ENH: power and sample size if null distribution is also noncentral

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Assessment

Difficulty
5/5
Estimated time
Over a week
Newbie friendliness
25/100
Issue type
Feature
Clarity
Needs clarification
Activity status
Stale
Tech stack
python
Domain
data

Research direction

Start by reading the current power classes and the discussion in pull request #6526, especially the linked comment. Map how null and alternative distributions are currently represented, then define the expected behavior for cases with noncentral distributions under both hypotheses and verify it against the power and sample-size use cases described here.

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Description

comp-stats type-enh

see
https://github.com/statsmodels/statsmodels/pull/6526#issuecomment-628328101

This is not really clear to me, but I guess that there is a general pattern that we will eventually need.

In the effect size and meta-analysis literature there are many cases where non-central distributions are important, e.g. standardized mean difference has a non-central t distribution, but meta-analysis works mostly with a normal approximation.

We get a noncentral t distribution if we don't remove the mean of the normal numerator, i.e.
A normally distributed random variable with unit variance and non-zero mean divided by a Chi-squared distributed random variable is noncentral t-distributed, where the noncentrality parameter is equal to the mean of the normal random variable.
https://en.wikipedia.org/wiki/Noncentral_t-distribution#Characterization

power computation would have noncentral distribution both under the null and under the alternative.

In equivalence testing for anova, we have noncentral F under the null and central F under the alternative (defined as all means equal).

The current power classes assume we have central distributions under the null and noncentral under the alternative, except for normal distribution case where both are the same.

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