The Use of Historical Control Information in Testing for a Trend in Proportions

Abstract

A problem commonly faced in biometric applications is to test for a trend in proportions in a 2 x k contingency table in which the k columns have a natural ordering. For example, a test compound may be administered at increasing dose levels to groups of test animals to determine if the proportion of animals developing a tumor increases with increasing dose level. Cochran (1954) and Armitage (1955) proposed a chi square statistic for testing for a trend in proportions. For problems with a large amount of historical data, a test for trend incorporating the historical data would be useful. This paper develops such a test, based on a statistic which is closely related to that of Cochran and Armitage. This method for binomial proportions is analogous to the method derived by Pocock (1976) for normallydistributed data. The problem will be developed in the context of an animal carcinogenesis experiment. Consider an experiment involving r+1 groups of test animals. One group serves as a control group, and the remaining r groups are given a test compound at increasing dose levels, dld2<<<<<dr The animals are followed for a fixed period of time, and the number of animals which develop a tumor at a particular organ is recorded for each group. Let ni denote the number of animals in the ith group, and let the number of these animals which develop a tumor during the course of the experiment be denoted by xi, i 0, 1, . . ., r. The results of this experiment can be summarized in a 2 x (r + 1) table such as Table 1. To test for ap increase in the proportions Pi = xJni with increasing dose level, Cochran and Armitage suggested the test statistic