How to perform a repeated measure CFA

Question

My purpose is to verify that a specific scale is unidimensional, and that the one dimensional structure is persistent over time. I have a 10 item scale, about 200 cases and score values at two different time points.

It would be simple if it was only one measure: I would split the data in half, I would perform PCA in one half and CFA in the other half, an approach that indeed is working well for the first measure. However, I would like to somehow provide a measure that the unique factor structure is persistent over time.

How can this be done?

Answer 1

You need to do a CFA with two factors - a factor for time 1 and a factor for time 2. Each factor is keyed to its respective items.

You need to fit two models. In Model 1, you do a basic CFA where each regession coefficient can seek its natural level. Factors 1 and 2 are correlated. If Model 1 fails, you might want to try Model 1a, by adding correlations between the error terms of corresponding items.

Now fit Model 2, to test for persistence over time.

In Model 2, constrain the regressions for each item to be the same at both time periods. Test Model 2 against Model 1. If Model 2 fits as well as Model 1 (Difference of deviances not significant under chi-squared), you can assume that you have a unidimensional factor at both time periods with the same relationship to the items on the measure.

Here it is using the usual model formulations:

For definiteness, consider item 2. At time 1.$$ F_1=\lambda_{21}x_{21}+ \epsilon_{21}$$ For item 2 at time 2. $$ F_2=\lambda_{22}x_{22}+ \epsilon_{22}$$. For Model 1a, posit $Cov(\epsilon_{21}, \epsilon_{22}) \neq 0$. For Model 1, these error terms are indepdendent.

Using this notation, in Model 2, you posit $\lambda_{21}=\lambda_{22}$, and likewise for all regression coefficients.