A floor for the draw

A basis drawn rather than chosen looked at the same problem from both ends. A balancing rule can hold only a few functions of two covariates, and an outcome can have any of seven shapes. The basis that maximises the worst shape’s protection at three functions guarantees 42.44% of every shape. Draw the basis at random from the right mixture and the guarantee rises to 61.33%, but it is a guarantee of an average: in 38.7% of draws the quadratic shape gets nothing at all, in 18.8% the product gets nothing, and any particular trial is one draw.

The essay called those the two ends of a range. A fixed basis maximises the floor. A drawn one maximises the expectation, with no floor at all. In between it imagined mixtures restricted to bases that each remove at least a stated share of every shape. Solving the same game over the restricted set, and tracing the expectation as the floor rises from zero to 42.44%, would give the whole trade in one curve, and all that would remain is for somebody to say where on that curve a trial should sit.

The curve has now been traced. What it shows is that the question was the wrong shape: there is not a curve to sit on.

A staircase with one real step #

The game is exact and the dictionary is finite. At three functions there are 2,024 bases, each with a computable share of each of the seven shapes removed. Fix a floor, keep only the bases whose worst shape clears it, and solve for the mixture that maximises the worst shape’s expected share. The admitted set changes only where some basis’s own floor sits, so the game need only be solved at each distinct floor, and at three functions there are just five of them.

The figure at the head of this essay is all five, and the same curve at four functions. The blue line at three functions:

Two things stand out.

The range from zero to 31.83% does not exist. No basis of three functions has a floor anywhere in there. Of 2,024 bases, 2,016 leave at least one shape with nothing at all, and the eight that do not all clear 31.83%. So a trial cannot “ask for a small floor”. It can ask for no floor, or for 31.83%, which is one over π and is the share of a product of splits that a function of one covariate reaches.

The cost comes all at once. Going from no floor to 31.83% costs 11.3 points of expectation, from 61.27% to 50.00%. Going from 31.83% all the way to 40.53% costs a further 0.27. The last step, to the single best fixed basis, costs 7.3.

So what the earlier essay described as a range has three useful positions and a couple of others that are dominated in everything but name. A floor of 40.53% at 49.73% is nearly identical in expectation to the floor of 31.83% at 50.00%, so nobody should prefer the latter once they have accepted the former. Someone who wants any floor at all should take that one. The real choice is between three points: the unconstrained draw, at no floor and 61.27%; a floor of 40.53% at 49.73%; and the fixed basis, at 42.44% on both.

Of those three, the middle one’s trade against the fixed basis is lopsided. It gives up 1.91 points of floor for 7.29 points of expectation.

How few bases are left #

The reason for the jump is structural. Seven shapes in three functions is a tight budget. Most bases cover two or three directions of a seven-dimensional problem and have nothing at all in the others, and a basis with nothing in a shape’s direction has a floor of exactly zero. A floor greater than zero is not a matter of degree: it means covering every shape, and only eight of the 2,024 bases can.

Which shapes are worth protecting has reported the discrete version of this for a fixed basis, that a finite maximin has no tie in it where a continuous one would. The staircase is the same fact about the mixture. A continuous mixture over discrete bases gives a continuous curve in the weights, for any fixed set of bases. But the floor constraint acts on individual bases, not on the mixture, and it switches bases on and off whole.

At four functions a trial can have both #

The picture changes qualitatively at four functions. There are 10,626 bases and 21 distinct floors, and the green line in the figure sits above everything.

The fixed basis at four functions floors at 63.66%, and in expectation it also gives 63.66% because it is a single basis. But seven different bases tie at that floor. A mixture over them keeps the floor exactly, since every basis in the draw clears 63.66% of every shape in every trial, and it raises the worst shape’s expected share to 76.53%.

That is thirteen points of expected protection, at no cost to the floor. At four functions the fixed basis is dominated. A trial should never use it, because there is an arrangement that is as safe in every trial and better on average. The only reason to keep it would be simplicity, and the mixture of seven bases is not complicated: a rule drawn from a short list once, before the first patient, and recorded.

The unconstrained draw offers only 2.67 more on the worst shape, at 79.20%, for an exposure that the earlier essay priced at one shape going wholly unprotected in 20.4% of draws. So 76.53% with a floor of 63.66% is nearly all of what is on offer, without the exposure.

That is a much better answer than the earlier essay’s question expected. It imagined a trial having to choose on a curve “what it is for”. At four functions no choice is necessary. At three, a choice is necessary and is simple, and at two, no floor is available at any price: every one of the 276 bases of two functions leaves some shape untouched, and the draw’s 40.8% is all that can be had.

What the mixture draws from #

The floor changes what is in the draw. The unconstrained game spreads over thirteen bases of very different kinds: some hold a split of each covariate, some hold a product of the covariates, some hold only functions of the first covariate plus a product. Any given draw leans hard on a few directions, and leaves others exposed. That is what makes the guarantee an average and the exposure real.

Under a floor of 31.83% the mixture collapses to two bases, in the proportions 0.79 and 0.21: the first covariate and its square with a product of splits, or the same pair with a product of the covariates. At the highest floor, it is the first covariate, its square, and a covariate-by-split function: the fixed basis.

There is a piece of design knowledge in that collapse. All eight bases that clear a floor at this size hold the first covariate’s square, and either the covariate or its median split, and then one interaction. So two of the three functions are spent on one covariate’s main shapes, and the third has to do all the interaction work by itself. The price falls on the shapes the third function cannot reach.

The same is visible shape by shape.

The unconstrained draw equalises: its worst five shapes all sit near 61.3%, which is what a maximin mixture does, and each has a real probability of being left out completely. The fixed basis does not equalise at all. It is spending two of its three functions on shapes that a trial may not care most about, because the floor forces it. A trial whose outcome is plausibly additive across two covariates gets 50% from the fixed basis and 61.3% from the draw, in expectation.

So a floor is not neutral about which shapes are served: it is a commitment to one covariate’s main shapes. A reader who thinks the outcome is likely additive might well read that as a reason to pass on the floor. Nothing in the game can decide that.

Why the floors are the numbers they are #

The steps are not arbitrary, and reading them explains the staircase better than the table does. Every floor is set by one shape, and it has always the shape the basis can reach only through something it does not hold.

The 63.66% at four functions is $2/\pi$. Of the seven tied bases, three hold the first covariate and have a median split of it as their worst shape; the other four hold the median split and have the linear shape as theirs. The two are correlated at $\sqrt{2/\pi}$ on a normal, and $2/\pi$ is the share of one that the other reaches. That is the same constant the reliability essay found, where a median split turned out to be a recording of its covariate at a reliability of $2/\pi$. Holding both would remove the tie, and holding both uses up a function, so at four functions a basis has to choose, and it pays $2/\pi$ either way. That is why there are ties at all: the seven differ in which of two equally costly mistakes they make.

The 31.83% at three functions is $1/\pi$, half of that, and it is the additive shape at a basis holding the median split: half of the additive shape is in the second covariate, about which the basis knows nothing. The 40.53% is $(2/\pi)^2$, the moderated and product shapes reached through a product of splits on both sides, each of which pays the reliability once.

So the staircase is an arithmetic of two constants, one half and $2/\pi$, and any dictionary holding both a covariate and its split would produce the same one. The discreteness does not come from small samples or from coarse grids. It is a fact about how many times a shape can be reached by way of something else. Two functions a covariate met the same parity from the other side, and it is what gives a design question of this kind its short list of answers. A floor is never a free parameter here. It is a count of how many reliabilities the worst shape has to pass through.

What a protocol can use #

The earlier essay asked somebody to say where on the curve a trial should sit. The measurement above makes the answer short enough to state.

At two functions there is nothing to decide. Draw or do not, but no floor exists.

At three functions, choose among three points. The unconstrained draw has the best average and some large per-shape exposures. The mixture at a floor of 40.53% gets almost all the floor of the fixed basis and 7.3 more points of expectation than it. The fixed basis is the simple one. The third of these is dominated unless simplicity counts for something.

At four functions, do not use a fixed basis. The mixture over seven tied bases is as safe and much better.

There is an unrecorded benefit too. A drawn basis asks something of the analysis: the draw is part of the randomisation and it has to be recorded. A mixture over seven bases carries that requirement at no more cost than a mixture over thirteen, and conditional on the draw the trial is an ordinary balanced trial on a fixed basis, so all the analysis this field has measured applies unchanged.

That is worth stating against what a programme of trials would see. Ten trials on the fixed basis at four functions would each deliver 63.66% of the worst shape, and on the corner shape they would deliver the same every time. Ten trials on the mixture would land on different bases, and on any given shape the deliveries would vary: the shape’s average would be 76.53% or better, and each individual reading at or above 63.66%. A reviewer comparing trials across a programme should expect that variation to be there and should expect it to be harmless. Nothing is hidden, so long as each trial’s draw is in its record.

And the guarantee has a caveat it should not lose. The 76.53% is an expectation over the draw, on the worst shape in expectation. A single trial’s realised share on any shape is one of the seven bases’ numbers. What the floor promises is that no such number falls below 63.66%. What the mixture promises is that the average across possible draws is 76.53%. A single trial experiences the first, and a programme of trials the second.

How the numbers were made #

The share of a shape a basis removes is exact, from the closed-form inner products of the dictionary’s product terms on a pair of independent standard normal covariates. All 276, 2,024 and 10,626 bases are enumerated. At each distinct floor the game is solved by fictitious play over 8,000 rounds, and its value is bracketed between the mixture’s worst shape and the opponent’s best response: that bracket is reported as a gap, and it is under half a point at every step, typically under a fifth. So 76.53% should be read as 76.5 ± 0.1, and small differences between neighbouring floors in the second table are not meaningful.

Three things are required of it. A floor raised cannot raise the expectation by more than that gap. The first non-zero floor at three functions must cost more than nine points. And several bases must tie at the top of the four-function curve. The claim that at the fixed basis’s own floor a mixture can do no better than the fixed basis is refused at four functions, by more than a point.

At three functions the solved 61.27% is a little under the earlier essay’s 61.33%, which ran longer and is the better number; the two agree to the game’s gap.

Still open: a dictionary that is not a list, and a floor on one shape #

Five functions. The step from three to four changes the answer’s character, from a forced choice to a dominated fixed basis. Whether that is general, so that from four functions on a mixture of tied bases always dominates, or whether it is an accident of the number of ties at this one size, is a question for the game at five, where 42,504 bases are still enumerable. The counted ties at the top floor are one number per size, and they decide it.

A floor that is not uniform. The floor here applies to every shape alike. A trial is unlikely to believe that. A floor of 90% on the shapes it fears and none on those it does not would admit a very different set of bases, and it is not clear whether the resulting curve is a staircase with a large first step, because that step comes from covering seven shapes at once. Weighted floors are the same game with a different filter, and a trial would have to bring the weights.

Which trade there was #

This essay takes a basis drawn from a mixture restricted to bases that keep a floor on every shape, on a dictionary of 24 product functions and seven shapes. Its claims are that the trade between floor and expectation is a staircase with a handful of steps; that no three-function basis has a floor between zero and 31.83%, and asking for one costs 11.3 points of expectation at once; that at three functions a floor of 40.53% gets 49.73% where the fixed basis gets 42.44% on both; that at four functions seven bases tie at the fixed basis’s 63.66%, and a mixture of them reaches 76.53%, so the fixed basis is dominated; and that at two functions no floor exists.

What stays out, as a decision: other dictionaries and other lists of shapes, on which the numbers would move though the staircase would not; finite samples, since every share is the population’s, and the rate that does not know the size deals with what a sample does to it; and which shapes a trial should fear, which it has to answer for itself. The boundary against a basis drawn rather than chosen is that it found the two ends and called the middle a range, and this one finds that there is almost no middle.