topic Is SAS telling something about V matrix in PROC MIXED? in Statistical Procedures

Is SAS telling something about V matrix in PROC MIXED?

Scarlett89 — Mon, 15 Aug 2016 08:38:15 GMT

Hi SAS Community,

I've got a non-positive definite G matrix. It is said that the estimates of fixed effects are still valid if the marginal covariance matrix V is positive definite. Unfortunately, I can't find anywhere if SAS system will tell me about it or will stop (it is telling only about G and R matrices and about infinite likelihood but not about the general matrix V). I though about saving the matrix with "ods output" and check somehow if it is positive definite but this is a block diagonal matrix and I have to write that I want 760 blocks (I have 760 subjects), since SAS shows only the blocks I am specifying. Is somebody familiar with that? Do I have to really prove V matrix or can I trust the results?

Darja

Re: Is SAS telling something about V matrix in PROC MIXED?

SteveDenham — Mon, 15 Aug 2016 15:34:54 GMT

Can you share your output? I suspect that you may have more random effects than are needed, and if this is the case, then the NPD of the G matrix should not be a problem--it's not like problems with the Hessian.

Steve Denham

Re: Is SAS telling something about V matrix in PROC MIXED?

Scarlett89 — Mon, 15 Aug 2016 15:51:26 GMT

Thank you very much for your answer. As random effects I have an intercept and two slopes, I can't remove any of them because it is what I have assumed in my analysis- that every subject is allowed to have random intercept and random slope. I suppose the problem is that I have somehow not enough observations after stratifying by group (one group is bigger than second and for bigger group D matrix is PD). Which part of the output or code would be helpful?

Re: Is SAS telling something about V matrix in PROC MIXED?

SteveDenham — Mon, 15 Aug 2016 16:51:04 GMT

Your PROC MIXED code, and any of the output up to and including the parameter estimates (both fixed and random), including the iteration history.

I think you have identified the source of the problem with an intercept and two slopes. I would expect to need only one, but seeing your code may make this easier to understand.

Steve Denham

Re: Is SAS telling something about V matrix in PROC MIXED?

lvm — Mon, 15 Aug 2016 19:06:45 GMT

As indicated by Steve, you usually only have to be concerned when you get a message about the Hessian being non-positive definite.

Re: Is SAS telling something about V matrix in PROC MIXED?

Scarlett89 — Tue, 16 Aug 2016 07:36:54 GMT

Here is my SAS Code:

proc mixed data=stat method=reml nobound;
class id bmi_mother sex country isced troubled;
model bmi_z_score=slope_before slope_after bmi_mother sex country isced troubled/s cl ddfm=kr;
random int slope_before slope_after/ type=un subject=id_no;
format sex sex_new. isced isc. country count. troubled tr. bmi_mother catA.;
by categ;
run;

I have a piecewise mixed model, so I am assuming two random slopes-slope before the event and slope after. I saved some output in Word for one group where G matrix was NPD and attached it here. I deleted the reference categories too, I hope it is ok, was not sure what I am allowed to post. The output with random slopes and intercepts is too big because I have 760 subjects, so in the document is G matrix, intercept and R, iteration history, as well as fixed effects estimates. Without stratifying, the G matrix is ok. I think the problem is that it is just not enough variability in the data or something bacause for this group I have less observations per subject. The results look good compared to the other models but I am still not sure about conclusions.

Re: Is SAS telling something about V matrix in PROC MIXED?

SteveDenham — Tue, 16 Aug 2016 12:03:51 GMT

I'm stumped as to how the G matrix is NPD, as there are estimates for every element of the covariance matrix. I am going to suggest re-running this in PROC GLIMMIX, so that the standard errors of the covariance elements might be presented.

proc glimmix data=stat  nobound;                                        
class id bmi_mother sex country isced troubled;
model bmi_z_score=slope_before slope_after bmi_mother sex country isced troubled/s cl ddfm=kr2;             
random int slope_before slope_after/ type=un subject=id_no;
format sex sex_new. isced isc. country count. troubled tr. bmi_mother catA.;
by categ;
run;

If this throws the same type error, try type=chol for the covariance structure, to use a Cholesky parameterization. This occasionally helps solve NPD problems, and if it doesn't, the values obtained should help point out the source of the near singularity.

Steve Denham

Re: Is SAS telling something about V matrix in PROC MIXED?

Scarlett89 — Tue, 16 Aug 2016 13:13:59 GMT

I think it has every estimate because I have specified nobound option, otherwise some variances are 0. The matrix in general is NPD apparently. I have to use proc mixed because it is the procedure of my master thesis but I think I have found that type=chol is the same as type=FA0(q) in proc mixed, so I have now:

proc mixed data=stat method=reml nobound COVTEST;
class id bmi_mother sex country isced troubled;
model bmi_z_score=slope_before slope_after bmi_mother sex country isced troubled/s cl ddfm=kr;
random int slope_before slope_after/ type=FA0(3) subject=id_no;
format sex sex_new. isced isc. country count. troubled tr. bmi_mother catA.;
by categ;
run;

Both groups have no warning about NPD G matrix anymore and the estimates are changed just a little bit. Does it mean that I still have estimated the unstructured matrix just with this factor-analysis (as far as I understood from the Internet) and everything is good? I am not familiar with this method unfortunately.. It took more iterations too. Could you please look on my results? I made SE as well, it is possible with COVTEST.

Re: Is SAS telling something about V matrix in PROC MIXED?

lvm — Tue, 16 Aug 2016 13:21:29 GMT

Your factor analytic structure, FA0(3), is still giving you an unstructured covaraince matrix. It is just contstraining it to be at least positive semi-definite. A valid covariance matrix must be positive semi-definite or posistive definite. It is always a very good practice to use FA0(#) or CHOL when one wants an unstructured G. BY the way, add the G and GCORR options to the RANDOM statement to get a direct display of G (in addition to the FA parameters that determine G).

Re: Is SAS telling something about V matrix in PROC MIXED?

Scarlett89 — Tue, 16 Aug 2016 13:31:13 GMT

Thank you so much for the explanation and your help!

Re: Is SAS telling something about V matrix in PROC MIXED?

SteveDenham — Tue, 16 Aug 2016 13:33:22 GMT

Please mark @lvm's reply as a correct answer.

Steve Denham

Re: Is SAS telling something about V matrix in PROC MIXED?

Scarlett89 — Tue, 16 Aug 2016 13:38:56 GMT

I did but thank you so much too, you helped me with the solution!

Re: Is SAS telling something about V matrix in PROC MIXED?

SteveDenham — Tue, 16 Aug 2016 13:39:16 GMT

This looks really good--I would go with this analysis, remembering that the variance estimates are the factor analytic estimates, and not the values in the G matrix. Follow @lvm's advice and add G and GCORR to the RANDOM statement. And pay no attention to the Z test probability values you get from the COVTEST option--you are right in using it to get standard errors.

Steve Denham

Re: Is SAS telling something about V matrix in PROC MIXED?

Scarlett89 — Tue, 16 Aug 2016 13:48:14 GMT

The very last question for understanding. So, basically, SAS do not say anything about V matrix in general and I have to search for different solutions in these situations? It just seems strange that SAS check the positive definiteness of G and R matrices but not the V matrix.

Re: Is SAS telling something about V matrix in PROC MIXED?

SteveDenham — Wed, 24 Aug 2016 11:12:49 GMT

A quick look back at theory says V = ZGZ' + R. Thus, we know everything we need to know about V once we know G and R, as Z is a design matrix. If either G or R has problems, then we know V will have the same problem, unless there is the very rare case where the matixes exactly offset. It is why you should always get a look at both of these, using the G or R options.

Steve Denham

Re: Is SAS telling something about V matrix in PROC MIXED?

Rick_SAS — Wed, 24 Aug 2016 11:24:42 GMT

To expand on @SteveDenham's remark, from a matrix perspective the mixed-modeling procedures in SAS are written to solve the general model Y=X*beta +Z*gamma + epsilon where gamma ~ MVN(0,G) and epsilon~MVN(0,R),

In this formulation, the input matrices are X, Z, G, and R. These correspond to the various statements in the mixed-modeling procedures. In the simplest examples, the MODELS stmt defines X, the RANDOM stmt defines Z, the options to the RANDOM stmt define G, and the REPEATED stmt defines R.

If you ASSUME that this model holds (including that G and R are valid SPD covariance matrices), then it FOLLOWS that the marginal model is Y~MVN(X*beta, V) where V=ZGZ` + R is positive definite. Thus you never need to check whether V is SPD when G and R are valid covariance matrices.

When you use the NOBOUNDS option, you are declaring that you don't want SAS to enforce PD during estimation. I'm sure there are valid statistical reasons why this might be desired, but from a mathematical perspective it really muddies the waters.