FDA Animal Toxicology Data Yields Sky-Scraper Distribution for Stratified Trend Test

The data for this example were provided by the U.S. Food and Drug Administration (FDA). Animals were treated with four dose levels of a carcinogen and then observed (at necropsy) for the presence or absence of a tumor type. The data were stratified by survival time (in weeks) into the four time-intervals 0-50, 51-80, 81-104, and terminal sacrifice. Since there were no tumors found in the first time-interval, this stratum may be excluded from data entry. The data for the remaining three strata are given below:

Since the scores are unequally spaced, we use the stratified Permutation test with general scores (Breslow and Day, Statistical Methods in Cancer Research, 1980, page 148) to determine if there is a dose-response relationship between the level of carcinogen and the presence of tumors. The test statistic is of the form:

where wj is the jth dose of carcinogen (w1 = 0, w2 = 1, w3 = 5, w4 = 50), and Xij is the number of animals with tumors at the jth dose level in the ith stratum. The p-value results from StatXact are tabulated below:

StatXact shows that there are large differences between the exact and asymptotic one-sided p-values, and they lead to different conclusions about the significance of the dose-response relationship. They also show that the usual practice of doubling the one-sided p-value is unnecessarily conservative with asymmetric distributions. But the most interesting finding of all is that the distribution of T has multiple towers. A normal approximation would be seriously misleading. This is shown below: