The collection

Every essay — page 14

Essays 313 to 336 of 436, in the same order.

What a chain cannot report

A thin admissible set falls into an arrangement and its mirror image, and the walk that samples it is uniform on half the reference distribution for ever while every diagnostic passes. That was found by enumerating fourteen units, and enumeration stops at about twenty-four. The test that does not enumerate is two chains — one from an assignment, one from its complement — compared on a statistic that changes sign under the complement, with a third chain from the same starting point to say whether a large reading is a fact about the set or about the length of the run. It agrees with the enumeration at every tolerance where the answer is known. At two hundred units it reports something else: the walk reaches the whole set at the tolerances a trial would use, and past a point the diagnostic stops agreeing with itself.

The test, checked where the answer is known. A fourteen-unit trial at eight tolerances. At each one the admissible set is enumerated — 1534, 886, 304, 158, 126, 116, 102, 84 assignments — and its components counted, which is only possible because 3432 equal splits of fourteen units can be walked. The dots are the test, which walks none of them: two chains, one started at an assignment and one at its complement, compared on a statistic the rule was not handed. Filled marks are tolerances the enumeration says leave the set in more than one piece. The test fires on every one of them and on none of the others, 0 misses and 0 false alarms.

A test rather than a survey

A thin admissible set falls into an arrangement and its mirror image, and the walk that samples it is uniform on half the reference distribution for ever. That was found by enumerating fourteen units, and enumeration stops at twenty-four.

4 figures · Randomisation, part 11

Three readings, one verdict. Every tolerance of a fourteen-unit trial, with three comparisons on each. The first is between a chain started at an assignment and a chain started at its complement, which is what a mirror split separates. The second is between two chains started at the same assignment on different streams, which nothing about the set can separate — so a large reading there says the run is too short and not that the set is in pieces. The third is the same comparison on the statistic's absolute value, which is symmetric under the complement and therefore blind to the split by construction. The verdict is the pattern rather than any one line: the split is called only where the first fires and the other two do not, which happens at exactly the tolerances the enumeration calls disconnected — 0.8, 0.75, 0.7.

The statistic that changes sign

A test for an unreachable half needs a quantity that tells one half from the other. Every symmetric reading of a mirror pair is identical, and a magnitude is the natural thing to reach for.

4 figures · Randomisation, part 12

A defect that is about size

The admitted share of a rerandomisation barely moves with the number of units. The number of admissible neighbours grows like the square of it, and that is what decides whether the walk can go everywhere.

4 figures · Randomisation, part 13

What the diagnostic says at two hundred units. The same test run 8 times on independent streams, at five tolerances of a two-hundred-unit trial, 40,000 steps each. At the loosest tolerance every run says the same thing — the walk reaches the whole set — and it keeps saying it as the set is thinned. Past a point the runs stop agreeing with each other: at the tightest tolerance here 6 of 8 report that the chains have not mixed and 2 report a split, which is a diagnostic disagreeing with itself rather than a property of the set. That disagreement is the honest answer at this size, and it is one nothing in this collection could give before: an enumeration stops at about twenty-four units.

The diagnostic at two hundred

Pointed at a trial size no enumeration reaches, the test gives three answers rather than one — and past a certain thinness it stops agreeing with itself, which is the honest reading and the one nothing could give before.

4 figures · Randomisation, part 14

A guarantee that needed a symmetry

What a balancing rule can remove of an interaction is decided by parity, and parity is a statement about a symmetry of the law rather than about the covariate. Keep the dependence and change the marginals — a Gaussian copula under a monotone transformation — and the two things every trial balances come apart. A median split is a function of the sign of the latent normal whatever the marginal is, so its exact zero survives every transformation to the last digit. A mean is odd only when the marginal is symmetric, and its zero is gone at a skewness of one. The separating case is a marginal that is heavy-tailed and symmetric, where every zero holds exactly: it is not normality the guarantees needed. And a threshold at a value on the covariate's own scale never had one at all.

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.

3 figures · Criterion, part 13

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.

4 figures · Criterion, part 14

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.

4 figures · Criterion, part 15

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.

4 figures · Criterion, part 16

Where a taper's case begins

A tapered block was measured against a plain one on the exact bias each implies, and the answer was that the taper's advantage has not arrived at any block length a hundred and twenty rows can afford — with the crossing at ℓ = 20 and a difference there of three tenths of a point, too small for the critical-value table to resolve. Two of those three statements are about the wrong quantity. What a sample reports at ℓ = 20 is four points rather than three tenths, because the autocovariances the window is applied to are themselves attenuated and the window that discards the long lags loses less of them; and the comparison is made at a shared block length where each window has its own best one. Read at each window's own setting, on the error rather than on the bias, the ordering reverses at a hundred and twenty rows.

The gap a sample shows

The exact difference between two block windows at a block length of twenty is three tenths of a point. What a hundred and twenty rows report is four and a third, because the autocovariances the window is applied to are attenuated too.

4 figures · Bootstrap, part 11

What a longer block buys and what it costs. A trapezoidal block at 120 rows, with the error split into the two things it is made of. The bias falls with the block length, because a longer block attenuates less, and it flattens at 23.2% because the sample's own autocovariances are short whatever window is applied to them. The spread rises with it, because a longer block means fewer of them. Their sum in quadrature has a minimum at ℓ = 16, which is not where either of the two has one. The faint line is the rectangle's total error, for scale: it is above the trapezoid's from ℓ = 12 onwards.

Bias is not the whole of it

A window that reaches zero at its ends attenuates less and uses less of each block. The block length that minimises its bias is not the one that minimises its error, and comparing two windows at one length compares one of them mis-tuned.

3 figures · Bootstrap, part 12

What each instrument costs to read. The number of draws each instrument needs to separate a rectangular block from a trapezoidal one at two standard errors, at a block length of 20 and 120 rows — measured from each instrument's own spread on the same draws. The implied variance needs 7.0 and the 95% point needs 20.2, a factor of 2.90 at this block length. There is a closed form beside it and it does not depend on either the scale or the size of the gap: the standard error of a p-quantile is √(p(1−p))/f(q) over √B where a standard deviation's is σ/√(2B), which at the 95% point of a nearly normal reference distribution is 3.30 times as many draws for the same statement. And the quantile route needs every one of those draws resampled, where the variance route needs none.

Measuring a variance rather than a quantile

A resample's implied long-run variance can be computed from the sample with no resampling in it at all. A critical value cannot, and the difference is a factor of three in the draws before any of the resampling is counted.

4 figures · Bootstrap, part 13

Where the taper's case begins, and it is not where the algebra says. The block length at which a trapezoidal block's implied variance stops being more biased than a rectangular one's, against the length of the sample. Computed exactly — from the law's own autocovariances, with no sampling in it — the answer is 19.2 and does not depend on the sample at all. What a sample of 120 rows reports is 13.3, and the reported crossing walks out towards the exact one as the sample grows: 13.3, 15.0, 16.4, 18.0. The mechanism is that the autocovariances the window is applied to are themselves attenuated, worst at the longest lags, and the window that discards those lags loses less of them.

The error no window repairs

Every block window's best estimate of a long-run variance is wrong by about forty per cent at a hundred and twenty rows, and the largest part of that is not a bias at all. Choosing the window moves a twentieth of it.

3 figures · Bootstrap, part 14

A covariance with no parameter

A regression's coefficients and its errors' dependence can be fitted together when the dependence is one number. When it is an estimated covariance there is nothing for “jointly” to mean — until a family is named, and then the family's own width is the parameter. Three things the measurement says, two of them the opposite of the guess: the plug-in's shortfall is the taper's rather than the data's and is nothing under two of four laws; nothing in the likelihood chooses the width, because nested families buy about a unit of it a lag and that is what a criterion charges; and a fit-only objective does not run away, because a unit diagonal fixes the trace.

A family before a fit

A regression's coefficients and one correlation can be maximised together. Replace the correlation with an estimated covariance and there is nothing left for "jointly" to mean — until a set of covariances is named, and the set turns out not to contain the truth.

9 figures · Dependence, part 10

Most of the rise is the optimiser's, and under one law it is not. The rise in log-likelihood from the tapered plug-in to the maximum over the same eight-lag band, beside what the same optimiser produces on a sample generated from the plug-in's own covariance — where the family is correctly specified by construction and there is nothing to find. Under AR(1) at 0.8 the raw rise is 5.72 and the manufactured baseline is 4.79, leaving 0.93 at 1.8 standard errors; under long memory the excess is 0.14, at 0.2. Under the moving average it is 11.87 at 19.4 standard errors, on every draw. The taper is a shrinkage, and it costs nothing where the sequence decays smoothly and a great deal where it stops dead.

The plug-in and the maximum

A tapered covariance estimate sits five and a half log-likelihood units below the maximum of the likelihood it is substituted into. Four fifths of that is what the optimiser would have found if nothing were missing.

9 figures · Dependence, part 11

A wider band is always a better fit. The likelihood maximised over the band, at five widths, averaged over 30 samples. A band at L lags is a band at L + 1 with the last entry held at zero, so the families are nested and the maximised likelihood cannot fall — it does not, on any draw. What it does is rise at 0.984 of log-likelihood a lag. A parameter that is doing nothing buys half a unit in expectation and Akaike's criterion charges one, so this is a criterion very nearly indifferent between every width on offer. The dashed line is what a charge of one unit a lag would exactly cancel. Nothing in the fit chooses a width, and what does choose one is a charge somebody has to pick.

Nothing in the fit picks the width

A wider band is always a better fit, and it is better by about one unit of log-likelihood a lag — which is the order of what a criterion charges for a parameter. Three defensible rules choose widths a factor of three apart.

8 figures · Order-selection, part 9

Fitting them together is worth something under one law. Four fits of the same regression under four dependences: least squares, the two-step plug-in every whitened rule in this collection runs, the coefficients and the band maximised together, and a whitening at the law's own covariance that nobody has. Under the moving average — the one law the band family contains — the joint fit beats the two-step by 0.0077 at 3.3 paired standard errors. Under the autoregression, long memory and the break it is a tie: 0.4, 1.0, 0.3 standard errors. That is the same ordering the likelihood gap gave, arrived at through the coefficients rather than through the objective.

What fitting them together buys

Maximising over the coefficients and the covariance together beats the two-step under one of four dependences and ties under the other three. It is the one the band family contains, and the likelihood said so before any coefficient was compared.

7 figures · Dependence, part 12

How long the list is

A tuning parameter chosen per candidate rather than once for the table costs something, and the window's figure was measured while the order's was not — because their lists are different lengths and matching them changes what each rule is. Matched at every length, the two cost the same. What separated them was not the list at all: a per-candidate whitening is a different error model for every candidate, the Gaussian likelihood has a term that says so, and the sieve's rule in this collection never carried it.

The number the comparison was missing. What it costs to choose the tuning parameter for every candidate separately rather than once for the table, under AR(1) at 0.8, paired on the draw. The window's figure is the one the earlier field reported; the order's is the one it named and did not make. They are the same size — 0.00401 against 0.00360, at 2.30 and 1.72 paired standard errors — and matching the lists at eight values leaves them the same size again. The prediction that the longer list would make the order's cost the larger of the two is not what happens; what happens is that the two rules cost the same once they are scored by the same criterion, which took a missing term to arrange.

The comparison that was not made

Choosing a whitening's window separately for every candidate costs 0.00401 of regret. The same question about an order was named and left, because the two lists are different lengths. The order's answer is 0.00360, and matching the lists changes almost nothing.

7 figures · Order-selection, part 10

The term that cancels, and the term that does not. The volume each candidate's whitening moves — log|Ω̂| — for a sieve of order 4 on one sample of 120 rows. Estimated once from the fullest candidate and used for the whole table, it is the same number for every candidate, so it drops out of every difference the criterion reads: that is why nothing in this collection has ever needed to carry it. Estimated from each candidate's own residuals it ranges over 23.87, which is more than a parameter is worth, and the criteria being compared are then fits made under different error models with no term saying so. The window's rule has carried this term since the estimated-covariance field and the sieve's never had it.

The volume a whitening moves

A sieve's whitening has a determinant and this collection's criterion for it never carried one. Shared across a table the term cancels exactly, which is why nothing ever noticed; used per candidate it is worth more than a parameter and the whole comparison turns on it.

5 figures · Criterion, part 18

What a longer list actually changes. How often the five candidates choose different tuning parameters, at a true null where every one of them contains the truth, so a disagreement is manufactured rather than discovered. The order's list is an interval of integers, and thinning it moves the rate smoothly from 0% at two values to 39% at thirteen. The window's is not an interval — it runs 0, 1, 2, 4, 8, 12, 20, 30 — so a thinned window list jumps depending on whether it happens to keep the width the criterion wants, between 0% and 42% with no order to it. So "the same length" was never quite the same thing for the two rules, and it is a smaller effect than the field it was invoked to explain.

A list is not a rule

How often five candidates disagree about a tuning parameter runs from nothing at two values on the list to two draws in five at thirteen. What the disagreement costs does not move at all.

7 figures · Order-selection, part 11

Two searches over one sample

A break point that has been looked for costs more than a count of parameters says. So does a window chosen from a list, and nothing here had charged for it. A rule that does both is running two searches over one sample, and the charges do not add: the pair manufactures a fifth less than the two apart. What follows is sharper than a correction to a sum — most of what a break search finds under correlated errors is the correlation, so once a whitening has been chosen from the same sample the break's own charge more than halves, and carrying the published one across switches the test off entirely.

Two searches, one sample

A searched break in a regression manufactures 34.7 of likelihood ratio where a count of coefficients says 11.1. A searched window manufactures 84.0. The two together manufacture 99.4, not 118.7.

7 figures · Break point, part 4

The charge that is not a sum

Charging two searches what each costs on its own is conservative, and conservative here means the test never fires. At the largest break measured it declares nothing, on every draw, while a calibrated threshold reaches 29%.

5 figures · Break point, part 5

A charge that depends on the rule

The break search's charge is 34.7 on its own and 15.4 once a window has been chosen from the same sample. Most of what a break search finds under correlated errors is the correlation, and a whitening has taken it already.

7 figures · Break point, part 6

The diagnostic after the trial

The test for whether a balanced-assignment walk can reach the whole admissible set is run on a covariate function, before any outcome exists. Run instead on the difference in arm means it is the same test — under the sharp null the outcome is a fixed column — and it is about the statistic the p-value is actually built from. Two findings: an outcome is a probe nobody chose, and on a set that is genuinely split three in ten of them see nothing at all; and the published two-sided p-value is exactly right on half a reference distribution, while a one-sided one is not.

Two diagnostics, one answer, two different moments. The two-chain statistic on a covariate probe and on the trial's own difference in arm means, at seven tolerances of a fourteen-unit rule, against the enumerated truth. Both are quiet wherever the set is one set and both fire wherever it is not, at every tolerance — which is what says the outcome probe is the same test rather than a resemblance of it. The difference between them is not accuracy and it is not power. It is when: the covariate probe can be run before a single outcome exists, when a practitioner can still loosen the rule or change the sampler, and it can be run again on a different function if it comes back quiet. The outcome probe runs after the trial, on the one column the trial produced, and what it can do with a positive verdict is repair the p-value rather than the design.

The statistic the p-value is about

The test for whether a balanced-assignment walk reaches its whole set is run on a covariate function chosen before the trial. Run on the difference in arm means it is the same test, and it is about the number the trial publishes.

8 figures · Randomisation, part 12

The probe a trial has is the probe a trial got. What the two-chain test says when it is run on the trial's own difference in arm means, over 24 outcomes on one fourteen-unit set. The set is in 2 mirror components — that is enumerated, not inferred — so every quiet reading is a miss. 29% of them are quiet. The reason is in the enumerated set rather than in the run: how far the two components are apart on a given probe ranges from 0.001 to 4.938 of a within-component spread across these outcomes, a factor of several thousand. Both covariate probes — chosen before any outcome existed, and replaceable if they had been quiet — report the split. An outcome cannot be chosen and cannot be replaced.

A probe nobody chose

On a set that is definitively in two pieces, seven of twenty-four outcomes report nothing at all. Every covariate probe reports it. What separates them is not accuracy — it is that one of them can be chosen and the other is what happened.

8 figures · Randomisation, part 13

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