PROC PLS: Reduced rank regression

[catch18]

     So with reduced rank regression in SAS, we identify response variables associated with our outcome of interest (Y) and model proc pls relationships between the independent variables and the selected responses. If there is an association between the derived factor/s from this initial modelling and the outcome variable (Y), could I conclude that the association is due to mediating effects of the response variables or to the independent effect of the derived factor?

Many thanks

[ChrisNZ]

Which procedure are you using?

[catch18]

Thanks @ChrisNZ.
I'm not sure I understand your question, but I used proc pls method=rrr.

[ChrisNZ]

Re-titled and moved to procedures forum.

[SteveDenham]

Umm, no to both I think.  For PROC PLS questions, @PaigeMiller seems to have the best handle in here. RRR maximizes the variance in the response variables that is explained by the X variables (simplified, I realize).  I suppose you could get a measure of mediation by looking at the change in the X factor coefficients as compared to a simple OLS regression, but that change is going to be affected by the (unknown) covariance between the Y variable and the additional variables included on the Y side.  There might be more on this, but once again, we've hit my wall on this subject.

SteveDenham

[PaigeMiller]

"Mediation" is not a concept I am familiar with. However, as far as I can see, the user has to interpret the results to mean "mediation is present" or "mediation is not present".

This is a great example of the importance of writing a meaningful subject line. The real question has nothing to do with PROC PLS or Reduced Rank Regression. The real question is about "Mediation" and you want people who are familiar with the concept of "mediation", so I suggest a brand new thread with something like this as the subject line: "Does My Fitted Model Show Mediation or Not?" And you would probably need to show the fitted model output from PROC PLS (if it isn't too large).

[catch18]

Thanks for your response @PaigeMiller 

I beg to differ though! The question was re-titled, but it has everything to do with RRR. I wondered whether the methodology as we have it from RRR creates factors which can have independent and direct associations with the  outcome of interest or whether any eventual associations with the outcome of interest will always be indirect?

I also didn't think I could do mediation analysis with the derived factors for the same outcome, based on the methodology of RRR.

Thanks

[PaigeMiller]

Ok, but as I understand it, "mediation" does not depend on the specific statistical method or PROC being used. Nevertheless, I can't address your question, and so you will have to wait until someone who does understand "mediation" comes along. And there is more of a chance of someone who understand "mediation" joining the discussion if "mediation" is in the subject line.

[catch18]

Mediation involves intermediary variables and the response variables in RRR could be considered as intermediary and so I felt doing a mediation analysis could be a duplication, but I needed clarification of the eventual relationship with the outcome of interest as explained above. Similarly for method=pcr as well.

Thanks

[SteveDenham]

Hi @catch18 

I had to go back to some basics on mediation, but I suspect that RRR is not a good method.  If you look at the Baron&Kenny stuff, you would do the following regressions: X-->Y, X-->M and X+M-->Y. Rather than the univariate regressions where most of the examples are, let's consider X, M and Y as vectors.  So, I would say do something like this: PROC PLS for X-->Y (which maximizes the covariance in the Y variables), PROC GLM for X-->M (which calculates a fixed covariance relationship), and then back to PROC PLS for X+M-->Y (and now you would look at how much the loadings on the X part of the X+M vector have changed.  This all hinges on "significance" or at least clear separation from the zero vector for the direct (X-->Y) and the mediated response (X-->M) regressions.

To me the hardest part of implementing the Barron&Kenny method would lie in the interpretation of X-->M using PROC GLM (the only place I could figure out how to do a MANOVA regression)..  Also, the standard Sobel test probably won't work for vectors--you would have to come up with something resembling Hotelling's Tsquared for the comparison.

So, I think it is possible to do this all with RRR, but it is going to be messy.  Would a mediation analysis on the first principal component of each of X, M and Y be an adequate approximation to the full multivariate analysis?

SteveDenham

[catch18]

Thanks for all the suggestions @SteveDenham. I'll look into it and read some more!

[catch18]

Many thanks for the response @SteveDenham