On the Exact Conditional Distribution for Peto's Combined Test for Carcinogenicity Assays

Abstract

The most frequently used approach to analyze data from carcinogenicity assays is the method introduced by Peto et al. [Peto, R., Pike, M. C., Day, N. E., Gray, R. G., Lee, P. N., Parish, S., Peto, J., Richards, S., Wahrendorf, J. (1980). Guidelines for simple, sensitive significance tests for carcinogenicity effects in long term animal experiments. In: Long Term and Short Term Screening Assays for Carcinogens: A Critical Appraisal. IARC Monograph on the Evaluation of the Carcenogenic Risk of Chemicals to Humans. Geneva: WHO, pp. 311–426, Annex to supplement 2]. These authors combine the logrank statistic used in the fatal context and the Mantel–Haenszel statistic appropriate for the incidental context into an overall test statistic. In this article, we investigate exact conditional distributions of this combined test statistic due to Peto et al. (1980), as well as that of its two components, within a rigorous mathematical framework. We derive exact distributions under different conditioning events, and we discuss their advantages and disadvantages. One of these conditioning events will lead to the well-known permutational distribution. These exact conditional distributions will be valid under the assumption of equal censoring distributions across the groups. An interesting result is that this assumption may be relaxed in the incidental context, at the expense of using a more restrictive condition than the one needed to obtain the permutational distribution. Clearly, less strict conditions lead to less discrete distributions and would therefore be preferable. However, relaxing the assumption of equal censoring across the experimental groups is an important goal because the censoring distributions usually differ across the experimental groups. We also show how these exact conditional distributions can be related to k × r tables and extended hypergeometric distributions. This is a very useful result because it allows one to apply efficient computer algorithms for k × r tables, such as those published by Mehta and Patel [Mehta, C. R., Patel, N. R. (1983). A network algorithm for performing Fisher's exact test in rxc contingency tables. JASA 78:427–434] or Mehta et al. [Mehta, C. R., Patel, N. R., Tsiatis, A. A. (1984). Exact significance testing to establish treatment equivalence with ordered categorical data. Biometrics 40:819–825]. Further results include limit theorems about the exact conditional distributions, which converge to the same limit as the unconditional distribution of the combined test statistic, as derived by Heimann and Neuhaus [Heimann, G., Neuhaus, G. (2001). On the asymptotic distribution for Peto's combined test for carcinogenicity assays under equal and unequal censoring. Biometrika 88:435–445] in an earlier article. We also discuss some previous proposals on how to obtain “exact” p values, and we show why these are wrong.

Key Words:

• Peto's combined test
• Logrank test
• Mantel–Haenszel test
• Carcinogenicity assays
• Exact conditional distributions
• Permutational distribution

Mathematics Subject Classification:

• 62 E15
• 62 G10

Acknowledgments