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

Maximin design — where it appears

The design whose worst efficiency over a stated range of parameter values is as large as it can be made. Its optimum usually attains that worst case at several points at once, which is a property of a convex problem and not of a discrete one.

Named by 18 essays across 11 fields — each of them below, with the objects they name alongside it.

A design that is right once, and one that is never wrong by much. Three designs for the Michaelis–Menten model, scored at every true value of K across a 16-fold range. The peaked curve is the two-point local design built at K = 1: 100% there and 66.7% at the worst point of the range. The curve just under it is the design that averages the criterion over a uniform prior on the same range, which is barely different — 67.9% at worst — because averaging is dominated by the middle of the range where the local design is already good. The flat line is the maximin design: 3 settings, never above 80.8% and never below 78.8%. Its worst case is 12.0 points better, and what it gives up is the 21.2 points at the one value the local design was built for.

The design for the worst case

A design for a non-linear model is optimal at a guess about the answer. Averaging over a prior repairs that on average; protecting the worst value in a range is a different problem, with a different answer, and it needs a third setting to reach it.

robust · Local design The zero was a fact about independence. What a balancing rule handed every main effect of both covariates removes of a pure interaction, as the covariates are allowed to move together. At ρ = 0 it is exactly nothing — at machine precision, at any number of main effects — which is the independent-covariate result and is correct. It is not small anywhere else: the product of the two covariates loses 64.0% of itself by ρ = 0.5, because h₁h₁ = h₀ + √2·h₂ and Mehler pairs h₂ with h₂ at ρ². Four interactions are drawn and none of them keeps the zero.

A zero that was an assumption

A rule handed every main effect of both covariates removes exactly none of a pure interaction. That is true at machine precision, it is a fact about independence, and it dies as the square of the correlation.

joint · Criterion Four designs, scored on the one parameter that was wanted. Every design scored by its Ds-efficiency for K at 13 true values across a 16-fold range. The peaked curve is the subset design built at the guess K = 1: 100% there and 42.1% at the worst point of the range. The flat curve is the maximin-Ds design, never above 64.8% and never below 61.2%. Between them is the maximin design for the pair — a robust design, protecting something else, and worth 41.9% at worst here. The lowest curve is the D-optimal design at the guess, which is what an experimenter who wanted K and looked up a design for the model would actually run: 29.1% at the worst point, against 61.2% available.

Protecting one parameter over a range

A design for a non-linear model is optimal at a guess. A design for one of its parameters over a range of guesses is a worst case of a ratio of two determinants, and it is not a special case of either problem it is made of.

guarantee · Local design Four designs, and what each of them guarantees. The worst Ds-efficiency each design achieves anywhere in a rectangle of parameter values 4 times wide in each coordinate. The design built at the guess guarantees 7.3%: it is perfect where it was built and nearly useless at one corner. The D-optimal design at the same guess guarantees 25.6% — it answers the wrong question everywhere and is therefore not concentrated on being right anywhere. The third is the one this field exists to measure: maximin over the first parameter's whole range, with the second held at its guess. It guarantees 7.0%, which is no better than the design that protects nothing. Protecting both is worth 45.9%, and it needs 3 settings to do it.

The worst case in two directions

A design that protects a range of one parameter is robust. Protect the range of one parameter while holding the other at a guess and the design is still robust, still has a guarantee, and guarantees no more than a design that protects nothing at all.

blind · Criterion What the worst case is worth, one function at a time. The smallest share each dictionary removes, over seven outcome shapes, at a correlation of 0.5. A rule balancing the mean of each covariate has a worst case of exactly zero — against the square, and against both products. Adding a median split to it, which is the second thing every trial balances, leaves the worst case at exactly zero, because a median split is odd and so is a mean. Adding the square instead moves it to 6.8%, and the extra functions after that move it to 7.4%. The worst case is decided by which parities the dictionary contains rather than by how many functions are in it.

What the extra function buys

A rule balancing the mean of each covariate has a worst case of exactly zero. Adding the median split — the other thing every trial balances — leaves it at exactly zero, and one square moves it.

dict · Criterion The optimum is a tie, and the tie is at both ends. The maximin design's efficiency across the range, and underneath it the prior that makes the averaged criterion as bad as possible. The efficiency curve is flat to within 2.1 points, and the minimum 78.74% is attained at K = 0.25 and 0.79 and 0.89 and 1.00 and 1.12 and 4.00 rather than at a single value: if it were attained once, the design could be moved towards that value and the worst case improved, so a tie is what having finished looks like. The bars are the least favourable prior's weights, computed by a completely different route — an averaging problem solved under the weighting that hurts most — and it puts its weight exactly where the ties are, reaching 78.63% against the direct search's 78.74%.

Where the minimum is attained

A design that protects a range is finished when its worst case is a tie. That is a checkable property rather than a description, it is why the search cannot climb a derivative, and it is the same corner the criteria field found at the end of the Φₚ family.

robust · Local design The guarantee, as the basis is allowed more functions. The lower line is the best worst case over the six named shapes for a basis of each size, found by scoring every subset of the dictionary — an exact answer, since the problem is finite. One function guarantees 2.3%, which is nearly nothing; three guarantee 59.0% and the basis that does it is the covariate, its square and its cube, with no indicator in it. The upper line is the same problem with the basis drawn rather than fixed, which is worth 2.09 times as much at two functions and 1.32 at three. The two lines converge because a basis large enough to protect everything has nothing left to randomise over.

Which shapes are worth protecting

Choosing a basis by its worst case is a finite problem with an exact answer. The answer has no tie in it, which a maximin optimum is supposed to have — and the tie comes back, along with twice the guarantee, when the basis is drawn rather than chosen.

basis · Blocking What a rule reads, against what the outcome uses. The variance of the treatment estimate relative to a coin's, for four things a rule might balance against three shapes the outcome might have, over 350 trials of 200 units. The diagonal is the easy part — a rule that reads the function the outcome uses removes about half the variance. What the table is for is the off-diagonal: reading the covariate alone is worth nothing against a quadratic (0.755), and reading all three is worth nearly as much against every shape as the matching rule is against its own (0.532, 0.493, 0.514).

Three functions of one number

A rule that balances the covariate is exposed to every shape the outcome might have. A rule that balances three functions of it costs two points of variance against the shape the first was built for and takes the worst case from a coin's to about half of it.

shape · Criterion A class that is a subspace has no guarantee below its own dimension. Each cell is the worst case over every unit-variance function in a class of dimension m, for a rule reading k functions: the smallest squared principal-angle cosine between the two subspaces. Wherever k is less than m the number is zero to machine precision, and that is not a weak guarantee but the absence of one — some direction of the class is orthogonal to the entire basis, and against an outcome in that direction the rule does exactly what a coin does. An experimenter who declines to name the shapes and asks instead to be protected against everything smooth is asking for the cells above the diagonal.

Where the guarantee is exactly zero

An experimenter who declines to name the shapes, and asks instead to be protected against anything in a class, is asking for a number that is not small but zero. Bounding the class is unavoidable, and the two ways of doing it choose different bases.

basis · Randomisation 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 A basis drawn rather than chosen. The share of the worst of seven shapes a balancing rule removes, over two covariates and a dictionary of twenty-four functions, for the best fixed basis of each size (lower line) and for a basis drawn at random from the least favourable mixture (upper line), at one to four functions. The fixed basis guarantees 0.0000, 0.0000, 0.4244, 0.6366: nothing at one function or two, because two of the shapes are pure interactions and two of them are main effects of different covariates, and no one or two functions touch all of them. The drawn basis guarantees 0.2042, 0.4084, 0.6133, 0.7920 in expectation over the draw — about a fifth of a shape for every function up to three. The marks above the upper line are the best reply to nature's mixture, 0.2051, 0.4098, 0.6142, 0.7931: the game's value lies between the two.

A basis drawn rather than chosen

On two covariates no fixed basis of one or two functions protects every shape an outcome might have — its worst case is exactly zero. Draw the basis at random from the right mixture and the guarantee is a fifth of a shape for every function. The price is per trial — at three functions, two draws in five leave one shape wholly unprotected.

product · Optimum The best worst cases among twenty-eight pairs. Every pair of two functions from eight — the covariate, its square, cube and fourth power, and cuts at −1, 0, 1 and 2 — held on both covariates, ranked by the smallest share it removes over seven outcome shapes at a correlation of 0.5. The best is the square and a cut at 1 at 7.00%; the mean and the square reach 6.84%. For almost every pair the shape holding the floor is the product of two median splits, which no pair removes more than a small part of.

Two functions a covariate

Every constraint costs admissible assignments at the same rate, so the decision a trial faces is a budget: which two functions of each covariate. Over twenty-eight pairs the worst case never picks the mean. At a correlation of a half it picks the square with a cut at one, beating the mean and the square by sixteen hundredths of a point on a floor held by a shape no pair removes a tenth of — and giving up 41% of a linear outcome to do it. Over the 127 lists of shapes a designer could write, twenty-three different pairs are best somewhere.

dict · Criterion The worst case against the number of functions, at four trial sizes. The largest share of a coin's imbalance the rule leaves in any of the four shapes, against how many functions it reads, at 50, 100, 200, 400 units. At 50 units: 103.5%, 38.6%, 37.2%, 37.0%, 39.8%, 41.7%. At 100 units: 105.9%, 37.8%, 38.9%, 30.3%, 27.0%, 24.6%. At 200 units: 95.9%, 44.2%, 42.8%, 32.9%, 28.4%, 11.7%. At 400 units: 108.5%, 41.8%, 41.3%, 32.1%, 26.1%, 5.8%. At fifty units every basis from two functions up leaves the worst shape between 37.0% and 41.7%; at four hundred, six functions leave it 5.8%.

How many functions a trial can hold

A rule that reads more functions of one covariate protects more shapes, and every function it reads costs the shapes it already held: the imbalance it leaves in a linear outcome falls like one over the trial size, with a constant that runs from about 3 for one function to about 16 for six. At fifty units that cost eats the protection, and the worst case over four shapes sits between 37% and 42% of a coin's from two functions to six. At four hundred it falls from 42% to 6%. By six functions the worst shape, at every size, is one the rule is reading.

shape · Criterion Where the design puts its runs as the noise stops being constant. The locally Ds-optimal design for the slow rate of the compartmental model at the guess (0.2, 1.2), on a horizon of 10, under each variance function: the settings and the share of runs at each. constant noise: 0.66 with 1.9% and 5.74 with 98.1%; noise follows the response, c = 1: 0.55 with 1.0% and 6.55 with 99.0%; noise follows the response, c = 3: 0.36 with 0.1% and 8.89 with 99.9%; noise follows the response, c = 10: 0.17 with 0.0% and 10.00 with 100.0%; noise follows the clock, c = 1: 0.49 with 4.2% and 4.39 with 95.8%; noise follows the clock, c = 3: 0.41 with 4.4% and 4.16 with 95.6%; noise follows the clock, c = 10: 0.34 with 4.4% and 4.03 with 95.6%.

A variance that follows the response

Every design for the two-rate compartmental model assumed the same noise at every sampling time. Let the noise grow with the response, as a combined additive and proportional error does, and the design for the slow rate runs to the tail of the curve, where the response and so the noise are smallest: at three additive standard deviations of proportional error at the peak, the constant-noise design is worth two thirds of the right one. The earlier essay expected a weight function to break the model's scale invariance. A variance that follows the response leaves it exact to six decimals; only one that follows the clock breaks it. And tailoring the design to the noise makes it more fragile to the rates: its worst case over the rectangle falls from 9.2% to 3.3%.

blind · Local design Three contrasts on one dataset, three different splits. The variance-minimising allocation for each of three ways of reporting the same two-arm comparison, against the first arm's proportion, with the second at 0.1. A risk difference wants the arm with the larger p(1 − p) to get more units; a log odds ratio wants it to get fewer, and the two curves are exact reflections of each other in the half line. A log risk ratio wants something else again. At a first-arm proportion of 0.6 they ask for 62.0%, 21.4% and 38.0% of the units. A trial reporting more than one of them cannot be optimal for either.

Two contrasts, one split

A risk difference wants 62.0% of the units in the first arm, a log risk ratio wants 21.4% and a log odds ratio wants 38.0% — on one dataset, with one pair of proportions. The difference's rule and the odds ratio's are exact reflections of each other, so no split can be near-optimal for both.

allocation · Allocation What a floor costs a drawn basis. The expected guarantee of a basis drawn from a mixture restricted to bases that each remove at least a stated share of every shape, solved at each distinct floor. 2 functions, of 276 bases: 40.8% at a floor of 0.0%; the best fixed basis floors at 0.0%. 3 functions, of 2024 bases: 61.3% at a floor of 0.0%, 50.0% at a floor of 31.8%, 50.0% at a floor of 34.7%, 49.7% at a floor of 40.5%, 42.4% at a floor of 42.4%; the best fixed basis floors at 42.4%. 4 functions, of 10626 bases: 79.2% at a floor of 0.0%, 77.1% at a floor of 31.8%, 77.1% at a floor of 34.7%, 77.1% at a floor of 35.1%, 77.1% at a floor of 35.6%, 77.1% at a floor of 37.0%, 77.1% at a floor of 38.3%, 77.1% at a floor of 40.5%, 76.5% at a floor of 41.0%, 76.5% at a floor of 42.4%, 76.5% at a floor of 42.8%, 76.5% at a floor of 42.9%, 76.5% at a floor of 43.5%, 76.5% at a floor of 44.7%, 76.5% at a floor of 45.3%, 76.5% at a floor of 46.0%, 76.5% at a floor of 46.8%, 76.5% at a floor of 47.2%, 76.5% at a floor of 50.0%, 76.5% at a floor of 56.0%, 76.5% at a floor of 63.7%; the best fixed basis floors at 63.7%.

A floor for the draw

A basis drawn at random guarantees more on average than any fixed one, at the price of leaving some shape wholly untouched in a large share of trials. Requiring every basis in the draw to keep a floor looks like a smooth trade between the two. It is a staircase, and the first step does nearly all the work. No basis has a floor between zero and 31.83%, so there is nothing to give up until a floor of that size is asked for. And at four functions, seven bases tie at the fixed basis's own 63.66%: a mixture of them keeps that floor exactly and adds thirteen points to the guarantee.

product · Optimum The extra cost of each follow-up schedule over the cheapest, across every correlation curve with adjacent visits at 0.7, when a patient costs 5 visits to recruit. With one baseline, 1 follow-up costs at most 28.0% more than the cheapest schedule, 2 follow-ups cost at most 11.1% more than the cheapest schedule, 3 follow-ups cost at most 24.1% more than the cheapest schedule, 4 follow-ups cost at most 34.8% more than the cheapest schedule, over every stable share from 0 to 0.7. The schedule whose worst case is smallest is 2 follow-ups; its worst case is at a stable share of 0.00. Under compound symmetry the cheapest is 3; under a stable share of 0.4 it is 1.

The correlation a pilot can see

Two models of repeated visits agree that adjacent visits correlate 0.7 and disagree about how many follow-ups a trial should buy: three under equal correlation, one when the correlation fades. Planning on the wrong one costs 28.0% or 9.9%. Two follow-ups cost at most 3.2% more than the best under either, and at most 11.1% under any curve with that adjacent correlation. A pilot of twenty patients at four visits chooses worse on average than that rule when the correlation fades; it takes about forty to draw level with it, and at an effect of half a standard deviation those forty patients' extra visits cost fifty times what the better schedule saves.

planned · Power What a trial's own first patients buy when they choose the follow-up schedule for the rest, powered for an effect of 0.2. Internal pilots measured on two follow-ups, the schedule that needs no pilot, whose correlation curve chooses the schedule for every later patient. Extra cost over the cheapest schedule at stable shares 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7: no pilot, two follow-ups throughout, 11.1%, 9.8%, 8.1%, 5.9%, 2.9%, 0.0%, 0.0%, 3.2%; an internal pilot of 20, 10.5%, 9.8%, 9.3%, 7.9%, 5.9%, 3.7%, 4.2%, 5.2%; an internal pilot of 40, 5.9%, 5.7%, 5.6%, 4.7%, 3.4%, 2.0%, 2.4%, 2.5%; an internal pilot of 80, 3.6%, 3.4%, 3.2%, 2.8%, 1.9%, 1.0%, 1.6%, 1.6%. Worst cases: no pilot 11.1%, 20 patients 10.5%, 40 patients 5.9%, 80 patients 3.6%.

A schedule the trial chooses for itself

A separate pilot cannot pay for itself in choosing a trial's follow-up schedule, because its patients are recruited, measured and discarded. A trial can instead measure its own first patients on the schedule that needs no pilot — two follow-ups — fit the correlation curve to them, and give everyone after them the schedule that curve makes cheapest. Those first patients are counted in the analysis and cost no extra visit. Forty of them cut the trial's worst-case excess cost across every curve with adjacent correlation 0.7 from 11.1% to 5.9% in a trial of two hundred an arm, and eighty cut it to 3.6%; twenty buy almost nothing. And because the curve is read from within-arm deviations, the choice cannot see the effect: the pilot's effect estimate is the same whichever schedule it chose.

planned · Power

Named alongside it

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

Basis functionsCovariate balanceRobustnessClosed formDesign criterionExperimental designInformation matrixInteractionProjectionEfficiencyThe non-linear modelAllocation rule

All concepts