Using Simple Alternative Hypothesis to Increase Statistical Power in Sparse Categorical Data

There are numerous statistical hypothesis tests for categorical data including Pearson's Chi-Square goodness-of-fit test and other discrete versions of goodness-of-fit tests. For these hypothesis tests, the null hypothesis is simple, and the alternative hypothesis is composite which negates the simple null hypothesis. For power calculation, a researcher specifies a significance level, a sample size, a simple null hypothesis, and a simple alternative hypothesis. In practice, there are cases when an experienced researcher has deep and broad scientific knowledge, but the researcher may suffer from a lack of statistical power due to a small sample size being available. In such a case, we may formulate hypothesis testing based on a simple alternative hypothesis instead of the composite alternative hypothesis. In this article, we investigate how much statistical power can be gained via a correctly specified simple alternative hypothesis and how much statistical power can be lost under a misspecified alternative hypothesis, particularly when an available sample size is small.