Field

What the design is asked to guarantee

Two halves of one question that were never separated: which parameters a design is for, and how much precision is enough. Protecting one parameter of a non-linear model over a range of its own values is a worst case of a ratio of determinants, and it is not a special case of either problem it is made of. Letting the experiment stop when it is precise enough is the adaptation with a theorem against it — the rule stops when its own noise estimate is low, so the interval it produces is short.

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.

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.

One rule keeps its promise and the other keeps its budget. Both stopping rules at five requirements, 1,500 experiments each, with a first stage of 5. The upper curve is the two-stage rule: 97.1%, 96.2%, 95.9%, 96.1%, 96.0% — at or above 95% at every point, which is a theorem rather than a tendency, because its interval is built from a spread estimated before the stopping point was chosen. It pays 2.06×, 2.01×, 2.00×, 1.99×, 1.99× the observations that knowing σ would need. The lower curve is the rule that re-estimates after every observation: 94.5%, 89.3%, 91.0%, 91.5%, 94.3%, on 0.98×, 0.87×, 0.88×, 0.93×, 0.96×. The second rule is the one anybody would run and the first is the one whose claim is true.

Stopping when it is precise enough

An experiment that runs until its estimate is precise enough is the natural design and the one with a theorem against it. Its two-stage cousin keeps its promise exactly, for every unknown spread, and pays twice the observations for it.

Four intervals at one stopping time. 3,500 experiments under the sequential rule with a first stage of 5 and a required half-width of 0.4, which is a demand that knowing σ would meet with 24.0 observations and which the rule meets with 20.8. The rule's own interval covers 90.3%. Replacing the fixed width by a t interval on the same data gives 92.0%. Keeping the rule's own random sample size and drawing a fresh sample of that size gives 89.8% at the fixed width and 95.5% for a t interval — so the sample size being random costs nothing, and the sample size being chosen by the data the interval is built from costs the rest. The spread estimated at the stopping moment is 17.1% below the truth, which is the same fact one level down.

The interval after a stop it chose

A rule that stops when the estimated precision is good enough stops on the samples whose estimate was small. Its interval covers 90% and claims 95%, and a fresh sample of the same random size covers 95.4%.

A third stage, between the two rules. Four rules for one requirement: a half-width of 0.4 at 95%, which knowing σ would meet with 24.0 observations, after a first stage of 5. Each bar is the mean number of observations as a multiple of that, and the note beside it is the coverage. The two-stage rule spends 2.03× and covers 95.42%, computed exactly. The three-stage rule — a second stage to half of what the first spread asks for, the spread re-read on everything, a third stage to the rest — spends 0.86× and covers 88.42%; given 8.50 more observations in every experiment's last stage it covers 95.00% on 1.20×. The rule that re-estimates after every observation spends 0.88× and covers 90.5% over 4,000 drawn experiments; it is the only row that has to be drawn, because it is the only one whose stopping point shares observations with the mean.

The fifth a third stage never reaches

A three-stage stopping rule keeps the two-stage rule's independence and the sequential rule's budget. With a first stage of five it covers 88.4% — worse than the rule it was meant to repair — because in a fifth of experiments the second stage buys nothing.

$Three repairs at one requirement. Coverage against spend at a half-width of 0.4 and a first stage of 5, all exact. A floor on the second stage at 5, 8, 10, 12, 15, 18, 21, 24 observations covers 88.4%, 91.4%, 92.4%, 93.1%, 93.9%, 94.6%, 95.3%, 96.0% at 0.86, 0.94, 0.97, 1.00, 1.02, 1.05, 1.09, 1.14 times the oracle's spend. The second stage reading with t, with its fraction dialled from 0.2 to 1, covers 88.1%, 89.0%, 90.6%, 92.1%, 93.4%, 94.4%, 95.2%, 95.9%, 96.4% at 0.84, 0.91, 1.02, 1.17, 1.34, 1.52, 1.70, 1.89, 2.08 times. A constant at the end covers 88.4%, 89.4%, 90.4%, 91.3%, 92.1%, 92.9%, 93.6%, 94.2%, 95.0%, 95.7% at 0.86, 0.90, 0.94, 0.98, 1.02, 1.06, 1.10, 1.14, 1.20, 1.26 times.$

A floor that reads the first stage

A three-stage stopping rule fails at a first stage of five because a low first reading asks for nothing more, and the only floors that repair it were set near the answer, by somebody who already knew roughly what the spread was. Inflating the first reading by Stein's own factor needs no such knowledge, and it does cut the experiments left stuck from a fifth to a fifteenth. But those still stuck cover less than before, the rule needs a different dial at every requirement, and a floor of 26 observations covers 96.5% at the spend where the rule covers 92.1%, even met by a spread it was not set for.

All essays