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

Monotone transformation — where it appears

An increasing relabelling of a variable, which preserves order and therefore every quantile. It moves means, variances and every statement about shape, and it leaves ranks, medians and any rule computed from them exactly where they were.

Named by 15 essays across 6 fields — each of them below, with the objects they name alongside it.

One margin rises; the other turns over. The two halves of the table's margin across the sweep: a lower-tail copula's own leak with a symmetric covariate, and a covariate skewed at 0.95 under a Gaussian copula. The marginal's leak rises at every step, from 3.727% to 36.056%. The copula's does not: it rises to 9.064% at a Spearman of 0.6 and falls to 8.219% by 0.7. It has to turn over, because at a rank correlation of one the two variables are a deterministic function of each other and there is no interaction left for a split to leak. So the margin of the table turns over before any cell in it does.

A margin that turns over

A skewed covariate's leak grows without limit as the dependence strengthens. A copula's own leak does not — it peaks at a rank correlation of 0.6 and falls. The margin of the table turns over before any cell in it does.

stronger · Adjustment One zero holds and one does not. Three rules, at a correlation of 0.5, against the skewness of the covariate. A rule balancing the mean of each covariate removes exactly nothing of their product when the marginal is symmetric — including the heavy-tailed symmetric one at skewness zero, which is what says the guarantee needs symmetry rather than normality — and removes up to 29.7% when it is not. A rule balancing a median split of each removes exactly nothing of the product of the splits under every marginal here, to 1e-30: both sides are functions of the sign of the latent normal, and a monotone transformation moves neither. A rule balancing a threshold at a value on the covariate's own scale removes between 4.9% and 22.5% — it never had a zero to lose, under any marginal at all.

A zero that rests on a symmetry

A balancing rule removes exactly none of an interaction between two odd functions, at every correlation. The argument needs the joint sign flip to preserve the law, and no real covariate is symmetric about anything.

skew · Criterion What a mean split leaves, with both halves varying. The share of a mean split's interaction that survives the rule balancing it, at every copula and every marginal, matched at a Spearman correlation of 0.40. The three radially symmetric copulas leave exactly nothing with a symmetric covariate and rise steeply with the skew. The two asymmetric ones start at 7.707% and go opposite ways: the lower-tail copula falls to 0.002% at a skewness of 0.95 — the two failures cancel almost exactly, and a guarantee both fields report as broken is restored — while the upper-tail one climbs to 40.288%. And the heavy-tailed symmetric covariate, which leaks exactly nothing on its own, doubles what the asymmetric copulas leak: 14.229% against 7.707%.

Two failures that cancel

A mildly skewed covariate under a lower-tail copula leaks 0.002% of an interaction where each failure alone leaks eight and seven per cent. Turn the copula over and the same pair compounds.

compound · Adjustment Where the two kinds of cut sit. Six covariates, each a monotone transformation of the same latent normal. The vertical line at zero is where every median split sits, on every one of them, because a monotone map preserves order: the median of the covariate is the image of the median of the latent normal. The marks on the curves are where a threshold at 1 on the covariate's scale falls — 1.000, 0.881, 0.875, 0.783, 0.713, 0.337 — and none of them is at zero. That is the whole of the difference. A function of the sign of the latent normal is odd, and a rule made of odd functions removes exactly nothing of an interaction between two of them; a threshold anywhere else is neither odd nor even and removes something.

A split survives what a mean does not

The two things every trial balances come apart on a skewed covariate. A median split is a function of the sign of the latent normal whatever the marginal is; a mean is not, and its exact zero is gone at a skewness of one.

skew · Criterion The same copula, turned over. A Clayton copula and its reflection, at the same Spearman correlation of 0.40 and the same Kendall tau of 0.275, against the covariate's marginal. With a symmetric covariate the two are the same number to nine decimals — 7.707% apiece — because the leak then depends on how much asymmetry the copula has and not on which way it points. Skew the covariate and they come apart: at a skewness of 2.26 the lower-tail copula leaves 3.431% and the upper-tail one 36.213%, a factor of 10.6. Both halves of the dependence are asymmetries and an asymmetry has a direction; a lower-tail copula concentrates the dependence where a right-skewed marginal is compressed and the two distortions partly undo each other, and an upper-tail one concentrates it where the marginal is stretched.

A symmetry that was not enough

A heavy-tailed symmetric covariate has a skewness of zero and leaks exactly nothing under three copulas. Under the two asymmetric ones it doubles the leak, from 7.707% to 14.229%.

compound · Adjustment A near-perfect cancellation, at one correlation. A covariate skewed at 0.75 under a lower-tail copula, at each of 7 rank correlations. The earlier field measures this cell at a Spearman of 0.4 and reads 0.002% where adding the two halves' own leaks gives 16.579% — a cancellation so near exact that it is that field's headline. Across the sweep the same cell reads 0.0084%, 0.0182%, 0.0097%, 0.0015%, 0.0864%, 0.4693%, 1.5327%. Its smallest value is at 0.4, in the interior, and by 0.7 it is 1007.40 times larger. The near-zero is where two curves cross, and they cross beside the one correlation that was measured.

The zero that was a crossing

A cell that leaks 0.002% where adding its two halves gives 16.6% is a field's headline. On a finer grid it passes through zero at a rank correlation of 0.38 — two hundredths from where it was measured.

stronger · Adjustment Three copulas that break nothing, and a factor of two between them. The three radially symmetric copulas, at a matched Spearman correlation of 0.40, against the covariate's marginal. All three leave exactly nothing with a symmetric covariate — that is the guarantee, and it holds to twenty decimal places. What they do to a skewed covariate is not the same at all: at a skewness of 2.26 a Frank copula leaves 12.118% where a Gaussian leaves 21.539% and a t on four degrees of freedom leaves 23.640%. A factor of 2.0 between two copulas that are both symmetric, both matched on rank correlation, and both harmless on their own. So the copula matters to the marginal's leak without breaking any symmetry of its own, which is a milder version of the same finding and applies to every trial rather than to the asymmetric ones.

A copula that halves a marginal

Three copulas break nothing on their own and put a factor of two between the same skewed covariate's leaks — 12.118% under a Frank against 23.640% under a t, at the same rank correlation.

compound · Adjustment Four cells change their answer. The four cells of the twenty whose excess changes sign as the dependence strengthens, over 7 recalibrations. Above the line the two failures compound — the cell leaks more than adding the copula's own leak and the marginal's — and below it they cancel. All four start above and end below, and all four are at the two most skewed covariates: skew 0.90 under heavy-tailed, skew 0.95 under heavy-tailed, skew 0.90 under upper tail, skew 0.95 under upper tail. Whether two failures of a dependence compound or cancel is therefore not a property of the pair. It is a property of the pair at a strength of dependence, and a fifth of the table changes its answer inside the range measured here.

An answer that changes

Eleven of twenty cells cancel and nine compound, at one rank correlation. Sweep the correlation and four of the twenty change sides — all four from compounding to cancelling, all four at the most skewed covariates.

stronger · Adjustment A cut at a quantile, and a cut at a value. Two rules that read identically in a protocol. One splits each covariate at its median; the other splits it at 1 on the covariate's own scale — a dose, a temperature, a clinical threshold. At a correlation of 0.5 the first removes exactly nothing of the interaction between its own two splits, under every marginal here, because a median split is a function of the sign of the latent normal whatever the marginal is. The second removes what the bars show, and it does so on a normal covariate too: the threshold sits at 1.000 on the latent scale rather than at zero, so it is 59.4% odd and 40.6% even. The exact zero was never about the cut; it was about the cut being at the median.

The cut that is not a quantile

A protocol that says split the covariate at a threshold and one that says split it at the median read the same and are different rules. One has an exact guarantee under every marginal and the other has none under any.

skew · Criterion Six forecasters, all calibrated, not equally useful. The resolution of six forecasters that are all perfectly calibrated, each reporting the true probability of the event given a signal that carries more or less of the latent state. The largest reliability anywhere in the family is 2.0e-33, so a calibration check passes every one of them. They are not equally good: resolution runs from exactly 0.00 for the forecaster that issues the base rate every time to 0.092758 for the one that sees everything, and their Brier scores run from 0.234237 — which is the world's own uncertainty, and the score of a table of base rates — to 0.141479. Calibration is a necessary condition that a constant forecast satisfies exactly.

Calibrated and useless

Six forecasters that are calibrated to 2·10⁻³³ run from resolution exactly 0 to 0.092758, and three forecasters with reliabilities from 0 to 0.013025 have areas under the ROC curve identical to every bit a double carries. Each measure is exactly blind to what the other one sees.

calibrate · Calibration 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 A guarantee that stops being a number. The worst case of each dictionary over six outcome shapes, at a correlation of 0.5, against the skewness of the covariate. Under a symmetric marginal every rule made of odd functions has a worst case of exactly zero, and the rule holding a mean and a median split of each covariate — the two things every trial balances — is one of them. Under skew that zero becomes 0.74%, 1.83%, 2.24%, 2.49%: small numbers, each of which depends on a marginal nobody stated. The guarantee has not improved by becoming positive. It has stopped being a guarantee, because it can no longer be written down without the covariate's distribution in it.

Balancing a skewed covariate

The worst case of the rule every trial runs goes from exactly zero to somewhere between a quarter of a per cent and two and a half. Which is small, and is a number that cannot be stated without the covariate's distribution in it.

skew · Criterion The table swept along the covariate instead. What a rule holding a mean of each covariate fails to remove of their interaction, at each of 4 copulas, as the covariate is skewed further and the rank correlation is held at 0.4. The sweep runs from a symmetric covariate at g = 0 to a skewness of 11.16, and the three settings the earlier table names — g = 0.3, 0.6 and 0.9 — are on it, where this sweep reproduces that table to the last digit. Every row rises and then falls: the lower-tail copula from 7.707% through 0.0015% and back to 5.587%, the upper-tail one to a maximum of 36.213%. So the quantity a trial is exposed to is not monotone in how skewed its covariate is.

The other dial

The table is swept along the strength of the dependence and never along the shape of the covariate. Swept along the shape at a fixed correlation, the same two copulas cross, the same way — and the near-zero cell turns out to be a minimum in both directions at once.

stronger · Adjustment Two 95% intervals for 2 of 20, under Jeffreys — Beta(½, ½). The equal-tailed interval runs from 0.0214 to 0.2839 and is 0.2625 wide; the shortest interval runs from 0.0093 to 0.2540 and is 0.2447 wide. Both hold 95% of the posterior, and the shorter one buys its 6.8% by moving its lower endpoint towards the denser side.

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.

bayes · Credible Four 95% intervals for the odds after 2 of 20. credible, transformed: 0.0218 to 0.3964. Wald, transformed: -0.0305 to 0.3012. delta method on the odds: -0.0512 to 0.2734. delta method on the log-odds: 0.0258 to 0.4789. The first two are the same intervals for the proportion with their endpoints put through the odds; the last two are fresh approximations made on the new scale.

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.

bayes · Credible

Named alongside it

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

Covariate balanceGaussian copulaInteractionMarginal distributionMedian splitSkewnessParityClosed formSymmetryCovariate adjustmentContinuous covariateCopula

All concepts