International Journal of Statistics and Probability
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.
Mutter, Louis and Kim, Steven B., "Using Simple Alternative Hypothesis to Increase Statistical Power in Sparse Categorical Data" (2017). Mathematics and Statistics Faculty Publications and Presentations. 17.