The structural equation model is commonly described as the combination of a measurement model and a structural model. These terms derive from the history of SEM as being a union of the factor analytic, or measurement, models of psychology and sociology and the simultaneous structural equations of the econometricians. In its pure form it therefore explicitly assumes that every variable that we can observe is an imperfect measure of some underlying latent causal variable and that the causal relationships of interest are always between these latent variables. As in many other things, purity is more a goal than a requirement. Using the example in Chapter 5 of the effect of air temperature on metabolic rate (Figure 6.1), the things that we can measure (the height of the mercury in the thermometer or the change in CO2 in the metabolic chamber) always contain measurement error (εi). The measurement model, shown by the dashed squares in Figure 6.1, describes the relationship between the observed measures and the underlying latent variables (the average kinetic energy of the molecules in the air and the metabolic rate of the animal). The structural model, shown by the dashed circle in Figure 6.1, describes the relationship between the ‘true’ underlying causal variables. If we have only one measured variable per latent variable and we assume that the measured variable contains no measurement error (i.e. the correlation between the measured variable and the underlying latent variable is perfect) then we end up with a path model. If we have a set of measured variables for each latent variable and we do not assume any causal relationships between the latent variables then we have a series of measurement models. If we have more complicated combinations, in which we assume causal relationships between the latent variables, then we have a full structural equation model. Therefore, if you have understood Chapters 1 to 5, you already know how to construct and test a structural equation model; you simply have to put the pieces together.

The goal of this chapter is therefore to deal with some technical details that I have ignored up to now. The first detail is the problem of identification.