Incorporating Historical Controls in Testing for a Trend in Proportions

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Abstract The testing of chemicals for carcinogenic activity in lifetime rodent bioassays has produced a large amount of data on a few strains of mice and rats. Because of the use of fixed protocols in repeated experiments, the opportunity presents itself for using this historical data in the analysis of a current experiment. What has been considered is the statistical incorporation of the historical control data into the current control group to increase the power of the test. This is especially relevant when dealing with tumors of a rare type where the toxicologist will place great significance on their occurrence in a treatment group. Experimentally it is assumed that three groups of about 50 animals each are tested at two treatment levels and a control. The response for each treatment group is assumed to be binomial. Further it is assumed that the probability of an animal with a tumor in the control group has a beta distribution that is determined from the historical control data. Several authors (Dempster, Selwyn, and Weeks 1983; Hoel 1983; Tarone 1982) have investigated the problem of testing for linear trend in proportions when historical information is available. In this article we give a locally most powerful test of trend for binomial response data by using a beta prior distribution for the historical control information. This generalized test is closely related to Tarone's statistic and generalizes the conditional test of Hoel. Under various conditions on the beta parameters (α, β), the asymptotic distribution of the test statistic is given under the null hypothesis H 0 of no trend in proportions and for a general sequence of alternative hypotheses that converge to H 0. Using these results, the asymptotic gain due to the incorporation of historical controls is given. For the usual bioassay design the asymptotic relative efficiency depends on α + β and is shown to be about 1.3 to 2.2, depending on the values of the beta parameters (higher efficiency for larger values of α + β). If the coefficient of variation for the beta-binomial is held constant, then large values of α + β correspond to small values of the mean of the beta-binomial distribution (i.e., rare tumors). The use of an exact conditional test is discussed, because for small beta-binomial means the approximations using asymptotic theory are somewhat unrealistic. Problems with the use of normality are illustrated numerically. Tables are given that compare the significance probability for selected outcomes for the exact and asymptotic tests. Also included are values for the Cochran—Armitage test, using no historical information. These tables show that the incorporation of the historical information greatly improves the power of the test for small means of the beta-binomial. The second observation is that the asymptotic test appears to exceed its nominal size. These observations were made using a mean of 1% for the beta-binomial. For large values of the mean, that is, 20%, the tests generally agreed. Thus the incorporation of historical data in general and the exact conditional test in particular is especially important where the spontaneous rates are small.