Structural Equation Models

Simply speaking, a structural equation model (SEM) is a combination of confirmatory factor analysis and path analysis. Structural equation modeling includes two sets of models – the measurement model and the structural model. The measurement model can be expressed as a factor model. Figure 1 is a model to measure cognitive ability using three variables – verbal ability, math ability, and speed ability (note each of them can be viewed as factors measured by lower level observed variables).

Figure 2 gives another example of measurement model – a model to measure health.

If one believes that health influences cognitive ability, then one can fit a path model using the factors – cognitive ability and health. Therefore, a structural model is actually a path model. Putting them together, we have a model in Figure 3. This model is called SEM model.

Example 1. Autoregressive model

In ACTIVE study, we have three variables – word series (ws), letter series (ls), and letter sets (ls) to measure reasoning ability. Also, we have data on all these three variables before and after training. Assume we want to test whether reasoning ability before training can predict reasoning ability after training. Then the SEM model in Figure 4 can be used. Not that we allow the factor in time 1 to predict the factor at time 2. In addition, we allow the uniqueness factors for each observed variable to be correlated. The R code for the analysis is given below.

First look at model fit. The chi-square value is 27 with 5 degrees of freedom. The p-value for chi-square test is almost 0. Thus, based on chi-square test, this is not a good model. However, CFI and TFI are both close to 1. The RMSEA is about 0.063 and SRMR is about 0.011. Considering the sample size here is large – N=1114, overall, we may accept this model is a fairly good model. Then we can answer our question. Because the regression coefficient from reasoning1 to reasoning2 is significant, reasoning ability before training seems to predict reasoning ability after training. In other words, those with higher reasoning ability before training tend to have higher reasoning ability after training.

Example 2. Mediation analysis with latent variables

In path analysis, we have fitted a complex mediation model. Since we know that ws1, ls1, and lt1 are measurements of reasoning ability, we can form a latent reasoning ability variable. Thus, our mediation model can be expressed as in Figure 5.

Given CFI = 0.997, RMSEA =  0.034 and SRMR = 0.015, we accept the model as a good model even though the chi-square test is significant. Based on the Sobel test, the total indirect effect from age to ept1 through hvltt1 and reasoning is significant.