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 11 essays across 9 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 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

Named alongside it

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

Basis functionsCovariate balanceDesign criterionProjectionRobustnessClosed formDesign measureEfficiencyEquivalence theoremExperimental designInformation matrixThe non-linear model

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