1. A Hierarchical Model of IQ
The final three-factor model of the WAIS-III can still be extended to a hierarchical model, which can be used to explain the strong correlations between our three sub measures of IQ.
2. Hierarchical Three-Factor WAIS
In this model, we have three main level latent variables that comprise the measurement model. This term describes the relationship between the measured variables and the latent variables. The second layer or level is sometimes called the structural model, which is the relationship between the latent variables. We have one overall IQ factor predicting the latent scores on verbal comprehension, working memory, and perceptual organization.
3. Model Specification
To add the second layer to the model, we are going to extend the syntax we learned for building latent variables from manifest variables. The first four lines comprise the updated WAIS model with the correlated error term. The last line adds a new latent variable by using equals tilde. However, instead of using manifest variables after the tilde, we use the names of the previously defined latent variables. Therefore, the order of the code does matter here, as you cannot use a variable name before you create the variable for latent variables.
After defining this model, we would continue to use the cfa() and summary() functions to analyze the model.
4. No Change in Model Fit
There is one surprising feature to note when using hierarchical models. Usually, a change in model specification results in a change in fit indices. In this example, adding the second level of latent variables did not change our fit indices as you can see by looking at the CFI and TLI for these two models.
5. Why Use Hierarchical Models
One reason why there often is no change in fit, is that we are simply shifting the parameters from covariances to loadings. That difference does not change the match of the model to the data because the parameters are still being estimated. Instead, we are declaring the direction or the reason why these parameters exist. For example, you can use a correlation or regression analysis when you have one independent and one dependent variable. Mathematically, these are the same, but the focus of explanation is different. By using a hierarchical analysis, we are suggesting that the correlations between these variables existed because they are all related to a separate underlying construct. Therefore, we are suggesting that general IQ is the reason for the subscores on the WAIS, just like regression predicts a direction for the relationship between the independent and dependent variables.
6. Let's practice!
In our last set of exercises, you will practice hierarchical models to expand your lavaan skills.