A one-sided nonparametric multiple comparison control percentile test: treatments versus control

Abstract

SUMMARY The control median test studied by Gart (1963) is generalized to a control percentile test in which there may be more than one treatment. Exact and asymptotic null distributions are obtained, the latter being that of the minimum of normal variates having a common correlation. Tables for the control median test are provided for small sample sizes. In applied research it is common to be confronted with the problem of deciding which, if any, of several treatments are better than a standard or control treatment on the basis of observations from groups of items to which each of the treatments will be applied. For example, in drug screening for a type of cancer an experimenter desires to know whether any of several doses of a drug is suited to alleviating the disease by prolonging survival. The dose levels to be examined, together with a standard treatment, are all assigned to different groups of test animals, and the observations consist of the survival times of the animals after tumour implantation and subsequent drug application. Hence, the ordered observations are obtained sequentially, and, since an inordinately long period of time may pass before all observations are recorded, an analysis is desired based on those in hand when a preselected number of control animals have died. For a situation such as this, the easily computed nonparametric test described below permits a rapid assessment of the null hypothesis that there is no difference among all the populations considered, relative to the alternative that the percentile of interest of at least one of the treatment populations is greater than that of the control. The statistic on which this test is based is the minimum of the numbers of observations in each treatment sample which do not exceed the preselected ordered observation in the control sample. Clearly, small values of the statistic militate against the null hypothesis in favour of the alternative. For the application cited above, suppose that the percentile of interest were the median and that the experimenter placed nO = 13 animals under standard treatment and n = 9 animals under each of p = 4 different doses of the drug. Then, having found x = 0 dead animals under some dose level when the seventh control animal died, he could, by means of Table 3 below, assert with a significance level of 3 67 % that that dose level is effective in prolonging median survival relative to the control treatment. Dunnett (1955) provided a one-sided experimentwise test for means in complete samples when the distributions involved are normal. In the realm of nonparametric many-to-one tests which permit early analysis of data, P. Nemenyi in an unpublished thesis mentioned by Miller (1966, p. 185), studied the joint, rather than control, median test based on data