Structural Equation Modeling (SEM) With Latent Variables James G. Anderson, Ph.D. Purdue University

Steps In Structural Equation Modeling Data preparation Model specification Identification Estimation Testing fit Respecification

Data Preparation Estimation of missing data Creation of scales and indices Descriptive statistics to include Examination for outliers skewness and kurtosis Transformation of variables

Measurement Model (1) Specifying the relationship between the latent variables and the observed variables Answers these questions: To what extent are the observed variables actually measuring the hypothesized latent variables? Which observed variable is the best measure of a particular latent variable? 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

Setting the Error Variance Error variance can be set to 0 if you have a single indicator of the latent variable and no information about its reliability Error Variance = (1-Reliability) Variance of the Observed Score if you know the reliability of the indicator

Creating a Latent Variable from Multiple Indicators Exploratory factor analysis can be used with multiple indicators of a construct to determine the number of factors and which indicators are associated with each factor. Confirmatory factor analysis can then be used to test the fit of the measurement model.

Example of a Complete Structural Equation Model We can specify a model to further discuss 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 3 shows there are four latent variables (depicted by ellipses), three exogenous variables (knowledge, Value and Satisfaction) and one endogenous (performance).

Variables Performance – Planning, Organization, controlling, coordinating and directing a farm cooperative Knowledge – Knowledge of economic phases of management directed toward profit-making Value-Tendency to rationally evaluate means to an economic end Satisfaction - Gratification from performing the managerial role

Structural Model (1) 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 Performance is related to Knowledge, Value and Satisfaction in a specific way. 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: Is Performance related to the three predictor variables? Exactly how strong is the influence of each variable on Performance? How well does the model fit the data?

Example of a Complete Structural Equation Model (2) Each of the four latent variables is assessed by two indicator variables. The indicator variables are depicted in rectangles.

Model Building If the original model does not provide an acceptable fit to the data, alternative models can be tested. The standardized residuals and modification indices can be used to determine how to modify the model to achieve a better fit to the data.

Covariance SEM involves the decomposition of covariances There are different types of covariance matrices: Among the observed variables Among the latent exogenous variables. Among the equation prediction errors Among the measurement errors

Covariance (2) Types of covariance Among the observed variables Among the latent exogenous variables IQ ACH HOME

Covariance (3) Among the equation prediction errors Religion Legal F1 Profess Error Experience E2 V2 F2 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.