[R-sig-ME] estimation of intercept in binomial glmer

[Ben Bolker]

On 13-12-03 04:35 PM, Gebregziabher, Mulugeta wrote:
> A helpful answer to this question is on page 363 of Fitzmaurice et al (2004): Applied longitudinal analysis. 
> The approximate relationship between Beta_glm and Beta_glmm is given as:
> Beta_glm=~Beta_glmm/sqrt(1+0.346*var(b_i))
> Where b_i is the random intercept in the glmm.
> Hope this helps!
> --------------------------------
> Mulugeta Gebregziabher, PhD
> Associate Professor of Biostatistics
> Department of Public Health Sciences
> Medical University of South Carolina

  Very useful. This Google books link:

http://books.google.ca/books?id=0exUN1yFBHEC&lpg=PA441&dq=fitzmaurice%20glm%20glmm&pg=PA477#v=snippet&q=attenuated&f=false

  gives the magic number as k^2=0.346 with k=16*sqrt(3)/(15*pi) ["The
derivation of this approximation is not important" (!!)]

> 256*3/(225*pi^2)
[1] 0.345843

I'm assuming we can get this from some kind of delta-method
approximation based on the logistic distribution (the logistic
distribution with scale parameter s has variance pi^2/3*s^2), but can't
produce it immediately without further work ...

 Section 12.2.2 of Agresti gives the marginal mean as logistic(c X beta)
where c = (1+0.6*sigma^2)^{-1/2}, referencing Zeger et al 1988

  Ben Bolker