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

A design that assumes less

A design for a non-linear model is optimal only at a guess about the answer. Two ways out: protect the worst parameter value in a range rather than the average, which is an optimum that sits on a tie rather than a slope; or stop guessing, run part of the experiment, and design the rest at the estimate — where the interesting question turns out to be what that does to the interval afterwards.
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.

 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.

 One experiment finding out where to look. A single run of the fully sequential design: 40 runs, the first 8 placed at the guess K = 1, then the model refitted and the design revised after every 2. The marks are the settings the runs were made at. The horizontal lines are where a design built at the truth K = 3 would have put them — 1.875 and 10.00 — and the rule walks onto them without being told: its estimate of K after the first eight runs was 2.694, and by the end 2.765 against a truth of 3. The whole experiment is 96.5% as efficient as the design that knew the answer, where running all 40 at the guess would have been 81.1%.

The design that stops guessing

Every repair so far protects a guess. The alternative is to run part of the experiment, estimate the parameter from it, and design the rest at the estimate — which recovers most of what a threefold wrong guess costs, and has a best moment to stop guessing that is earlier than anyone expects.

 What the interval covers, after a design that read the data. 800 experiments of 12 runs, all at the same truth. The first pair is a design fixed in advance; the second is one whose settings were chosen from the first stage's own outcomes. If choosing the design from the data broke the inference, the second pair would sit below the first, and it does not — 91.3% against 92.6%. What does move the coverage is the shape of the interval rather than the design: the Wald interval assumes the estimate is normal around its own standard error and falls short under both designs, and the profile interval, computed from the same residual sums of squares with no derivative in it, covers 94.3% and 94.9%.

What a design chosen from the data costs

Two fields on this site measured what happens when a rule reads the data, and the error rate broke both times. A design that reads the data to decide where to put its runs breaks nothing — and the control that proves it also finds what the real shortfall is.

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