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I'm building binomial proportion confidence intervals for a patient dataset containing the frequency of home nursing visits during the week prior to hospital admission. The freq. of home visit categories are integer values of 1, 2, 3,... and observations for a particular patient can be assumed to be independent from other patients.

It looks something like this for a dataset of 100 patients:

No. Visits:   Frequency:      % Total:
    1             10             10%
    2             50             50%
    3             30             30%
    5             10             10%
Total:           100            100%

So in this example I'm building separate 95% CIs for $p_1 = 0.10$, $p_2 = 0.50$, etc. with $n=100$.

I'm using the Rule of Three to compute 95% CI $= [0, 3/n]$ for frequency of visits (in this case No. Visits $= 4$) which were not observed in the dataset. However it occurred to me I could mechanically use the Rule of Three to compute 95% CIs $= [0, 3/n]$ extending for No. Visits $= 6$ all the way up to infinity.

Is this reasonable?

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    $\begingroup$ In case anyone is unfamiliar with this, OP means this "Rule of Three". $\endgroup$
    – Glen_b
    Commented Mar 19, 2013 at 1:12
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    $\begingroup$ Does this related question assist you? I recommend the Wilson-score based "quasi-rule-of-3.84" which is $\text{CI}(0.95) = [0 , 3.841459/(n+3.841459) ]$. $\endgroup$
    – Ben
    Commented Jul 30, 2021 at 23:25
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    $\begingroup$ It's hard to see how it would be useful to construct CI's for an infinite sequence of events that didn't happen, especially because if you actually use them all, you ought to correct for multiple comparisons, which would make them all equal to the interval $[0,1]$! $\endgroup$
    – whuber ♦
    Commented Aug 7, 2023 at 21:37
  • $\begingroup$ @Ben Yes, thank you, I'll read the responses. $\endgroup$
    – RobertF
    Commented Aug 10, 2023 at 2:11

2 Answers 2

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Here is a paper on the rule of three in the clinical setting: https://onlinelibrary.wiley.com/doi/10.1111/j.1445-2197.2009.04994.x

The bottom line is that the rule of three works quite poorly, but there are better alternatives available at the click of a button!

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    $\begingroup$ The abstract is unclear: it seems to admit the rule of three works just fine in the cases where it's supposed to be applied, namely "the initial complication rate is zero." Since this is behind a paywall and you are a co-author, I would like to invite you to take this opportunity to elaborate on this conclusion and clarify what it actually says. $\endgroup$
    – whuber ♦
    Commented Aug 7, 2023 at 22:23
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    $\begingroup$ Well, the conclusion is that the rule of three should not be preferred to other methods for any reason other than being trivially calculable. The fact that clinicians use it as a default even with non-zero cases is to be discouraged. If I recall correctly, we suggested that Wilson's scores intervals were preferable among the standard methods, but any other than the Wald intervals are OK. (Sorry about the paywall. I don't any more have a copy myself, and my university turned off my library and email after I retired after 29 years.) $\endgroup$
    – Michael Lew
    Commented Aug 8, 2023 at 21:25
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    $\begingroup$ The "even in non-zero cases" is the key, because (of course!) it applies only to when the outcome is a count of zero. Because that is theoretically justified (and works well, IMO), the distinction between a correct and an incorrect application of the rule is a crucial one. $\endgroup$
    – whuber ♦
    Commented Aug 9, 2023 at 13:43
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    $\begingroup$ @Whuber I agree. However, it is my opinion that telling people to use a method that works only in some circumstances is less optimal than telling them to use a method that always works. People who are not focussed on statistics tend to get confused about the conditions necessary for methods to be reliable and therefore ignore them. $\endgroup$
    – Michael Lew
    Commented Aug 9, 2023 at 22:03
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This are number of home-nursing visits during one week prior to hospitalization, so if there is a maximum of one visit per day (probably unreasonable) the maximum would be seven. I think the way up to infinity is unreasonable!

You could summarize by saying that the events 4 visits and 6 or more visits where unobserved, and use the rule of three for those. For a descriptive analysis, that could be enough, but maybe think about smoothing the probability estimates via a model or otherwise.

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