Diagnostic plots for assessing the frailty distribution in multivariate survival data

In biomedical studies, frailty models are commonly used in analyzing multivariate survival data, where the objective of the study is to estimate both the covariate effect and the dependence between the multivariate survival times. However, inference based on these models are dependent on the distributional assumption of frailty. We propose a diagnostic plot for assessing the frailty assumption. The proposed method is based on the cross-ratio function and the diagnostic plot suggested by Oakes (1989). We use kernel regression smoothing with bandwidth choice by cross-validation, to obtain the proposed plot. The resulting plot is capable of differentiating between the gamma and positive stable frailty models when strong association is present. We illustrate the feasibility of our method using simulation studies under known frailty distributions. The approach is applied to data on blindness for each eye of diabetic patients with adult onset diabetes and a reasonable fit to the gamma frailty model is found.