|Date: ||Tue, 12 Apr 2011 22:00:06 -0400|
|Reply-To: ||Jason Schoeneberger <jschoeneberger@CAROLINA.RR.COM>|
|Sender: ||"SAS(r) Discussion" <SAS-L@LISTSERV.UGA.EDU>|
|From: ||Jason Schoeneberger <jschoeneberger@CAROLINA.RR.COM>|
|Subject: ||Block Diagonal Covariance Matrix for GLMM|
|Content-Type: ||text/plain; charset="us-ascii"|
I've been trudging through a good volume of literature around General Linear
Mixed Models, but have yet to find an understandable explanation of the
contents within a block diagonal covariance matrix for a GLMM with a binary
outcome variable under a logit link. Maybe it's because no good way to
explain it exists. Goldstein's (2003) text provides a very simple example
of a block diagonal matrix for the linear mixed model, where the diagonal
consists of the variance of higher level random effects plus the level-1
residuals, with higher-level random effects on the off diagonals. Of course,
each block is associated with each 'subject'.
With binary dependent variables, the level-1 error component is subsumed in
the variance of the outcome denoted pi(1-pi) and the variance depends on the
expectation of y, thus for binary data the variance is directly dependent
upon the outcome variable. I'm looking for insight because I'm trying to
formally state why there is a lack of information when analyzing binary
data, and the subsequent need for more cases to achieve certain levels of
power, etc. In Longford (1993) he discusses Information about Variation and
provides an approximation calculation for determining roughly how many more
records would be needed for examining a binary outcome compard to normal.
In this calculation, he refers to 'w' as a generalized weight common to all
observations, where for normal distributions the variance=1 so w=1. He
states the information matrix is a function of the generalized crossproducts
of w(Xj, yj)^T Zj, where the cluster size are w multiples.
For binary data he provides an example where w=1/6 (for binary data with
p=.79 or .21). Does anyone know how 1/6 = .79 or .21?
Any direct insights or nudges toward other references is greatly