Z-residual diagnostic tool for assessing covariate functional form in shared frailty models

ABSTRACT

Survival analysis often involves modeling hazard functions while considering frailty to account for unobserved cluster-level factors in clustered survival data. Shared frailty models have gained popularity for this purpose, but assessing covariate functional form in these models presents unique challenges. Martingale and deviance residuals are commonly used for visually assessing covariate functional form against continuous covariates. Nevertheless, their subjective nature and lack of a reference distribution make it challenging to derive numerical statistical tests from these residuals. To address these limitations, we propose ‘Z-residuals’, a novel diagnostic tool designed for shared frailty models, leveraging the concept of randomized survival probability and introducing both graphical and numerical tests. To implement this approach, we develop an R package to compute Z-residuals for shared frailty models. Through extensive simulation studies, we demonstrate the high power of our derived numerical test for assessing the functional form of covariates. To validate the effectiveness of our method, we apply it to a real data application concerning the modelling of survival time for acute myeloid leukemia patients. Our Z-residual diagnosis results reveal the inadequacy of log-transformation of the covariate, highlighting the limitations of other diagnostic methods for effectively assessing covariate functional form in shared frailty models.

KEYWORDS:

• Survival analysis
• clustered survival data
• shared frailty models
• covariate functional form
• residual diagnostics
• Z-residuals

Acknowledgments

The authors greatly appreciate the editor, associate editor, and reviewers for their invaluable feedback and suggestions, which have significantly enhanced the quality of our work.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research is supported by the Discovery Grants from the Natural Sciences and Engineering Research Council of Canada (NSERC).