The window that closes

A person about to measure a tree wants one number before setting out: how many junctions are enough. The usual answer is an arithmetic in which precision is the whole difficulty — pick the width the claim needs, divide by the noise, square it, and the count falls out.

That arithmetic assumes the answer sits on the truth and only wobbles. Once the radii carry error it does not. The fit runs downward and it runs downward as the square of the error, so the sample has two things to overcome and only one of them yields to counting.

What enough has to mean when the answer moves #

Two criteria, and neither is unusual on its own.

The first is that the answer be narrow enough to say something. An interval whose half-width is under ±0.25 sits a quarter of the way from one candidate rule to the other, which is the loosest thing anybody would write down as three rather than two; it must also exclude 2 outright, since an interval containing the rival has not chosen.

The second is that the answer be honest: the interval still contains the 3 the tree was built at. A claim that is narrow and wrong is worse than no claim, and the whole of the difficulty here is that narrow and wrong is what the arithmetic tends towards.

Why the two criteria pull apart #

Each is a statement about the same interval and they move in opposite directions as the sample grows.

Narrowness improves with counting. Honesty does not: the interval’s centre is displaced by an amount that depends on the error and on the daughter ratios and on nothing else, so every junction added pulls the edges in towards a centre that stays put. Past some count the near edge crosses the truth and the interval, still perfectly correctly computed, stops containing it.

That is the window. It opens where precision arrives and shuts where the displacement overtakes the width, and whether it exists at all is a property of the instrument rather than of the fieldwork.

The sweep #

Twelve sample sizes from five junctions to a thousand, seven error levels from half a per cent to ten, a hundred and fifty replicate samples at each combination, all on the band of daughter ratios from 0.6 to 1 — the junctions that carry the information and the only ones worth designing a morning around.

Every sample comes from a tree built at exactly 3, so the truth is known and the interval’s coverage is a fact rather than an estimate. At zero error the fit returns 3.000 with no spread at all, which is the control the rest is read against.

Two per cent, which is the assumption everywhere else #

Two per cent is callipers on a clean branch and it is the figure the rest of this thread has quietly assumed throughout.

At seventy-five junctions the answer is 2.960, interval [2.89, 3.02]. That is a good result by any ordinary reading: tight, excludes the rival rule comfortably, contains the truth.

At two hundred junctions the interval still contains 3. Above two hundred it does not. The sample has not become less careful and nothing has gone wrong; the edges have simply arrived.

A beautifully precise wrong number #

At a thousand junctions the same fit returns 2.959 ± 0.011, interval [2.94, 2.98].

That is the sentence worth sitting with. A thousand junctions is a season’s work. The answer is quoted to three decimal places, the interval spans a twenty-fifth of the distance between the two rules, and it excludes the exponent the tree was actually built at.

Nothing in the output says so. The interval is a correct statement about how much the answer moves when the measurement is repeated, and the measurement really does move that little. What it is silent about is where the centre sits, which is the same silence a confident twig sample exploits.

The spread is pure precision, and that is the problem #

The half of the answer that behaves is worth measuring rather than assuming, because the whole argument rests on it.

Multiply the spread by the square root of the sample and the product should be flat if the spread is nothing but sampling variation. It is. At two per cent the constant is 0.354 and it holds across the entire ladder from five junctions to a thousand to within about a fifth; the worst line on the chart varies by under a quarter, and it is the one where five junctions barely have an answer to scatter.

What that buys #

A flat line is a licence to extrapolate. The constant runs from 0.092 at half a per cent to 1.003 at ten, so the spread at any sample size at any of those errors is one division, and the displacement is measured directly at each error and does not depend on the count.

So the crossing point — the sample size at which the two are equal — can be computed rather than found by bisection, and it can be computed at counts the sweep never ran. That is why the numbers below reach 1,226 junctions when the ladder stops at a thousand.

The crossing, from 1,226 junctions down to two #

The count at which the displacement overtakes the spread falls faster than the error rises, because the displacement goes as the square of the error and the spread goes as the error itself.

At half a per cent it is 1,226 junctions — a machined section under a microscope, and a count nobody will ever reach, which is the same as saying the difficulty does not arise. At one per cent it is 305. At two, 75. At three, 32. At five, ten. At seven, four. At ten per cent, two.

Two junctions #

The last row is worth reading twice. At ten per cent of error on each radius — a radius read off a photograph with no scale bar at the junction — the third junction measured already makes the report more confident and no more correct.

That is not a statement that ten per cent is too coarse to work with. Coarse instruments can still answer questions; what they cannot do is answer them by accumulation. The count that would have been the remedy is exhausted before the fieldwork has begun.

Three per cent, where the window is twenty junctions wide #

Between the comfortable rows and the impossible ones there is a row where the window is real and narrow.

At three per cent, twenty junctions is the fewest that state a claim and seventy-five is the most that still contain the truth. At fifty the answer is 2.914, interval [2.80, 3.02], which does everything asked of it.

Measuring a hundred is worse than measuring fifty #

Which is the sentence this essay exists for, and it is not a paradox.

A hundred junctions at three per cent gives a narrower interval than fifty, centred in the same place, and the narrower interval no longer holds the truth. Two people measure the same tree with the same callipers, one of them works twice as hard, and the harder worker publishes the answer that is wrong.

There is no error in the second person’s arithmetic to find, which is what makes this different from a mistake. The extra fifty junctions did exactly what extra junctions do.

Five per cent, where there is no window #

At five per cent the two bars do not overlap.

Fifty junctions is the fewest that state a claim; thirty is the most that still contain 3. Below fifty the interval is too wide to exclude the rival rule, and above thirty it has already excluded the truth. There is no count between them because there is no count between them.

What the shut row looks like from inside #

The uncomfortable part is that the shut row does not announce itself.

At fifty junctions and five per cent the answer is 2.773 with an interval of [2.61, 2.95]. A reader is being handed a number that excludes 2 by a wide margin and sits a little under 3 — which reads as a real tree departing slightly from the hydraulic optimum, and is exactly the finding somebody would be pleased to have.

A branch that is not round is a five-per-cent instrument. That is the ordinary field case rather than the pathological one, and the ordinary field case is where the window is shut.

The crossing against the window it should sit inside #

The two halves can be drawn on one another, and the picture says which of the two constraints is doing the work at each error.

Where the window is open the crossing sits comfortably inside it. Where the window is shut the crossing has moved below the count at which precision arrives at all: the answer is already displaced by more than its own width before it is narrow enough to state.

Where the criterion came from #

A threshold that decides a result should be stated so it can be argued with, and this one is stated in the sweep’s own settings rather than chosen after the numbers arrived.

Half-width under ±0.25 is a quarter of the distance from Murray’s law to Da Vinci’s. Excluding 2 is what choosing between them means. Containing 3 is coverage. The central ninety per cent is the interval the whole thread quotes.

None of the three is unusual, and that is deliberate: a window that shuts only under an exotic criterion would be a fact about the criterion.

What a stricter or looser criterion would do #

Both directions are worth following, because a reader’s first instinct is that the window can be reopened by relaxing something.

Loosen the width and the lower edge of the window falls — fewer junctions state a claim. The upper edge does not move at all, because coverage has nothing to do with the width demanded. So a looser criterion widens the window at exactly one end, and the end it widens is the end nobody was struggling with.

Tighten the width and the lower edge rises while the upper edge again stays put, so the window shuts at a lower error. The whole family of criteria moves one edge. The other edge is set by the displacement, and the displacement is set by the instrument.

The shape is the opposite of the usual one #

Ordinary sample-size arithmetic has one edge: a count below which the question cannot be answered and above which it can, with more always at least as good as less.

Here there are two, and the upper one is the binding one. That inverts the advice a person carries into the field. Measure as many as time allows is a sound instruction when the only enemy is scatter and a mildly harmful one when the answer is displaced; the instruction that survives is measure the count the instrument supports, and spend the rest of the morning on the instrument.

Which is what the error level buys #

Halving the error quadruples the count the sample can bear, because the displacement goes as the square while the spread goes as the error itself. The rows show it exactly: 1,226 junctions at half a per cent, 305 at one, 75 at two — a factor of four at each step.

Nothing else in the design does that. Choosing better junctions helps for a different reason and is priced elsewhere; adding replicates prices the mean and not the sample; choosing a better estimator does not touch a displacement that is a property of the geometry rather than of the fitter.

What the sweep does not sweep #

Three things are held fixed and each of them bounds the claim.

The band is 0.6 to 1 throughout — the best junctions a tree offers. Every count here is therefore the most favourable count, and a sample taken as it comes does worse at every error.

The tree is built at exactly 3. A tree built at 2 is displaced less, which is the whole of why the two rules close on each other rather than staying a fixed distance apart.

And the error is Gaussian, relative, independent on each radius and identical at every junction. A swelling at a fork is none of those things.

What it does not say about real trees #

The window is a fact about an estimator handed a stated kind of noise. It is not a measurement of any tree, and no tree here was measured.

What transfers is the shape. Wherever an estimate carries a displacement that does not fall with the sample, the count has an upper edge, and the edge can be computed from two quantities a person already has: how far the answer moves when the measurement is repeated, and how far it sits from the truth on a case where the truth is known. The second is what a synthetic control is for, and it is the same argument the survey protocol makes about refusals.

The number a report should carry #

One line, and it costs nothing: the count at which the displacement would overtake the spread, at the error the study claims for its own instrument.

It needs the instrument’s error and the daughter ratios measured, both of which are in hand before any exponent is fitted. If the reported sample is under it, the interval means what it says. If the sample is over it, the interval is narrower than the answer is wrong by, and the reader has been told nothing about that by any other quantity in the paper.

Why this is not the retention rate again #

The branching thread already carries a diagnostic — attempted against retained — and it is worth being clear that this is a second one rather than the same one restated.

Retention catches junctions the noise made impossible, which is a selection acting on the sample. The crossing catches a displacement acting on the fit, and it fires on samples where retention is a hundred per cent. On the 0.6 to 1 band at two per cent, essentially every junction is admissible and the fit is still displaced. Two failures, two diagnostics, and the clean retention rate is no evidence at all about the second.

What would refute it #

The claim is that the spread scales as the reciprocal root of the sample while the displacement is flat in it, so the two cross.

Either half is refutable. A displacement that fell with the sample would put this whole essay wrong, and it would show up as a bend in the constant measured above; the constant is flat to a fifth across a factor of two hundred in sample size, which is what says the scaling is real rather than assumed over the range that was run.

And a spread that fell faster than the root would push the crossing out. It does not: the worst departure anywhere on the ladder is under a quarter, at the error and count where five junctions have almost nothing to scatter.

The one row that is genuinely safe #

At one per cent the window runs from five junctions to beyond a thousand, and the crossing sits at 305.

So a person who can measure a radius to one per cent, on comparable forks, with a sample of a few dozen, is in the regime the ordinary arithmetic describes and can stop reading. That is a real and reachable standard — a cut section, a photograph with a scale bar in the plane of the junction, a calliper used twice — and it is the recommendation this essay ends up making, which is a recommendation about equipment rather than about effort.

Why the informative band and not a whole tree #

Everything above is measured on comparable forks, and a reader might reasonably ask why the count was not priced on the junctions a tree actually offers.

Because on those junctions there is no window to price. The displacement on twigs is larger by orders of magnitude at every error, so the upper edge falls below the lower one before the first row of the ladder, and the answer to how many is none of them. That is a result the thread already has and it is reached here from the other direction, which is worth recording because two routes to one number is the only kind of confirmation available in an arithmetic like this.

What is left open #

The sweep prices a sample against a known truth, which is a luxury no fieldworker has.

What it does not supply is the inverse: given a measured interval and a stated instrument error, the interval the truth actually sits in. That is a correction rather than a diagnostic, it needs the displacement coefficient at the daughter ratios of the sample in hand, and the coefficient is computable — so the correction is available in principle and is not attempted here.

The reason for stopping short is that a correction invites being trusted. A diagnostic that says this sample cannot answer is safe to be wrong about in one direction only.

What a person with callipers should do #

Three lines, and they fall out of the rows rather than from advice.

Measure the error first, on something whose radius is known, and report it. Take the comparable forks and leave the twigs. Then take the count the error supports — a few dozen at one per cent, a few dozen at two, twenty to seventy-five at three — and stop.

At five per cent, stop earlier: not at a count but at the instrument, because the question at that error is no longer about counting. The window that closes is a window on the sample, and what closes it is the measurement.