5.1 Lecture
Often, we work with scales that have a validated or hypothesized factor structure. In the former case, the scale structure has been validated through previous psychometric studies. In the latter case, we may have conducted an EFA to estimate the factor structure on prior data, or theory/intuition may suggest a plausible structure. Regardless of how we come to expect a given factor structure, such situations represent confirmatory modeling problems, because we are attempting to empirically confirm an a priori expectation. Hence, exploratory methods like EFA are not appropriate, and we should employ confirmatory modeling techniques.
This week we consider one such technique: confirmatory factor analysis (CFA). As the name suggests, CFA is related to the EFA methods we discussed last week in that both methods are flavors of factor analysis. However, the two methods address fundamentally different research questions. Rather than attempting to estimate an unknown factor structure (as in EFA), we now want to compare a hypothesized measurement model (i.e., factor structure) to observed data in order to evaluate the model’s plausibility.
5.1.1 Recordings
Overview
We’l start with Caspar’s lecture recording. These slide should give you a good overview of the important ideas.
Note:
When Caspar discusses the complexity of the second-order CFA model, it’s easy to misunderstand his statements. We need to be careful not to over-generalize.
- In general, a second-order CFA is not more complex than a first-order CFA.
- Actually, in most practical applications, the opposite is true.
- A second-order CFA is more complex than a first-order CFA, when the factors
in the first-order CFA are uncorrelated.
- This is the situation Caspar references in the recording when claiming that the second-order model is more complex.
- We hardly ever want to fit such first-order CFA, though.
- The default CFA fully saturates the latent covariance structure.
- If the factors in the first-order CFA are fully correlated (according to
standard practice), and we include a single second-order factor, the following
statements hold.
- If the first-order CFA has more than three factors, the first-order model is more complex than the second-order model.
- If the first-order model has three or fewer factors, the first- and second-order models are equivalent (due to scaling constraints we need to impose to identify the second-order model).
- The second-order model cannot be more complex than the first-order model (assuming both models are correctly identified and no extra constraints are imposed).
- The above statements may not hold in more complex situations (e.g., more than
one second-order factor, partially saturated first-order correlation structure,
etc.).
- You can always identify the more complex model by calculating the degrees of freedom for both models.
- The model with fewer degrees of freedom is more complex.
The next five recordings supplement Caspar’s overview. These recordings dig deeper into some of the more tricky ideas that tend to cause the most confusion. In particular, these recordings focus on model estimation, identification, and scaling.