  In some randomized clinical trials, patients may die before the measurements
of their outcomes. Even though randomization generates comparable treatment and
control groups, the remaining survivors often differ significantly in
background variables that are prognostic to the outcomes. This is called the
truncation by death problem. Under the potential outcomes framework, the only
well-defined causal effect on the outcome is within the subgroup of patients
who would always survive under both treatment and control. Because the
definition of the subgroup depends on the potential values of the survival
status that could not be observed jointly, without making strong parametric
assumptions, we cannot identify the causal effect of interest and consequently
can only obtain bounds of it. Unfortunately, however, many bounds are too wide
to be useful. We propose to use detailed survival information before and after
the measurements of the outcomes to sharpen the bounds of the subgroup causal
effect. Because survival times contain useful information about the final
outcome, carefully utilizing them could improve statistical inference without
imposing strong parametric assumptions. Moreover, we propose to use a copula
model to relax the commonly-invoked but often doubtful monotonicity assumption
that the treatment extends the survival time for all patients.

Applications;Methodology