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Consider the inference '60% events +/- 3%' from a random sample of a binary frame. The complement is '40% non-events +/- 3%.

If expressed together (i.e. '60% events AND 40% non-events), would the two point estimates be considered two simultaneous inferences requiring adjustment of their interval sizes?

Or ... since the complement (non-events) is wholly correlated to the main estimate (events), does the lack of independent information mean the concept of adjusting intervals for simultaneous inferences not apply?

MCornejo
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    The usual adjustment for uncorrelated confidence intervals would set the combined interval at $\pm 4.2\%.$ But think about it: would you assert, with a straight face, that "the total number of events is $100 \pm 4.2\%$ of all events?" – whuber Oct 18 '18 at 15:53
  • *How* I adjust the interval sizes is of no importance: the issue concerns how much sense it makes in this circumstance to report any nonzero interval at all! – whuber Oct 18 '18 at 17:11
  • When you report the total as as fraction of the total, there is no sampling error: the value is 100%, period. That, at any rate, is what I understand by your use of the term "complement": by *definition,* the 40% of your example is 100% minus the 60%. Mathematically, when a random variable $X$ has values between $0$ and $1$ and $Y$ is defined as $Y=1-X,$ the variance of $X+Y$ is not the sum of the variances of $X$ and $Y$ separately: it's *zero*. – whuber Oct 18 '18 at 17:21
  • I trust the preceding dialog has at least established that the errors in the two estimates must be identical but of opposite signs! – whuber Oct 18 '18 at 19:05

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While the proportion estimate and its complement occur simultaneously, adjustment of confidence or interval size for multiple inferences does not apply.

This is because such adjustments apply to random variables and in this case there is only one random variable, the main proportion estimate. Rather than being random, the estimate of the the complement is completely and deterministically dependent on the first estimate.

Thus, the accuracy (i.e. correct/incorrect) of either confidence interval will always be that of the other. No adjustment is necessary.

My thanks to Magdalena Murawska for the pointing out the relevance of stochastic vs deterministic processes.

MCornejo
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