PROC PLS package documentation on the RRR approach

[LindaP]

Hello,

I am doing a diet related analysis with multiple health outcomes and would like to use the RRR approach as previously outlined by Hoffmann et al.

However, I need a better understanding on what the PROC PLS package is exactly doing for the RRR approach. Reading the documentation on SAS I only managed to see the one for PLS. 

SAS - PROC PLS Package

Four Factor extraction methods are available in the SAS software – PROC PLS Package:

1. PLS – uses partial least squares

2. SIMPLS – SIMPLS method

3. PCR – Principal Component Regression

4. RRR – reduced rank regression (uses OLS)

From SAS documentation I was able to locate a detailed explanation for PLS but did not

succeed in seeing the documentation for RRR although it was explained that RRR in SAS

uses OLS and is the most stable compared to the other 3 methods

(https://support.sas.com/documentation/onlinedoc/stat/151/pls.pdf). In his article,

Hoffman compared RRR vs PLS and concluded that RRR is more robust compared to PLS

(https://doi.org/10.1093/aje/kwh134).

Kindly assist me with the reference to the detailed mathematical annotation/reference article used when creating the RRR package in the PROC PLS package if available.

[PaigeMiller]

Your favorite internet search engine should find a number of articles on Reduced Rank Regression.

With regard to this:

Hoffman compared RRR vs PLS and concluded that RRR is more robust compared to PLS

if your only criterion for success if the fit of the response variables, then I agree. Sometimes there are other criteria, and if your criteria for success is model stability and robustness to multicollinearity among the x-variables, or modeling and understanding the variation in the X-variables, then I disagree.

[PaigeMiller]

---

@PaigeMiller wrote:

With regard to this:

Hoffman compared RRR vs PLS and concluded that RRR is more robust compared to PLS

if your only criterion for success if the fit of the response variables, then I agree. Sometimes there are other criteria, and if your criteria for success is model stability and robustness to multicollinearity among the x-variables, or modeling and understanding the variation in the X-variables, then I disagree.

---

Yesterday, I said the above. However, I'd like to modify what I said to: Hoffman never uses the word "robust". He makes no claims about RRR being more "robust". What he did show was that for the data he was using, RRR predicted more of the response variation than the other methods. But again I add: Sometimes there are other criteria than getting the highest amount of response variation predicted, and if your criteria for success is model stability and robustness to multicollinearity among the x-variables, or modeling and understanding the variation in the X-variables, then methods other than RRR will be useful.

You also said:

it was explained that RRR in SAS uses OLS and is the most stable compared to the other 3 methods

Hoffman never uses the word "stable" either. I would claim other methods are more "stable" than RRR, but I don't have research to prove that.

[PaigeMiller]

I was able to locate a detailed explanation for PLS but did not succeed in seeing the documentation for RRR although it was explained that RRR in SAS uses OLS and is the most stable compared to the other 3 methods

(https://support.sas.com/documentation/onlinedoc/stat/151/pls.pdf)

I don't see where the claim is made in that paper that RRR is the most stable compared to the other 3 methods, and would disagree based upon my understanding of the word "stable" in a modeling context.

[LindaP]

On Page 7607; Cross Validation - None of the regression methods implemented in the PLS procedure fit the observed data any better than ordinary least squares (OLS) regression.  On the description 7604 - In reduced rank regression,
the Y-weights qi are the eigenvectors of the covariance matrix YY of the responses predicted by ordinary
least squares regression; the X-scores are the projections of the Y-scores Yqi onto the X space.

If you have a link to the detailed documentation of the RRR kindly share.

[PaigeMiller]

---

@LindaP wrote:

On Page 7607; Cross Validation - None of the regression methods implemented in the PLS procedure fit the observed data any better than ordinary least squares (OLS) regression. 

---

Yes, I would expect OLS to fit better. However, dimension reduction techniques provide value, even if the fit is not as good. One value is that PLS is robust against multicollinearity in the X variables, while OLS can be severely affected by multicollinearity in the X variables. There are other benefits to dimension reduction techniques as well.

If you have a link to the detailed documentation of the RRR kindly share.

Internet search finds many such documentation.

[LindaP]

Thank you Paige. Kindly share if you have a specific reference that would be helpful? I wrote this question here because I have exhausted internet sources within my reach and could not find a definitive answer.

[PaigeMiller]

I don't have a reference. I'm sure there are plenty of documents that explain RRR out there. If you can't find a definitive reference, please be specific about what the documents on the internet are not providing.