1. Multidimensionality: What does it mean?
So, now you're working with multidimensional data - but what does that mean? What does the number of factors tell you, and why are you getting conflicting information from theory and empirical analyses?
2. Factors = constructs
The answers are in the relationship of theory to statistics. A construct is a hypothesized attribute that can't be directly observed or measured. Examples of commonly studied constructs include self-determination, reasoning ability, and extraversion.
3. Factors = constructs
Typically, measures are designed to measure specific constructs. While constructs are theory-driven, factors are their mathematical counterparts. Each factor corresponds to a construct.
4. Interpreting confirmatory analyses
When you conduct a confirmatory analysis, you evaluate the strength of the hypothesized relationships between items and the constructs they were designed to measure.
Model fit statistics provide information about how well the hypothesis fits the data, and factor loadings quantify the relationships between items and constructs for reporting and interpretability. We'll talk about this more in the following chapters.
5. Interpreting exploratory analyses
Without theory, you conduct exploratory analyses guided by mathematical values like eigenvalues. You'll continue the exploration by running a multidimensional EFA to see how the items relate to each of those factors.
While all of this information can be very helpful in a situation where you don't know much about your data, a lack of theoretical grounding can make these results very difficult to interpret. You can make an educated guess from the factor loadings resulting from your EFA, but this can be challenging.
6. Running a multidimensional EFA
Running a multidimensional EFA works just like a unidimensional EFA. All that you have to change is the nfactors argument.
In the exercises, you'll go with the number of factors recommended from the scree plot, which was six. As you'll remember, the theory behind the dataset recommended five factors. We'll discuss this disagreement in more detail later and offer some empirical ways to compare different models.
Remember, to view a summary of results from the model object, all you have to do is enter the model object on its own line. No need to use the summary argument!
7. Factor loadings
As before, you'll be interested in the factor loadings, which represent each item's relationship to each underlying construct. However, the factor loadings look pretty different for a multidimensional EFA!
Here are the loadings for the first 15 items. As you can see, the six factors are represented by the six columns. You'll notice the factors have been assigned arbitrary names, and that they're not in the order you'd expect. This is due to the rotation that happens during the mathematical process behind a multidimensional EFA.
You'll also notice that some item/factor pairings don't have loadings. The results automatically exclude negligible factor loadings for ease of interpretation.
Interpreting these loadings can be challenging. Unlike a CFA, the factors don't have assigned meanings, so the user is left to infer these meanings based on the patterns of item loadings. For example, we can guess that factor MR5 is closely related to agreeableness since the A items all have strong relationships. We can also guess that item A1 should be inversely scored due to its negative relationship with the factor.
8. Factor scores
Individuals' factor scores also look different. Rather than a single factor score, each person now has a factor score estimated for each of the six factors. As before, any missing data means that factor scores will not be estimated, as we can see with person 65237.
These factor scores shouldn't be interpreted until you have a good working hypothesis for what each of the factors in your EFA represents. Once you have that hypothesis, you can make inferences about the constructs.
9. Let's practice!
Okay! Now it's your turn to run a multidimensional EFA.