I'm conducting a simulation study to compare different methods of constructing CIs for a single proportion. The traditional (Wald/Asymptotic) CI can estimate the upper and lower bounds above 1 and below 0, respectively. In these cases, researchers usually truncate the value to the appropriate bound. My question is, if I want to calculate interval width, do I calculate it before or after truncation? By interval width, I am referring to the upper bound - lower bound.
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$\begingroup$ The answer likely depends on how you intend to interpret "interval width:" what do you have in mind? $\endgroup$– whuber ♦Commented Oct 16, 2019 at 18:43
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$\begingroup$ @whuber I should have clarified that. By interval width, I am referring to (upper bound - lower bound). $\endgroup$– Emma JeanCommented Oct 16, 2019 at 18:44
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1$\begingroup$ I think you should also motivate why you want to compare the widths. It seems odd to report an interval that includes values outside the parameter space. $\endgroup$– HStamperCommented Oct 16, 2019 at 18:48
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$\begingroup$ I want to use it to compare how "conservative" the intervals are. I've seen interval width used in other papers that compare CIs. The Wald instance is the only interval that I am looking at that can fall outside 0 and 1 so I'm not entirely sure how to handle it. I could throw out the "invalid" intervals but I think that is not realistic. $\endgroup$– Emma JeanCommented Oct 16, 2019 at 18:50
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2$\begingroup$ Do it both ways, then, and compare your results. One of the methods will give the same coverage and same power as the other but will yield a smaller average width, allowing you to ignore the other method. $\endgroup$– whuber ♦Commented Oct 16, 2019 at 19:44
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This is an interesting question. Initially, would argue that, using classical theory, you should truncate since the CI (or confidence set) should be a subset of the parameter space (or family set).
However, an example caught me thinking, and I do not know if that is the best approach. Imagine that the true proportion is $\mathbf{p} = 0.95$, and you have two methods two construct CI's. The first method produces a CI of $(0.88,0.96)$, while the second returns $(0.94,1.04)$. Both CI's contain $\mathbf{p}$, the first one is shorter than the second without truncation, but is larger if you truncate.
If the parameter is near the boundary and you have a method "biased towards" the boundary and other one which is not, the comparation may not be a "fair" one if you truncate. @whuber suggestion seems the most appropriate.
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2$\begingroup$ My thoughts exactly, Lucas! Thank you for sharing your ideas. $\endgroup$ Commented Oct 16, 2019 at 21:00