Within-cluster correlation in lognormal shared frailty models 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
 A: The Kendall's $\tau$ formula for within-cluster correlation you cite for a gamma frailty is based (1) on the the general relationship between $\tau$ and the assumed frailty distribution and (2) the specific properties of a gamma distribution constrained to mean 1, often used to model frailty. Hougaard discusses this in Analysis of Multivariate Survival Data.
Kendall's $\tau$ for a frailty distribution can be written (Hougaard, equation 7.23) as:
$$\tau= 4 \int_0^{\infty} sL(s)L''(s) ds -1 $$
where $L(s)$ is the Laplace transform of the frailty probability distribution. For the specific case of a gamma distribution having a mean value of 1 and thus parameterized solely by the variance $\theta$, you get the result you cite, $\tau=\theta/(2+\theta)$. (See Hougaard's Exercise 7.5, written in terms of $\delta = 1/\theta$.) The $2$ in the denominator is determined solely by the properties of a gamma distribution having a mean of 1.
Unfortunately from your perspective (Hougaard, Section 7.6):

The lognormal distributions have also been used as frailty distributions. In this case, the Laplace transforms are theoretically intractable and therefore probability results have to be evaluated by means of an approximation or numerical integration. Simple explicit results for dependence measures like $\tau$ and Spearman's $\rho$ are not known.

Hougaard's Appendix Section A.3.5 discusses a way to approximate the Laplace transform for a lognormal distribution. That shouldn't be too hard in principle to apply to get an approximation to $\tau$  as a function of lognormal frailty variance, but it's beyond my comfort zone for providing an answer. The R parfm package has a function for calculating $\tau$ for parametric frailty models, whose code might provide an approach.
