Weighted log rank type statistics for comparing survival curves when there is a time lag in the effectiveness of treatment

In certain long-term treatment trials, one expects some lag period before the treatment is fully effective. This paper presents two weighted log rank type statistics designed to have good efficiency over a wide range of lags: a maximin efficiency robust statistic and a simplified version of this statistic. Both statistics can be computed as easily as the log rank statistic. Asymptotic efficiency calculations, supported by small sample simulations, show that the two proposed statistics are substantially more efficient than the conventional log rank statistic in certain lag situations with comparatively little efficiency loss relative to the log rank statistic when no lag exists. We recommend the maximin statistic for situations where a lag is expected but cannot be specified precisely in advance.