Coverage of four nominal 95% intervals, n = 30

Computed exactly by summing over all 31 possible counts, not simulated. The Wald interval drops to 26.0% and is jagged everywhere; Clopper–Pearson never falls below 95% and pays for it in width.

Each stage draws a thousand points from the current proposal, keeps the top tenth, and refits the normal to them, each weighted by the target density over the proposal's. Fitting the mean alone, the levels are 1.24, 2.99, 4.59, 5.00 and the fit reaches 5 at stage 4, ending at N(5.199, 1). Fitting mean and spread from the same seed, the spread goes 1.000, 0.456, 0.244, 0.096, 0.054, 0.043 over the first six stages, roughly halving each time, and after 30 stages the level has stalled at 3.728 with a spread of 1.0e-6, never reaching 5.

Counted over 2,000 runs at a thousand final draws and 600 at ten thousand. Mean and spread fitted: 1.1% and 0.8%. Mean and spread fitted, each update smoothed at 0.7: 1.4% and 1.0%. Where the full fit is heading, N(5.186, 0.181²): 93.7% and 90.8%. Spread fitted, floored at 0.71: 93.8% and 94.2%. Mean fitted, spread held at one: 94.5% and 94.2%. The full fit, a tenth of draws from N(5, 1): 92.6% and 93.8%.

The second moment of the weights integrated from 5 to the typical largest of R draws, as relative error per draw. N(5.187, 0.181²), the normal closest to Z conditioned past 5: 0.77, 0.78, 0.80, 0.82, 0.85, 0.89, 0.95, 1.18. N(5.187, 1): 2.38, 2.38, 2.38, 2.38, 2.38, 2.38, 2.38, 2.38 — at 100, 300, 1,000, 3,000, 10,000, 30,000, 100,000, 1,000,000 draws. The first has an infinite second moment whose divergence starts at 5.549, inside the reach of a thousand draws (5.745); the second's is finite.

Coverage of the nominal 95% textbook (Wald) interval at 40 trials, conditional on the count being at most k or at least 40 − k, computed exactly over the counts for every true proportion. With k = 2 its highest value anywhere is 88.96%; With k = 5 its highest value anywhere is 94.90%; With k = 8 its highest value anywhere is 95.57%. A set whose highest value stays below 95% is a set of counts on which the label is wrong whatever the proportion is.

For each tail set of counts at most k or at least n − k, the highest coverage the nominal 95% interval reaches at any true proportion, conditional on the count being in the set. The textbook interval stays below 95% up to k = 4, 5, 7 at 20, 40, 100 trials — at forty, 76.01%, 88.96%, 92.84%, 94.31%, 94.90% for k = 1 to 5, and at k = 1 between 75.25% and 77.30% at all three sizes. The score interval at forty reaches 99.31% or more on every tail set, so none refutes it.

Above: the prior found by solving the betting game exactly — 41 atoms of weight spread across the whole range, the heaviest 7.2% at a proportion of 0.601. Below: for each of the 41 possible counts, the posterior probability under that prior that the nominal 95% interval built from the count contains the proportion. Every score (Wilson) interval is at least 96.46% credible, which is the certificate that no set of counts can bet against its label. Under the same prior the textbook intervals range from 0.00% at a count of zero to 96.83%.

For each interval and number of trials, the upper bound is the highest credibility some prior guarantees every interval in the family, and the lower bound the lowest credibility some prior holds every interval to; each is the exact value of a betting game. Score (Wilson): upper 98.39%, 97.46%, 97.05%, 96.71%, 96.51%, 96.46%, 96.03%, 96.04%, 95.77%, 95.56%; lower 92.00%, 93.09%, 93.17%, 93.08%, 93.25%, 94.09%, 94.22%, 94.18%, 94.36%, 94.76%. Agresti–Coull: upper 98.71%, 97.76%, 97.25%, 96.91%, 96.65%, 96.59%, 96.27%, 96.19%, 95.85%, 95.60%; lower 94.25%, 93.77%, 94.13%, 94.17%, 94.21%, 94.55%, 94.91%, 94.77%, 94.72%, 94.96%. Exact (Clopper–Pearson): upper 99.50%, 99.02%, 98.57%, 98.26%, 97.95%, 97.74%, 97.36%, 97.10%, 97.01%, 96.61%; lower 97.16%, 96.92%, 96.53%, 96.32%, 96.13%, 96.23%, 96.04%, 96.12%, 96.05%, 95.95% — at 10, 15, 20, 25, 30, 40, 50, 60, 80, 100 trials. The score and Agresti–Coull bounds straddle 95% at every size; the exact interval's lower bound is above it at every size.

Between proportions of 0.2 and 0.8 the arrangement coin's coverage stays within 0.121% of 95%, where mid-p's strays by 2.76%. Its worst coverage is 92.60%, at a proportion of 0.140, near the ends of the range where the count leaves few arrangements; mid-p's is 92.94%. Averaged over the proportion, 95.63% against mid-p's 96.11% and Clopper–Pearson's 97.71%.

The randomised interval for a Poisson mean, with its coin read from where in the window the events fell, covers exactly 97.5% at every expected count between 0.025 and ln 40 = 3.689 and exactly 95% past it, when a count of zero is given the exact interval's [0, ln 40]; its worst is 95.00%. Given mid-p's [0, ln 20] instead, it falls to 92.50% just past 3.00. Mid-p itself reaches 91.66% at 3.03; the exact interval never falls below 95.62% and averages well above 95%.

The worst coverage of the arrival-time interval at any expected count, against the upper limit of the interval reported when no event is seen: 97.5% less e raised to minus that limit, for limits below ln 40, and exactly 95% from ln 40 = 3.689 on. Mid-p's limit, ln 20 = 2.996, leaves the worst at 92.5%; the exact interval's, ln 40, is the shortest that leaves none; the score interval's, the square of 1.96 = 3.841, is longer than it needs to be and buys nothing more on the minimum.

Worst coverage over the whole range of the proportion or the expected count. The coin read from the order of the trials, whose count of none has one face and so reports mid-p's interval: 92.51% at 10 trials, 92.60% at 20 trials, 92.72% at 50 trials. With the counts of none and of all given the exact interval's one-sided limit instead: 94.56%, 94.79%, 94.93% — short of 95% only because a count of one still has n faces rather than a continuum. Read from arrival times, which give a count of one a continuum: 92.50% with mid-p's limit at none and 95.00% with the exact one.

With the rate rising linearly across the window, the coin read from the last arrival leaves the average rate below its interval 2.07% of the time and above it 3.34% at the largest drift, where the rate starts at zero, against 2.50% and 2.50% with none; its coverage is 94.59%. The folded coin, read from each event's distance to the window's middle, misses 2.50% and 2.50% at every drift: a linear trend moves no probability between distances from the middle.

Expected width as a share of the exact (Garwood) interval's. The arrival-time interval, with the exact limit at a count of zero: 95.8% at 0.25, 93.2% at 0.5, 90.6% at 1, 89.6% at 2, 90.0% at 3, 91.3% at 5, 92.7% at 8, 93.8% at 12. Mid-p: 83.4%, 85.0%, 87.1%, 89.4%, 90.6%, 92.0%, 93.2%, 94.1%. The score interval: 102.4%, 101.1%, 99.3%, 97.2%, 96.3%, 95.5%, 95.4%, 95.6%.

C(20, k) arrangements at each count: one at no successes, 20 at one, 190 at two, 184,756 at 10. At a proportion of 0.05 the counts of none and one carry 73.6% of the probability.

Reading the recorded order only, the test of p = 0.5 has size 4.999%; choosing the most favourable of 2, 4, 8, 16 orderings, 6.659%, 7.275%, 7.678%, 7.681% — up to the 7.681% of rejecting whenever any coin value would. For p = 0.3: 4.969%, 5.179%, 5.179%, 5.179%, 5.179%, against a ceiling of 5.179%.

The average proportion is 0.4 throughout. With no drift the arrangement coin misses 2.50% below and 2.50% above. At a drift of 0.3, 1.25% and 2.83%; its total coverage is 95.92%, a drawn coin's 96.54% and mid-p's 97.57%.

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.

How many times as many draws each companion is worth, on the same 4,000 simulated samples of 40 observations. The coverage of the interval is estimated with the observed count as its companion, whose expectation is 12 exactly; they correlate at 0.2665 and the companion is worth 1.08 times the draws. The expected width is estimated with p̂(1 − p̂) as its companion, whose expectation is 0.20475 exactly; they correlate at 0.9977 because the width is a monotone function of it, and the companion is worth 214 times the draws — 856 thousand simulated samples' worth of precision from four thousand.

Computed exactly by summing over all 31 possible counts, not simulated. The Wald interval drops to 26.0% and is jagged everywhere; Clopper–Pearson never falls below 95% and pays for it in width.

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.

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.

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.

Measured over 20,000 samples. The t interval covers 94.9% and the z interval 90.7%. The difference is the price of pretending the standard deviation was known.

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.

The t interval covers exactly 95% in every cell. Estimated at 1,000 replications its cells read 93.9% to 96.5%, and 2 of the twenty are flagged by their own ±1.96 standard errors.

Bars: cells of 1,000 t intervals on five normal observations, each from its own seeds. Line: Binomial(1000, 0.95)/1000, computed. The counted spread is 1.038 times the binomial variance, and the chi-square across the binomial's eight bins is 5.83 on 7 degrees of freedom.

Computed exactly from the binomial: one correct cell of 1,000 replications is flagged with probability 5.83%, so twenty cells flag at least one 69.9% of the time, and a hundred 99.8%. More replications do not repair it — at 10,000 a cell twenty flag 63.3% — while reading each cell at Bonferroni's level holds twenty to 7.6%. The point with its bar is a hundred counted tables: 72%.

A cell's true coverage below 95% by δ points is detected 80% of the time, at a two-sided 5%, after ((1.96·√(0.95·0.05) + 0.84·√(p₁(1 − p₁)))/δ)² replications. A thousand replications see 2.03 points; ten thousand see 0.62. The marked points are checked by exact binomial summation.

Each bar is the probability of one count, shaded by which interval built on that count contains 0.2. Both cover 95.1% of samples, only Wilson 0.0%, only Clopper–Pearson 1.6%, neither 3.3%. The correlation between their hits is 0.810, so on shared draws the variance of their difference is 4.891 times smaller than on independent ones.

Each dot is one simulation's estimate of the difference in coverage, whose exact value is -1.636 points. On shared draws the estimates spread with standard deviation 0.398 points (exact 0.401); on independent draws 0.880 (exact 0.887). The bracket under each row is ±1 exact standard deviation.

Each point is one pair of the Wald, Wilson, Agresti–Coull and Clopper–Pearson intervals at one sample size (20 or 50) and one true proportion (0.05 to 0.50), computed exactly. 44 of the 120 settings are omitted because the two intervals cover exactly the same counts. The ratio ranges from 0.859 to 5.619; 5 settings sit below one, where shared draws make the comparison noisier. The curve is 1/(1 − ρ), which the ratio would equal if the two coverages had the same variance.

Dots are counted over 400 simulations with ±2 standard-error bars; ticks are exact. Wilson–Clopper–Pearson, paired, shared draws: 92.5% (exact 92.65%); Wilson–Clopper–Pearson, unpaired, shared draws: 100.0% (exact 100.00%); Wilson–Clopper–Pearson, unpaired, independent draws: 93.5%; Wilson–Wald, paired, shared draws: 93.3% (exact 94.97%); Wilson–Wald, unpaired, shared draws: 90.0% (exact 93.07%); Wilson–Wald, unpaired, independent draws: 96.8%.

Each line is one run's running estimate; the dashed band is where the Wilson interval of the running estimate still contains 95%, and a run stops, marked, the first time it leaves the band. 8 of these twenty stop before 10,000 replications. The exact probability of stopping, from the recursion over the count, is 29.54%.

Exact, from the recursion over the binomial count. Checked every 100 replications a correct 95% interval has been flagged 35.61% of the time by 10,000; every 250, 29.54%; every 1,000, 18.83%. The point at 10,000 is 4,000 counted runs checked every 250: 29.13%.

Exact. Stopping at the first of forty looks where the estimate reads at least 95%, an interval whose true coverage is 94% is reported as reaching 95% 37.21% of the time, and one at 93% 12.28%. Run to a fixed 10,000 replications, the 94% interval reads 95% with probability 0.0009%. The point at 94% is 4,000 counted runs: 37.43%.

Bars: the chance the run ends reporting a coverage of at least 95%. a fixed 10,000 replications: 0.0009%, reporting 94.00% on average (exact); a fixed 1,500: 5.45%, reporting 94.00% on average (exact); stopped when it settles: 6.15%, reporting 93.99% on average (counted, 1496 used on average); stopped when it reaches 95%: 37.21%, reporting 94.59% on average (exact).

Exact. Reading the Wilson interval at z = 2.7628 — 0.573% a look — holds forty looks to 4.990% for a correct interval, where 1.96 flags 29.54%. Against an interval covering 94.5% the boundary detects 46.84%, a single analysis at 10,000 detects 62.67%, and the naive monitor 78.97% — most of it the same false alarms.

One seed each. Plain simulation draws nothing past 5 in 100,000 and estimates zero throughout. At 100,000 draws the proposal N(5, 1) reads 1.009 of the truth, N(9, 1) 1.041, N(4.5, 0.25²) 1.039 and N(5, 0.3²) 1.008. Values above 2.2 are drawn at the top edge.

The relative error per draw of importance sampling for P(Z > 5) with a N(m, 1) proposal, √(exp(m²)·Q(5 + m)/p² − 1), in closed form. It is smallest, 2.376, at m = 5.097, where 565 draws reach a 10% relative error. At m = 4 it is 3.347, at 7 6.343, at 8 21.52 and at 9 119.5. Plain simulation, m = 0, is 1868. The point at 5 is counted from 600 runs of 10,000 draws.

Each curve integrates φ(x)²/q(x) from 5 to the typical largest of R draws, exactly, and reports the relative error per draw it implies. N(4.5, 0.25²) and N(5, 0.3²) both have infinite second moments. The first one's visible error rises from 5.456 at 1,000 draws to 9.488 at 100,000; the second one's moves from 1.135 to 1.148, and does not begin to climb until well past any run counted here. N(3, 1) and N(5, 1) are finite and settle at 7.768 and 2.383. Points: the same quantity read off each run's own sample variance, the median over runs — N(4.5, 0.25²) at 1,000: 6.81, N(4.5, 0.25²) at 10,000: 8.67, N(4.5, 0.25²) at 100,000: 10.83, N(5, 0.3²) at 1,000: 1.15, N(5, 0.3²) at 10,000: 1.15, N(5, 0.3²) at 100,000: 1.15.

Counted over 2,000 runs at 1,000 draws, 600 at 10,000 and 150 at 100,000, with ±2 standard-error bars. N(5, 1): 94.3%, 94.7%; N(3, 1): 92.7%, 93.3%; N(4.5, 0.25²): 85.2%, 86.0%, 81.3%; N(5, 0.3²): 94.3%, 95.0%, 95.3%; N(9, 1): 19.3%, 54.5%. The last is below the plotted range.

For each proposal: the share of its draws landing past 5, the effective sample size of its weights as a share of all draws and of the landed draws, and the counted coverage of its nominal interval. N(5, 1): 50.00% landed, 14.96% and 29.9%, coverage 94.7%; N(5, 0.3²): 50.00% landed, 43.02% and 86.3%, coverage 95.0%; N(3, 1): 2.28% landed, 1.62% and 71.6%, coverage 93.3%; N(4.5, 0.25²): 2.28% landed, 1.20% and 52.9%, coverage 86.0%; N(9, 1): 100.00% landed, 0.02% and 0.0%, coverage 54.5%.

Read against the expected number of successes the four sample sizes draw the same curve near the boundary. The worst coverage is 83.50% at n = 10, 83.71% at n = 30, 83.79% at n = 100, 83.81% at n = 1000, each at an expected count near 0.177, and the limiting depth is e^(−0.1765) = 83.82%.

The coverage is the probability of the counts whose intervals contain the true proportion. Just below an expected count of 0.1765, 0 covers, and the coverage is 84.10%.

More trials do not bring the worst coverage of the first three to 95%. At 1,000 trials: Wilson 83.81%, Jeffreys 89.77%, Wilson, repaired near zero 92.72%, Agresti–Coull 94.36%. Wilson's approaches e^(−0.1765) = 83.82% and stops there.

The worst coverage over every proportion beside the average over a uniform one, with the average expected width. Clopper–Pearson: worst 95.05%, average 97.34%, width 0.299. Blaker: worst 95.00%, average 96.31%, width 0.283. Wilson: worst 83.71%, average 95.24%, width 0.271.

Both exact intervals stay at or above 95% everywhere; Clopper–Pearson sits higher. Wilson crosses the line in both directions. Coverage below 90% is drawn at the floor of the axis.

The chance the whole interval lies below the truth and the chance it lies above, at every proportion. The worst below is 2.50% and the worst above is 2.50%. Clopper–Pearson guarantees each side separately.

Clopper–Pearson never falls below 95% and runs up to 99.80%. The mid-p interval, which is the randomised interval with its coin fixed at one half, runs from 92.94% to 99.80%. The randomised interval covers 95% at every proportion, to within the 0.043% of the numerical integration over the coin.

The same data give a lower end anywhere from 0.0564 to 0.0323 and an upper end from 0.3783 to 0.3187, depending on a uniform draw nobody observed. The coin at one half gives the mid-p interval, 0.0396 to 0.3561. Clopper–Pearson's 0.0321 to 0.3789 contains every one of them.

Averaged over a uniform proportion. Wilson 0.2708, worst coverage 83.71%; randomised 0.2732, worst coverage 95.00%; mid-p 0.2759, worst coverage 92.45%; Blaker 0.2833, worst coverage 95.00%; Clopper–Pearson 0.2990, worst coverage 95.05%. The randomised interval covers exactly 95% at every proportion and is 0.9% wider than Wilson's.

The chance that a 95% interval contains a replication's estimate is 95.00% when the original landed on the truth, 82.99% one standard error away and 48.40% two away. Averaged over where originals land it is 83.42%, and 5.00% of originals capture a replication less than half the time.

A replication a tenth of the size lands inside 44.54% of the time, one of the same size 83.42%, and one ten times the size 93.83%. Only an infinitely large replication reaches 95%.

Only originals that reached p < 0.05 are kept. At 8% power their intervals capture a replication's estimate 52.49% of the time; at 52% power, 83.78%. Unselected intervals capture 83.42% whatever the power.

The replications share the original, so they miss together. None of 10 lands outside with probability 31.40%, against 16.32% if they were independent; at least half land outside with probability 8.72%, against 1.51%.

The intervals are separated, and the test of the difference gives p = 0.0056. Two 95% intervals with equal standard errors just touch at p = 0.0056.

Two 95% intervals with equal standard errors touch at p = 0.0056, and at a ratio of ten at p = 0.0319. Two ±1 standard error bars touch at p = 0.157 and 0.274. Neither kind of bar marks 5%.

For independent estimates with equal standard errors, touching intervals mark a difference of 2.77 standard errors. At a correlation of 0.5, 3.92; at 0.9, 8.77 — a p-value below 6e-10 already at 0.8.

At no true difference the ordinary test rejects 5.00% of the time and non-overlap occurs 0.56% of the time. At a true difference of 2.8 standard errors — 80% power for the ordinary test — the intervals fail to overlap 51.1% of the time.

Every group has the same true mean and the same standard error. With ten groups, some pair of 95% intervals fails to overlap 14.61% of the time, some pair of 83.4% intervals 62.74%, and some pair of ±1 standard error bars 92.32%.

Five hundred pairs. With the true effect identical in both, exactly one of the two is significant in 50.0% of pairs; among those, the difference between the two is significant in 9.7%. The dashed lines mark 1.96 on each axis; the diagonal lines mark a significant difference.

Two studies split — one significant, one not — with probability 2π(1 − π), peaking at 50% at half power. Among splits the difference is significant 9.8% of the time near that peak. With 2 subgroups of the same effect, at least one is significant and at least one is not with probability up to 49.9%.

At the sample that gives 80% power for the overall effect, a difference between the halves as large as the overall effect is detected 28.8% of the time, and one half as large 10.8%. Detecting them at 80% needs 4 and 16 times the sample.

The shading is the chance that exactly one of two independent studies is significant, over the plane of their true z means. It is highest on a cross centred where either mean is 1.96 and lowest in the corner where both are large, whatever the difference between them. The difference test's power is constant along lines parallel to the diagonal: 50% on the inner pair and 80% on the outer. Two studies at means 8 and 4 — one effect half the other — split 2.1% of the time, and the difference test finds the halving 80.7% of the time.

The two studies' true z means differ by 2 throughout and only their midpoint moves. The verdicts disagree 73.2% of the time at a midpoint of 2 and 0.00% of the time at 7; beyond a midpoint of about 4 both are nearly always significant. The test of the difference finds it 29.3% of the time at every midpoint.

A study sized for 80% power overall, split into two halves whose effects differ by a multiple of the overall effect. A split between the halves' verdicts is 1.46 times as likely when they differ by the whole effect as when they do not, and never more than 1.86 times as likely across the range. A significant test of the difference is 5.77 times as likely at the whole effect and 16.0 at twice it.

Five hundred pairs, correlated at 0.447 — the square root of its share — because the subgroup is part of the whole. The whole trial is significant and the subgroup is not in 57.5% of trials; the subgroup differs significantly from the rest in 5.0%. The dashed lines mark 1.96 on each axis and the slanted lines mark a significant difference between the subgroup and the rest.

The whole trial has 80% power when the effect is the same everywhere. At a subgroup of a fifth of the trial, "significant overall, not in the subgroup" happens 57.5% of the time when the subgroup has the same effect, 62.6% when it has half and 58.3% when it has none. The curves separate only at shares above a third, and there the reading becomes less likely when the subgroup has no effect.

Power against a subgroup that has no effect, when the rest has the effect that gives the whole trial 80% power if it were uniform. The right comparison is the subgroup against the rest. Treating the whole-trial and subgroup estimates as independent divides by too large a standard error: its size falls far below 5% and its power below the right test's at every share.

Four hundred patients, 80% power overall, characteristics whose latents correlate 0 (their split indicators 0.000). Some interaction test reaches 5% in 5.10% at 1, 9.13% at 2, 17.43% at 4, 34.63% at 8, 47.13% at 12, 56.23% at 16; the equicorrelated-normal integral gives 5.00%, 9.75%, 18.55%, 33.66%, 45.96%, 55.99%. A Bonferroni reading of the eight holds 5.57%.

The whole trial estimates 0.215 with a 95% interval from 0.019 to 0.411. Of the sixteen subgroups, 13 have intervals that include zero; the true effect in every one of them is 0.280.

Over 2,411 trials significant overall, none of the 16 subgroups is non-significant in 6.51%; the median count is 7.

Eight independent characteristics are 8.00 independent tests; at an indicator correlation of 0.333 (latents at 0.5) the integral gives 6.74 and the count 7.07; at 0.713 (latents at 0.9), 4.03 and 4.01.

A trial of 400 with 80% power for its main effect, eight characteristics, the first modifying the effect. At h = 0, 0.25, 0.5, 0.75, 1 the real interaction crosses 5% in 4.2%, 10.3%, 29.2%, 56.6%, 81.0% of trials; Bonferroni's 2.734 in 0.4%, 1.8%, 8.7%, 26.6%, 53.6%; the exact threshold for this overlap, 2.727, in 0.4%, 1.9%, 8.8%, 27.0%, 53.8%.

Over 2,000 trials a point. At h = 0, 0.25, 0.5, 0.75, 1: latent correlation 0, 12.3%, 20.6%, 39.3%, 63.6%, 82.4%; latent correlation 0.5, 11.7%, 19.7%, 35.6%, 56.3%, 76.1%; latent correlation 0.9, 12.3%, 16.6%, 28.5%, 41.1%, 54.4%. With no real interaction one characteristic in eight is the largest by chance.

Latent correlation 0.9, the first characteristic modifying the effect at h = 1. Interaction |z| by characteristic: 1: 2.67, 2: 2.16, 3: 1.55, 4: 2.49, 5: 1.99, 6: 1.33, 7: 2.58, 8: 2.88. The Bonferroni threshold is 2.73; the 5% threshold is 1.96.

Over 2,000 trials a point, at h = 0, 0.25, 0.5, 0.75, 1: another characteristic crosses in 3.1%, 6.2%, 14.3%, 29.8%, 50.2%; the largest crossing is another characteristic in 2.9%, 5.7%, 11.9%, 20.4%, 25.7%; only the real one crosses in 0.1%, 0.6%, 2.9%, 8.7%, 11.8%.

Expected interaction z when characteristic 1 modifies the effect at h = 1: 2.80, 2.00, 2.00, 2.00, 0.00, 0.00, 0.00, 0.00 — the real row 2.80, its three nested neighbours 0.713 of that, the four unrelated characteristics nothing. One seeded trial of 400 reads 1.91, 1.90, 2.06, 1.95, 0.14, 0.33, 0.39, 2.08.

Over 2,000 trials of 400 a structure at h = 1. eight nested: largest row 54.4%, pattern as if independent 53.6%, joint fit 45.6%; four nested + four unrelated, real one nested: largest row 63.5%, pattern as if independent 63.8%, joint fit 50.1%; four nested + four unrelated, real one unrelated: largest row 83.4%, pattern as if independent 74.3%, joint fit 82.2%; eight unrelated: largest row 82.4%, pattern as if independent 81.6%, joint fit 82.4%. The shape fitted with the overlap's own correlation names the largest row on every trial.

Over 2,000 trials of 400 a point, at h = 0, 0.25, 0.5, 0.75, 1: eight nested, 12.3%, 16.6%, 28.5%, 41.1%, 54.4%; four nested, four unrelated, 9.4%, 14.4%, 30.3%, 47.3%, 63.5%; eight unrelated, 12.3%, 20.6%, 39.3%, 63.6%, 82.4%. One in eight is chance.

The smallest set holding 90% of the posterior over which characteristic is the modifier. Average size at h = 0, 0.25, 0.5, 0.75, 1: eight nested, 7.22, 7.06, 6.63, 5.89, 4.96; four nested, four unrelated, 6.84, 6.67, 6.16, 5.27, 4.10; eight unrelated, 6.75, 6.63, 6.22, 5.30, 3.86. It holds the real modifier in 93.0% and 95.5%, 92.8% and 97.4%, 93.5% and 98.9% of trials at h = 0.5 and 1.

The unconditional coverage of three intervals for the ratio of two Poisson rates, each built conditionally on the total and each reporting the whole of (0, ∞) when the total is zero, at the conditional proportion 0.5, against the expected total count. With the arrival times in both windows as the coin the interval covers exactly 95% plus a twentieth of e raised to minus the expected total: 96.84% at an expected total of one, 95.25% at three, 95.00% from about ten. The exact conditional interval covers 100.00%, 99.79%, 97.82% and 96.55% at forty; mid-p 99.11% at three and 95.89% at ten.