Analysing sequential events in clinical trials

Data from clinical trials where the endpoint is a single survival time are readily analysed by standard methods, most commonly using a semi-parametric proportional hazards approach. However, when the outcome involves two sequential survival times, standard methods may not be applicable.

We consider methods appropriate for the analysis of survival data in clinical trials where there are two distinct, sequential and opposing survival endpoints and where inferences about the second event are of particular interest. Two motivating examples of randomized clinical trials with different designs provide important illustrations of the methodology in practice.

Bivariate log-normal survival models are proposed as useful way of modeling such data. These models can be simply implemented in two stages, each of which is a univariate log-normal survival analysis. Different approaches to the analyses are described according to whether a second randomized treatment assignment is made at the time when the first event occurs and the second phase of the study commences. In the absence of a second randomization, the bivariate log-normal model adjusts for selection into the second phase of the study.

The investigation of 'treatment sequences' should, wherever possible, be handled by repeat randomization, which can then be followed by valid, unbiased analyses. However, in many clinical trial scenarios, this is simply not possible. In this case, the best approach is to consider the data as arising from an observational study, whilst controlling for all appropriate covariates.

The approach we describe is appropriate for log-normally distributed data but could be generalised to handle other distributions, although the process of model fitting would be less straight-forward.