A width nobody varied

Five numbers in this collection say how much two pairs of eyes differ, and every population result here is a consequence of them.

The claim #

The population model carries four spreads, each written down with the range the literature reports for it, and until now not one of them had ever been moved. The file that holds them said, in a comment above the table, that the ranges were there so a result could be re-read at the pessimistic end — and named a function that did not exist.

• The spreads are declared, not derived. The lens as an age range of twenty to seventy; the macular pigment at a standard deviation of 0.13 on a median of 0.35; the cone outer segment’s optical density at nine per cent of 0.4; and the three pigment peaks at about 1.5 nanometres each.
• Moving them is cheap and nobody had. Two hundred members of a population cost nine milliseconds to draw, and the whole table of responses below took under a second.
• The answers move by between nothing and a factor of 2.6 across the reported ranges, and which end of that they land on is not guessable from how large the effect is.
• One of them moves by exactly nothing, and it is the one about the median member — which is what says the rest are measuring a spread rather than quietly re-fitting something.
• And the ranges themselves are a declared input. Nothing below quotes a study. What is computed is the response, which is a property of this collection’s own machinery.

The five, and what each of them is #

The population here is not a metaphor: it is two hundred sets of cone absorptances, each computed from a pigment template at that member’s own five parameters, carried to tristimulus values through one fixed matrix so that a difference between two members is a difference in what their cones caught.

Four of the five are random in the sense that nothing observable predicts them, and the fifth is not. The lens yellows with age, monotonically and predictably, so the largest single source of disagreement in the whole model is the one thing about an observer that is written on their passport. It enters as a range of ages rather than as a standard deviation, and that difference matters more than it looks.

The macular pigment and the cone optical density are quoted as standard deviations because that is how the individual-observer literature reports them, and both are measured by psychophysics on small samples. The three pigment peaks are measured by microspectrophotometry and by genetics, and those two methods agree with one another better than the psychophysical ones do.

So the four are not equally well known, and that is the whole reason this essay exists. A spread of 0.13 that different studies put anywhere between 0.10 and 0.20 is a different kind of number from a peak wavelength known to a nanometre.

What moving them does #

The instrument is an elasticity: multiply one width by a factor, recompute a published number, and take the ratio of the two proportional changes. An elasticity of one means an answer that doubles when the width doubles; an elasticity of a tenth means an answer that barely notices.

It is taken over a stated finite span rather than as a derivative, and the reason is the subject rather than the arithmetic. A derivative answers what if this width were infinitesimally different, which nobody is asking. The disagreement between studies is a factor of one and a half to two, so the honest instrument is a finite difference over a factor of that size — here a factor of 1.5625 between the two evaluations, taken symmetrically in the logarithm so that the span is not quietly biased towards the wide end.

The results run from 0.03 to 0.92 and none of them is one. Widening the population does not simply widen the answers: the disagreement over a narrowband match rises as the 0.25 power of the lens range and the 0.29 power of the macular spread, so a population twice as varied disagrees about a fifth more.

That is worth pausing on, because the intuition runs the other way. A reader told that a spread is uncertain by a factor of two will expect the conclusions drawn from it to be uncertain by something of that order. They are not: a median disagreement of 9.34 ΔE00 becomes 11.1 at the wide end and 8.1 at the narrow one. The reason is that a colour difference is a distance in a space with three compressed axes, and a distance between two members is a difference of two integrals over the same spectrum, so a good deal of any widening cancels before it reaches the answer.

The exception is anything that is a shape rather than a magnitude. The confusion-point clouds have elasticities of 0.80 on the lens range, which is nearly proportional, because a copunctal point is where two nearly parallel planes meet and a small change in a cone response moves it a long way. Every quantity in the table that is about where a point sits responds three times as strongly as the quantities that are about how far apart two colours look.

The one that moves by nothing #

One row of the table is empty, and it is empty exactly rather than nearly.

The median member’s distance from the 1931 standard observer — the residual on the fitted matrix, which is the honest caveat on the whole population model — has an elasticity of zero on every one of the four widths. Not 10⁻⁶. Zero, in every digit, because the construction that scales a width leaves every median exactly where it was and this quantity is a statement about the median member alone.

That row is the control, and it is the check that would have caught the obvious way of getting this wrong. An implementation that scaled the age range about its stated median rather than about the range’s own midpoint would have made a wider population a systematically older one; the ageing lens is monotone, so that shift would have shown up in every elasticity below as a width effect while being a change of centre. The control would have reported it immediately, because a shifted median moves the median member.

Where the published number sits inside its own range #

The ranges are one answer and their shape is another, and the shape turns out to be simple.

Taking each claim’s optimistic and pessimistic ends and asking where the published value falls between them: 40.2 per cent for the median disagreement, 51.4 for the tail, 43.1 for the cloud radius, 56.3 for the protan point, 48.8 for the deutan and 59.3 for the tritan. Every one within ten points of the middle, with a mean of 49.9.

That is neither guaranteed nor decorative. The claims are nonlinear functions of the four widths — a population is drawn, absorptances come from a template, tristimulus values follow, and a percentile is taken at the end — so a range that is symmetric in the widths has no obligation to come out symmetric in the answer. It does, which says the claims are close to linear in the logarithm of each width across the ranges the literature reports.

That is the condition under which an elasticity means anything, and it is the assumption every ranking built on these widths quietly makes. Measuring it costs nothing and it had not been measured. The response could easily have been strongly convex, in which case a single number for how a claim responds to a width would be a statement about one end of its range rather than about the range.

There is no systematic direction in it either. Three of the six moving claims sit slightly towards their pessimistic end and three slightly towards their optimistic one, so the published set is neither a lucky reading nor a cautious one — it is the middle of what the declared spreads allow, which is where a set of medians ought to land and is not where it had to.

Both ends at once #

The reading the model’s own documentation promised is the one in the picture at the top: all four widths at the wide end of their reported spans together, and all four at the narrow end together.

The spread of a narrowband match moves by a factor of 1.36 between the ends. The size of the tritanope’s confusion cloud moves by 1.91. And the three σ-distances that the claim about published transforms rests on move by 1.64, 2.62 and 2.00 — which is enough to matter, and is the subject of its own essay.

One number summarises each row, and it is the factor by which the whole quadrature of the four uncertainties multiplies the answer. For the median disagreement it is 1.27; for the ninety-fifth percentile 1.13; for the tritan cloud 1.55; and for the deuteranope’s σ-distance 1.76. Nothing in this collection is quoted to three significant figures on the strength of a model whose own inputs carry a factor of 1.76, and the figures here now say so where they used to be silent.

It is not a confidence interval and is not presented as one. The four ends are not quantiles, the widths are not independent draws, and taking all four to their extremes at once is a deliberately pessimistic reading rather than a probable one. What it is good for is a reader who distrusts all five numbers at once, which is a reasonable thing to be.

What was computed, and how #

The construction has one property that everything above depends on, and it is asserted rather than assumed: scaling a width moves no median.

The macular pigment’s and the cone density’s medians are untouched by construction, since only the standard deviation is multiplied. The three pigment peaks likewise. The lens is the awkward one, because it enters as a range: its span is scaled about the range’s own midpoint, so the mean lens density is held while the width moves.

That construction has a limit and the limit is stated rather than hidden. Above a multiplier of 1.8 the young end of the age range would go below zero years, the range clamps at zero, and the midpoint rises — so past that point the scaling is a population that is both wider and older, and its elasticity is two effects added. Every factor reported here is checked against that line rather than assumed to be inside it.

The matrix that carries a member’s cone responses to tristimulus values is fitted once, to the reference member, and every member then uses it. Two populations at different widths therefore share their bookkeeping exactly, which is what makes them comparable in one table. A version that re-fitted per population would have produced a table of numbers that each meant something slightly different.

Where the model stops #

An elasticity is not a probability. It says how the answer responds to a width, not how likely the width is to be wrong. Turning one into the other needs a distribution over the widths, and this collection does not have one — which is precisely why the ranges enter as a stated span with the arithmetic done in the open rather than as a prior.

The four are varied one at a time and then together, and never against each other. Real spreads are correlated: a study that recruits an unusually wide age range probably measured its macular densities on the same unusual sample. A correlated set of widths is a different calculation and would give a smaller total than the independent one, so what is reported here is the conservative direction.

That last picture is the one to look at before believing any of the rest. The lens and the macular pigment are absorbing filters with almost all their absorption below 500 nanometres, and the two long-wavelength pigments are what a deuteranope’s confusions are about — so the two running opposite ways is a check on the machinery rather than a discovery. A version of this model in which the lens moved the deuteranope’s point would be wrong somewhere, and this is where it would show.

Two hundred is a sample too. The population is drawn from a stated seed and is reproducible to the last digit, which is a different property from being representative: another two hundred people drawn the same way would give slightly different numbers, and how different is a question about Monte Carlo error rather than about the widths. It is small here — the medians of two independent draws of two hundred agree to about a per cent — but it is a second uncertainty stacked under the first, and a reader adding error bars should know both are present.

And the pigment template is not the observer. The population is built on a nomogram at stated peaks rather than on the physiological cone fundamentals, which is why nothing here is ever quoted as a difference from the standard observer. Every number above is a spread within the population or a disagreement between two of its members, and both are invariant to which member is called the reference.

The generalisation #

The defect this essay begins from is not that the widths were wrong. They are ordinary, defensible numbers. The defect is that a caveat lived in a comment, where nothing could reach it — the file said a result could be re-read at the pessimistic end, and named the function that would do it, and there was no such function anywhere in the tree.

That is the same shape as three other things this collection has found in itself. A claim in prose that nothing tests is a sample of size zero, and a promise in a comment is worse, because it reads as evidence that somebody checked.

The general form is worth stating as a habit rather than as an observation. A modelling input written down with a range beside it is an invitation, and an invitation nobody accepts becomes a claim. After enough phases the range stops being read as here is what is uncertain and starts being read as here is what was considered, which is the opposite of what it says.

The same instrument has been turned on the collection’s other constructed constants, and the two rankings are worth setting side by side.

Who found it, and when #

The individual-observer variation this model is built on is old and well documented. That the lens yellows with age has been known since the nineteenth century and quantified through the twentieth; the macular pigment’s variation was measured through the middle of it; the polymorphism in the long-wavelength pigment was found genetically in the 1980s and explains part of why that peak’s spread is the least normally distributed of the five.

What is new here is not any of those measurements. It is the arithmetic of asking what this collection’s own conclusions do when they move, and that arithmetic needed nothing but a multiplier and a rerun.

The habit of stating a spread and never varying it is not peculiar to this site. A standard’s tabulated observer is a single set of numbers with no variance attached; a specification written as a colour difference is a statement about a pair that turns out to be a statement about a person; and the individual observer functions the CIE publishes are used far more often to check whether an effect exists than to ask how much of a published number they move.

The colour literature does propagate observer variation, and the CIE has published a set of individual observer functions for the purpose. What is rarer, there and everywhere else, is propagating the uncertainty in the variation — the second-order question of how well the spread itself is known. It is the question a sensitivity analysis is for, and it is usually skipped for the reason it was skipped here: the first-order answer is interesting enough to publish.

Where the ladder goes next #

The table above ranks the four widths by how much of the answer each carries, and that ranking has been quoted here since the population was built: the lens, at eighty-one per cent, well clear of everything else.

Ranking them by how much doubt each puts on the collection’s conclusions gives a different first entry, by a margin small enough that the honest summary is that the two leading terms are interchangeable. That is a more useful question than it looks, because it is the one with an action attached — it says which measurement would be worth making better, and the answer is not the one the attribution points at.