1. Investigating model fit
Just like with an EFA, you'll also need to investigate and report on model fit statistics for a CFA.
2. Default fit statistics
By default, the only absolute fit statistic the summary() function prints is the log likelihood test. You can take a look at that, but as you'll recall, the p-value from this test is often significant due to sample size even though the desired result is a lack of significance.
3. Changing the options
In order to get some more meaningful fit statistics, you can set the fit.indices option to include some common fit statistics. The CFI, GFI, and RMSEA are all absolute fit statistics. By using the options() function, you are setting the fit.indices option in the global environment, so the specified indices will always be displayed.
The RMSEA should be familiar to you from the EFA chapters. We'll add the GFI, or Goodness of Fit Index, and the CFI, or Comparative Fit Index, which are other fit statistics available from the fit.indices option.
The CFI and GFI are conceptually similar to the TLI we discussed in the previous chapter, and the guidelines for "good" values are correspondingly the same. You'll remember from the previous chapter that an RMSEA less than 0.05 is generally considered good model fit.
4. Absolute model fit
As promised, you can see that the summary output starts with the likelihood ratio test, which is significant. In theory, we want this to be a non-significant result, but in practice, that rarely happens. Below the likelihood ratio test, you'll see the fit statistics we added.
The Goodness of Fit Index here is approximately 0-point-853. A common guideline for this index is that you want it to be greater than 0-point-9.
Below that, you'll see the RMSEA. Remember that ideally, you want this value to be less than 0-point-05.
For the last absolute fit index, you'll see the CFI. The calculation for this index is similar to the TLI, and it should be interpreted similarly. Ideally, this value would also be greater than 0-point-9.
5. Relative fit
The relative fit statistic in the output is the BIC, which is also included in the summary output by default. This isn't much good when you're looking at a single model, but it can be used to compare models.
In addition to looking at the summary output, you can also access it by viewing the BIC list element of the result object from the summary() function. This can come in handy if you want to store the BICs from multiple models in a dataframe.
6. Relative fit: comparing models
For example, let's look at the BICs for the theory-based CFA and the EFA-based CFA you created from the EFA results. Remember that the lower BIC is preferred, so these results indicate that the theory-based model fits better.
Relative fit indices such as the BIC are only useful when comparing nested models that are fit to the same dataset. You shouldn't ever try to compare relative fit indices from non-nested models or from results calculated from different datasets.
7. Let's practice!
That's it for your brief overview of how to access and interpret fit statistics from a CFA. Now that you have an understanding of fit statistics let's take these skills and put them into practice.