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

Peto et al. (1980) proposed a combined test statistic for carcinogenicity s tudies in a pooled analysis of incidental and fatal tumours. For the analys is 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 cr eate these intervals as was suggested by Peto et al. (1980), in practical a pplications one uses a fixed number of prespecified intervals. For this met hod we derive the asymptotic distribution of the combined test statistic un der the null hypothesis, under both equal and unequal censoring distributio ns 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 nu ll 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 f ew intervals, we propose a corrected variance estimator yielding an asympto tic normal distribution with standard variance in any case.