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Question

Can someone answer (in as non-technical terms as possible) whether or not frailty models and robust sandwich variance estimators are trying to solve the same problem in different contexts? That problem being that estimated standard errors will be underestimated if correlation between groups is not accounted for.

Secondary to that, I know that invoking a robust sandwich estimator does not affect point estimates, is that the general idea with frailty models too?

Some context

I know for instance that in R's Survival package when estimating the survival function via Kaplan-Meier, one can supply the argument cluster to Survfit, which will invoke a robust sandwich variance estimator.

On the other hand when performing cox proportional hazard regression, one can supply either a cluster argument or a frailty argument. I have tried both with a test dataset and a single variable and I note that outcomes are similar for both the estimated coefficient and the estimated standard error on the coefficient. This, along with some reading around the problem, led me to the belief that frailty models and robust sandwich variance estimators are trying to solve the same problem.

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It might be better to say that they appear in the same context but answer slightly different questions. These approaches are used when there are correlations among observations that can't be accounted for just by the covariates included in the model. In a Cox model with only a single type of event experienced no more than once by an individual, those could be within-group correlations, e.g., within hospitals. If there are repeated events, those could be within-individual correlations.

The difference between them is essentially the distinction between generalized estimating equations (GEE) and mixed models, discussed in some detail on this page, and also in this answer. The regression coefficients produced by the 2 types of models can differ, as they model the data from different perspectives.

Handling of the cluster() term in the R survival package is similar to what's done with GEE. You are interested in modeling some type of population-average effects, but you need to adjust coefficient covariance estimates to take into account the correlations within individuals.

The frailty() term leads to modeling inter-individual or inter-group differences directly, similar to the random effects in a mixed model. In a Cox model, the frailty is an individual- or group-specific multiplicative factor in the hazard. Starting with an assumption about the distribution of the frailty among individuals, the modeling then estimates together the Cox coefficients (common to all individuals) and the frailties. That leads to "conditional" estimates of regression coefficients, conceptually closer to what would be used to predict individual outcomes.

Frailty-type models in survival now seem to be deprecated in favor of the coxme, or Cox mixed-effects, package.

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