Standard errors of random effects in GLIMMIX

[cb23_york]

Dear SAS community,

I have a fundamental question regarding the calculation of the standard errors (or rather the standard error predictions) of cluster specific random effects in the GLIMMIX procedure.  I note the following from the manual that states "The numbers displayed in the Std Err Pred column of the "Solution for Random Effects" table are not the standard errors of the gamma-hat displayed in the Estimate column; rather, they are the square roots of the prediction errors gamma-hat-i - gamma-i, where gamma-hat-i  is the predictor of the th random effect and gamma-i is the ith random effect."

Unfortunately I'm missing something in my understanding.   Firstly, I'm interested in the standard errors of the random effects, so why give me the square roots of the prediction errors?  Secondly how on earth are these calculated?  We produce an estimate, gamma-hat-i of gamm_i, but how is it possible to know what gamma-i is to obtain the difference? 

I'm also a little surprised that there seems to be so little documentation about this calculation.

Any help about the motivation or intuition for this would be greatly appreciated.

best wishes,   Chris

[lvm]

Details can be found in the section "Estimated PRecision of Estimates" in the GLIMMIX manual. There is a lot of detail. Perhaps a better description can be found in the reference book: SAS for Mixed Models, 2nd edition (by Littell et al. 2006) -- which is not on-line. Some of your confusion may be just wording. Many argue that the term "standard error" should not be used for an empirical random-effect term (actually, and EBLUP). SE should be used for the square root of the (estimated) variance of an estimated parameter (i.e., as in fixed effects). The specific formula will depend if Kenward-Roger method is used for adjusting all the variance-covariance estimates.

[cb23_york]

Thanks Ivm, I can now see how the Std Err Preds are produced.  I am still struggling a little with the intuition as to why they are produced this way rather than the standard way that, say, the SE of the Beta estimates are produced.  Searching on the terms you provided and looking at the Stata manuals on how it calculates SEs it would seem that the motivation underpinning the SAS approach is that we cannot appeal to the Central Limit Theorem and make a safe (ish) assumption regarding the distribution of the parameter (i.e. that its Normal).  Is this understanding correct?

[lvm]

This is not related to central limit theorem. The point is that random effects are NOT parameters to be estimated; they are random variables. Thus, one is "predicting" each gamma, not estimating it. The concept of standard error is usually applied to estimated parameters, not predicted random variables. Many authors do loosly use the term standard error for the square root of the estimated variance of the predicted random effect (the EBLUP). SAS is calculating these variances (and their square roots) in the "standard" way (for random effects). You use these values in the Random Effects Solution table for "confidence" intervals, and so on.

[Ruxandra]

I am new on this forum and this is rather a question on the same theme  than a reply.

Does it make sense to test that 2 random effects are significantly different?

I.e. to be more specific: assume that you have subjects clustered within a institution and you have subjects characteristic (x1) and institution characteristic (x2). There is an interest in comparing the highest performing institution with the lowest after adjusting for the mentioned characteristics.  My outcome is continuous therefore proc mixed as in:

proc mixed;

class institution;

model y=x1 x2;

random institution/s;

estimate "Institution 1 vs 2" | institution 1 -1;

estimate "institution 1 vs 3" |institution 1 0 -1;

run;
The estimates above seem to be comparing the random effects for institution and therefore the institutions but not sure if this makes sense.

Any opinion?

Thanks,

Ruxandra