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What does the width of the confidence interval imply about the statistical power of a given study? I specifically want to figure out if a narrow confidence interval reasonably excludes a lack of statistical power to detect a given effect size as an explanation for a large p value.

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If the CI is narrow in some meaningful way (e.g. the width would not be a meaningful effect size), then typically the analysis was well powered. Exceptions could include cases, where the assumptions of the used analysis are violated (e.g. using Poisson regression in the presence of substantial overdispersion, ignoring observed or unobserved factors influencing group assignment in an observational study, informative censoring in standard survival analysis etc.), the study design is flawed (i.e. at best provides a biased estimate of the estimand of interest) or the results are misinterpreted.

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