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Tests for a trend in proportion based on mixtures of beta distributions incorporating historical controls
Author(s) -
Tiwari Ram C.,
Zalkikar Jyoti N.
Publication year - 1999
Publication title -
environmetrics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.68
H-Index - 58
eISSN - 1099-095X
pISSN - 1180-4009
DOI - 10.1002/(sici)1099-095x(199901/02)10:1<1::aid-env332>3.0.co;2-k
Subject(s) - null hypothesis , statistics , negative binomial distribution , binomial distribution , statistical hypothesis testing
Carcinogen bioassay involves the exposure of laboratory animals to one or more levels of the test substance and an evaluation of the tumor occurrence rates in the treated groups relative to those in the unexposed controls. The historical control groups obtained from other similar studies also provide information concerning the rate of spontaneous tumor occurrence and may be useful in analyzing the results of the current experiment. This is especially relevant when the tumor is of rare type where the toxicologist places great significance on its occurrence in a treatment group. Under the assumption that historical control and concurrent control tumor rates are identically distributed, several authors have investigated the problem of statistical incorporation of the historical data into the current control group while testing for an increasing trend in the proportions across the treatment groups. Often this is not the case, as the historical data are gathered from studies subject to profound sources of variation, such as animal age, animal body weight, and pathologists which occur even under the same protocol. In this article we relax this assumption and develop locally most powerful tests for a trend in binomial response data by using the mixtures of beta distributions as models to accommodate the extra‐binomial variation in the historical controls. The asymptotic distributions of the test statistics are obtained under the null hypothesis of no trend in proportions and also for a sequence of Pitman alternatives converging to the null hypothesis. Using these results, the application of the tests with the experimental as well as hypothetical data sets is demonstrated. The robustness of these tests under small perturbations in the historical control data is studied. The study shows that the tests based on the mixtures of beta distributions are robust in the sense that their p ‐values do not fluctuate under small perturbations in the historical data, unlike some of the existing test procedures. Thus while incorporating the historical data, the mixture distributions are more appropriate than the existing procedures for modelling the variation in the historical controls. Copyright © 1999 John Wiley & Sons, Ltd.