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Abstract:
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In biomedical studies, it is often of interest to test the alternative hypothesis that the means of three or more groups follow a monotonic trend such as u1>u2>u3 against the composite null 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-grade cancer and aggressive cancer subjects increases or decreases throughout the three groups. Various methods are available for testing monotonic alternatives. However, they 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 show that these methods do not control type I error when the group means are unequal and thus fail to disentangle a monotonic trend from a trendless pattern. To test the broader null, we develop a greedy testing method motivated by the generalized likelihood ratio test. We show the greedy test effectively controls type 1 error under the entire null space and achieves higher power than several naive methods. We illustrate our method on a real data analysis to study microbial associations with pathogen susceptibility.
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