Construction of multinomial distributions under mean constraints and its application in power analyses

In designing clinical studies, we often need to conduct power analyses for ordered categorical response when a desired treatment effect is expressed in terms of a mean change in category scores, such as disease severity. It is not obvious what alternative distributions should be for multinomial cases to test against a null distribution. This article first describes how a change in mean score can be translated into a shift in multinomial probability distributions. Then systematic schemes to construct multinomial distributions that meet such mean value constraints are proposed, along with applications to power analyses of ordered categorical data. Specifically, sample size calculations are presented and compared among the three commonly used test procedures: chi-square test, ridit analysis, and a likelihood-based score test under proportional odds model, with our proposed constructions. Finally, upper and lower bounds for such sample sizes are presented through single objective optimization methods.