According to Peter Austin (ref. below), in a gamma shared frailty model (i.e. a Cox regression model with cluster specific random effects which are iid logarithms of gamma distributions), the within-cluster correlation of subjects can be calculated as $\theta\, /\, (\theta + 2)$, with $\theta$ being the variance of the random effect as given in the summary of a
coxph object in R.
I'm not quite sure where this formula stems from. Formally, the 2 corresponds to the residual within-cluster variance, but since there is no residual part in Cox regression models, I'm not sure how to derive that. (Sorry, I'm a novice at frailty models...) And since I don't really understand this, I'm fully at loss with my real question which is the following:
How can the within-cluster correlation be determined for a lognormal shared frailty model, where the random effects are normally distributed? And can it be calculated from the R summary of a
coxme object at all?
Peter Austin. A Tutorial on Multilevel Survival Analysis: Methods, Models and Applications. International Statistical Review (2017), 85, 2, 185–203 doi:10.1111/insr.12214