PCA for regression on repeated measures (difference scores)

I have 2 (20x10) data matrices, Z1 and Z2, which correspond to a 10-dimensional questionnaire acquired in two different days: T1 and T2, on 20 subjects. I want to reduce dimensionality of the questionnaire, using PCA. I am interested in conducting a regression analysis:

Y ~ 1 + X1 + X2 + ... + Xn


where Y is the difference in brain activity between day1 and day2 (T1-T2), n is the number of retained principal components, and predictors X1...Xn represent the difference in the questionnaire scores between T1 and T2.

My question

Does it make sense to run a PCA on the difference between the two questionnaires (Z1-Z2)? Alternatively, which is the best approach in this case, if my primary interest is to reduce dimensionality before the regression analysis?

Useful info: 20 subjects filled a 10-dimensional questionnaire at time 1 (T1) and time 2 (T2). Brain activity was also measured at T1 and T2. Dimensions of the questionnaire are not independent (there is correlation among them).

(In Can I do a PCA on repeated measures for data reduction? the solution was to use MFA, but, in my opinion, it makes no sense here, because I want the difference between questionnaire scores at T1-T2, and not a single score on repeated measures.)

• Just for the context: what happened between T1 and T2? – amoeba Dec 7 '16 at 12:10
• One hour of behavioral task. In each trial of the task, subject were asked to detect if two stimuli were identical or not. – smndpln Dec 7 '16 at 12:16
• I edited the title to try to make it more specific - see if you like it. – amoeba Dec 14 '16 at 13:20

Does it make sense to run a PCA on the difference between the two questionnaires (Z1-Z2)?

I don't see why not. If you want to use differences Z1-Z2 as your predictors, then compute these differences and then do whatever regression-type analysis you prefer. Principal component regression (PCA + regression) is one possibility.

I don't see any problems associated with "repeated measures" here.

[...] which is the best approach in this case, if my primary interest is to reduce dimensionality before the regression analysis?

[...] the primary purpose is to evaluate which regressor mostly explains the independent variable (if any).

It is not very clear what your "primary purpose" really is.

If you want to build a good regression model, I would suggest to look into penalized/regularized regression methods, such as ridge or lasso or elastic net. It sounds as if you are interested in which specific predictors affect your outcome, and for that you need L1 penalty, but you also have not enough data and probably high collinearity, so you need L2 penalty as well; this suggests elastic net.

If you do want to "reduce dimensionality" as a primary purpose (this is a different purpose!), then PCA can make sense.

• Primary purpose is a good regression model, and at the beginning I thought dimensionality reduction to be mandatory. That's not true, and elastic net is probably the best choice, effectively. I'm new to this method, and I hope to understand how to choose the correct alpha. Do you have any suggestion? – smndpln Dec 14 '16 at 15:59

What is the primary purpose? Prediction? Or do you have some prior on which / how many factors and their interpretation?

What about using a regression tree? That's typically an efficient way.

But with 20 obs and 10 vars you might be hard pressed to do a statistical analysis. You are close to a case study.

• There shouldn't be questions in your answers. Anyway, the primary purpose is to evaluate which regressor mostly explains the independent variable (if any). 20 is actually an above average number for neuroimaging studies. – smndpln Dec 11 '16 at 14:31
• Secondly, how would you use the regression tree? Conducting two analysis on T1 and T2, separately? – smndpln Dec 11 '16 at 14:49
• I would create a new regressand, S, defined as T1-T2 and then proceed as normal. I don't quite see why it's not simply possible to run PCA on this? – Superpronker Dec 11 '16 at 14:55