Nonparametric Tests for the Umbrella Alternative in a Mixed Design for a Known Peak

Abstract

When an assumption from a parametric test cannot be verified, a nonparametric test provides a simple way of conducting a test on populations. The motivation behind conducting a test of the hypothesis is to examine the effect of a treatment or multiple treatments against one another. For example, in dose-response studies, monkeys are assigned to k groups corresponding to k doses of an experimental drug. The effect of the drug on these monkeys is likely to increase or decrease with increasing and decreasing doses. The drug’s effect on these monkeys may be an increasing function of dosage to a certain level, and then its effect decreases with further increasing doses. An umbrella alternative, in this case, is considered the most appropriate hypothesis for these kinds of studies. Tests statistics are proposed to test for the umbrella alternative in mixed designs consisting of combinations of a Completely Randomized Design (CRD), a Randomized Complete Block Design (RCBD), an Incomplete Block Design (IBD) and a Balanced Incomplete Block Design (BIBD). Powers obtained were based on a variety of cases. Different proportions of blocks to different sample sizes of a Completely Randomized Design portion were considered. In all treatments, equal sample sizes for the Completely Randomized Design were considered. Furthermore, an equal number of blocks of a randomized complete block design to an Incomplete Block Design and Balanced Incomplete blocks were considered. Studies in a Monte Carlo simulation were conducted using SAS to vary the design and to estimate the test statistic powers to each other. The underlying distributions considered were normal, t and exponential.