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Intervals, counted

An interval that claims 95% is making a checkable statement about a procedure. Build every possible sample and count. The interval taught first fails, the failure is worst where proportions are most often reported, and more data does not monotonically help.

Coverage of four nominal 95% intervals, n = 20. Computed exactly by summing over all 21 possible counts, not simulated. The Wald interval drops to 18.2% and is jagged everywhere; Clopper–Pearson never falls below 95% and pays for it in width.

What the 95% refers to

An interval that claims 95% is making a checkable statement about a procedure, not about the interval in front of you. Build every possible sample and count, and the interval taught first turns out to cover 87.6% of the time.

Coverage against sample size, true proportion 0.15. Coverage does not improve monotonically. n = 19 covers 93.8% while the larger n = 20 covers 81.9%. The sample space is discrete, so the endpoints jump as n changes.

More data is not monotonically better

Coverage of an interval for a proportion does not improve smoothly as the sample grows. It oscillates, and there are larger samples that cover materially worse than smaller ones — a sample of twenty covers twelve points worse than a sample of nineteen.

Twenty 95% intervals for a proportion that really is 0.35. 2 of the twenty miss the true value. The 95% is a property of the procedure across repetitions — no single interval has a 95% chance of anything, because it either contains 0.35 or it does not.

Twenty intervals and one expected miss

The 95% belongs to the procedure, not to the interval in front of you. Twenty intervals from twenty samples make that visible in a way no definition does, and the one that misses is not a mistake.

Expected width against coverage, n = 30, p = 0.15. The Wald interval is the shortest and covers 94.2%. Clopper–Pearson covers 98.3% and is 13% wider. Shortness is not a virtue on its own — an interval of zero width is the shortest of all.

The shortest interval is the one that misses

Four intervals for the same data, with their widths and their coverage measured together. The narrowest is the one that fails its stated level, which is exactly why it looks the most appealing.

Where the bootstrap works and where it does not. Uniform data on [0, 1]. For the mean the percentile bootstrap covers 93.5%. For the maximum it covers 0.0%, because a resample can never contain a value larger than the largest one observed, so the interval cannot reach above it.

Where the bootstrap lies

Resampling is the most generally useful trick in the subject and it has a failure mode that is easy to state: it cannot see past the data. For a statistic that lives at the edge of the sample, coverage collapses from 95% to almost nothing.

Student's t on 5 degrees of freedom, against the normal. The two-sided 95% critical value is 2.571 for t(5) and 1.960 for the normal — 31% wider. Using the normal at this sample size makes every interval too short by that much.

The correction for not knowing the spread

The t distribution exists because the standard deviation is estimated rather than known. At eight observations, using the normal instead makes every interval 12% too short — and the coverage that follows can be measured rather than argued about.

Three promises, and no procedure keeps all three. Average coverage and worst-case coverage for four 95% intervals for a proportion at n = 40, computed exactly. Their expected widths are 0.2418, 0.2417, 0.2472, 0.2641 in the same order. The textbook interval and the score interval have the same expected width to four digits — 0.2418 and 0.2417 — and worst-case coverages of 55.31% and 92.21%. The exact interval never breaks its promise and is 9.3% wider than the score interval to do it. Each of the three columns orders the four procedures differently.

An interval that covers and says nothing

A procedure returning the whole line 95% of the time and the empty set otherwise has coverage exactly 95% at every parameter value. Two real intervals at forty observations have expected widths of 0.2418 and 0.2417 and worst-case coverages of 55.31% and 92.21%.

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