Z-residual diagnostics for detecting misspecification of the functional form of covariates for shared frailty models

In survival analysis, the hazard function often depends on a set of covariates. Martingale and deviance residual are most widely used for examining the validity of the function form of covariates by checking whether there is a discernible trend in their scatterplot against continuous covariates. However, visual inspection of martingale and deviance residuals is often subjective. In addition, these residuals lack a reference distribution due to censoring. It is therefore challenging to derive numerical statistical tests based on martingale or deviance residuals. In this paper, we extend the idea of randomized survival probability (Li et al. 2021) and develop a residual diagnostic tool that can provide both graphical and numerical tests for checking the covariate functional form in semi-parametric shared frailty models. We develop a general function that calculates Z-residuals for semi-parametric shared frailty models based on the output from the \texttt{coxph} function in the \texttt{survival} package in R. Our extensive simulation studies indicate that the derived numerical test based on Z-residuals has great power for checking the functional form of covariates. In a real data application on modelling the survival time of acute myeloid leukemia patients, the Z-residual diagnosis results show that a model with log-transformation is inappropriate for modelling the survival time, which could not be detected by other diagnostic methods.