A defect that is about size

The fourteen-unit trial whose admissible set falls into two mirror components is a real defect and it raises an obvious question that the enumeration itself can answer. Is it about the thinness, or about the size?

The two are separable, and they are separable because of a result this collection already has: the acceptance rate of a rerandomisation is a fact about the basis rather than about the number of units. So a tolerance can be held fixed, the trial grown, and the share admitted will barely move while everything else does.

There is a reason to expect the answer before running anything, and stating it first makes the sweep a test rather than a survey. A walk moves by swapping one treated unit for one control unit, so the number of moves available to it grows like the square of the trial size while the share of them that are admissible does not grow at all. A graph whose vertices have few neighbours falls into pieces and one whose vertices have many does not, and everything else about the problem — the basis, the tolerance, the covariate — enters only through the share.

The sweep #

A box at 0.8 on four balancing functions, at twelve, fourteen, sixteen, eighteen and twenty units. Every equal split enumerated — 924 at twelve, 3,432 at fourteen, 12,870 at sixteen, 48,620 at eighteen and 184,756 at twenty, which is where this collection’s enumerations stop.

The admitted share is one in 27, one in 30, one in 30, one in 20 and one in 18. Over a fourteen-fold growth in the number of assignments the thinness moves by less than a factor of two, and what movement there is comes from the design’s own sampling variation rather than from the size.

The number of pieces is 4, 2, 1, 1 and 1.

Thinness stays put and reachability does not. At twelve units the set is in four pieces, two of which are single assignments with no admissible neighbour at all. At fourteen it is in two. At sixteen and above it is in one, at a tolerance that leaves the same share admitted.

Why it is size #

The mechanism is a count and it can be written down before anything is enumerated.

An assignment of n units into two equal arms has (n/2)² single swaps available to it: pick a treated unit, pick a control unit, exchange them. That is 36 at twelve units, 49 at fourteen, 64 at sixteen, 81 at eighteen and 100 at twenty — growing like a quarter of the square of the trial size.

If admissibility were independent of which swap is proposed, the expected number of admissible neighbours would be that count times the admitted share: 1.3, 1.7, 2.1, 4.0 and 5.6.

The events are not independent — a swap of two similar units barely moves the imbalance, so an assignment’s neighbours are strongly correlated with it — so the product is a back-of-an-envelope rather than a theorem. But it puts the crossing exactly where the enumeration puts it. A graph whose vertices have on average fewer than two neighbours is a graph in pieces, and a graph whose vertices have four is not.

That is the whole of the finding. The defect is a sparse-graph phenomenon, and what makes the graph sparse is that a small trial has few swaps rather than that a thin set has few members.

The same crossing at three tolerances #

One sweep at one tolerance establishes a coincidence. Three establish a mechanism, and the enumeration is cheap enough to run all three.

At a box of 0.9 every one of the five sizes is in one piece, and the smallest expected degree anywhere in the sweep is 2.3.

At 0.8 — the sweep above — the degrees are 1.3, 1.7, 2.1, 4.0 and 5.6, and the components go 4, 2, 1, 1,

1. The collapse to one piece happens between an expected degree of 1.7 and one of 2.1.

At 0.7 the sets are much thinner — one admissible assignment in 92, 41, 50, 38 and 32 — and the degrees are 0.4, 1.2, 1.3, 2.1 and 3.1. The components go 4, 2, 4, 1, 1. The collapse happens between a degree of 1.3 and one of 2.1, which is two units later in the trial size and at the same degree.

Three tolerances, three different sizes at which the set becomes connected, and one crossing point. The admitted share at the crossing differs by a factor of three between the sweeps — one in 30 against one in 38 — and the expected degree at it does not differ at all.

The quantity that predicts connectivity is the degree and not the share, and the two are related by the swap count, which is a function of the trial size alone. That is as close to a mechanism as an enumeration over five sizes can get.

The set grows three hundredfold while the share does not move #

The two columns are easiest to separate by converting the shares into counts. One in 27 of 924, one in 30 of 3,432, one in 30 of 12,870, one in 20 of 48,620 and one in 18 of 184,756 give admissible sets of about 34, 114, 429, 2,431 and 10,264 assignments.

A factor of three hundred, against a share that moves by less than two and with no trend in it.

That is the sweep’s whole result in one line, and it says which quantity the defect is a function of. The set is in four pieces when it holds thirty-four assignments and in one when it holds four hundred and twenty-nine. Nothing about how thin the set is changed between those two; what changed is how many things are in it, and therefore how many neighbours each of them has.

The independence count is right about the trend and wrong about the level #

The expected-neighbour figures — 1.3, 1.7, 2.1, 4.0 and 5.6 — are computed as though admissibility were independent of which swap is proposed, and it is worth asking how far that model can be trusted, because the whole prediction rests on it.

It gets the transition right. The set falls into pieces where the expected degree is below two and is one piece from 2.1 upwards, which is where a graph stops being able to afford isolated vertices.

It gets the level badly wrong, and in the direction the essay’s own next sentence points at. The fourteen-unit design this collection enumerates in full has 116 admissible assignments and 49 single exchanges out of each, of which 936 are admitted — an average of 8.07 admissible neighbours apiece, against the independence model’s 1.7.

A factor of nearly five, and the reason is that admissibility is clustered rather than scattered. A swap of two units with similar covariate values barely moves the imbalance, so an assignment already inside the box has most of its neighbours inside it too; the share of proposals from an admissible state that are admitted is 936/5,684 = 16.5%, against the 3.3% share of all splits that are admissible at all.

Which strengthens rather than weakens the conclusion, and it says what the independence count is for. It is not an estimate of the degree; it is a floor that scales correctly. Both it and the true degree grow like a quarter of the square of the trial size times a share that does not move, so the ratio between them is a property of the tolerance and the covariate rather than of the size — which is exactly what makes the count usable for the extrapolation this field wants, and useless for predicting how many pieces a particular set falls into.

Two quantities that were being read as one #

The reason this is worth a field rather than a paragraph is that the two quantities have been conflated everywhere, including in this collection.

The cost of a usable draw — how many evaluations the walk spends per independent draw, and where it stops being cheaper than hunting — does collapse onto thinness. The crossing sits at about one admissible assignment in two hundred whatever dial produced the thinness, because a hunt’s cost is exactly 1/p and a walk’s cost per usable draw stays between 82 and 394 evaluations across forty settings.

Connectivity does not collapse the same way. The same essay reports it directly: 15.2% admitted with three components, 11.0% with two, 9.9% with one — a share that goes up as the pieces go up, which is the wrong direction for a thinness explanation and is the right one for a size explanation, because those three cells are at different trial sizes.

One property of an admissible set is scale-free and the other is not, and reading the second off the first is the mistake this essay exists to prevent.

Why the count is only a count #

The expected degree is a product of a swap count and an admitted share, and the two events it multiplies are not independent. It is worth being precise about which direction the dependence pushes, because the number lands so close to the answer that a reader is entitled to suspect it of being fitted.

An assignment’s neighbours are the assignments one swap away, and a swap of two units with similar covariate values barely moves the imbalance. So an admissible assignment’s neighbours are more likely to be admissible than a random assignment is — the swap graph is locally clustered, in the way a geometric graph is and a random one is not.

That pushes the true degree above the product, which would move the crossing to a smaller trial size than the count predicts. It does not, and the reason is the other half of the same clustering: a locally clustered graph needs a higher average degree to be connected than a random one does, because its edges are spent inside neighbourhoods rather than between them. The two effects run opposite ways and the product lands between them.

None of that is derived here and none of it is needed. What the count is for is to say that the crossing should be at a fixed degree rather than at a fixed share, and that prediction is what the three sweeps check.

What breaks first #

The four-piece case at twelve units is worth looking at rather than counting, because it is a different failure from the two-piece one.

At twelve units and a box of 0.8 the set has 34 admissible assignments in four components of sizes 16, 16, 1 and 1. The two components of sixteen are a mirror pair, as at fourteen. The two singletons are assignments with no admissible neighbour at all: a chain started at one of them proposes a swap, is refused, stands still, and repeats for ever. Its stationary distribution is a point mass on its starting state, its acceptance rate is zero, and its output is one assignment repeated B times.

That failure is loud rather than silent — an acceptance rate of zero is noticed — which is worth saying, because it means the dangerous case is the intermediate one. A set in two pieces has an ordinary acceptance rate and reports half a distribution; a set with isolated points has an acceptance rate that says something is wrong. The defect that hides is the one that leaves the chain able to move.

The thinnest sweep shows a third shape. At a box of 0.7 the sixteen-unit set has 256 admissible assignments in four components, where the fourteen-unit set at the same tolerance has 84 in two. The number of assignments has trebled and the number of pieces has doubled, which is the opposite of the trend the whole essay is about — and it is a reminder that the enumeration is over one draw of the units. A different draw of sixteen standard normals gives a different design, a different admissible set and a different swap graph, and nothing here averages over that.

So the sweeps are five points on one realisation rather than five expectations, and the crossing at a degree of about two is a statement about the realisations enumerated. That is the honest scope, and it is why the essay’s conclusion is a prediction to be tested rather than a rule to be applied.

What it says about two hundred units #

The sweep stops at twenty units because the enumeration does. What it licenses is an expectation rather than an answer.

At two hundred units the swap count is 10,000, and at an admitted share of one in eighty — which is what a tolerance a trial might actually use produces — the expected number of admissible neighbours is over a hundred. On the count above, that is a dense graph and there is no reason to expect pieces.

An expectation is not a measurement, which is why the test exists and why it is pointed at two hundred units. What this essay contributes to that is the prediction the test is checked against: the answer at two hundred should be connected, and a diagnostic saying otherwise is more likely to be reporting its own run length.

What a practitioner does with it #

The sweep produces something a trial designer can use before running anything, and it is worth writing out because it is one line of arithmetic.

Count the balancing functions, choose a tolerance, and estimate the admitted share by rejection sampling — which costs a few thousand draws and is what the sampler has to do anyway to find a starting point. Multiply it by (n/2)². If the answer is comfortably above two, the swap graph is dense and the walk almost certainly goes everywhere. If it is near or below two, it does not, and the instrument is the way to find out.

At two hundred units with three balancing functions and a tolerance admitting one assignment in eighty, the answer is over a hundred and nothing is at risk. At sixteen units with four functions and a tight tolerance it is 2.1 and everything is. The rule is not a rule about trial size and it is not a rule about tolerance; it is a rule about their product, which is the quantity nobody was computing.

That is the practical content of the whole field, and it took an enumeration to find and an instrument to check.

What is claimed here, and what is not #

This essay takes whether the walk’s unreachable half is about thinness or about size. The claims are that at a fixed tolerance the admitted share moves by less than a factor of two between twelve and twenty units, from one in 27 to one in 18; that the number of components over the same range goes 4, 2, 1, 1, 1; that the number of single swaps available grows like a quarter of the square of the trial size, from 36 to 100; that the expected number of admissible neighbours therefore crosses two between fourteen and sixteen units, which is exactly where the components collapse to one; and that the twelve-unit set contains assignments with no admissible neighbour at all, which is a louder failure than the mirror split.

What stays out, and is named as a decision: other bases and other tolerances. One rule at one tolerance is swept over five sizes here. A full sweep over the two dials and the size would be a three-dimensional enumeration and the enumeration is the binding constraint; what is done instead is to establish the mechanism on one line of it and to build an instrument that does not enumerate for everywhere else.

Also out: a threshold. Nothing here says a trial of sixteen units is safe. The expected-degree count puts the crossing near two neighbours, and where a particular design falls relative to that depends on the units drawn — which is a measurement per trial rather than a rule.

The boundary against the essay that found the split is that it establishes the defect at one size and this one asks what the size was doing. The answer is: most of it.

The checks, and the refusals that make them mean something #

Two claims are gated. The admitted share is required to move by less than a factor of six across the sizes in the sweep, which is what makes the comparison a comparison at fixed thinness rather than a comparison of two things at once. And the smallest size is required to be disconnected and the largest connected, because a sweep in which nothing changed would be a sweep about nothing.

The refusal carried from the field’s first essay still stands and is what makes this one necessary: a chain’s own diagnostics are refused as evidence of reachability, so a practitioner at a size the enumeration cannot reach has nothing to read but the instrument.