Parametric and non-parametric tests for the overall comparison of several treatments to a control when treatment is expected to increase variability.

Abstract

We consider the problem of making an overall comparison of several treatments to a control where experimental units are randomly assigned to either the 'control' group which receives no treatment or to one of k-1 'treatment' groups. We assume that the effect of the treatments is, if anything, a location shift possibly accompanied by an increase in scale relative to that of the control group. The ANOVA F test loses considerable power in such circumstances. A modification of the ANOVA F test has been proposed which uses the variance estimate from the controls in place of the usual pooled variance estimate. However, this modification has shortcomings when k exceeds two and the variances of the treatment groups are not inflated. We develop a combination procedure to avoid the pitfalls of the modified and usual F tests. We then propose parametric and non-parametric implementations of a likelihood ratio test that more efficiently incorporates the assumptions of this problem, yielding a test with a high power profile over a large range of normal alternatives. We use simulations to compare the power of the competing tests against several alternatives for normal and non-normal data.