On the asymptotic distribution for Peto's combined test for carcinogenicity assays under equal and unequal censoring

Abstract

Peto et al. (1980) proposed a combined test statistic for carcinogenicity studies in a pooled analysis of incidental and fatal tumours. For the analysis of the incidental tumours one needs to divide the time span of the study into subintervals. Rather than using the data‐driven method approach to create these intervals as was suggested by Peto et al. (1980), in practical applications one uses a fixed number of prespecified intervals. For this method we derive the asymptotic distribution of the combined test statistic under the null hypothesis, under both equal and unequal censoring distributions and under alternatives. It turns out that the decomposition of the time axis into a fixed number of intervals can cause a biased normal limiting null distribution with nonstandard variance. This effect may be negligible if the number of intervals is large. On the other hand, if there are only a few intervals, we propose a corrected variance estimator yielding an asymptotic normal distribution with standard variance in any case.