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Building on this question:

Confidence interval and sample size multinomial probabilities

In a binomial confidence interval, a 90% two-sided CI corresponds to a 95% one-sided CI. The question above illustrates that each multinomial proportion can be treated as a binomial (that category, vs. not that category) and a CI constructed using the same formulas that work for a binomial. What I am curious about is whether the relationship between the two-sided and one-sided CIs holds true in the multinomial application of the binomial CI formulas.

Do I just have to follow the procedure in the linked post to establish 90% two-sided CIs, upon which I have 95% confidence that the true probability of each value is strictly greater than the lower bound or strictly less than the upper bound of that two-sided CI? Are there any other considerations?

Also, from my understanding, the Wald approximation is not great particularly for probabilities close to 0 or 1, and Clopper-Pearson is better. Is there some existing Python implementation from say statsmodels that statisticians rely on for this purpose?

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  • $\begingroup$ The focus of this question is not clear. The multinomial CIs are Binomial CIs. Are you perhaps indirectly trying to ask about the wisdom of applying corrections for multiple tests that you might be tempted to make? $\endgroup$
    – whuber ♦
    Commented May 3 at 19:48
  • $\begingroup$ If the multinomial CIs are binomial CIs, then that answers the second paragraph of my question in the affirmative. I am indeed also asking about any corrections that may need to be made given assessing multinomial probabilities implies running multiple binomial tests. I'm also asking about what to do to ensure that the intervals for probability values close to 0 and 1 are accurate. $\endgroup$
    – user11629
    Commented May 3 at 19:54
  • $\begingroup$ I don't see why multinomial confidence intervals are binomial. It's a joint interval on multiple parameters similar to multiple testing problem. statsmodels has two-sided but no one-sided intervals statsmodels.org/dev/generated/… (I have not looked at it in 8 years.) $\endgroup$
    – Josef
    Commented May 3 at 20:11
  • $\begingroup$ (correction) pointwise intervals are binomial, $\endgroup$
    – Josef
    Commented May 3 at 20:15
  • $\begingroup$ statsmodels.org/dev/generated/… also only has two-sided intervals for one proportion. However, AFAICS, all methods except inverting binom test are "central", equal-tail probabilities, so the one-sided interval is the same as one side of the two-sided interval for doubled alpha (2*alpha) $\endgroup$
    – Josef
    Commented May 3 at 20:23

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