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

Ds-optimality — where it appears

The design criterion that optimises for a nominated subset of the parameters, treating the rest as nuisance, with its own equivalence theorem at the subset's size. Composing it with a worst case over a parameter range is not a special case of either, which is where the design field's arithmetic stops being a catalogue.

Named by 8 essays across 3 fields — each of them below, with the objects they name alongside it.

Two designs for one model, and the weights are not equal. Where the runs go, for the same two-parameter model at K = 1 and a ceiling of T = 10. The D-optimal design for both parameters is the familiar one: half the runs at 0.8333 and half at the ceiling. The design for the half-saturation constant alone moves the lower setting down to 0.6040 — where the response curve is still bending, which is where K is visible — and, unlike every design in the two fields before this one, it does not split the runs evenly: the weights are exactly 1/√2 and 1 − 1/√2, 0.7071 and 0.2929, at every K, V and T. The equal weights of D-optimality were a consequence of asking about both parameters at once, and nobody had to notice while that was the only question being asked.

An efficiency that is a ratio

A design chosen for a model is not a design chosen for the parameter somebody wanted. Asking for one of two parameters moves the runs, unbalances the weights, and costs the other question exactly 15.07% — at every setting, because it is algebra.

guarantee · Criterion The end of the family is the member the algorithm cannot reach. Every member of the Φₚ family optimised on the same 11² candidates, and two eigenvalues of each answer. The upper curve is the smallest eigenvalue of the information matrix — the quantity E-optimality maximises — which rises from 0.0993 at the D end to 0.1994 at p = 64. The lower curve is the gap between that eigenvalue and the next one up, which falls from 0.0611 to 0.0011. A smallest eigenvalue is not differentiable where it is repeated, and the family is driving the gap to zero: the one criterion here whose meaning fits in a sentence is the one whose optimum sits on a corner of its own surface. The candidate grid is on a slider and it answers a narrower question than it looks. On this square region, refining an odd grid from seven to eleven moves nothing at all — the optimum's support is the corners, the edge midpoints and the centre, and every odd grid from five up contains all of them. An even grid has no centre point and cannot reach the answer at any member of the family. On a disc, where the boundary passes through no grid point, refinement does move it.

The family behind the letters

A, D and E are not three ideas. They are three points of one family with a single dial, and running the dial from one end to the other doubles the smallest eigenvalue of the information matrix while closing the gap above it fifty-three-fold — which is the family driving its own last member to the place where it stops being differentiable.

criteria · Criterion One curve, three designs, and the disagreement is the weights. The compartmental response exp(−θ₁t) − exp(−θ₂t) at the guess θ = (0.2, 1.2), with three designs underneath it. Each mark is a setting and its height is the share of the experiment spent there. The D-optimal design splits the runs equally between two settings — that is the determinant's answer and it is equal for every model of this kind. The design for the first parameter alone puts 1.9% of the experiment at its early setting and the rest at its late one, because the early runs are there to identify the nuisance and nothing more. The design that protects a 4-fold rectangle needs 3 settings and spends 9.1% at the earliest of them.

The guess with two numbers in it

Every optimal design for a non-linear model is optimal at a guess. Where the model has one parameter that moves the settings, that guess is a number and everything about it comes out in closed form; where it has two, three constants become functions and one of them becomes zero.

blind · Local design 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 Where a Ds-optimal design puts its runs. The Ds-optimal measure over 121 candidate settings on a square region. It keeps 9 of them and discards the rest, and the 9 it keeps are the settings a catalogue would have offered without any of this arithmetic. What the search adds is the weights: 0.2500, 0.1250, 0.0625, which nine equal runs cannot express.

The two terms anybody wanted

D-optimality estimates all six parameters of a quadratic as precisely as possible. Nobody wants that. An experimenter looking for a maximum wants the two curvature terms, and the design that gives them is not the D-optimal one — it is a quarter of the runs at the centre, exactly, and the D-optimal design is 75.3% efficient for the question that was actually asked.

criteria · Criterion 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 main effects cost the rest of the model. The compound optimum of α·log Ds(main effects) + (1 − α)·log D on an 11-by-11 grid of the square, at 14 weights from α = 0 to 0.999, placed by its D-efficiency against the D-optimal design and its main-effect efficiency against the factorial. The D-optimal design (α = 0) gives the main effects 74.3% of what the factorial gives. At α = 0.5 the design keeps 94.6% of D and gives 85.6%; at 0.9, 64.9% and 96.8%; at 0.99, 31.3% and 99.7%. The factorial itself (open, at the left edge) gives the main effects everything and the model nothing: its D-efficiency is zero. The design optimal for the two quadratic terms (red) gives the main effects 50.0% at a D-efficiency of 83.6%.

A subset the model cannot use

Nominate the two main effects of a quadratic response surface as the parameters of interest and the optimal design is the 2² factorial — four corners, a quarter each — which leaves the full model's information matrix at rank four of six and the curvature with nothing at all. That is why the field's algorithm would not converge on it. Approached through a compound criterion instead, the trade is explicit: 96.8% of the main effects' best information costs the model 35% of its D-efficiency, and the last three points cost nine tenths of what is left about the curvature.

criteria · Criterion What a stated power leaves of the main effects. The main-effect efficiency of the best design reaching a power of 0.8 at level 0.05, as N|δ|² rises through 50, 60, 70, 80, 100, 130, 170, 220, 300, 400, 600, 1000. a saddle, two-degree test, the compound path: out of reach, out of reach, 0.806, 0.841, 0.881, 0.914, 0.937, 0.952, 0.966, 0.975, 0.984, 0.990. a bowl, two-degree test, the compound path: 0.782, 0.823, 0.851, 0.871, 0.898, 0.923, 0.941, 0.955, 0.967, 0.976, 0.984, 0.990. a bowl, its own one-degree contrast, the compound path: 0.827, 0.859, 0.880, 0.896, 0.918, 0.937, 0.952, 0.964, 0.973, 0.980, 0.987, 0.992. a bowl, a factorial with centre points: 0.914, 0.930, 0.940, 0.948, 0.959, 0.969, 0.976, 0.982, 0.987, 0.990, 0.993, 0.996.

A design chosen for a test

A screening experiment that wants its main effects and a check on curvature has been told to trade them with a weight nobody can state. It can state a power. Read against a power of 80%, the frontier collapses to one curve in runs times the squared size of the curvature, and a large enough trial pays almost nothing. But the curve belongs to a test. A factorial with a few centre points beats the entire frontier on a bowl, spending 4% of its main-effect information where the frontier spends 10%, and against a saddle of the same size it has exactly its 5% level.

criteria · Criterion

Named alongside it

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

Design measureD-optimalityEquivalence theoremOptimal designInformation matrixNuisance parameterExperimental designThe non-linear modelCurvatureClosed formCompartmental modelE-optimality

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