Regression standardization and attributable fraction estimation with between-within frailty models for clustered survival data

The between-within frailty model has been proposed as a viable analysis tool for clustered survival time outcomes. Previous research has shown that this model gives consistent estimates of the exposure–outcome hazard ratio in the presence of unmeasured cluster-constant confounding, which the ordinary frailty model does not, and that estimates obtained from the between-within frailty model are often more efficient than estimates obtained from the stratified Cox proportional hazards model. In this paper, we derive novel estimation techniques for regression standardization with between-within frailty models. We also show how between-within frailty models can be used to estimate the attributable fraction function, which is a generalization of the attributable fraction for survival time outcomes. We illustrate the proposed methods by analyzing a large cohort on preterm birth and attention deficit hyperactivity disorder. To facilitate use of the proposed methods, we provide R code for all analyses.