How do I test whether one predictor is significantly better than another? Is Hotelling's T the best option?

Question

Overview:

I want to test if "emotional numbing" is a significantly better predictor of "lower intimate relationship functioning" than "reexperiencing" or "hyperarousal"

It was suggested to me to use stepwise regression to analyze the data. However, one of my dissertation committee members suggested Hotelling's T.

My hypothesis:

• Hypothesis 1: Emotional numbing will be a better predictor of lower relationship functioning than re-experiencing or hyperarousal symptoms.
• Hypothesis 1a: Emotional numbing will be a better predictor of lower dyadic adjustment than re-experiencing or hyperarousal symptoms.
• Hypothesis 1b: Emotional numbing will be a better predictor of intimacy than re-experiencing or hyperarousal symptoms.
• Hypothesis 1c: Emotional numbing will be a better predictor of communication than re-experiencing or hyperarousal symptoms.

Questions:

• Is Hotelling's T the most appropriate way to determine whether emotional numbing is a significantly better predictor than the other 2 variables?
• Or is there is a better statistical test for this task?

Answer 1

Avoid stepwise variable selection. One good approach, although it changes the hypothesis, is to ask whether one set of predictors is adequate, i.e., that another set of predictors does not add significant predictive information to the first set. Then reverse the roles. For example, if you have two competing predictors x1 and x2, test whether x1 adds to x2 and whether x2 adds to x1. If x1 adds to x2 but x2 does not add to x1, then the conclusion is fairly clear that x1 is important and adequate, i.e., that there is no strong evidence that x2 is needed, and we conclude that x1 is "better" than x2.

Answer 2

How do you define better? (lower MSE, more correct classifications, etc.)

Are your variables binary, multinomial, ordered, continuous, ...?

Do you want to know how they predict above and beyond other variables? or just by themselves?

Hotelling's T-squared is usually used to compare multiple response variables to either a hypothesized set of values or compare between 2 groups. I don't see how it would be useful for comparing predictors (but maybe your committee member knows a different way, you could ask for additional information).