A nonparametric test for the association between longitudinal covariates and censored survival data

Many biomedical studies focus on the association between a longitudinal measurement and a time-to-event outcome while quantifying this association by means of a longitudinal-survival joint model. In this article we propose using the $LLR$ test - a longitudinal extension of the log-rank test statistic given by Peto and Peto (1972) - to provide evidence of a plausible association between a time-to-event outcome (right- or interval-censored) and a time-dependent covariate. As joint model methods are complex and hard to interpret, it is wise to conduct a preliminary test such as $LLR$ for checking the association between both processes. The $LLR$ statistic can be expressed in the form of a weighted difference of hazards, yielding a broad class of weighted log-rank test statistics known as $LWLR$, which allow a specific emphasis along the time axis of the effects of the time-dependent covariate on the survival. The asymptotic distribution of $LLR$ is derived by means of a permutation approach under the assumption that the censoring mechanism is independent of the survival time and the longitudinal covariate. A simulation study is conducted to evaluate the performance of the test statistics $LLR$ and $LWLR$, showing that the empirical size is close to the nominal significance level and that the power of the test depends on the association between the covariates and the survival time. A data set together with a toy example are used to illustrate the $LLR$ test. The data set explores the study Epidemiology of Diabetes Interventions and Complications (Sparling and others, 2006) which includes interval-censored data. A software implementation of our method is available on github (https://github.com/RamonOller/LWLRtest).