GLMM Credibility
Are they saying that phi is the "within variance" and equals the EPV, and the random coefficient sigma's "between variance" the VHM? It seems like as the ratio phi/sigma increases z = n / (n + k) would get smaller and we'd move away from the GLM estimate towards the grand mean, since I assume the GLM beta estimate is what we're credibility weighting: GLMM random coefficient = z*GLM_beta + (1-z)*(random_var_grand_mean). The text implies that as phi/sigma increases we move towards the GLM estimate. I must have something backwards somewhere? I could see them asking us to calculate a random coefficient using credibility and the GLMM parameters.
Comments
As the ratio of phi to sigma increases we move toward the GLM estimate. We can think of phi (the within variance) as approximately n and sigma (the between variance) as approximately k in the formula Z = n / (n+k). As sigma (the variance between the categories of a random effect variable) decreases, n+k becomes smaller and hence Z increases.