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Introduction to structural equation modeling

1.1 Introduction

The origins of structural equation modeling (SEM) stem from factor analysis (Spearman 1904; Tucker 1955) and path analysis (or simultaneous equations) (Wright 1918, 1921, 1934). Integrating the measurement (factor analysis) and structural (path analysis) approaches produces a more generalized analytical framework, called a structural equation model (Jöreskog 1967, 1969, 1973; Keesling 1972; Wiley 1973). In SEM, unobservable latent variables (constructs or factors) are estimated from observed indicator variables, and the focus is on estimation of the relations among the latent variables free of the influence of measurement errors (Jöreskog 1973; Jöreskog and Sörbom 1979; Bentler 1980, 1983; Bollen 1989).

SEM provides a mechanism for taking into account measurement error in the observed variables involved in a model. In social sciences, some constructs, such as intelligence, ability, trust, self-esteem, motivation, success, ambition, prejudice, alienation, conservatism, and so on cannot be directly observed. They are essentially hypothetical constructs or concepts, for which there exists no operational method for direct measurement. Researchers can only find some observed measures that are indicators of a latent variable. The observed indicators of a latent variable usually contain sizable measurement errors. Even for variables that can be directly measured, measurement errors are always a concern in statistical analysis. Traditional statistical methods (e.g. multiple regressions, analysis of variance (ANOVA), path analysis, simultaneous equations) ignore the potential measurement error of variables included in a model. If an independent variable in a multiple regression model has measurement error, then the model residuals would be correlated with this independent variable, leading to violation of the basic statistical assumption. As a result, the parameter estimates of the regression model would be biased and result in incorrect conclusions. SEM provides a flexible and powerful means of simultaneously assessing the quality of measurement and examining causal relationships among constructs. That is, it offers an opportunity to construct the unobserved latent variables and estimate the relationships among the latent variables that are uncontaminated by measurement errors.

Other advantages of SEM include, but are not limited to, the ability to model multiple dependent variables simultaneously; the ability to test overall model fit, direct and indirect effects, complex and specific hypotheses, and parameter invariance across multiple between-subjects groups; the ability to handle difficult data (e.g. time series with autocorrelated error, non-normal, censored, count, and categorical outcomes); and the ability to combine person-centered and variable-centered analytical approaches. The related topics on these model features will be discussed in the following chapters of this book.

This chapter gives a brief introduction to SEM through five steps that characterize most SEM applications (Bollen and Long 1993):

1. Model formulation (Section 1.2 ). This refers to correctly specifying the structural equation model that the researcher wants to test. The model may be formulated on the basis of theory or empirical findings. A general structural equation model is formed of two parts: the measurement model and the structural model.
2. Model identification (Section 1.3 ). This step determines whether there is a unique solution for all the free parameters in the specified model. Model estimation cannot be implemented if a model is not identified, and model estimation may not converge or reach a solution if the model is misspecified.
3. Model estimation (Section 1.4 ). This step estimates model parameters and generates a fitting function. Various estimation methods are available for SEM. The most common method for structural equation model estimation is maximum likelihood (ML).
4. Model evaluation (Section 1.5 ). After meaningful model parameter estimates are obtained, the researcher needs to assess whether the model fits the data. If the model fits the data well and the results are interpretable, then the modeling process can stop after this step.
5. Model modification (Section 1.6 ). If the model does not fit the data, re-specification or modification of the model is needed. In this instance, the researcher makes a decision regarding how to delete, add, or modify parameters in the model. The fit of the model could be improved through parameter re-specification. Once a model is re-specified, steps 1 through 4 may be carried out again. The model modification may be repeated more than once in real research. In the following sections, we will introduce the SEM process step by step.

Finally, Section 1.7 provides a list of computer programs available for SEM.

1.2 Model formulation

In SEM, researchers begin with the specification of a model to be estimated. There are different approaches to specify a model of interest. The most intuitive way of doing this is to describe one's model using path diagrams, as first suggested by Wright (1934). Path diagrams are fundamental to SEM since they allow researchers to formulate the model of interest in a direct and appealing fashion. The diagram provides a useful guide for clarifying a researcher's ideas about the relationships among variables and can be directly translated into corresponding equations for modeling. Several conventions are used in developing a structural equation model path diagram, in which the observed variables (also referred to as measured variables, manifest variables, or indicators) are presented in boxes, and latent variables or factors are in circles or ovals. Relationships between variables are indicated by lines; the lack of a line connecting variables implies that no direct relationship has been hypothesized between the corresponding variables. A line with a single arrow represents a hypothesized direct relationship between two variables, with the head of the arrow pointing toward the variable being influenced by another variable. The bi-directional arrows refer to relationships or associations, instead of effects, between variables.

An example of a hypothesized general structural equation model is specified in the path diagram shown in Figure 1.1. As mentioned previously, the latent variables are enclosed in ovals and the observed variables are in boxes in the path diagram. The measurement of a latent variable or a factor is accomplished through one or more observable indicators, such as responses to questionnaire items that are assumed to represent the latent variable. In our model, two observed variables (x 1 and x 2) are used as indicators of the latent variable ?1, three indicators (x3-x5) for latent variable ? 2, and three (y 1-y 3) for latent variable ? 1. Note that ?2 has a single indicator, indicating that the latent variable is directly measured by a single observed variable.

Figure 1.1 A hypothesized general structural equation model.

The latent variables or factors that are determined by variables within the model are called endogenous latent variables, denoted by ?; the latent variables, whose causes lie outside the model, are called exogenous latent variables, denoted by ?. In the example model, there are two exogenous latent variables (? 1 and ? 2) and two endogenous latent variables (?1 and ?2). Indicators of the exogenous latent variables are called exogenous indicators (e.g. x 1-x 5), and indicators of the endogenous latent variables are endogenous indicators (e.g. y 1-y 4). The former has a measurement error term symbolized as d, and the latter has measurement errors symbolized as e (see Figure 1.1).

The coefficients ß and ? in the path diagram are path coefficients. The first subscript notation of a path coefficient indexes the dependent endogenous variable, and the second subscript notation indexes the causal variable (either endogenous or exogenous). If the causal variable is exogenous (?), the path coefficient is a ?; if the causal variable is another endogenous variable (?), the path coefficient is a ß. For example, ß12 is the effect of endogenous variable ? 2 on the endogenous variable ? 1; ?12 is the effect of the second exogenous variable ? 2 on the first endogenous variable ?1. As in multiple regressions, nothing is predicted perfectly; there are always residuals or errors. The ?s in the model, pointing toward the endogenous variables, are structural equation residual terms.

Different from traditional statistical methods, such as multiple regressions, ANOVA, and path analysis, SEM focuses on latent variables/factors rather than on the observed variables. The basic objective of SEM is to provide a means of estimating the structural relations among the unobserved latent variables of a hypothesized model free of the effects of measurement errors. This objective is fulfilled through...