# Chapter 4: Structural Equation Models

### Introduction

In factor analysis, the focus of interest is on how observed indicators act as measures of latent factors, on average levels and variability of the factors, and on assigning values to the factors for individuals. In structural equation models (SEMs), the focus is extended and shifted toward answering questions about relationships (associations) between the constructs of interest, both latent and observed, rather than their measurement. This is illustrated by Figure 4.1, which shows a path diagram for the SEM which is considered in the examples of this chapter. This model has two core elements:

First, the *structural model* describes the relationships between the constructs of interest. Here those constructs are the five latent variables in the model, and the structural model for them is given by the substantive theoretical model introduced in Chapter 1 (see Figure 1.1). In the diagram, one-headed arrows indicate conditional distributions (regression relationships) where a response variable depends on an explanatory variable, while two-headed arrows indicate correlations (or residual correlations) for variables which are considered to be on an equal footing. Here the two factors *Effectiveness* and *Procedural Fairness* are on an equal footing and are used as only explanatory variables (such variables are also known as *exogenous variables*), while the factors Obligation to Obey, Moral Alignment and Co-operation are response variables to the exogenous variables and/or to each other (i.e. they are *endogenous variables*). Note that here Obligation to Obey and Moral Alignment are "intervening" or "mediating" variables which are response variables in some relationships (here given the exogenous variables) and explanatory variables in others (here for Co-operation). This kind of ordering of the variables and the assignment of different roles for them in the structural model is derived from substantive theory for the constructs.

Second, the *measurement model* describes how any latent constructs are measured by observed indicators. Here we use the fifteen survey indicators introduced in Chapter 1, and assume that each latent factor is measured by three of the indicators as shown in Figure 4.1. The measurement model for each factor is a factor analysis model for its three indicators.

More generally, some or even all of the explanatory and/or response variables in the structural model of a SEM may be observed rather than latent variables. For example, here we might have included the respondents' age or ethnic group as additional explanatory variables for the latent factors. For such variables a separate measurement model is redundant and is omitted, because the constructs are assumed to be directly observable.