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

Compartmental model — where it appears

A response built from a rise and a decay with a rate each, whose two parameters both move the settings at which it is best measured. Because both parameters move the optimal settings, the guess a design has to be built at is a point in a rectangle rather than a single number.

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

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, 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 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

Named alongside it

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

The non-linear modelOptimal designDesign measureDs-optimalityEfficiencyEquivalence theoremInformation matrixLocal optimalityMaximin designNuisance parameterRobustnessScale invariance

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