Testing monotonic trends among multiple group means against a composite null

In biomedical applications, it is often of interest to test the alternative hypothesis that the means of three or more groups follow a strictly monotonic trend such as {micro}1 > {micro}2 > {micro}3 against the null hypothesis that the group means are either equal or unequal but are not monotonic. This is useful, for example, for detecting biomarkers whose level in healthy, low-risk cancer and aggressive cancer subjects increases or decreases throughout the three groups. Various trend tests are available for testing monotonic alternatives. However, existing methods are designed for a highly restrictive null hypothesis where all group means are equal, which represents a special case of the null space in our problem. We demonstrate that these methods fail to control type I error when the group means may be unequal under the null. To test this broader null hypothesis, develop a greedy testing method which has an intuitive interpretation related to two-sample t tests. We show both theoretically and through simulations that the proposed method effectively controls type 1 error throughout the entire null space and achieves higher power than a naive implementation of multiple t-tests. We illustrate the greedy trend test method in real data to study microbial associations with parasite-related pathology in zebrafish.