Estimating variability of a center covariate in a mixed model

["Hunsicker, Lawrence" <lawrence-hunsicker@uiowa.edu>]

Colleagues:

I have a question about how to estimate properly the variance of a center level covariate coefficient in a logistic mixed model.  Let me first give you a sense of the question.  I am interested in whether there is center level variability in the frequency with which a particular diagnostic test is ordered on patients, after correcting both for individual patient level covariates and also other center level covariates.  I am therefore using glmmPQL to do a mixed logistic model.  The following is the S-Plus code for the situation where there are no center covariates added:

any.glmm<-glmmPQL(fixed = any ~ SEX + RACE + CARFAIL + CVA + PVASC + HYPER + DIABPRIM + INC.AGE,

        random = ~1|PROVUSRD, family = binomial(link=logit), data = ""> dispersion = NULL, subset = !is.na(PROVUSRD),

        niter = 10, verbose = T)

This code runs fine, and I can determine the significance of the random effect by running the same model without the random effect and then doing a likelihood ratio test.  I would then like to add center level covariates as effects fixed across all centers. 

But I am also interested in determining which other center covariates are significant “adjusters” of the residual center effect.  Herein lies my question.  It is my understanding that f I put the center covariates in the random model _expression_, I am telling the program that I want to indicate that not only is the intercept drawn from a random distribution, but also the slopes of each of the covariates put in the random model.  In the past I have just put the center covariate in as a fixed covariate in the fixed model.  I suspect that this estimates correctly the optimized value of the center effect coefficients.  But my instinct tells me that this will then determine the variance of the coefficients estimate using the d.f. of the total number of patients, where I suspect that the d.f. should be related instead to the total number of centers.  I am likely to have overestimated the precision of the center level covariates.

What is the correct way to put in center level covariates and get an accurate estimate both of their optimized value, and also of their variance?

(Incidentally, is there a convenient way to determine the variance of the estimated center random intercept STD?  I’d love to be able to give 95% confidence intervals for the estimated STD of the random effect.)

Many thanks to any that can help me with this.

Larry Hunsicker