Normal Distribution
nothing is called 95% until it has been counted
Concept

Confidence interval — where it appears

A range computed from data by a rule that would contain the truth in a stated share of the experiments that might have been run. The statement is about the rule rather than about the interval in hand, which is why coverage can only be counted over repetitions.

Named by 49 essays across 22 fields — each of them below, with the objects they name alongside it.

One experiment, with the blocks getting smaller as the target comes into range. A single run at a requirement of 0.25, with the block sizes 5, 5, 11, 25, 11, 8, 3, 2 and a total of 70 observations in 8 blocks. The rule stops when the observations in hand reach z²σ̂²/d², with σ̂² pooled from the within-block contrasts — an estimate that moves as the run goes on, so the target moves too. Early blocks are large because the target is far away and cannot be overshot; late ones are small because a block is the granularity of the answer. The interval afterwards is built from the 8 block means and from nothing the rule looked at, and it has 7 degrees of freedom against the rule's 62.

A block size that changes

The blinded rule's exactness never needed the blocks to be the same size. Letting the size be chosen from the contrasts as the run goes on leaves the coverage exactly where it was — and runs straight into an identity that says what a schedule can and cannot buy.

pace · Stopping The construction survives a difference of two weighted means. Coverage of δ̂ ± t√(S_D²/H) on b − 1 degrees of freedom, over 900 runs at a requirement of 0.3, where δ̂ is the block differences weighted by h_b = (1/m_A + 1/m_B)⁻¹ and H is their total. The theorem the one-mean field rests on goes through with h_b in place of the block size, and the reason is that the weights a weighted least squares decomposition needs are the inverse variances — which is exactly what h_b is. The stopping rule reads only within-arm within-block contrasts, so it is a function of nothing the interval reports, whatever it does with the block sizes. Each bar is within 2.9% of the level it claims.

A width promised for a difference

The exact fixed-width interval was built for one mean. Two arms make the target 42.7 units of effective size and each unit costs four observations, so the same promise about a difference costs 169.4 rather than 42.7 — and the theorem survives untouched with the harmonic size in place of the block size.

contrast · Width Three intervals, one shortfall. What each of three intervals actually covers, at four rules and two block windows, over 300 samples of 120 rows. All three are built from the same resamples on the same draws, so a difference between them is a difference in what is done with the resampled series. Not one of the twenty-four cells reaches the ninety-five per cent it promises. The studentised interval runs from 75.7% to 92.3%, the percentile interval — the earlier field's — from 80.0% to 89.7%, and a normal interval on the same scale from 81.7% to 89.0%. The standard repair for a percentile interval's shortfall does not repair it.

An interval that carries its scale

A percentile interval inherits the resampled distribution's skewness and its scale error together. The standard repair is one extra variance per resample. It was named and not run, so this runs it.

student · Bootstrap A quantile is the dearer reading, everywhere. The error each rule and window delivers on the two error readings, over 400 draws. The lower pair of lines is the implied long-run variance — the instrument the earlier field uses — and the upper pair is the 95% point of the standardised resampled mean, read against the finite-sample truth of 3.889 found by simulating the law directly. The quantile costs more at every one of the eight cells: at the plug-in rule it is 59.1% against 45.3% for the taper. That is not a defect in the bootstrap; a quantile is a statement about the shape of a distribution as well as its scale, and a fixed number of resamples estimates a tail worse than a variance. What matters for the comparison is that the two orderings between the windows are not the same, which the margins figure is about.

The instrument and the reading

Every comparison between two block windows in this collection is an error in an implied long-run variance. Nobody reads a long-run variance. Read on the 95% point a test uses, the same bootstrap costs half as much again.

readout · Bootstrap 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.

intervals · Coverage Three mechanisms leave the slope alone; one does not. The bias of the complete-case slope under each of four missingness rules, counted over 4000 studies of 200 rows at 35.0% missing, with the closed form printed beside each count. Missingness that depends on nothing, on the regressor, or on the second covariate leaves the slope exactly where it was — the closed forms are zero to machine precision and the counts are -0.0005, -0.0005 and -0.0011 against standard errors of about 0.0018. Missingness that depends on the outcome moves it by -0.1635, which is 27.3% of the slope being estimated. The same share of rows is lost in every case.

Three mechanisms and one dataset

Four rules for which outcomes go missing, each calibrated to lose the same 35% of the rows and each leaning on what it reads with the same coefficient. Three leave the fitted slope exactly where it was, and the one that reads the outcome moves it by 0.163531.

missing · Missingness Where a replication's estimate lands against a 95% interval, replication the same size. 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.

Five times in six

A 95% interval is read as a 95% chance that a replication's estimate will land inside it. With the spread known and a replication of the same size, the chance is 83.42% — five times in six — because both estimates are uncertain. An original that landed two standard errors from the truth captures a replication 48.40% of the time; a replication a tenth the size lands inside 44.54% of the time; and among significant originals from studies with 17% power, 66.94%.

alongside · Repetition No block size is best at both things the procedure claims. Two claims and one dial. The honest interval's half-width falls as the blocks get smaller, because the interval's degrees of freedom are the number of blocks: 0.2602 at blocks of two against 0.2933 at blocks of sixteen. The fixed-width claim — that the mean is within 0.25 of the truth — gets more reliable as they get larger, because the sample size is less variable: 92.40% against 95.00%. Both are computed from the same runs, and the second is reproduced to within a tenth of a point by E[2Φ(d√N/σ) − 1], which needs the sample-size distribution and nothing else. The schedules sit at the bottom left: as narrow as the smallest fixed block and as few observations, with the rule's spread estimate on half as many degrees of freedom again.

Two degrees of freedom, one total

The block size is a dial, and the two things a fixed-width procedure claims move in opposite directions along it. Divide the width by the square root of the sample size and one of them turns out to depend on the number of blocks and on nothing else.

pace · Width What studentising costs. How much wider the studentised interval is than the percentile one, cell by cell, over 300 draws, with what each cell gains in coverage beside it. Averaged over the eight cells the interval is 2.09 times as wide and covers 0.46 points better. At the two rules that choose short blocks the two intervals are within a fifth of each other; at the oracle's length, where a resample holds two or three whole blocks, the studentised interval is 4.37 and 5.04 times as wide. A repair that doubles the width and buys half a point is not one a reader could not have had by widening the interval it replaced.

What studentising costs

Averaged over eight cells the studentised interval is 2.09 times as wide as the percentile one and covers 0.46 points better. At the block lengths the rules choose, the scale it divides by rests on two or three numbers.

student · Bootstrap Three sets of weights, five designs, and no estimator that is exact everywhere. Coverage of the same interval under three weightings. h_b is the inverse variance when the arms share a variance or the allocation is constant; equal weights are right when every block has the same two counts; the estimated precision weights are right in the limit and exact nowhere, because the decomposition needs the weights to be the constants they are only estimating. In the corner — two variances, changing sizes, changing allocation — the two exact estimators are the ones that miss, at 98.45% and 95.65%, and the one with no theorem behind it is at 95.05%. That is the whole statement: there is an exact estimator under either condition, and none under both.

Which weights are the inverse variances

There is an exact estimator when the two arms share a variance and another when every block has the same two counts, and between them they cover every trial anybody designs on purpose. In the corner where neither holds, both cover 98.45% instead of 95%, and the only estimator at its level is the one with no theorem behind it.

contrast · Nuisance Unrepresentative in every respect but the one that matters. Three properties of the complete cases as the chance of being observed leans harder on the regressor, in closed form, at 35.0% of outcomes missing throughout. The mean of the regressor among the rows kept climbs from 0.0000 to 0.5528 against a population mean of zero, and the mean of the outcome from 0.0000 to 0.3980 above its own. The bias in the fitted slope is exactly zero at every one of the ten settings, because selection acting on the regressor alone leaves the conditional law of the outcome given the regressor untouched and least squares conditions on exactly that. The sample is wrong about almost everything and right about the one quantity being estimated.

Dropping the incomplete rows

Push the missingness until the rows that survive have a covariate mean of 0.543905 against a population zero and a variance of 0.5041 against one, and the fitted slope is still exactly right. Where the rule reads the outcome instead, the same sweep takes coverage to 2.42% at eight hundred rows.

missing · Missingness Three bands called 95%, at n = 20. Half-widths in sample standard deviations: 0.468 for the mean, 2.14 for one future observation, 2.75 to hold 95% of the population. The first two differ by exactly the square root of n + 1, which is 4.58 here.

A tenth as wide, and both of them right

The interval for a mean and the interval for one future observation are both labelled 95%, and at a hundred observations one is 10.05 times the other — exactly the square root of n + 1. Read the narrow one as the wide one and it covers a new value 15.7% of the time.

estimated · Bands Two 95% intervals 2.772 standard errors of the difference apart, standard errors in the ratio 1. 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 intervals that overlap

Two 95% intervals that just touch are read as a difference at the edge of significance. With equal standard errors their difference has p = 0.0056, not 0.05; two intervals can overlap by 58.6% of an arm and still differ at exactly 5%; standard-error bars that just touch mark p = 0.157; and when the two estimates are correlated at 0.8, touching intervals conceal a difference of 6.2 standard errors. Read as a test, non-overlap needs 1.66 times the sample for the same power.

alongside · Repetition Two arms leave one degree of freedom per block unaccounted for. Each point is one run. The one-mean field's identity is (b − 1) + (N − b) = N − 1, and every schedule moves along that line rather than off it. Two arms give the rule N − 2b and the interval b − 1, which come to N − b − 1 — short of the N − 2 two arms leave by exactly one per block, since a block's arm counts absorb one degree of freedom each and only one of the two directions carries the difference. The hollow points add what the block sums are worth, b − 1 more, and land on the total. The missing degrees of freedom are not lost; they are in a place the interval has to be shown it may read.

The degrees of freedom in the sums

One arm partitions N − 1 exactly. Two arms give the rule N − 2b and the interval b − 1, which is short by one per block — and the missing ones are in the block sums, which are correlated with the differences at −0.79 and are usable anyway.

contrast · Blocking Four rules of four change sign. The margin between the two block windows in points of coverage, under each of four rules, on each of three intervals built from the same resamples, over 300 draws. Positive is the tapered window covering better. On the percentile interval the taper wins at all four rules, by 5.33, 1.67, 5.00 and 4.00 points, which is the earlier field's own reading. On the studentised interval the rectangle wins at all four, by 4.33, 7.00, 4.33 and 2.67. And a normal interval, which uses no resampling at all, puts the two within a third of a point at every rule — so the disagreement is manufactured entirely by what is done with the resamples.

The ordering reverses again

One field found two of four rules changing sign between two readings of one resampling. Turn the same resamples into a studentised interval instead of a percentile one and all four change sign.

student · Bootstrap 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.

intervals · Repetition A normal mean with a flat prior — one interval, two readings. Both intervals are [1.878, 3.446]. The frequentist reading is that the procedure captures the truth 95% of the time; the Bayesian reading is that the parameter is in this interval with probability 0.95. The endpoints are identical to machine precision.

Where the two schools agree

With a flat prior on a normal mean, the credible interval and the confidence interval are the same interval, endpoint for endpoint. Knowing exactly when that stops being true is more useful than either camp's general argument.

bayes · Credible A filled value is not an observation. What a 95% interval for the slope actually covers after each way of handling 35.0% missing outcomes, counted over 4000 studies of 200 rows. Dropping the incomplete rows covers 95.93%. Filling with the observed mean covers 13.85%, because the estimate itself has moved. Filling with a fitted value covers 80.85% against a closed prediction of 79.73%: the estimate is right and the reported standard error is short by a factor of 0.6567 against a predicted 0.6500, because the residual sum of squares is divided by the whole sample's degrees of freedom. Adding residual noise recovers the spread and covers 85.78% against a predicted 84.62%, since the interval still ignores the variance of having imputed at all.

One imputation is not an observation

Three ways of filling a missing outcome, under a mechanism that makes dropping the rows beyond reproach. Filling with the observed mean covers 13.85%, filling with a fitted value covers 80.85%, adding noise covers 85.78%, and the thing all three were meant to improve on covers 95.93%.

missing · Missingness Robust, at the sample sizes it is reached for. Counted coverage of five 95% intervals for a slope, at six sample sizes, under an error variance leaning towards the edges of the design (γ = 0.8), over 20000 draws at the small end. The model-based interval sits at about 87.06% everywhere and does not improve with the sample, because it is a claim about a variance it is not estimating. The robust ones do improve: HC0 covers 88.73% at 20 rows, 92.70% at 50 and 94.93% at 1,000. Its promise is asymptotic and its use is not, and the gap between those two facts is this picture. The leave-one-out correction read against a t on n − 2 is the only line that is near its promise at the small end: 94.55% at 20 rows.

Robust is not free

A robust standard error's promise is asymptotic and its use is not. Its 95% interval covers 88.73% at twenty rows, and under mild heteroskedasticity it is the worse of the two intervals until a hundred.

sandwich · Misspecification The estimator has no upper bound on what it costs. What a thinning overlap does to a stabilised inverse-probability estimate of an average effect of 1.0000, over 600 samples of 600 at each of six settings. The spread rises from 0.1965 to 0.8122 and the root mean square error from 0.1964 to 0.9219, so at the thin end the error is very nearly the whole of the quantity being estimated. The lower line is the share of the arm's weighted total the single largest observation owns, averaged over the same draws: 0.69% to 9.78%, and in the worst single draw of the sweep 82.75%. Coverage of the 95% interval goes from 93.7% to 55.5%.

The region with no comparison

A trimmed interval covers the average effect over everybody 90.8% of the time at six hundred rows and 41.0% at nine thousand six hundred, while covering the average effect over the units it kept 94.3% and 96.0% throughout. An interval that gets worse as the sample grows is an interval about something else.

weights · Weighting One cohort of 40, and two intervals around the end of its curve. A single simulated study of 40 subjects with exponential survival at rate 0.35, dropout at rate 0.15 and follow-up to 6 — the first seed from 8811 upward whose plain band reaches below −0.05, chosen to show the failure rather than its frequency. The step curve is Kaplan–Meier and the smooth curve the truth. The plain band, the estimate plus and minus 1.96 Greenwood standard errors, first dips below zero at t = 3.78 and reaches −0.052; early on it also rises to 1.023, above one. At t = 5 the estimate is 0.069 with 1 subject still under observation, the plain interval runs from −0.052 to 0.191 and the log-log interval from 0.006 to 0.251, against a truth of 0.174. The log-log band is built on a scale that cannot leave [0, 1], and it bends away from the edge rather than through it.

The interval at the end of the curve

The interval most software prints around a survival curve covers 89.7% at five years, where 3.3 of forty subjects are still being watched and where the curve is actually read. The same variance carried on a log–log scale covers 94.8% there — and the failure was never the width.

survival · Censoring Twenty cells of an interval that is exactly 95%, 1,000 replications each. 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.

A coverage table with its own error

Twenty cells estimating the coverage of an interval that is exactly 95%, at a thousand replications each, read from 93.9% to 96.5% — and a table like that flags at least one of its correct cells on 69.9% of honest runs. Ten times the replications does not repair it: at ten thousand the same table still flags one 63.3% of the time.

method · Seeds What a schedule is allowed to read, and what happens when it reads more. The construction allows the block sizes to be anything at all as long as they are functions of the within-block contrasts, which are independent of every block mean. A schedule that shrinks the block whenever the between-block spread is running above what the contrasts say is a direct attempt to hold down the quantity the interval will be built from, and it succeeds: the estimate lands at 0.8373σ² against the honest 0.9831, and the coverage goes with it. Reading the running mean instead pushes the other way and over-covers — which is not a repair, it is the same violation with the sign reversed, and the level is no longer a property of the procedure at all.

A schedule that reads the mean

The block sizes may be anything at all provided they are functions of the contrasts. Two natural schedules break that, in opposite directions — and the most natural mistake of the three is not a schedule at all but a stopping rule, at 86.87% coverage and fewer observations.

pace · Stopping What a second break adds. Over 200 draws, the likelihood ratio a search over one break point reports, and how much more a search over an ordered pair adds on top of it. Under AR(1) at 0.8, which has no break at all, the first search manufactures 5.697 and the second adds 4.278. Under a law with exactly one break — where a second one is as absent as the first was in the row above — the first search reports 9.442 and the second still adds 5.800. Searching for something that is not there costs the same whether or not something else was there to find.

A second break on a flat profile

Searching a hundred and twenty rows for one change point manufactures five units of likelihood. Searching for a second manufactures four more, on a series that has at most one — and on a profile whose whole range is under seven.

charged · Break point Four intervals at one stopping time. 3,500 experiments under the sequential rule with a first stage of 5 and a required half-width of 0.4, which is a demand that knowing σ would meet with 24.0 observations and which the rule meets with 20.8. The rule's own interval covers 90.3%. Replacing the fixed width by a t interval on the same data gives 92.0%. Keeping the rule's own random sample size and drawing a fresh sample of that size gives 89.8% at the fixed width and 95.5% for a t interval — so the sample size being random costs nothing, and the sample size being chosen by the data the interval is built from costs the rest. The spread estimated at the stopping moment is 17.1% below the truth, which is the same fact one level down.

The interval after a stop it chose

A rule that stops when the estimated precision is good enough stops on the samples whose estimate was small. Its interval covers 90% and claims 95%, and a fresh sample of the same random size covers 95.4%.

guarantee · Stopping Where the maximum is, from 15 runs. One dataset, one fitted quadratic, and two answers to "where is the best setting". The delta method reports 0.80 ± 0.46, a finite interval it will report whatever the data does. Fieller's set is 0.49 to 1.76, because the curvature here has t = -4.04. The true optimum is at 0.75.

The optimum is a ratio, and its interval is sometimes the whole line

The best setting is −b₁/2b₂: a ratio of two estimates whose denominator is a curvature the design can often barely see. The delta method reports a finite interval every time and covers 68.8% where the curvature is weak; Fieller's set covers 95% and says so by being unbounded.

surface · Optimum What the interval covers, after a design that read the data. 800 experiments of 12 runs, all at the same truth. The first pair is a design fixed in advance; the second is one whose settings were chosen from the first stage's own outcomes. If choosing the design from the data broke the inference, the second pair would sit below the first, and it does not — 91.3% against 92.6%. What does move the coverage is the shape of the interval rather than the design: the Wald interval assumes the estimate is normal around its own standard error and falls short under both designs, and the profile interval, computed from the same residual sums of squares with no derivative in it, covers 94.3% and 94.9%.

What a design chosen from the data costs

Two fields on this site measured what happens when a rule reads the data, and the error rate broke both times. A design that reads the data to decide where to put its runs breaks nothing — and the control that proves it also finds what the real shortfall is.

robust · Local design The rate falls geometrically; the count does not fall at all. The share of equal splits of two hundred units that a tolerance of 1 coin-spreads admits, against the number of functions the tolerance is stated for, with the closed form (2Φ(1) − 1)^k drawn beside it. The rate falls by about two thirds with every constraint. The admissible count is that rate times C(200, 100), and it goes from 2^195 to 2^192 — it does not fall in any sense a trial cares about. What the falling rate costs is sampling: 9,878 draws to collect a thousand admissible ones at six constraints, against 1,465 at one.

What a reference distribution costs to sample

A randomisation test on a trial too large to enumerate has to sample its reference distribution, at 1/p attempts per draw and a p-value resolved to 1/(B + 1). Six constraints cost 9,878 attempts per thousand draws, and a thousand draws resolve p to 9.99·10⁻⁴ and not one digit finer.

product · Reference Free until the sums stop seeing what the differences see. Coverage with and without the block sums pooled into the interval's variance estimate. With one effect and one level they are free. With an effect that varies between blocks they are still free, because a block's sum picks that variation up exactly as its difference does. With a level that varies they make the interval 37% wider and conservative. And where the effect falls as the level rises — a ceiling, and not an exotic thing to suppose — the sums carry none of the between-block variation while the differences carry all of it, the pooled estimate is short, and the interval that uses it covers 88.75% on a width 20% narrower than the honest one.

What a two-arm rule may not pool

A spread computed "within the block" without the arm label carries a share of the effect, so the trial runs 173 observations at a null and 282 at an effect of 1.5. The stopping rule is reading the thing it exists to measure, and the phrase that produced it is one word long.

contrast · Allocation What the interval actually covers. The coverage of the two-sided interval each rule and window builds, over 400 samples of 120 rows, against the 95% it promises. Not one of the eight reaches it: the best is 91.0% and the worst is 80.8%, on a promise of 95%. So the first thing this instrument says is that the choice between the two windows is a choice inside a range that is already four to fourteen points short, which neither of the other two readings can express at all. The second is the ordering: the tapered window covers better under every one of the four rules, by 5.00, 2.50, 5.75 and 4.00 points — including at a length written into a protocol and at the rule of thumb, where the implied variance says the rectangle wins.

What the interval covers

Eight rules and windows, and not one of them reaches its promised 95%. The range is 80.8% to 91.0%, and the choice between two block windows is a choice inside a shortfall that is four times larger.

readout · Bootstrap One penalty, read along two dials. How much wider the studentised interval is than the percentile one, at every sample size and every block length on the grid, with the number of whole blocks each cell leaves written beneath. Read across a row and the block length changes; read down a column and the sample size does. The penalty is nearly a function of the block count alone: the cells at 15 blocks read 1.16, 1.20, 1.17, 1.15, 1.13, 1.10 across three sample sizes and three block lengths, while the cells at one block length read anything from 1.10 to 3.95. The largest penalty on the grid is 3.95, at the cell with 3 whole blocks in it.

The count or the length

A block length and a block count are one number read two ways at one sample size. Read at three, the studentised interval's width penalty tracks the count — with an R² of 0.9911 against a closed form that has no length in it — and its coverage tracks the length.

student · Bootstrap The extra 1/m, and the correction nobody quotes. What a pooled 95% interval covers against the number of imputations, counted over 2000 studies of 200 rows at 35.0% of outcomes missing. Rubin's rules — total variance W̄ + (1 + 1/m)B, read against a t distribution on (m − 1)(1 + W̄/((1 + 1/m)B))² degrees of freedom — cover 94.10% at two imputations and reach their promise by 5, at 95.25%. Dropping the (1 + 1/m) factor takes two imputations to 93.10%; using a normal quantile instead of the degrees-of-freedom correction takes it to 92.55%; dropping both takes it to 91.45%. The median degrees of freedom at two imputations is 12.95, which is why the second correction is the larger.

The variance between imputations

Pooling several filled datasets covers 94.10% at two imputations and reaches its promise at five, where a single fill covered 85.78%. The correction everybody quotes is the smaller of the two doing the work — 1.00 ± 0.22 points against 1.55 ± 0.28.

missing · Missingness Either model is enough; neither is not. The bias of three estimators of an average effect of 1.0000, over 600 samples of 600 units with the assignment rule at strength 1, in each of the four cells made by getting each nuisance model right or wrong. The wrong model in both cases is one that omits the second covariate, which the outcome and the assignment both depend on. The outcome model alone is off by 0.8064 whenever it is the wrong one; weighting alone is off by 0.8190 whenever the propensity model is. The augmented estimator built from both is off by -0.0085, -0.0083 and -0.0016 in the three cells where at least one of them is right, and by 0.8118 in the fourth — which is between its two components rather than better than either.

Either model, but not neither

The augmented estimator's bias is −0.0085, −0.0083 and −0.0016 wherever one nuisance model is right, against components off by 0.8064 and 0.8190. One step past the overlap sweep it is the least biased estimator on the table at 0.0857 and the worst on it at 1.9265.

weights · Weighting Which samples Wilson and Clopper–Pearson each cover, n = 50, p = 0.2. 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.

The same draws for both methods

Two intervals computed on the same simulated datasets give a difference in coverage whose variance can be 4.891 times smaller than on separate datasets — or, for a pair that covers different samples, 1.164 times larger. Which one a comparison gets is an exact sum over the counts each interval covers, and a standard error that ignores the sharing covers 100.00% for one pair and 93.07% for the other.

method · Seeds Ordered stagewise: the outcomes at least as extreme as stopping at 160 observations with z = 3.3. Each column is one look of an O'Brien–Fleming trial; above the boundary a trial stops there. Highlighted are the outcomes that count as at least as extreme as the observed one when outcomes are ordered stagewise: at 80, z ≥ 4.56 (probability 2.54 × 10⁻⁶ with no effect); at 160, z ≥ 3.30 (probability 4.82 × 10⁻⁴ with no effect); at 240, none; at 320, none; at 400, none. The two-sided p-value is 9.69 × 10⁻⁴.

The outcomes a trial could have stopped with

A trial that stops at its second look with z = 3.3 has a two-sided p-value of 0.000969, 0.000987, 0.00187 or 0.0421, depending on how the outcomes it could have stopped with are ordered. One of the four orderings does not change when the looks the trial never reached are replanned, and the same one gives a trial that ran to the end with z = 6 a p-value of 0.0256.

sequential · Stopping The interval every package reports first does not cover. Counted coverage of two 95% intervals for the 100-block return level of a normal parent, against the length of the record they were fitted from, over 300 records at each length. The level they are about is known in closed form, so this is coverage of a number rather than agreement between two estimates. The delta-method interval covers 80.3% at 25 blocks and reaches only 89.0% at 200; the profile-likelihood interval sits between 94.0% and 94.7% throughout. The gap is not a small-sample effect that lengthening the record removes — it narrows by 8.7 points for an eightfold longer record.

Two intervals for one return level

Two 95% intervals read off the same fits of the same records, against a level known in closed form. The symmetric one covers 80.3% at twenty-five blocks and reaches only 89.0% at two hundred — and 99.24% of its misses are the interval sitting entirely below the truth, which is not the endpoint anybody expects to fail.

extreme · Extremes The damage does not stay in the term that was left out. Where each coefficient lands when the model that fills the missing outcomes and the model that analyses them disagree, over 1500 studies of 200 rows at 35.0% missing and 20 imputations. An imputer that omits a covariate the analysis fits attenuates that covariate's coefficient by exactly the missing fraction — -0.1405 counted against a closed -0.1400 — and pushes the coefficient it did impute on the other way by exactly the product of the omitted coefficient, the covariates' correlation and the missing fraction: 0.0402 counted against 0.0420. Both closed forms come out of the same two-by-two solve. Matching models leave both alone, and so does an imputer that knows more than the analysis.

An imputation model the analysis does not contain

A model that fills the gaps without a covariate the analysis fits attenuates that covariate's coefficient by exactly the missing share, 0.4 to 0.26, and moves the one it did carry by exactly γρf, 0.6 to 0.642. The reverse case is supposed to inflate the interval, and at four strengths of the extra knowledge it does not.

missing · Missingness The price of insurance is noise, not coverage. What a robust standard error costs under a constant error variance — the case where the model-based one is exactly right — at five sample sizes over 20000 draws at the small end. It is not coverage: the leave-one-out interval read against a t on n − 2 covers 95.52% at 20 rows against the model-based 95.06%. It is a wider interval, by a factor of 1.0689 at 20 rows falling to 1.0049 at 250, and it is a variance estimate 2.57 times as variable at 20 rows and 1.85 times at 250 — against a denominator that is exactly V²·2/(n − 2), so only the numerator is counted. The uncorrected estimate is the cheaper of the two at small samples and the dearer at large: 1.31 against 1.76.

Right for the wrong reason

A robust standard error costs no coverage where the risk is absent — 95.52% against 95.06% at twenty rows. It costs a 6.89% wider interval and a variance estimate 2.572 times as variable, and the pre-test that would avoid paying recovers 15.9% of what the insurance is worth.

sandwich · Misspecification Twenty runs simulating an exactly 95% interval, checked every 250 replications. 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%.

A simulation that stops when it looks settled

A simulation of an interval that covers exactly 95%, checked every 250 replications for a significant departure and stopped when it finds one, flags that correct interval on 29.54% of runs. Stopped instead as soon as its estimate reaches 95%, it reports an interval that covers 94% as meeting its level on 37.21% of runs. Stopped when the estimate stops moving, it reports the right number — and has quietly chosen to run about fifteen hundred replications.

method · Seeds The squared estimate 1 standard errors from the flat point, exact and linearised. At δ = √n·μ/σ = 1 the exact law of the squared estimate has mean 2.00, variance 6.00 and skewness 2.177; the delta method's normal has mean 1.00, variance 4.00, no skewness, and 30.85% of its mass below zero, where a square cannot go. The Kolmogorov distance between them is 0.3085.

Where the derivative is zero

The delta method reads a standard error off a tangent line, and at a flat point the tangent says the spread is zero. The interval built on it for a squared mean covers 99.991% there and 85.978% one and a half standard errors away, with nearly every miss on the same side — and the law it should have used is a χ², not a normal.

normal · Clt Four intervals at 3 blocks of 32 rows. What four 95% intervals for the mean of a first-order autoregression at 0.7 cover, and how wide they are on average, at 120 rows cut into 3 whole blocks of 32, over 240 draws with 200 resamples each under the rectangle. The normal interval, the block-means variance with 1.96, covers 82.1% at a width of 0.708. The percentile interval covers 82.1% at 0.632 and the studentised one 90.4% at 2.495. The fourth resamples nothing: it is the normal interval with 1.96 replaced by Student's t on 2 degrees of freedom, and it covers 94.2% at 1.555, 0.62 times the studentised interval's width.

The interval with no resampling in it

Replace 1.96 in a normal interval on the block-means variance with Student's t on one fewer degrees of freedom than there are whole blocks, and resample nothing. Across twenty-four cells it covers at least as often as the studentised bootstrap interval at every one, by 0.42 to 10.42 points; it is narrower wherever seven blocks or fewer are left; and at fifteen blocks of 32 it covers 95.0%, which no resampled interval on the grid reaches.

student · Bootstrap An estimate reported as a function of an assumption. What the slope really is, against a shift in the outcomes nobody saw — line from the closed form, dots counted over 2000 studies of 200 rows at 35.0% missing. Every point on this line produces exactly the same observed data, and the complete-case estimate is the flat line at 0.5996 regardless. The truth moves at -0.2845 per unit of shift, which is a function of the missingness model and the missing fraction and of nothing that can be estimated: across the swept range the true slope runs from 0.8845 to 0.3155, a span of 0.5691 against a value of 0.60 in the world where the shift is zero. Reporting the line is the honest form of the answer.

The mechanism the data cannot see

Two worlds produce identical covariates, identical patterns of what is recorded and identical recorded outcomes, to the last bit. Their true slopes are 0.6 and 0.315452, and the truth moves at 0.284548 per unit of an assumption nothing in the data can inform.

missing · Missingness The rows are held fixed; only the clusters move. Counted coverage of four 95% intervals for a slope, at five cluster counts with the row count held at 300 throughout and a within-cluster correlation of 0.1, over 6000 draws apiece, with the sizes equal. The interval that counts rows covers 53.42% at 5 clusters — a second closed form says 2Φ(z/√D) − 1 = 54.44% for a design effect of 6.900, and reads nothing about clusters at all. The cluster-robust interval read against a normal covers 74.43% there and 94.20% at 100 clusters; read against a t on G − 1 it covers 85.08% and 94.47%. The number of independent things is the cluster count, and every quantity here is blind to how many rows were typed.

The count that is not the rows

Three hundred rows in five clusters of sixty carry 6.9000 times the variance an independent-rows calculation reports, and the interval that counts rows covers 53.42%. The same five unequal sizes laid out two ways give design effects of 9.3158 and 5.4652.

sandwich · Misspecification The two standardised means, and the shape of Fieller's set for their ratio at a = 3, d = 1. Each dot is one pair (zx, zy) drawn around (3, 1). Outside the horizontal band |zy| > 1.96 Fieller's set is a bounded interval, with probability 17.01%; inside the band and outside the disc of radius 1.96 it is everything outside an interval, 75.03%; inside the disc it is the whole line, 7.96%. On 40,000 counted draws Fieller covers ρ = 3.00 95.21% of the time and the delta interval 82.48%.

A ratio whose interval has to be the whole line

The delta interval for a ratio of two means covers 95.61% when the denominator is eight standard errors from zero and 1.10% at a ten-thousandth of one, and ten times as wide it still covers only 3.48%. Linearising is not the fault. Gleser and Hwang proved that every interval that is always finite fails the same way, so an interval that keeps its promise has to be the whole line some of the time.

normal · Clt Estimating P(Z > 5) = 2.8665×10⁻⁷ with plain draws and with four proposals. 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 draws aimed at the tail

The chance a standard normal exceeds 5 is 2.8665×10⁻⁷, and a plain simulation needs 349 million draws to estimate it to within ten per cent. Draws aimed at the tail and weighted back need 565. Aimed slightly too narrowly, the same method has an infinite variance, an interval that covers 86.0% and gets worse with more draws, and an effective sample size that reads healthier than a proposal that works.

method · Seeds Every finding Benjamini–Hochberg made in thirty families, with its interval, at real effects of 2. 85 findings, sorted by their estimate. 12 of their ordinary 95% intervals miss the true effect, every one of them on the far side; 6 of the wider false-coverage-rate intervals miss.

Intervals for the findings

Benjamini–Hochberg's findings usually go out each with its ordinary 95% interval. With ten real effects of two standard errors among twenty tests, 11.59% of those intervals miss their effect, every miss on the far side, and the interval around the most prominent finding covers 72.36% of the time — 2.38% when the effects are one standard error. Intervals widened for the number of findings hold the share that miss under 5%.

multiplicity · Multiplicity A peak where the recorded data have none. The profile log-likelihood of a selection model in cy, the coefficient that lets the chance of being recorded depend on the outcome itself, for one study of 800 rows whose missingness is at random, with residuals normal; every other parameter is maximised at each fixed value. The model assumes the outcome is normal given the covariates. The curve peaks at cy = 0.35, where the fitted slope is 0.839, and the values of cy within the 95% cut run from −0.13 to 0.75; the likelihood-ratio statistic against cy = 0 is 1.47. The study was drawn with cy = 0.00. With the outcome's law left free, every value of cy fits the recorded rows equally well and this curve would be flat: its curvature is the normal assumption.

The assumption that identifies the mechanism

A selection model estimates how strongly an outcome decides whether it is recorded — the quantity two identical datasets showed no statistic can see — and it does so by assuming the outcome is normal. Where that holds and the outcome does decide, it repairs a slope complete cases put at 0.4318 to 0.5795. Where the missingness is at random and the residual is merely skewed, it reports selection that is not there, moves the slope from 0.5971 to 1.0319, and rejects missingness at random in 72.5% of studies.

missing · Missingness 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%.

intervals · Coverage The law is the eigenvalues, and nothing else. The mean and the skewness of n(ĝ − g) at the stationary point, measured over 40,000 draws, against the closed forms ½ Σλ and 2√2 Σλ³ ⁄ (Σλ²)^(3⁄2). The worst disagreement anywhere is 0.028. In one variable the second-order law is a single χ² and its sign is the sign of g″; here it is a weighted sum with the Hessian's eigenvalues as weights, so a bowl and a valley differ in both moments and a saddle has both equal to zero.

A flat point with more than one direction

At a stationary point of a function of several means the second-order law is ½ Z′HZ, so the bias is half the Hessian's trace — 2.008 for a bowl, 5.028 for a valley, and −0.006 for a saddle, where the eigenvalues cancel. The saddle's coverage is the worst of the three.

normal · Clt

Named alongside it

The objects these essays reach for when they reach for this one.

CoverageMonte CarloClosed formDegrees of freedomSample sizeStandard errorDependenceFixed-width intervalStopping ruleBinomial proportionBlock bootstrapBlock length

All concepts