**Date & Time:** Thursday, June 23, 2011 :: 3:30 p.m.

**Venue:** Ramanujan Hall

**Title:** A simple nonparametric test for testing treatment versus control

**Speaker:**Prof. Madhava Rao K.S.,
Department of Statistics, University of Botswana,
Private Bag 00705, Gaborone

**Abstract:**
In testing whether a treatment has an effect or not, the experimenter is often obliged to use the same
subjects for control and treated groups. In such a case it is generally unrealistic to assume independence and one is led
to tests of bivariate symmetry. In the nonparametric setup, Hollander (1971) proposed a conditionally distribution-free
test for bivariate symmetry based on the sample distribution function. The hypothesis of interest considered by him
was that the bivariate random variables

(X , Y )are exchangeable. In this paper we propose a simple nonparametric test for testing the hypothesis of exchangeability of X and Y . The motivation for the proposed test is based on the fact that the hypothesis of exchangeability can be shown to be equivalent to the hypothesis of conditional symmetry about an axis and therefore a test of the former hypothesis can be addressed through a test of the latter hypothesis. The proposed test is based on a measure of deviance between the observed counts of pairs of bivariate samples that are found in suitably defined clusters of sets and their mirror images. The exact null distribution of the test statistics is tabulated for sample size from n = 5 up to n = 30. The null mean and null variance of the test statistics are obtained and the asymptotic null distribution of the proposed test statistics turns out to be chi-square. The empirical power of the proposed test is evaluated by simulating samples from suitable class of bivariate distributions. The empirical findings suggest that the test performs reasonably well against various classes of asymmertic bivariate distributions as compared to its parametric (Bell and Haller (1969) and non- parametric competitors (Hollander(1971) and Wilcoxon(1945)).