1. Heywood Cases on the Latent Variable
In the last two chapters, you learned how to specify, interpret, and edit your latent variable models. In this chapter, we will cover how to handle common errors that occur when examining multifactor latent variable models.
2. Heywood Cases
You may encounter a Heywood case when specifying models. These are named after H.B. Heywood who first defined them in 1931. They occur when the estimation of the correlation between variables is over the absolute value of 1, or out of bounds. Second, Heywood cases can also occur when negative variances are estimated for the model. Since the variance formula is squared, negative variances should not happen. Generally, correlation errors happen on the latent variables and negative variances occur in the error terms. In our model picture, the double headed arrows on the manifest variables represent error variances.
3. A Latent Variable Example
You previously built a model with the EPI dataset in the psych package. In my example, I have scrambled the items on each latent variable to illustrate what happens when your model is misspecified. When we run this code, lavaan will provide a warning message in blue indicating that the correlations are not positive definite. Positive definite errors indicate that at least one variable is probably a combination of the other variables. This message could mean that the correlations are over 1 or that they are so close to 1 that the solution is unlikely.
It is tempting to run the cfa() and summary() functions together quickly, and the lavaan output can be quite long. Always check for warnings after the cfa() function because the summary() function does not show them again. They can be easy to miss.
4. How to Find the Error
While you can run the suggested inspect() function, it is just as easy to check the summary of the model in the covariances section. I can tell that all three of these variables are highly correlated, but especially neuroticism to lying. Since these variables are so highly correlated, we can try collapsing them into one latent variable, instead of two.
5. How to Fix the Error
To combine latent variables, you simply move the questions to one latent variable, instead of two. In my model specification, I moved all the manifest variables for neuroticism and lying into one combined latent variable and renamed this latent for clarity. When we run the cfa() function now, we no longer get an error.
6. How to Fix the Error (2)
The covariances summary indicates that extraversion and our combined latent variable are highly negatively correlated, and we need to decide if the correlation is too high or if we should try a one factor model of personality.
7. Let's practice!
In the exercises, you will build a model exploring a scale for animal adoption and adjust that model to remove a latent variable Heywood case.