The simple answer is that the intervention factor is included in the model as a dummy variable (usually 1 if intervention, 0 if no intervention). The beta for that term can be tested against 0. If significant, intervention is significant. By the way, the result would be equivalent to using ANOVA.

Thanks very much, James, for getting back to me. Is there a worked example of this being done that I could look at?

Here is on of the most straight-forward explanations I have seen:

https://www.statology.org/dummy-variables-regression/

I was unclear about one part of your question. Do you mean the eigenvectors/loadings from the analysis of T1 will be used to calculate the aggregate scores for T2? Or do you mean a new PCA will be conducted on T2? There are issues with either of these. It is unclear to me which of these James E. McLean is assuming. There is also the issue that since you are talking about measuring impact, it would be good for you to describe your design more thoroughly so we know, for example, about your control condition.

Thanks James. I'll check the link that you posted. Yes, I understand it would be the same as an anova, using T1 and T2 factor scores. But I was unsure how this was computed using it multiple regression.

Thanks Daniel for considering my question. I don't think a new pca needs to be done for the T2 scores. I think you calculate the T2 mean factor scores for the 3 factors identified and then use multiple regression with the T1 mean factor scores for each of the factors to check for any significant change to measure the impact of the intervention.

Whatever methods (not clear if you are using PCA or factor analysis) you use for T1 you will be optimizing for those data, so it is likely the solution will not fit as well for T2. What is your research question?