Every essay — page 2
Tests, and the second number
A p-value alone cannot be read: the same 0.04 means different things at different sample sizes, and nothing at all without knowing how many analyses were available. Under a real effect it is a draw from a distribution several orders of magnitude wide, a replication of a result at 0.05 succeeds exactly half the time under any model centred on that result, and the standard ways of combining several p-values disagree about which evidence counts. Every figure here carries the number that makes it interpretable.
A p-value that is not flat is not a p-value
Under a true null, p-values are uniform. That is stronger than saying the test rejects 5% of the time, it constrains the whole distribution rather than one point of it, and it catches implementation errors that a rejection rate sails past.
What a p-value does not say
The same p of 0.04 corresponds to a large effect in ten observations and a negligible one in two thousand. A p-value alone cannot be interpreted, and the number that makes it interpretable is almost never printed beside it.
The winner's curse
Filter honest studies down to the ones that reached significance and the effects they report are systematically too large. At low power the inflation is a factor of two, nobody has done anything wrong, and the selection did all of it.
Twenty analyses of nothing
Twenty honest, correct analyses of data with no effect in it find something significant 57% of the time. Nobody p-hacked, every individual p-value is right, and the reported one is the smallest of twenty.
The p-value a replication gets
Under a true null a p-value is flat. Under a real effect its distribution is closed form and wide — a study with 80% power returns anything from 4.4×10⁻⁵ to 0.13 in eight runs of ten — and the chance that an exact replication of a p = 0.05 result is significant again is exactly one half, under both of the models people use without naming them.
Two ways to combine p-values
Fisher's and Stouffer's combinations are both exactly right when every null is true, for the single reason that each p-value is flat. Under a real effect they disagree about which evidence counts: with Stouffer held at 50% power across ten studies, Fisher is the more powerful while the signal sits in six or fewer of them and the less powerful from seven.
The smallest of three combinations
Reporting whichever of Fisher's, Stouffer's and Tippett's combinations is smallest is a test of its own, and on ten studies of nothing it rejects 9.66% of the time — not 5%, and nowhere near the 15% the three sizes add to, because the statistics are correlated at up to 0.903. Read at 2.448% each it is exact, and then it trails the best single combination by at most 7.45 points and leads the worst by at least 10.30.
Studies that share a control
Ten comparisons against one control arm of the same size are correlated at exactly one half, and combined as though they were independent, Stouffer's sum rejects 24.23% of sets in which nothing works, Fisher's 17.88%, while Tippett's smallest p-value falls to 3.72%. Repaired with the correlation the design states, each is exact again — and the smallest of the three is then inflated less than for independent studies, 7.05% against 9.66%, because sharing makes the three combinations agree. The two inflations the dependence was expected to stack do not stack.
Reversals that are not errors
Simpson's reversal, the base rate, regression to the mean. Each is normally taught with one famous table. A table is a point; these are swept, so how much of the space behaves that way and how large it can get both have answers — including why two correct analyses of one baseline disagree, how far a group enrolled on a noisy reading falls with nothing done to it, and why the correlation predicts the regression of the extremes only when the population is normal.
Simpson's reversal is a region, not a table
The treatment wins in both groups and loses overall. That is normally shown with one famous table, which cannot answer the two questions a reader has — how often, and how large. Swept, it turns out to occupy 31% of the allocation space.
What a positive test is worth
A test that is 90% sensitive and 95% specific sounds accurate. For a condition affecting one person in a thousand, 98% of its positive results are wrong, and a worse test on a commoner condition beats a better test on a rare one.
Regression to the mean
Select the worst performers, measure them again, and they improve. Select the best and they decline. No intervention is required for either, the size of the apparent effect is predictable from the correlation alone, and it is the reason so many things appear to work.
Two analyses of one baseline
Two groups read at baseline and again at follow-up, with no change for anybody. Subtracting the baseline reports a group difference of −0.0014 and adjusting for it reports 0.4008 — and each analysis is exactly right about one reason the groups started apart and wrong by 0.40 about the other.
The measurement that got them enrolled
Enrol the top tenth of one screening reading and give them nothing, and they fall by 0.702 standard deviations at follow-up. Measured from a fresh reading taken after enrolment they fall by nothing. Averaging ten screening readings still leaves 0.101, and it takes twenty-one to get under 0.05.
A lead that a heavy tail keeps
Four populations whose readings all correlate at exactly 0.6, and whose least-squares slopes all read 0.6. Select the top one per cent on one reading and measure them again: they keep 60% of their lead if the true scores are normal, 76.1% if they are Laplace, 78.4% if they are a t on four degrees of freedom — and 44.3% if they are uniform. The correlation predicts the regression of the extremes for one shape of population only.
The slope of a density nobody can see
Tweedie's formula corrects a reading by the slope of the readings' own log-density, and a study has its readings. Estimated from a thousand of them, the correction for the top one per cent beats the correlation's linear rule on 84.0% to 98.0% of studies from heavy-tailed populations and on 75.0% to 81.5% from a bounded one — and costs an error of 0.09 to 0.12 where the population is normal and the rule was already exact. At 250 readings the log-spline loses to the rule it replaces, and at 16,000 the same log-spline gets worse on a power tail.
A correction built on a retest
Tweedie's correction for regression to the mean needs the error variance, and a study gets it from a few people measured twice. Twenty of them are enough: with normal errors the estimated correction then beats the correlation's rule on 73% to 77% of studies of a thousand readings from a heavy-tailed population, though its variance is uncertain by a third. What twenty cannot reveal is the error's shape. Laplace errors of the same variance make the correction overstate the lead a top group keeps by about 0.17, the plain rule then wins on most studies, and a study needs about two hundred retest pairs before a check on its own differences notices.
The prior, doing visible work
A credible interval says the thing everyone wants a confidence interval to say, and it needs a prior to do it. So the prior is treated as a component with a measurable effect: it is worth a stated number of observations, and the interval it produces has a coverage that can be summed over the sample space like any other.
What a prior is worth
A prior is not a philosophical position, it is a component with a stated size. For a proportion it is worth exactly a + b observations, which turns "how much does the prior matter" from an argument into a subtraction.
What a credible interval covers
A credible interval makes the statement everyone wants and does not claim to have a coverage. It has one anyway, it can be summed over the sample space exactly, and on a reasonable prior it beats the interval taught first.
![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.](/f/a-normal-mean-with-a-flat-prior-one-interval-two-readings-feabb882.svg)
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.
The base rate was always Bayes
The screening arithmetic everybody finds counter-intuitive is a posterior update with a prior of one in a thousand. Naming it that way turns a famous puzzle into an instance of a rule, and makes the sequential version obvious.
The shortest interval, and the one that does not move
Two 95% intervals come out of every posterior and they are not the same set. The shorter one is shorter by 4.86% on average and 22.41% at its best, it covers 86.72% where the other covers 95.68%, and it is not even the shortest once the parameter is written a different way.
An interval for something else
An interval for the odds is free — put the endpoints through the odds and the coverage does not move, exactly, for any interval at all. The method everyone uses instead computes a new standard error on the new scale, and at twenty trials that costs four points of coverage, produces negative odds, and has no value at all when nothing was observed.
When the prior is confident and wrong
A prior worth thirty-five observations, centred in the wrong place, produces a 95% interval that covers nothing at all — and reports a width 5% narrower than an honest one. It takes seventeen thousand observations to repair, not thirty-five, and the worst study to run is the one whose sample size equals the prior's weight, exactly.
A prior wrong about its own spread
A prior can be centred exactly where the truths lie and still be wrong about how far they spread. Ten times too confident — worth a hundred observations about proportions whose population is worth ten — its 95% credible intervals cover 48.2% of the population's studies at twenty trials, and nothing in the interval shows it: it is 0.456 of a flat prior's width, against 0.436 for a prior whose hundred observations' confidence is earned. A prior with tails too thin misses the one study in ten that comes from somewhere unusual a third of the time, and a flat component of two per cent brings that study's coverage from 65.6% to 84.0%.