Weighted logrank tests to detect a transient improvement in survivorship

In investigating whether an intervention has an effect on survival, we exploit the flexibility of weighted logrank tests for constructing optimal tests sensitive to specified patterns of changes in the ratios or differences of the hazards as a function of time. The tests presented seem appropriate if either the intervention on trial imposes changes in life style, such as a diet modification or an exercise regimen, or if the effect of therapy itself is limited in duration, such as in the case of drugs for AIDS that may induce drug resistant viruses. For such problems, we postulate no difference between the hazards of the two groups initially, increased discrepancies as time goes on, and an eventual leveling-off of the discrepancies in hazards perhaps due to compliance problems or to adoption of the behavior change under study by individuals in the control group. We suggest a quadratic weight function for such problems and show how to evaluate the power of the proposed tests as well as their efficiency relative to other logrank tests. Other uses of quadratic weight functions, for example for evaluating effects with lag times, and for some parametric procedures are also described. Additive and multiplicative models are presented and both discrete and continuous times are considered.