In studies that involve censored time-to-event data, stratification is frequently encountered due to different reasons, such as stratified sampling or model adjustment due to violation of model assumptions. Often, the main interest is not in the clustering variables, and the cluster-related parameters are treated as nuisance. When inference is about a parameter of interest in presence of many nuisance parameters, standard likelihood methods often perform very poorly and may lead to severe bias. This problem is particularly evident in models for clustered data with cluster-specific nuisance parameters, when the number of clusters is relatively high with respect to the within-cluster size. However, it is still unclear how the presence of censoring would affect this issue. We consider clustered failure time data with independent censoring, and propose frequentist inference based on an integrated likelihood. We then apply the proposed approach to a stratified Weibull model. Simulation studies show that appropriately defined integrated likelihoods provide very accurate inferential results in all circumstances, such as for highly clustered data or heavy censoring, even in extreme settings where standard likelihood procedures lead to strongly misleading results. We show that the proposed method performs generally as well as the frailty model, but it is superior when the frailty distribution is seriously misspecified. An application, which concerns treatments for a frequent disease in late-stage HIV-infected people, illustrates the proposed inferential method in Weibull regression models, and compares different inferential conclusions from alternative methods.
