# Structural Equation Modeling

Often, researchers are interested in variables that cannot be directly observed or measured (e.g., beliefs, intelligence). These unobserved variables are known as “latent” constructs or factors. We try to measure these unobserved constructs through observable variables. For example, there are underlying memory, language, perceptual organization, speed of processing, and executive functions among all human beings that underlie their performance on a wide range of neuropsychological observed performance tests. SEM is a family of analytical methods that are designed to specify relations between latent constructs and the underlying observed indicators (measurement model) as well as test the causal relations between latent constructs (structural models). These techniques include confirmatory factor analysis (CFA), path analysis, full structural models, latent growth models, and many other variations of these techniques.

CFA is a special form of factor analysis and is used to test the a priori hypothesis that certain observed variables capture or measure a latent construct. An advantage of factor analysis, in general, is that five different measures assessing well-being can be reduced into a latent variable of well-being that has less error and more reliability than any of the individual variables that it comprises.

Path analysis or causal modeling tests, at the simplest level, use linear modeling techniques to examine the casual relationships between manifest variables (observed variables), latent variables (unobserved variables), or a combination of the two. While we have already discussed the premise that correlation does necessarily mean causation, one can examine the logical flow of relationships. As an example in Figure 16.1, we examine the effects of age, presence of an ApoE + blood genotype, accumulation of abnormal amyloid levels in the brain, volume of the hippocampus on brain MRI, and resultant memory performance on the Auditory Verbal Learning Test Delayed Recall (AVLTDEL) among subjects with mild cognitive impairment. By using regression models and standardized beta weights that simultaneously adjust for the effect of each variable on each other, we discover that having a positive ApoE4+ blood genotype is related to increases in abnormal amyloid levels in the brain as well as reduced hippocampal volume. However, this increase in amyloid does not seem to relate to poorer AVLTDEL performance. Rather, it seems that there are direct effects of reduced hippocampal volume as well as direct effects of ApoE4+ status on AVLTDEL performance, as well as an indirect effect of ApoE4+ status on hippocampal volume, which in turn is related to cognitive performance. This model controls for the direct and nondirect effects of age on AVLTDEL performance. Because ApoE4 status is genetically determined at birth, and age cannot be caused by any of the biological measures, and cognition cannot cause biological changes in the brain, this type of modeling provides clues as to how different risk factors may affect each other and the resultant effect on cognitive performance. Please note that el, *e2,* and

Figure 16.1 A graphical representation of a path analysis model.

*e3* within the oval shapes refer to the error terms in the model. All of the variables in the model are in boxes because they are directly observed variables. Latent variables, if added to the model, would be denoted by oval shapes.

In SEM, a series of simultaneous equations is assessed using path-tracing rules. The discrepancy between data-derived covariance matrix and the model-derived covariance matrix forms the basis for estimating how well the conceptual model or tested model fits the data. On the basis of this discrepancy function, a wide variety of fit measures are being used; however, there is little consistency in choice of fit indexes or criteria for their evaluation. Regardless of the fit index that is being used, researchers should not forget the most fundamental rule that there is no *true* model, and the best model is the one that is most parsimonious, substantively and theoretically meaningful, and that can be cross-validated and replicated reasonably well in another population (MacCallum & Austin, 2000).