8 Summary

In this chapter we have reviewed briefly the use of path analysis to represent certain linear and genetic models. We have discussed the conventions of path analysis, and shown how it may be used to derive the covariance matrices predicted under a particular model. We emphasize that the systems described here have been chosen as simple examples to illustrate elementary principles of path analysis. Although these examples are somewhat simplistic in the sense that they do not elucidate many of the characteristics of which structural equation models are capable, familiarity with them should provide sufficient skills for comprehension of other, more advanced, genetic models described in this text and for development of one's own path models. However, one aspect of structural models which has not been discussed in this chapter is that of multiple indicators. While not strictly a feature of path analysis, multiple indicator models, -- those with more than one measure for each dependent or independent variable -- warrant some attention because they are used often in genetic analyses of twin data, and in analyses of behavioral data in general. Our initial regression examples from Figure 5.2 assumed that we had only a single measure for each variable (systolic blood pressure, sodium intake, etc), and could ignore measurement error in these observed variables. Inclusion of multiple indicators allows for explicit representation of assumptions about measurement error in a model. In our regression example of Figures 5.2d and e, for example, we might have several measures of our independent ($x$) variables, a number of measures of sodium intake (e.g., diet diary and urinary sodium), multiple measures of exercise (e.g., exercise diary and frequency checklist), and numerous measures of obesity (e.g., self-report body mass index, measures of skinfold thickness). Likewise, we might have many estimates of our dependent $\eta$ variables, such as repeated measures of blood pressure, and several tests for coronary artery disease. Figure 5.4 expands Figure 5.2a by illustrating the cases of (a) one variable per construct, (b) two variables per construct, and (c) three or more observed variables per construct.

Figure 5.4: Regression path models with multiple indicators.

Covariance and variance expectations for multiple indicator models such as those shown in Figure 5.4 follow without exception from the path tracing rules outlined earlier in this chapter. However, the increase in number of variables in these models often results in substantial increases in model complexity. One of the important attractions of Mx is its flexibility in specifying models using matrix algebra. Various commands are available that allow changing the number of variables with relative ease. It is to the Mx model specification that we now turn.