introduction to cfa sem sept 19 2k5

8/12/2019 Introduction to CFA SEM Sept 19 2k5

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CFA SEM:Modeling Causal Processes

BUSI6280


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The term structural equation modeling

conveys two key aspects of the procedure:

That the causal processes under study are

represented by a series of structural (i.e.,regression ) equations, That these structural relations can be

modeled pictorially to enable a clearerconceptualization of the theory understudy.


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Why Use SEM ?

SEM lends itself well to the analysis of data forinferential purposes.

Whereas, traditional multivariate procedures areincapable of either assessing or correcting formeasurement error, SEM provides explicitesti mates of these parameters.

SEM procedures can incorporate both unobserved

(i .e. latent) and observed var iables.


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The factor analytic model (EFA or CFA) focuses solelyon how the observed variables are l inked to theirunder lying latent f actors .

Factor analysis is concerned with the extent to which

the observed variables are generated by the underlyinglatent constructs and thus str ength of the regression paths f rom the factors to the observed var iables (the factorloadings) are of primary interest.

Although inter-factor relations are also of interest, anyregression str ucture among them is not considered in thefactor analytic model.

Purpose of Factor Analysis


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Measurement model: latent variables and their observedmeasures (i.e., the CFA model)

Structural model: Model with links among the latentvariables.

Full (Complete) Model : a measurement model and astructural model

Recursive model: Direction of cause is from one direction

only Non-recursive model: reciprocal or feedback effects (oftendifferent from one another).

Type of Models


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Three different scenarios or models

(Jreskog 1993)

Strictly Confirmatory (SC)

Alternative Models (AM) Model Generating (MG)


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The SC Scenario

The researcher postulates a single modelbased on theory , collects the appropriatedata, and then tests the fit of thehypothesized model to the sample data.

From the results of this test, the researcher

either rejects or fails to reject the model. Nofur ther modif ications to the model are made.


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The AM Scenario

The researcher proposes several alternative(i.e., competing) models , all of which are

grounded in theory .

Following analysis of a single set of empirical

data, the researcher selects one model asmost appropr iate in representing the sampledata.


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The MG Scenario

The researcher, having postulated and rejected atheoretically derived model on the basis of its poor fit tothe sample data, proceeds in an exploratory (rather thanconfirmatory) fashion to modify and re-estimate themodel.

The primary focus here is to locate the source of misfitin the model. Jreskog noted that, althoughrespecification may be either theory- or data-driven, theul timate objective is to f ind a model that is both substantivelymeaningful and statistically well f i tting.


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SEM procedures alternative computer programs

AMOS-Arbuckle, 1995 EQS-Bentler, 1995 LISCOMP-Muthn, 1998

CALIS-SAS Institute, 1992 RAMONA-Browne, Mels, & Coward, 1994 SEPATH-Steiger, 1994

LISREL program is the most widely used, 1970s


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Exogenous latent variables are synonymous with independentvariables; they cause fluctuations in the values of otherlatent variables in the model.

Changes in the values of exogenous variables are not

explained by the model. Rather, they are considered to beinfluenced by other factors external to the model.

Endogenous latent variables are synonymous with dependentvariables and, as such, are influenced by the exogenous

variables in the model, either directly, or indirectly.

SEM - Language


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By convention, observed measures are represented by Roman

letters and latent constructs by Greek letters:

Those that are exogenous are termed X-variables . Those that are endogenous are termed Y-variables .

The measurement model may be specified either in termsof LISREL exogenous notation (i.e., X-variables), or interms of its endogenous notation (i.e., Y-variables).

The exogenous latent constructs are termed as (xi).

The endogenous latent constructs are termed as (eta).

SEM - Language


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SEM LanguageThe Measurement Model

x is a q x 1 vector of observed exogenous variables y is a p x 1 vector of observed endogenous variables. is an n x 1 vector of latent exogenous variables is an m x 1 vector of latent endogenous variables. is a q x n matrix of coefficients ( ij) linking x and

. is a q x 1 vector of random disturbance term

(errors of measurement) associated with x vector. is a p x 1 vector of random disturbance term

(errors of measurement) associated with y vector.


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SEM LanguageThe Structural Model

(gamma) is an m x n matrix of coefficients ( ij ) thatrelates the n exogenous factors to the m endogenous

factors .

B(beta) is an m x m matrix of coefficients ( ij ) that relatesthe m endogenous factors to one another.

(zeta) is an m x 1 vector of residuals ( i) representingerrors in the equation relating and .

(phi) is an n x n matrix of coefficients ( ij) thatcaptures the variance/covariance between s. (psi) is the m x m matrix of covariance between s.


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The Structural Model

Measurement Model for the X-variables (1):x= x+

Measurement Model for the Y-variables (2):y= y+

Structural Equation Model (3):=B + +


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The following minimal assumptions are presumed to hold for the system of equations

is uncorrelated with (construct) is uncorrelated with (construct) is uncorrelated with and (construct) , , and are mutually uncorrelated.

E() = 0 E() = 0 E() = 0 E() = 0

E() = 0 (I-B) is nonsingular so that (I-B) exists. This makes theequation 3 to be written in the reduced form.


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Symbol Representation

Unobserved (latent) Factors Observed Variable Path coefficient for regression of

observed variable on unobservedfactors

Path coefficient for regression ofone factor on another.

Residual error (disturbance) in prediction of unobserved factors

Measurement error associatedwith observed variable.


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Summary of Matrices, Greek Notation, and Programs CodesMatrix Program Matrix

Greek Letter Matrix Element Code Type

Measurement Model

Lambda-X x x LX Regression Lambda-Y y y LY Regression Theta delta Q TD Var/cov Theta epsilon Q e e TE Var/cov

Structural Model Gamma GA Regression Beta BE Regression Phi PH Var/cov

Psi PS Var/cov Xi (or Ksi) --- --- Vector Eta --- --- Vector Zeta --- --- Vector Var/cov = variance-covariance


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The Structural Model - 1 predicted by 1

x11

x21

x31

1

X1

X2

X3

2

3

1

1

Y1

Y2

1 1

2

y11

y21


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An important corollary of SEM is that the

variances and covariance of dependent (orendogenous) variables, whether they beobserved or unobserved, are neverparameters of the model ; these areexplained by the exogenous variables.

In contrast, the var iance and covariance of

independent variable are impor tantparameters that need to be estimated.


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MEASUREMENT (CFA) MODELS

1 X1 11

2 X2 21 1

3 X3 31

CFA Part

1 1 11 Y1 1

21Y2 2

CFA Part


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CFA Model

error ReadSCASC

error WriteSC

error TalkSC

SSCerror InteractSC


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CFA with Greek Notation

1 x1 11 1

2 x2 1121

3

x3

32

2 4 x4 42


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Regression Equations (Xs)

1. x 1 = 111 + 1 2. x 2 = 211 + 23. x 3 = 322 + 3 4. x 4 = 422 + 4Or in matrix form

X = x +


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The parameters of this model are x,,and Where:

x represents the matrix of regressioncoefficients related to the s (described earlier).

(phi) is an x symmetrical variance-covariance matrix among the exogenousfactors.

(theta-delta) is a symmetrical q x q variance-covariance matrix among the error ofmeasurement for the q exogenous observedvariables


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The general factor analytic model

can be expanded as:X = x +

x1 11 0 1 x2 21 0 1 2

= +

x3 0 32 2 3 x4 0 42 4


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is the Loadings Matrix

The matrix is often termed the factor-loading matrix because it portrays the

pattern by which each observed variable islinked to its respective factor.


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The Ys

1. y 1 = 111 + 1 2. y 2 = 211 + 2

3. y 3 = 322 + 3 4. y 4 = 422 + 4

Or in matrix form

Y = y +


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Matrix Notation for Loadings

with Regression ModelY = y + y1 11 0 1

y2 = 21 0 1 + 2y3 0 32 2 3y4 0 32 4


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A just-identified model is one in whichthere is a one-to-one correspondence between thedata and the structural parameters.

Number of data variances and covariances equalnumber of parameters to be estimated.

However, despite the capability of the model toyield a unique solution for all parameters, the just-identified model is not scienti f ically interesting

because it has no degrees of f reedom and

therefore can never be rejected.


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Overidentified Model

An overidentified model is one in which thenumber of estimable parameters is less than the

number of data points (i.e., variance, covariance ofthe observed variable). This situation results in positi ve degrees of

freedom that allows for rejection of the model,

thereby rendering it scientific use. The aim inSEM , then, is to specify a model such that i tmeets the cr i ter ion of over identif ication.


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Underidentified Model

An underidentified model is one in which thenumber of parameters to be estimated exceeds the

number of variances and covariances.As such, the model contains insufficientinformation (from the input data) for the

purpose of attaining a determinate solution of

parameter estimation; that is, an inf ini te numberof solutions are possible for an under identi f iedmodel.


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Suppose there are 12 observed variable, this means thatwe have 12(12+ 1)/2= 78 data points.

Suppose that there are 30 unknown parameters.Thus, with 78 data points and 30 parameters to beestimated, we have an overidentified model with 48degrees of freedom.

It is important to point out, however, that thespecification of an over identi f ied model is a

necessary, but not suff icient condition to resolve theidenti f ication problem. Indeed, the imposition ofconstraints on particular parameters can sometime be

beneficial in helping the researcher to attain an

overidentified model


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No Scale Set for Constructs

Linked to the issue of identification is therequirement that every latent variable have

its scale determined. This requirementarises because these variable are unobservedand therefore have no definite metric scale;


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Assume CFA Model with 12 variables (items) and4 factors (3 items per factor) .

We can assume that there are 12 regression coefficient ( s) There are 12 error variance (s). There are 4 factors variances (which may be standardized

and therefore set to 1).

There are 6 covariances between factors.

If the factor variances are not set to 1 then then one of theparameters for each factor can be fixed to a value of1.00 (they are therefore not to be estimated). The rationaleunderlying this constraint is tied to the issue of statisticalidentification. In total, then, there are 30 parameters to beestimated for this CFA model .

introduction to cfa sem sept 19 2k5

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