occurred by chance.

In contrast, the Generalized Least Squares method converges on an estimated

variancecovariance matrix that minimizes the sum of the squared

differences between the elements in the estimated and the samples variance -

covariance matrix.

Mathematically, both of these criteria for convergence involve minimizing a

mathematical function through successive approximations. Both methods are

good if the variables are distributed normally and the sample size is adequate.

Tabachnick and Fidell suggest using an estimation method called the scaled

Maximum Likelihood method if non-normality can not be corrected. EQS will

do this if required and gives a corrected Chi squared called the Satorra-

Bentler scaled 2 (TF, p. 713).As well, adjustments to the standard errors of

the path coefficients are calculated so as to correct their statistical

significance.

(I will discuss using dichotomous variables later.)

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Testing the Adequacy of the Theoretical Model

(including Goodness-of-Fit Indices)

The Basic Chi Squared Test for Goodness-of-Fit

Once the SEM program has converged on a solution, a 2 statistic is calculatedthat tests how well the estimated variance - covariance matrix fits the actual

variance - covariance matrix among the measured variables (its value is taken

from the value of the mathematical function that is minimized to achieve

convergence). The degrees of freedom for this statistic are equal to the amount

of unique information in the sample variancecovariance matrix minus the

number of parameters that need to be estimated. If there are p measured

variables, then the total degrees of freedom, p*

= p ( p + 1) / 2.

In the ski trip satisfaction study, for example, there are five measured

variables so the total number of degrees of freedom are 5 x 6 / 2 = 15. Given

that the path diagram shows that the researcher wants to estimate five path

coefficients and six variances, 2 is tested with 15 - 5 - 6 = 4 degrees offreedom.

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Because the desired result is for a good fit between the estimated and the

sample variancecovariance matrices, the researcher hopes for a non-

significant 2 . However, 2 values are dependent upon sample size such thateven very small differences are significant when the sample size is large. The

result is that a number of Goodness-of-fit indices have been developed to

correct for this problem and many of them are part of the output in the EQS

program.

Testing the Significance of the Path Coefficients

In the Bentler-Weeks SEM procedure, the unstandardized path coefficients

are normally distributed so that when they are divided by their standard

error a Z score is obtained. It is these Z scores that provided a test of

whether the path coefficient is significantly different from zero.

As the unstandardized path coefficients are on a different scale from one

another, researchers often report the standardized path coefficients that vary

from -1 to 0 to +1 and which indicate the relative strengths of the causal paths

in the model in the same way as a standardized regression coefficient. Notice

that the standardized path coefficients from the latent variables to their

measured counterparts are factor loadings. Indeed, if the measurement

model specifies the entire model then the SEM procedure has executed a

confirmatory factor analysis as will be illustrated shortly in an example.

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Goodness-ofFit Indices

Given the problems associated with using the lack of significance of a 2statistic to indicate a models goodness-of-fit to the data, many other indices

have been developed. Unfortunately, there is no consensus on which of these

indices are the best ones to use, so SEM programs avoid this problem by

outputting most of these indices. Usually, they all show that the model is or is

not a good fit, so it is a matter of preference which one the researcher reports.

Hu & Bentler (1999) argue, however, that researchers should usually report

one residual based fit index and one comparative fit index. The following

section discusses some of the more commonly used goodness-of-fit indices of

both types.

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Comparative Fit Indices

In SEM the theoretical model can be compared to a just identified model that

contains all possible causal paths. A just identified model can not be tested,

but the matrix equation will give an exact and unique mathematical solution.

If one more parameter needs to be estimated, then the mathematical solution

is indeterminate (there are not enough equations to yield a solution). More

commonly the theoretical model is compared to a model in which all the

variables are independent of one another. Here all path coefficients are set to

zero and only the values in the variancecovariance matrix for the exogenous

variables are estimated.

These two extremes illustrate the fact that models vary along a continuum

from a model in which all the variables are independent of one another (only

the variances of the exogenous variables are estimated) to the just identified

model which can be specified but which can not be tested because there are no

degrees of freedom left. The theoretical models goodness-of-fit is estimated by

comparing it with one or the other of these extremes.

A simple and often used comparative goodness-of-fit index is the Bentler

Bonett Normal Fit Index (NFI) defined as:

NFI = 2indep - 2model / 2indep

This index varies from 0 to 1 and values greater than 0.9 indicate a good fit.

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However, for relatively small samples ( < 200) the NFI underestimates the fit

of a model. Thus, it has been replaced by the Comparative Fit Index (CFI)

which uses a noncentrality parameter, , which is an index of modelmisspecification; the larger the value of the greater the misspecification,with = 0 indicating that the estimated model is perfect.

The Comparative Fit Index is defined by the following equation:

CFI = 1 - (model / indep )where = (2 - df)

This index also varies from 0 to 1 and is a better estimate of goodness-

of-fit for smaller samples. Models are a good fit if CFI > .95.

Another popular comparative fit index is the Root Mean Square Error

of Approximation (RMSEA) which, compares the model to a just identified

model. This statistic is defined as

RMSEA = ( Fo / dfmodel )

And Fo = (2model - dfmodel) / N , with Fo set to zero if its value is negative.

Fo = 0 indicates a perfect fit, so small values of RMSEA are desired. A good

fitting model is indicated if RMSEA < 0.06. However, like the NFI, this

statistic tends to reject models that fit well when the sample size is small.

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Residual-Based Fit Indices

A popular index of goodness-of-fit that has intuitive appeal is the Root

Mean Square Residual (RMR) index. This statistic is based upon the average

of the squared differences between each element of the sample variance -

covariance matrix and the corresponding element of the estimated variance

covariance matrix.

RMR = ( 2 ij ( (sj - j)2

/ p ( p - 1) )

,

where p is the number of measured variables and sj and j are thecorresponding variances and covariances from the two matrices.

Models that fit well have small RMR values, but these values are dependent

upon the scale of the original measured variables in the model. Therefore, a

standardized Root Mean Square Residual index (sRMR) has been developed.

The values for sRMR range from 0 to 1 with small values indicating a good fit

(small residuals). sRMR < .08 indicates that the model is a good fit.

(In the social psychology literature the CFI and the sRMR are fit indices that

are often reported.)

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Model Identification

Can the Theoretical Model be Tested?

In order to test any model, it has to be over-identified. This means that there

is a unique solution to the mathematical procedure which results in estimates

for all the parameters that are allowed to vary freely in the model, and that

there is at least one degree of freedom available to test this model using chi

squared. If there are p measured variables, then the total number of degrees

of freedom is, p*

= p ( p + 1) / 2. The number of parameters that need to be

estimated must, therefore, be less than this number.

However, if the model describes relationships among latent variables as well

as the relationships among these factors and measured variables, then the

SEM program may still not be able to converge on a solution. This is because

both the structural model and the measurement model must be over-identified

in order to test the entire models goodness-of-fit.

Once p*

has been calculated, the next step is to establish whether the

measurement model is likely to be over-identified. If there is one latent

factor in the model, there needs to be three variables measuring this construct

and their errors must be uncorrelated. If there are two or more latent factors,

the same conditions apply provided that each set of three measured variables

only load on one factor and that the factors are allowed to covary. Sometimes

two indicators per factor is sufficient under these conditions provided that

none of the variances and covariances among the factors are zero.

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Note that, in order for the factor to have meaning, one of the measured

variables is used to scale the latent variable by setting the path coefficient to 1

(the variance of the latent variable is the same as the measured variable).

This is called a marker variable. Failure to set the scale of a factor is one

common error which results in identification problems.

Looking at whether the structural model is over-identified is the next step in

this process. Provided there is only one latent variable or that the latent

variables are recursive and their disturbances do not correlate, this part of the

model is likely to be identified.

Notice that the phrase likely to be over-identified is used when discussing

both the measurement model and the structural model. This is because the

guidelines just reviewed do not guarantee that a particular model can be

tested. Establishing this with certainty is complex, so perhaps the best

strategy is to apply these guidelines and then run the analysis. The EQS

program signals when this problem has arisen by indicating that some

parameters are linearly dependent on other parameters.

(NOTE: Dun et al. (1993) suggest using p*

= p (p + 1) / 2 and then running the

analysis to see if problems arise. If they do, more parameters can be given

fixed values to deal with the problem.)

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Using Parcels of Measured Variables When the

Sample Size is Small

Sometimes applied researchers are forced to do structural equation modeling

with a relatively small N (less than 200). In this instance, the researcher must

balance the need to include enough measured variables to adequately specify

the measurement model with the need to restrict the number of parameters

estimated by the model as a whole. The solution is to create a small number of

parcels made up by averaging the responses to several of the original

measured variables (e.g., questionnaire items) with the minimum of three

parcels per latent variablethe number usually required for any SEM

program to run properly.

Parcels of items (measured variables) are constructed for each latent

variable using the following item-to-construct balance method:

Consider parceling measured variables (items) measuring a construct into

three parcels. First a factor analysis is done on all the items measuring the

construct. Then the three items with the highest factor loadings are used to

anchor the three parcels. The three items with the next highest loadings are

added to the anchors in reverse order, and so on. Together these three parcels

are used as the manifest variables in the SEM analysis rather than the original

items (see Little, Cunningham, Shahar, & Widaman, 2002). This reduces the

number of parameters that the SEM program needs to estimate for the

measurement model. However, this should only be done if N is small (usually

< 200). Otherwise it is better to use the original measured variables in the

analysis as multiple indicators of the construct.

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EXAMPLE OF A SIMPLE CONFIRMATORY FACTOR ANALYSIS

USING THE EQS PROGRAM

Confirmatory Factor Analysis is a factor analytic technique that is design to

test theory that specifies an underlying structure to a construct. Instead of

discovering the underlying factor structure in a post hoc fashion through the

exploratory factor analysis techniques covered earlier in this course, the

function of this type of factor analysis is to confirm that the theorized factor

structure underlying a construct is plausible. When SEM is used purely for

confirmatory factor analysis, the theory that defines a construct in a certain

way is tested, but the relationship of that construct to other constructs is not

explored. This means that SEM is used to test a measurement model.

Self-concept is a complex multi-dimensional psychological construct that

psychologists have been interested in since our discipline began. In the

example, a two factor theory of academic self-concept is specified which

suggests that it is comprised of two underlying and interrelated components

reflecting different aspects of the self: English self-concept (ESC) and maths

self-concept (MSC). Each of these components can be measured in several

ways and that the responses to these measures are caused by these two factors

in the manner specified by the path diagram on the next page. SEM tests this

overall model as well as the specified causal paths which represent hypotheses

derived from this academic self-concept theory. The study is a secondary

analysis of published data summarizing the responses of 996 adolescents to a

self-concept test battery. The authors of the study have provided the

variance-covariance matrix and so EQS is used to analyse the data in this

matrix.

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The Path Diagram Specified by Theory

E12*

V3

V9

V10

ESC*

E3*

E9*

E10*

V4

V11

V12

MSC*

E4*

E11*

1.0

1.0

*1.0

*

1.0

1.0

1.0

*

1.0

*

1.0

*

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The EQS Syntax File ( *.EQS)

/TITLE

Self-concept: Confirmatory Factor Analysis Example

The ti tle statement can be several l ines long and can contain explanatory notes

on the decisions that r esul ted in the syntax being used.

/SPECIFICATIONS

VARIABLES= 12; CASES= 996;

DATAFILE='c:\data\Eqs\807 2004\SEM notes.CFA example.byrne.ess';

MATRIX=COR;

ANALYSIS = COV;

METHOD=ML;

The specif ications commands give the computer details on the number of

vari ables in the data set (VAR), the sample size (CASE),the location of the data

matri x (DATAF I LE -- a * .ESS is a EQS data fi le), the type of data matr ix being

analysed (MATRIX = COR or COV); the basis for the analysis (ANALYSIS =

COR or COV), with the default being the variancecovariance matri x, and the

iterative estimation procedure (METHOD). Al l subcommands are separated by

semi -colons (a general EQS syntax rule except for data matrices).

The data is in the form of a correlation matri x with standard deviationsV1 V2 V3 V4 V5 V6 V7 V8 V9 V10 V11 V12

1.0000 0.0710 0.2890 0.1700 0.8630 0.4786 0.2522 0.2160 0.1560 0.1280 0.1770 0.1350

0.0710 1.0000 0.2450 0.2530 0.2660 0.3060 0.2619 0.7675 0.2420 0.3070 0.2475 0.3040

0.2890 0.2450 1.0000 0.0120 0.2270 0.2990 0.2389 0.3430 0.7050 0.8543 0.0660 0.0270

0.1700 0.2530 0.0120 1.0000 0.2000 0.2250 0.3460 0.3472 0.0140 0.0690 0.8640 0.8280

0.8630 0.2660 0.2270 0.2000 1.0000 0.8310 0.7100 0.2160 0.1900 0.1310 0.2700 0.1880

0.4786 0.3060 0.2990 0.2250 0.8310 1.0000 0.2537 0.2830 0.2100 0.1740 0.2570 0.1870

0.2522 0.2619 0.2389 0.3460 0.7100 0.2537 1.0000 0.2545 0.1440 0.1396 0.2426 0.0367

0.2160 0.7675 0.3430 0.3472 0.2160 0.2830 0.2545 1.0000 0.2690 0.2900 0.2489 0.3057

0.1560 0.2420 0.7050 0.0140 0.1900 0.2100 0.1440 0.2690 1.0000 0.7627 0.1420 0.0280

0.1280 0.3070 0.8543 0.0690 0.1310 0.1740 0.1396 0.2900 0.7627 1.0000 0.0960 0.14600.1770 0.2475 0.0660 0.8640 0.2700 0.2570 0.2426 0.2489 0.1420 0.0960 1.0000 0.8060

0.1350 0.3040 0.0270 0.8280 0.1880 0.1870 0.0367 0.3057 0.0280 0.1460 0.8060 1.0000

The second last row is the standard deviations of the vari ables

14.1000 12.3000 10.0000 16.1000 9.3000 14.9000 9.4000 15.3000 11.3000 15.7000 11.5000 12.4000

The last row contains the means which, in thi s example, are set to zero.

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EQS Syntax (Continued)

/LABELSV3 = ESC1; V4 = MSC1;

V9 = ESC2; V10 = ESC3; V11 = MSC2; V12 = MSC3;

F1 = ESC; F2 = MSC;

These commands give more meaningful variable labels than those used by the

computer. The syntax also helps you wr ite the equations which need to use the

computer labels. Dont forget the semi -colons.

/EQUATIONSV3 = F1 + E3;

V9 = *F1 + E9;

V10 = *F1 + E10;

V4 = F2 + E4;

V11 = *F2 + E11;

V12 = *F2 + E12;

These equations specify the model that is being tested. Each equation is

separated by a semi-colon. I f you run EQS using a path diagram (EQS

diagrammer f unction), these equations wil l be generated fr om the path diagramautomatically.

/VARI ANCES

F1 TO F2 = *;

E3 TO E4 = *; E9 TO E12 = *;

/COVARIANCES

F1 TO F2 = *;

/END

These commands specify the matr ix of var iances and covariances. Notice thatthe covari ation among the error terms are not specif ied implying that they are set

to zero (the defaul t). The /END statement tell s the computer to begin the

analysis.

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The EQS Output File (*.OUT)

The f irst page of the output repeats the syntax that was just presented.

TITLE: Self-concept: Confirmatory Factor Analysis Example

COVARIANCE MATRIX TO BE ANALYZED: 6 VARIABLES (SELECTED

FROM 12 VARIABLES) BASED ON 996 CASES.

This line reminds the researcher that the model contains 6 of the original 12

variables in the data set. Successive runs of EQS could specify dif ferent subsets

of data for dif ferent analyses.

Then the output gives the enti re variance-covar iance matr ix among the variables

used in the analysis.

ESC1 MSC1 ESC2 ESC3 MSC2

V 3 V 4 V 9 V 10 V 11

ESC1 V 3 100.000

MSC1 V 4 1.932 259.210

ESC2 V 9 79.665 2.547 127.690

ESC3 V 10 134.125 17.441 135.311 246.490

MSC2 V 11 7.590 159.970 18.453 17.333 132.250

MSC3 V 12 3.348 165.302 3.923 28.423 114.936

MSC3

V 12

MSC3 V 12 153.760

I f a correlation matr ix was analysed, the computer reminds the researcher

because this is not a good idea.

CORRELATI ON MATRIX TO BE ANALYZED:

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BENTLER-WEEKS STRUCTURAL REPRESENTATION:

NUMBER OF DEPENDENT VARIABLES = 6

DEPENDENT V'S : 3 4 9 10 11 12

Here the endogenous (dependent) variables in the model are identified.

NUMBER OF INDEPENDENT VARIABLES = 8

INDEPENDENT F'S : 1 2

INDEPENDENT E'S : 3 4 9 10 11 12

Here the exogenous (independent) variables in the model are identified.

NUMBER OF FREE PARAMETERS = 13

This is the number of parameters the researcher is estimating in thi s analysis

(stars in the equations plus the stars in the vari ancecovar iance matrix , or

equivalently the number of stars on the path diagram).

The total number of degrees of freedom in this data set is calculated by the

formula p*

= p (p + 1) / 2 where p is the number of measured variables. In

this example, there are 6 measured variables, so the total number of degrees

of freedom are 6 x 7 / 2 = 21. As the researcher wishes to estimate 13

parameters, the degrees of freedom remaining that can be used to test the

goodness-of-fit of the model is 21 - 13 = 8.

DETERMINANT OF INPUT MATRIX IS 0.99223E+11.

Clearl y mul ticoll ineari ty is not a problem in this data set.

AVERAGE ABSOLUTE STANDARDIZED RESIDUALS = 0.0182

AVERAGE OFF-DIAGONAL STANDARDIZED RESIDUALS = 0.0255

The computer then pri nts out the residual variancecovariance matri x and the

standardized residual var iancecovar iance matri x (not shown). Following each

matri x is an average of al l the residuals and all the off diagonal residuals. These

averages should be small if the model f its the data well . The average of the off

diagonal residuals is given because smal l residual covariance values are more

crucial for the model to be a good fi t.

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LARGEST STANDARDIZED RESIDUALS:

V11, V9 V12, V10 V4, V3 V 9, V4 V12, V3

0.076 0.069 -0.064 -0.054 -0.044

The computer then pr ints out the 20 largest residual values (thi s is an extract) sothat the researcher knows which relationshi ps are not modelled very well . For

example, the fi rst and largest residual in th is table shows that the model does

not explain a relationship between an index of Engl ish self concept (V9) and

math self -concept (V11) as well as other relationships in the samples correlation

matri x. Whether the researcher wil l use this information or not depends on the

overall f it of the model and the size of these residual covariances (look at

standardized residual s > .10).

DISTRIBUTION OF STANDARDIZED RESIDUALS

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

! !

20- -

! !

! !

! !

! ! RANGE FREQ PERCENT

15- -

! ! 1 -0.5 - -- 0 .00%

! ! 2 -0.4 - -0.5 0 .00%

! * ! 3 -0.3 - -0.4 0 .00%! * ! 4 -0.2 - -0.3 0 .00%

10- * - 5 -0.1 - -0.2 0 .00%

! * * ! 6 0.0 - -0.1 12 57.14%

! * * ! 7 0.1 - 0.0 9 42.86%

! * * ! 8 0.2 - 0.1 0 .00%

! * * ! 9 0.3 - 0.2 0 .00%

5- * * - A 0.4 - 0.3 0 .00%

! * * ! B 0.5 - 0.4 0 .00%

! * * ! C ++ - 0.5 0 .00%

! * * ! -------------------------------

! * * ! TOTAL 21 100.00%----------------------------------------

1 2 3 4 5 6 7 8 9 A B C EACH "*" REPRESENTS 1 RESIDUALS

Th is hi stogram shows that the residuals are centred on zero (100% are between

0.1 and 0.1) and are symmetr ical. This information indicates that the model

does not contain a ser ious specif ication error.

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GOODNESS OF FIT SUMMARY

INDEPENDENCE MODEL CHI-SQUARE = 5093.587 ON 15

DEGREES OF FREEDOM

This first chi square test should be signif icant as it test the hypothesis that thevari ables are independent of one another (one of the standards of comparison

used by some of the comparative goodness-of -f it indices).

CHI-SQUARE = 266.589 BASED ON 8 DEGREES OF FREEDOM

PROBABILITY IS LESS THAN 0.000001

This is the basic chi square value that tests the goodness-of-f i t of the model.

Because the sample size is large (N = 996), the fact that th is statistic i s

signif icant does not mean that the model i s a poor fi t.

BENTLER-BONETT NORMED FIT INDEX= 0.948

BENTLER-BONETT NONNORMED FIT INDEX= 0.905

COMPARATIVE FIT INDEX (CFI) = 0.949

These fit i ndices and parti cularly the CFI suggest that the model i s qui te a good

f it as they are all around the 0.9. The CFI should be greater than 0.95 for the

model to be a good fi t.

ITERATIVE SUMMARY

PARAMETER

ITERATION ABS CHANGE ALPHA FUNCTION

1 69.848500 1.00000 .48517

2 5.597344 1.00000 .28123

3 .887007 1.00000 .26795

4 .069267 1.00000 .26793

5 .012263 1.00000 .26793

6 .000785 1.00000 .26793

This output shows how the function specif ied by the estimation method

converges on a minimum value. Noti ce that the chi square testing the goodness-

of-f it of the model is equal to the minimum function value mul tipl ied by (N-1):

0.26793 x 995 = 266.589 (with in rounding error).

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The computer then wr ites the equations for the endogenous variables with the

estimated parameters.

MEASUREMENT EQUATIONS WITH STANDARD ERRORS AND TEST

STATISTICS

ESC1 =V3 = 1.000 F1 + 1.000 E3

MSC1 =V4 = 1.000 F2 + 1.000 E4

These fir st set of equations are the ones in whi ch the paths of one of the

indicators (a marker) for the four factors was fi xed at a value of 1. This gives

the underlying factor the same scale as the marker variable.

ESC2 =V9 = 1.010*F1 + 1.000 E9

.031

32.704@

This equation shows that the unstandardized path coeff icient between the

Engli sh self -concept factor and the specif ic measure of Engl ish self -concept

(ESC2) is 1.010. The standard error for th is statistic is 0.031 and the Z score is

1.010 / .031 = 32.704. As this Z score is greater than 1.96, it is signi fi cant as

indicated by @.

ESC3 =V10 = 1.705*F1 + 1.000 E10.040

42.362@

MSC2 =V11 = .696*F2 + 1.000 E11

.014

49.339@

MSC3 =V12 = .720*F2 + 1.000 E12

.016

44.579@

Sometimes, one estimated parameter is linearly dependent upon the others. The

computer wil l run the analysis, but wil l print out an error message concerning

this linear dependency. Do NOT tr ust the output if thi s message appears. A less

extr eme form of li near dependence is shown by a parameter estimate having a

very smal l standard error (relative to standard errors found in past research).

Consider eliminating thi s var iable and rerunning the analysis.

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The computer then pr ints the estimated variances and covariances of the

exogenous variables with standard errors and Z scores. The estimates for the

measur ed var iables shou ld be inspected to see if they seem reasonable given your

knowledge of these variances and covari ances from past research. Sometimes

the estimation procedure produces an odd solution that does not conform to the

f indings fr om past research. I n this instance, it is probably wise to give more

credence to the resul ts of past research until the resul ts of the analysis are

replicated.

VARIANCES OF INDEPENDENT VARIABLES (EXTRACT)

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

I F1 - ESC 78.690*I

I 4.539 I

I 17.338@I

I E3 - ESC1 21.310*I I

1.524 I I

13.980@I I

As an extr eme, the SEM program can estimate negative variances for measured

vari ables. This is fl agged by the computer with an error message saying that

these variances can not be estimated. I n this instance, the anal ysis is seriously

f lawed and the researcher wil l need to reassess whether the use of SEM to

analyse the data set is warranted. As well, the estimated variances of the latent

exogenous factors can be negative. The computer wil l not allow this to happen

and wil l constrain the variance to zero (or a lower bound estimate that is

posi tive). I f you see thi s error message, ser iously question the validity of the

results.

mailto:17.338@Imailto:17.338@I

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STANDARDIZED SOLUTION: R-SQUARED

ESC1 =V3 = .887 F1 + .462 E3 .787

MSC1 =V4 = .941 F2 + .338 E4 .886ESC2 = V9 = .793*F1 + .610 E9 .628

ESC3 = V10 = .963*F1 + .269 E10 .928MSC2 =V11 = .918*F2 + .397 E11 .842

MSC3 =V12 = .879*F2 + .476 E12 .773

These standardized path coeff icients from the measur ed vari ables to the latent

factors are the ones that are most usuall y wri tten onto the path diagram in

publ ished reports (they are calculated fr om the unstandardized path coeff icientsafter the analysis is completed). Note that thi s table can not be calculated if some

of the estimated var iances for the exogenous variables are negative.

I n this example, the standardized path coeff icients are the factor loadings of the

measured var iables on the latent factors. The squared mul tiple correlati ons (the

square of the path coeff icient) are estimates of the proportion of the var iance of

the measur ed var iables which is shared with the underlying factor. Thi s is a

commonali ty estimate for the variable on the factor and, equivalentl y, an

estimate of i ts reli abili ty. For example, the reli abili ty of the Math Self -Concept

Scale (V12) is .773. This number indicates the proportion of variance in the

measured variable that measures the under lying Maths Self -concept construct

(F2). (This number is NOT a good reli abil ity estimate if the error terms for the

measur ed variables are correlated. )

Notice that in thi s table the standardized path coeff icients for the marker

vari ables (and the error terms) that were fixed in the equations now have a value

dif ferent f rom 1 due to the standardization procedure. I f you need their statistical

signif icance, set another measur ed var iable as the marker and rerun the

analysis. This wil l give you the same value for the path coeff icients because thetwo solu tions are equivalent. The standardized variances are not printed as the

computer sets all variances equal to 1.

The corr elations among the latent factors is given in the last table of the output.

I n th is case it is 0.091 indicating that English and Math Academic Self -Concept

are relatively independent of one another.

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Additional Syntax

The following syntax allows you to output additional goodness-of-fit indices.

/PRINT

FIT = ALL;

For this example, some of the output from this command is:

ROOT MEAN SQUARED RESIDUAL (RMR) = 5.133

STANDARDIZED RMR = 0.032

ROOT MEAN SQ. ERROR OF APP.(RMSEA)= 0.180

90% CONFIDENCE INTERVAL OF RMSEA ( 0.162, 0.199)

Post Hoc Adjustment of the Theoretical Model:

Addition and Subtraction of Parameters

In the above example, the value of the goodness-of-fit indices suggest that the

model is a good fit because they meet the established criteria. However, in

some analyses the solution looks quite good but could be improved (the fit

indices approach the criteria for a good fit, but do not meet these criteria). In

this situation, the researcher can conduct post hoc tests which suggest which

parameters should be estimated rather than fixed and which parameters can

be removed (set to zero). Then the modified model can and should be tested

on a new sample. These post hoc procedures capitalize on chance and so

Tabachnick and Fidell suggest only adding or subtracting a few paths one at a

time. As well they advocate using a conservative significance level (p < .01)

for selecting modifications to the parameters specified by theory.

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Lagrange Multiplier Test

This post hoc procedure indicates which parameters could be added to the

model (estimated) to improve its goodness-of-fit based upon the current

sample. Both univariate and multivariate tests are conducted but the

multivariate test is the more important one as it identifies the parameters that

could be added into the model in a stepwise fashion similar to forward

selection in multiple regression. To run this procedure use the syntax:

/LMTEST.

Using this syntax in the context of the current example yields the following

output:

MULTIVARIATE LAGRANGE MULTIPLIER TEST BY

SIMULTANEOUS PROCESS IN STAGE 1

PARAMETER SETS (SUBMATRICES) ACTIVE AT THIS STAGE ARE:

PVV PFV PFF PDD GVV GVF GFV GFF BVF BFF

This line indicates the type of parameter matr ices that were active at this stage of

the analysis. The f irst letter i ndicates the matri x containing the suggested

parameter (P = ; G =; and B = B) and the remaining letters indicate thetype of var iables involved (V = measur ed variables, F = factors, E = errors, and

D = disturbances).

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CUMULATIVE MULTIVARIATE STATISTICS UNIVARIATE INCREMENT

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

HANCOCK'S

SEQUENTIAL

STEP PARAMETER CHI-SQUARE D.F. PROB. CHI-SQUARE PROB. D.F. PROB.

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

1 V4,F1 14.601 1 .000 14.601 .000 8 .067

2 V3,F2 25.807 2 .000 11.206 .001 7 .130

This part of the output indicates the possible changes along with a2test which,if signif icant, indicates that the model wil l be improved. For example the

analysis suggests adding a path fr om F1 (the Engl ish self -concept factor to V4

(supposedly a measure of M ath self -concept). Whil e statisticall y this makes

sense, theoretically it may not. I ndeed, whether the researcher actuall y makes

this change and recomputes the model depends on a thoughtful analysis of the

theoretical impli cations. One or two theoreticall y meaningful changes may well

improve the model suf f iciently to become a good fi t and th is strategy is usual ly

better than implementing all the changes without regard for theory.

Whenever changes are made, it is important to rerun the analysis on the

modif ied model so as to check on its goodness-of-f it and to examine the impact

of the changes on all the parameter estimates. I ndeed, if the modif ications

resul t i n parameter estimates that are not consistent with past research, the

researcher may decide that these modif ications are not worth making at all .

Af ter all , this is a post hoc procedure relying on purely statistical cr iteria. There

is no guarantee that the changes it suggests are changes that improveunderstand of the phenomena under study.

The Wald Test

This post hoc procedure is used to delete parameters (set them to zero)

and so make the model more restrictive. It is usually done after parameters

have been added using the Legrange test as adding paths changes the

parameter estimates. The following syntax is used to activate this procedure:

/WTEST

Because parameters are being set to zero, results of the 2 test should be non-significant. The output is similar to the Lagrange test. In this instance, no

paths were dropped when this test was conducted.

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Comparing Nested Models: The Chi Square Difference Test

Sometimes different theories (or different versions of the same theories)

specify two models such that one model is nested inside the other. In this

instance the two models can be directly compared to see if the larger model

(the one with more paths) significantly improves the goodness-of-fit (or,

equivalently, if the added restrictions significantly reduces the goodness-of-

fit). This comparison is achieved by subtracting the 2 values for the twomodels. The result is another 2 statistic with degrees of freedom equal to thedifference in the degrees of freedom for the two models. This procedure

requires the estimation of two models, but its advantage is that it is theory-

based and provides evidence that directly bears upon the relative merits of the

two theories. In my view, this is a whole lot better than fixing a model in a

post hoc fashion using the Legrange Multiplier and/or the Wald tests.

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EXAMPLE OF TESTING A CAUSAL MODEL

USING THE EQS PROGRAM

This path diagram specif ies a simple theoretical model of job satisfaction (an

endogenous latent variable) which was tested on 122 employees in an i ndustr ial

sales force. The exogenous latent variables in this model are achievement

moti vation and self-esteem.

SAT1

SAT2

ACH1

ACH2

SE1

SE2

F1

ACH*

SE*

1.0

E4*

*E5*

1.0

1.0E6*1.0

*

E7*1.0

*

E3*1.0

1.0

E2*1.0*

D1*

1.0

*

*

1.01.0

*

1.0

1.0

*

1.0

*

1.0

1.01.0*

1.0

*

*

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The syntax used to test this theoretical model is derived directly from the

path diagram:

/TITLE

PERFORMANCE AND JOB SATISFACTION IN AN INDUSTRIAL SALES

FORCE

/SPECIFICATIONS

CASES = 122; VARIABLES = 8; MATRIX=CORRELATION;

ANALYSIS=COVARIANCE; METHOD = ML;

The number of parameters being estimated is 15, so the sample size (CASES =

122) is a li ttle smal l (i t should be at least 15 x 10 = 150).

The ANALYSIS command specif ies that the variance-covariance matri x should

be analysed. This matr ix is created fr om the corr elation matri x and the standard

deviations of the variables contained in the command statement /STA below.

/LABELS

V2 = SAT1; V3 = SAT2; V4 = ACH1;

V5 = ACH2; V6 = SE1; V7 = SE2;

F1 = JOBSAT; F2 = ACH; F3 = SE;

/EQUATIONS

V2 = F1 + E2;V3 = *F1 + E3;

V4 = F2 + E4;

V5 = *F2 + E5;

V6 = F3 + E6;

V7 = *F3 + E7;

F1 = *F2 + *F3 + D1;

The start values can be specif ied (f rom past research) for some or all of the

parameters in these equations (they are given as numbers to the left of the stars.

e.g., 0.5*F2). Specifying these start values is more crucial if you have a smal l

sample size.

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/VARIANCES

F2 TO F3 = *;

E2 TO E7 = *;D1 = *;

/COVARIANCES

F2,F3 = *;

These sets of statements specif y the var iances and covariances for the

exogenous variables. Covari ances are set to zero by defaul t, so it is onl y

necessary to state that the covar iance between F2 and F3 needs to be estimated.

/MATRIX

1.000

.418 1.000

.394 .627 1.000

.129 .202 .266 1.000

.189 .284 .208 .365 1.000

.544 .281 .324 .201 .161 1.000

.507 .225 .314 .172 .174 .546 1.000

-.357 -.156 -.038 -.199 -.277 -.294 -.174 1.000

/STANDARD DEVIATIONS

2.09 3.43 2.81 1.95 2.08 2.16 2.06 3.65

This is the way a matrix of correlations with standard deviations (so as to

create a covariance matrix for the computer to analyze) is specified in the

command file.

/END

This statement ends the commands and tells the computer to begin the

analysis.

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The EQS Output

The output starts out by repeating the syntax. Then the variancecovariancematr ix among the measur ed variables is given, followed by:

BENTLER-WEEKS STRUCTURAL REPRESENTATION:

NUMBER OF DEPENDENT VARIABLES = 7

DEPENDENT V'S : 2 3 4 5 6 7

DEPENDENT F'S : 1

NUMBER OF INDEPENDENT VARIABLES = 9

INDEPENDENT F'S : 2 3

INDEPENDENT E'S : 2 3 4 5 6 7

INDEPENDENT D'S : 1

NUMBER OF FREE PARAMETERS = 15

NUMBER OF FIXED NONZERO PARAMETERS = 10

The number of degrees of f reedom are p (p + 1) / 2 = 6 x 7 / 2 = 21. Therefore,

the degrees of freedom that can be used to test the goodness-of-f i t of the model is21 - 15 = 6.

DETERMINANT OF INPUT MATRIX IS 0.82146E+04

This shows that there is no problem with multi coll ineari ty.

The computer then pri nts out the residual variancecovariance matri x and the

standardized residual matr ix . The summary of the values in the standardized

matr ix shows that the residuals are smal l i ndicating that the model f i ts the data

well:

AVERAGE ABSOLUTE STANDARDIZED RESIDUALS = 0.0113

AVERAGE OFF-DIAGONAL STANDARDIZED RESIDUALS = 0.0159

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The hi stogram of the residuals show that they are smal l and centred around zero

(over 97% li e in the range 0.1 to0.1).

GOODNESS OF FIT SUMMARY

CHI-SQUARE = 3.915 BASED ON 6 DEGREES OF FREEDOM

PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS 0.68813

COMPARATIVE FIT INDEX (CFI) = 1.000

This part of the output shows that the model i s a very good fi t and the chi square

is not signi f icant. The CFI also shows this and is the statistic to report given the

small sample size.

The computer then wr ites the equations for the endogenous variables with the

parameter estimates and their signif icance.

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MEASUREMENT EQUATIONS WITH STANDARD ERRORS AND

TEST STATISTICS

SAT1=V2 = 1.000 F1 + 1.000 E2

SAT2=V3 = .929*F1 + 1.000 E3

.189

4.931@

ACH1=V4 = 1.000 F2 + 1.000 E4

ACH2=V5 = 1.006*F2 + 1.000 E5

.361

2.784@

SE1=V6 = 1.000 F3 + 1.000 E6

SE2=V7 = .879*F3 + 1.000 E7

.222

3.965

JOBSAT=F1 = .733*F2 + .547*F3 + 1.000 D1

.376 .234

1.949@ 2.335@

The output also estimates the variance and covariances of the exogenous

vari ables, including the corr elation between F2 and F3 as 0.396.

STANDARDIZED SOLUTION: R-SQUARED

SAT1= V2 = .743 F1 + .669 E2 .553

SAT2= V3 = .843*F1 + .537 E3 .711

ACH1=V4 = .622 F2 + .783 E4 .387

ACH2=V5 = .587*F2 + .810 E5 .344

SE1= V6 = .770 F3 + .638 E6 .593

SE2= V7 = .709*F3 + .705 E7 .503

JOBSAT= F1 = .349*F2 + .357*F3 + .808 D1 .347

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Fur ther Comments on Structural Equation M odell ing with EQS

The Equivalency of Solutions Using Different Marker Variables

If there is more than one latent factor in your path diagram, then changing

the variable used as the marker variable for a factor results in an equivalent

solution (the size of the path coefficients are the same). However, if one

manifest variable is a much better index of the construct than the others, it is

best to use it as the marker variable as the solution tends to be more stable.

Setting the variance of exogenous factors to 1.0 (standardizing the variance)

rather than one of the paths to a marker variable is another option if you do

not want to specify a variable as a marker variable. This also results in an

equivalent solution.

The bottom line is that your choice of marker variable or your choice to set

the variance of a exogenous factor to 1.0 and not specify a marker variable

depends upon what you are interested in theoretically. For example, if you

want to scale a factor to a well known and highly reliable instrument, then you

should make this measured variable your marker variable. If all measuresare equivalent (and perhaps of unknown reliabilitye.g., face valid measures

of the construct) and you are not concerned with scaling a latent exogenous

factor, but rather want to know how all the measures load on this factor, then

set the variance of this exogenous factor to 1.0.

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Including Dichotomous Variables in the Theoretical Model

Truly nominal variables define different groups of respondents withoutranking them (see TF, section 14.5.7, p. 730). Provided the sample size is large

enough, the SEM strategy is to test the theoretical model within each group

separately. If the model is supported in both samples, then your results

generalize across samples. For example, the model is supported for both men

and women; within white, immigrant, and Aboriginal samples; etc

A more advanced form of SEM not covered in this course or TF, starts by

testing the model within each sample and then doing a multiple group analysis

in order, for example, to test the invariance of the factorial structure of a

theoretical construct across groups. Simply put, the analysis constrains

certain parameters within each group to be equal (e.g., the size of the path

from a measured variable to a latent construct is the same for both men and

women) and examines whether the goodness-of-fit is still as good as the

goodness-of-fit of the model when these constraints are not applied.

If the sample size is small, you can include a nominal variable in the path

diagram as one or more dummy variables. Clearly these dummy variablesare not normally distributed so you have to use a robust estimation procedure.

If you are using ordinal data which reflects a underlying continuous variable

(e.g., age: 1 = young, 2 = middle aged, 3 = old), you must estimate the size of

the correlations that would have been obtained if you had actually measured

the continuous variable directly. These estimates are called polychoric

correlations (between two ordinal variables) or polyserial correlations

(between an ordinal and an interval variable) and they form the basis of the

analysis. This goes far beyond the scope of this course and is only briefly

mentioned in TF (section 14.5.6, p. 734).

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Multivariate Kurtosis

If you run EQS using raw data, the program will print out Mardias

coefficient which indicates multivariate kurtosis (use p < .001 to determine if

kurtosis is a problem). The normalized estimate for this coefficient allows you

to see if the variables in the data set are normally distributed or not. For

large sample sizes, the values of the normalized coefficient correspond to Z

values and so large values indicate some non-normality is present due to

kurtosis.

The computer also prints out the 5 cases that have the largest normalized

estimate (and which contribute the most to the overall value of Mardias

coefficient). If one or two of these cases have much larger normalized

estimate values than the others, consider dropping these cases and re-running

the analysis. However, it is better not to drop these cases immediately, but to

check the variables for univariate and multivariate outliers and to adjust the

variables that have a non-normal distribution using transformations. This is

usually done with the SPSS program. EQS reads SPSS data files and converts

them into ***.ess files with ease.