Permutation Tests Using Estimated Distribution Functions

Abstract

Abstract In this article we develop permutation tests for estimated distribution functions. The tests are formed by averaging a functional of estimated distribution functions that are calculated from independent sampling units, where the units may be a single response, a set of repeated responses, or a censored response. We study primarily two functionals—the difference in means functional and the Mann-Whitney functional, and two types of responses—repeated conditionally independent responses and censored responses. For repeated responses, the permutation test using the difference in means functional produces a permutation form of the corresponding mixed-effects test. A new permutation test is developed when we apply the Mann-Whitney functional to the repeated responses. This is a case in which the rank-transform method does not work. On the other hand, for right-censored or interval-censored data, we obtain permutation forms of standard rank tests using the Mann-Whitney functional (or weighted forms of the functional), and the difference in means functional gives new tests. The latter tests generalize the permutation t-test and the mean-based permutation tests to censored data. These permutation tests are valid for all sample sizes and do not need weights for stabilization like the weighted Kaplan-Meier statistics. We perform the permutation tests on two examples, one with repeated measures and one with interval-censored responses.