Normal Distribution
nothing is called 95% until it has been counted
a-rule-that-may-not-see-the-mean

One curve, three designs, and the disagreement is the weights

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 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.

What the procedure may not readslider: how many times wide the protected rectangle is, 4 positionswide3 views

What else it draws

The same object, drawn to answer the other questions the essays put to it.

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 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.

Exact coverage, at every block size. Coverage of the interval each rule reports, at a nominal 95%, over 2,500 runs each with a standard error of 0.44 points. The blinded rule stops on the within-block contrasts and reports an interval built from the block means, and those two are independent whatever the rule does — so the interval is an ordinary t interval on b − 1 degrees of freedom and its coverage is exact. It is exact at every block size drawn. The interval a practitioner writes at the purely sequential rule's stopping time covers 91.72%, and Stein's two-stage rule is exact for the same reason as the blinded rule and spends 2.10 times the observations to be so. The bars are truncated at 86% so the differences can be seen.

Coverage of the interval each rule reports, at a nominal 95%, over 2,500 runs each with a standard error of 0.44 points. The blinded rule stops on the within-block contrasts and reports an interval built from the block means, and those two are independent whatever the rule does — so the interval is an ordinary t interval on b − 1 degrees of freedom and its coverage is exact. It is exact at every block size drawn. The interval a practitioner writes at the purely sequential rule's stopping time covers 91.72%, and Stein's two-stage rule is exact for the same reason as the blinded rule and spends 2.10 times the observations to be so. The bars are truncated at 86% so the differences can be seen.

Where it is used

4 essays draw this figure, each at the numbers its own argument is about, so the same picture answers 4 different questions.

All 80 figures

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