Structural Equation Modeling (SEM) With Latent Variables

James G. Anderson, Ph.D.

Purdue University

Steps In Structural Equation Modeling

1. Model specification

2. Identification

3. Estimation

4. Testing fit

5. Respecification

Measurement Model (1)

• Specifying the relationship between the latent variables and the observed variables

• Answers the questions:1) To what extent are the observed variables actually

measuring the hypothesized latent variables?2) Which observed variable is the best measure of a

particular latent variable?3) To what extent are the observed variables actually

measuring something other than the hypothesized latent variable?

Measurement Model (2)

• The relationships between the observed variables and the latent variables are described by factor loadings

• Factor loadings provide information about the extent to which a given observed variable is able to measure the latent variable. They serve as validity coefficients.

• Measurement error is defined as that portion of an observed variable that is measuring something other than what the latent variable is hypothesized to measure. It serves as a measure of reliability.

Measurement Model (3)

• Measurement error could be the result of:– An unobserved variable that is measuring some

other latent variable– Unreliability– A second-order factor

Structural Model

• The researcher specifies the structural model to allow for certain relationships among the latent variables depicted by lines or arrows

• In the path diagram, we specified that Ability and Achievement were related in a specific way. That is, intelligence had some influence on later achievement. Thus, one result from the structural model is an indication of the extent to which these a priori hypothesized relationships are supported by our sample data.

Structural Model (2)

• The structural equation addresses the following questions:– Are Ability and Achievement related?– Exactly how strong is the influence of Ability

on Achievement?– Could there be other latent variables that we

need to consider to get a better understanding of the influence on Achievement?

Example of a Complete Structual Equation Model

• We can specify a model to further duscuss how to diagram a model, specify the equations related to the model and discuss the “effects” apparent in the model. The example we use is a model of educational achievement and aspirations.

• Figure 2 shows there are four latent variables (depicted by ellipses) two independent, home background (Home) and Ability and two dependent, aspirations (Aspire) and achievement (Achieve).

Example of a Complete Structual Equation Model (2)

• Three of these latent variables are assessed by two indicator variables and one latent variable, home background, is assessed by three indicator variables. The indicator variables are depicted in rectangles.

Covariance

• SEM involves the decomposition of covariances

• There are different types of covariance matrices:

1) Among the observed variables2) Among the latent exogenous variables.3) Among the equation prediction errors4) Among the measurement errors

Covariance (2)

• Types of covariance1) Among the observed variables2) Among the latent exogenous variables

Set the covariance between IQ and HOME to 0

IQ

HOMEACH

Covariance (3)

3) Among the equation prediction errors

Set the error covariance between Legal and Profess free

Religion

Experience

Legal Error

Profess Error

V1 F1E1 E3

V2 F2E2 E4

Total, Direct and Indirect Effects

• There is a direct effect between two latent variables when a single directed line or arrow connects them

• There is an indirect effect between two variables when the second latent variable is connected to the first latent variable through one or more other latent variables

• The total effect between two latent variables is the sum of any direct effect and all indirect effects that connect them.