Inclusion of clustering within a Cox PH model

Question

I am working with multi-center data and modelling time-to-event data. I have included a cluster term within my model to account for similarities between patients that were treated at the same center. The model was fit using the coxph function from the survival library in R.

I know that this results in robust SEs and resembles a GEE approach where the co-variance of individuals is accounted for and that effect estimates (coefficients) are common across clusters. My question however, is does the inclusion of this cluster term result in different baseline hazards being estimated for each cluster? I believe this is referred to as a frailty model if it does.

Answer 1

When you include a cluster() term within the formula of coxph() from the survival package you only correct the standard errors of the log hazard ratios using the grouped jackknife method that accounts for clustering. You still have a single baseline hazard.

In frailty models you include a frailty term to account for the clustering. In this case the model looks like this:

$$h_{ij}(t) = h_0(t) \omega_i \exp(x_{ij}^\top \beta),$$

where $h_0(t)$ is the baseline hazard, $\omega_i$ is the frailty term for the $i$-th cluster, $x_{ij}$ is the covariate vector for unit $j$ in cluster $i$, and $\beta$ the corresponding log hazard ratios. This model still has a single baseline hazard function $h_0(t)$, but perhaps you could call $h_0(t) \times \omega_i$ the baseline hazard function for cluster $i$. These hazard functions still obey the proportional hazards assumption.

If you want to have different baseline hazard functions per cluster that are not necessarily proportional, then you could stratify per cluster using the strata() in the formula argument of coxph().