The Z statistic at T = 1.0 can be written as Let Z 3 be the normal (0,1) approximation to the logrank test in the interval between T = 0.75 and T = 1.0 Let Z 2 be the normal (0,1) approximation to the logrank test in the interval between T = 0.50 and T = 0.75. Let Z 1 be the normal (0,1) approximation to the logrank test at T = 0.50. This implies that the Z statistic at information time T = 1.0 is the weighted sum of the Z statistics within discrete information times (t i, t j) within (0, 1) where the weight is the square-root of the information in time (t i, t j) As summarized in Fleming and Lin it has been shown that sequentially computed logrank statistics have independent increments which are asymptotically normal. The logrank test statistics is the sum of the observed number of events minus the expected number of events for the active treatment at each time t i of an event divided by the (sum of the variances at each time t i) 0.5. The author reports there are no competing interests to declare When the treatment effect does not extend over the entire observation period the backward logrank testing procedure is more powerful compared to the power of a single logrank test at the end of the study. Critical values and power are obtained by simulation for repeated logrank tests at prespecified times and α distribution. A backward sequential logrank procedure, with testing at the end of study and if not significant then sequentially at prespecified earlier time points with a prespecified distribution of Type-1 error, is proposed. Therefore, it would be advantageous to incorporate a shorter duration of treatment effect into the study design. If the analysis using all events over the observation time does not show a significant benefit, then post hoc analyses at earlier times, which show nominal significance, would only be viewed as post hoc and suggestive of an effect. In survival studies the treatment effect on the hazard ratio may not extend over the entire observation time, yet the effect over a shorter time may be medically important.
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