In randomized clinical trials with a time-to-event outcome, the intervention effect is usually quantified by a hazard ratio. While its interpretability is debatable, it also relies on the proportional hazards assumption. Alternative measures could be more relevant, such as the difference in restricted mean survival time (ΔRMST) between the intervention and control groups up to time t*. The intervention effect measured by the ΔRMST is not relying on the proportional hazards assumption and is easily interpretable as the expected survival duration gain due to intervention over t*. In cluster randomized trials (CRTs), social units are randomized to intervention or control groups, inducing a correlation between the survival times of the subjects of a same cluster. In a previous work, we proposed the use of pseudo-values regression for the ΔRMST estimation in CRTs. Pseudo-values regression consists in computing pseudo-values for each individual and considering them as the dependent variable of a generalized linear model fitted by generalized estimating equations (GEE). From a simulation study, we concluded that this method well estimated the variance and controlled the type I error, under both proportional and non-proportional hazards assumption, when there is a sufficient total number of clusters (≥50). However, we observed an inflated type I error rate with less than 50 clusters which could be explained by the fact that the method relies on a GEE approach, known to increase the type I error rate in case of a limited number of clusters. Here we aim to assess and compare several approaches to correct the small-sample bias of the variance estimator in pseudo-values regression for ΔRMST in CRTs with small numbers of clusters.

We evaluate the performances of four bias-corrected variance estimators developed by Mancl and DeRouen (2001), Kauermann and Carroll (2001), Fay and Graubard (2001) and Morel et al. (2003) to directly correct the small-sample bias of the variance estimator in GEE. In addition, we combine the four small-sample correction methods with a Student distribution to account for the variability of the standard error estimator, instead of the usual normal distribution of the Wald test statistic. We compare the methods by using a simulation study with several scenarios (number of clusters, mean cluster size, coefficient of variation of the cluster sizes, intervention effect, degree of clustering). The simulation study performances are assessed through the relative bias in estimating the variance, the type I error, the coverage rate, the relative bias in estimating the intervention effect and the power.

The relative bias in estimating the variance, in absolute value, for the four bias-corrected variance estimators, do not exceed 10% in most of the scenarios, but could achieve 20% when the number of clusters is very limited. Across all scenarios, the bias-corrected variance estimators combined with the Student distribution show better performance compared to the normal distribution. The type I error rates are closer to the 5% nominal rate and the coverage rates are closer to 95%. All the methods lead to negligible relative bias in estimating the intervention effect.

This work opens the way for estimating a ΔRMST in presence of correlated time-to-event induced by CRT study design. We specifically propose the use of pseudo-values regression to correctly assessed the intervention effect and its variance in case of a limited number of clusters.

Cluster randomised trial, Time-to-event outcome, Restricted mean survival time, Pseudo-values, Small samples

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