Knowing the macula's density frees two sessions, not the line on its peak

An older eye’s colour matches carry more than the standard observer has numbers for, and the fits that read them have grown one number at a time. The matches carry their own lens fitted rod weight, lens age and macular density from a single session. A band of extra absorption in an ageing lens added an amplitude and a centre, and a second lamp pair finds a lens band’s centre showed the centre needed a second session. Then a macula whose peak sits a few nanometres from the model’s 460 nm turned out to be read as a lens band near 438 nm by any fit that had no number for it — the finding of a macula three nanometres short is a lens band.

A second session keeps a band and the macula apart gave the fit all six numbers and found two sessions enough to find both departures, at a price paid by the band nearest the macula: at 445 nm the band’s amplitude and the macula’s shift were correlated at 0.82. A line on the macula’s flank buys less than a repeated session then tried a third session with a narrow line at every position from 465 to 510 nm and found no position bringing that correlation under 0.78. The line at 465 nm, on the macula’s peak, made the band cheapest — its standard error within a fifth of its band-only value — and left the correlation at 0.86.

Its closing section offered an explanation and a way out. The macular density soaks up what the band and the macula’s peak would otherwise share, it argued, and the density can be measured without colour matches at all: heterochromatic flicker photometry finds the macular pigment directly, by how much more blue light than green is needed to null a flicker in the fovea against the periphery. With the density known, the fit has five numbers, and a line near the peak would have only the band and the peak to tell apart. The prediction was that fixing the density takes the correlation below 0.6 with the line at 465 nm and leaves it near 0.8 with two sessions, because two sessions’ lamps never separated the density from the peak.

The correlation falls where the density carried it, not where the line sits #

With the macular density known exactly and fixed, the band–shift correlation for a band at 445 nm and an unmoved macula falls from 0.83 to 0.62 under two sessions, from 0.81 to 0.68 with the second session repeated, from 0.80 to 0.61 with the cyan line at 487 nm — and only from 0.86 to 0.74 with the line at 465 nm, on the macula’s peak. The same holds with the macula moved three nanometres either way: two sessions end at 0.61 or 0.62, the line on the peak at 0.73 or 0.74.

• The prediction is reversed, not merely missed. The line on the peak was to fall furthest and two sessions least; two sessions fall furthest and the line on the peak least.
• The density was the road. In two sessions’ fit it is correlated at −0.73 with the band and −0.75 with the shift; closing it breaks most of the trade.
• What the density cannot carry stays. The line on the peak had already measured the density well, so most of what remains of its trade is direct, between the band and the peak at the short flank.

Each pair of bars is one design of sessions, the upper bar with the density fitted alongside the other five numbers and the lower with it known. Every design falls, and they fall by very different amounts. Two sessions and the lines at 487 and 500 nm fall furthest, by about two tenths, to 0.62, 0.61 and 0.63; the line at 465 nm falls by an eighth and ends highest, tied with the line at 475 nm. The order changes less than the sizes do: the line on the peak goes from second-highest to highest, and the lowest correlation in the whole chart belongs to the cyan line at 487 nm, the design the flank-line essay concluded was not worth building.

The density was the road the trade ran along #

Why a known density helps two sessions most is visible in the fit with the density free. A colour match is one number that all six of the fit’s numbers move at once. A little more macular pigment absorbs a little more light between about 420 and 500 nm; so does a slightly taller lens band at 445 nm; so does a macula whose peak sits a little short, putting more of its absorption at the band’s wavelengths. When the matches cannot quite tell these apart, the fit can trade them in a ring: raise the density, lower the band, move the peak, and arrive at nearly the same matches.

Under two sessions the fitted density is correlated at −0.73 with the band’s amplitude and −0.75 with the macula’s shift. Both are negative because both a taller band and a peak moved towards the band’s wavelengths are things the fit can buy back with less pigment. Two strong correlations through one number make a third: the band and the shift, each tied to the density, are tied to each other through it. Fix the density and that part of their correlation has nothing left to run through.

The third sessions differ in how much of the road they had already closed. The repeated session weakens both paths, to −0.61 and −0.68, and so do the lines at 465 to 487 nm, a little; the lines at 500 and 510 nm leave them as strong as two sessions do, or stronger. But the band–shift correlation under the line at 465 nm was the highest of any design before the density was fixed, at 0.86, and the paths through the density under that design are no weaker than under the line at 487 nm. The difference is in what is left once the road is shut: under the line at 465 nm, most of the correlation was never routed through the density at all.

That residue is the direct trade the earlier essay’s alternative named. A band at 445 nm and a macula moved short both put absorption on the macula’s short flank, and a line at 465 nm, sitting on the peak, measures that flank in nearly the proportions both of them change it. A line on the peak measures the band and the peak each better and the difference between them no better, so the two remain entangled even when nothing else is allowed to move. Two sessions’ lamps sit further from the peak; what they could not separate was mostly the density from each departure, and that is exactly what the outside measurement supplies.

A known density costs the band less than its own fit #

The correlation is one half of what the macula costs the band; the other is the band’s own standard error, compared with its value in a fit that does not have the macula’s peak to estimate at all. With the density known, two sessions’ band error falls from 1.78 to 1.21 times its band-only value; the repeated session’s from 1.41 to 1.12; the line at 465 nm’s from 1.16 to 0.84. Every design falls by between a fifth and a little over a third.

Three designs end below one, which needs a word. The band-only value is itself a two-session fit with the macular density free and the macula’s peak not fitted; it pays for estimating the density from the matches. A five-number fit given the density for nothing can beat it, and with the line at 465 nm, which measures the band’s region best, it does by a sixth. So the comparison that matters is between designs at the same footing, and on that footing the order is the familiar one: the line on the peak still gives the band its smallest error, 0.84 times; the line at 475 nm 0.93; two sessions the largest, 1.21. A known density does not reorder what each design is good at. It reorders which design still has a trade left between the two departures.

The macula’s peak gains as well, because the density and the peak are themselves traded: a little more pigment at 460 nm looks, at most of the session’s wavelengths, like a peak moved slightly so as to put more pigment where those wavelengths are. Two sessions place the peak to 1.29 nm with the density fitted and to 0.85 nm with it known; the line at 465 nm goes from 0.90 to 0.65, and every design ends between 0.65 and 0.86 nm. Knowing how much pigment there is turns out to be worth as much to the peak’s position as any third session was.

The matches already know the density about as well as flicker does #

The calculation so far gives the fit the density exactly. No measurement does that, and how well a real one does decides whether any of this is available. Heterochromatic flicker photometry’s precision depends on the instrument and the observer, but repeatability studies commonly report a scatter of a few hundredths of a density unit at the peak — three to five in many of them — and older eyes, with their yellower lenses and smaller pupils, are at the worse end.

The six-number fit already estimates the density, and its own precision is the comparison. With two sessions the matches measure the macular density to a standard error of 0.027; with any third session tried, to between 0.017 and 0.025. An outside measurement adds to that; it does not replace it. Combined properly, two independent estimates of one number are weighted by how well each is known, and an outside measurement with twice the matches’ error carries a fifth of the weight.

With a measurement good to 0.05, two sessions’ correlation moves from 0.83 to 0.80; good to 0.03, to 0.77; good to 0.02, to 0.73. It falls below 0.7 only when the outside measurement is better than about 0.015 — roughly twice as precise as the matches themselves — and reaches 0.67 at 0.01 and 0.63 at 0.005. The three other designs follow the same shape, each flat to the right of the matches’ own precision and falling to the left of it. On the chart’s logarithmic axis the useful part of every curve lies at measurements finer than the dashed line.

Read one design at a time, the band’s error tells the same story as the correlation. Under two sessions it is 1.78 times its band-only value with the density fitted, 1.67 with a measurement good to 0.05, 1.55 with one good to 0.03, and 1.30 only at 0.01. Under the line at 465 nm the measurement first matters at about 0.02, and the band reaches its band-only value there. The gains are real at every precision, but at the precision flicker photometry routinely reaches they are a few hundredths of correlation and a tenth of the band’s error — about what one more repeated session of matches buys.

So the explanation offered by the flank-line essay was right in its mechanism and wrong in its arithmetic. The density does carry much of the trade, and fixing it does break much of the trade; but an outside measurement only helps the fit in proportion to how much better it is than what the matches already say, and the matches say a good deal. The macular density is the number six-number fits reach for first, as a lens of the wrong shape lands in the macula found from the other side, precisely because so many wavelengths in so many matches depend on it.

Far from the macula, a known density ties the band to the peak #

The census has two other bands, and they do not follow the one at 445 nm. At 430 nm fixing the density barely moves the correlation — 0.78 to 0.75 under two sessions, 0.86 to 0.87 with the line at 465 nm. At 415 nm it raises it: from 0.58 to 0.84 under two sessions, and from 0.66 to 0.79 with the line on the peak. Under every design and every macula position, the band at 415 nm and the macula’s shift are more tightly bound with the density known than with it fitted.

The reason is the same ring of trades read from its other side. A band at 415 nm sits below the macula’s short flank, where a lens band and the macular density both absorb but a shift of the macula’s peak changes little. With the density free, the fit trades the far band mostly against the density, and the shift is left to the wavelengths nearer the peak, where the band does not reach — the two departures are fairly independent because each has its own partner. Fix the density and the far band’s partner is gone. What remains that can imitate a little more absorption at 415 nm is a macula moved short, pushing its flank down towards the band, and the fit now trades the band against that.

That makes the outside measurement a mixed gift for an analysis of real older observers. For a band near the macula it breaks a trade; for a band far from it, it creates one, and an analysis that fixes the density should expect the far band’s error to depend on the macula’s peak in a way it did not before. The band’s own standard error at 415 nm still falls with the density known, from 1.23 to 0.75 times its band-only value under two sessions, so the gift is net positive for precision; it is the separability that moves.

What an analysis of older observers should take from this #

A flicker measurement of the macular density is worth adding to a six-number fit, and it is worth very little unless it is better than the matches’ own estimate. At 0.03 or 0.05 it buys what a repeated session of matches buys; at 0.01 or better it breaks most of the band–shift trade for a band near the macula under two plain sessions. An experimenter choosing between a better flicker instrument and a specially designed third lamp — on top of the two sessions a lost lens band goes to age or the macula found placing a band’s centre to about five nanometres — should choose the instrument, and should report the flicker measurement’s precision beside the matches’, because the gain depends on the ratio.

The line on the macula’s peak and an outside density are not complements. The line on the peak was the best third session when the density was fitted, because it measured the density well; with the density known, much of its advantage disappears. The design that gains most from a known density is the plain one, because its trade was mostly through the density. And for every design, a band near 445 nm and a macula moved short keep a direct trade, between 0.61 and 0.74 in correlation, that no measurement of the pigment’s amount can buy out. Reporting both numbers with their correlation, as a second session keeps a band and the macula apart urged, stays the honest course.

How the fits were re-read #

The observer is the band essays’ at sixty: rods feeding the S pathway with weight one half, a lens of the model’s shape scaled by age, a lens band — a Gaussian fifteen nanometres wide, a tenth of a density — at 415, 430 or 445 nm, and a macula of density 0.35 whose 42-nanometre Gaussian is centred at 460 nm moved by −3, 0 or +3. A session is a set of asymmetric matches over the surface family between two lamps; two sessions are daylight against the phosphor LED and the LED against a three-emitter lamp, and each third session is the LED against a four-emitter lamp whose narrow line sits between 465 and 510 nm, or the second session again.

The six-number fit’s covariance at the true observer comes from its Fisher information — the same matches, step and setting noise as the fits themselves, reproduced to a part in a billion where the earlier essays ran the fits. Inverting the covariance recovers the information. A density known exactly is that information with the density’s row and column removed, inverted again; a density measured to a standard error σ is the information with 1/σ² added to the density’s diagonal entry, inverted. A measurement with a standard error of a thousand gives back the free fit to a part in a million, which is the check that the two readings are the same calculation. The band-only value is the earlier essays’ two-session five-number fit.

What this does not reach #

Information, not experiments. Every number here is read from the Fisher information at the truth. Simulated experiments with a known density, fitted from noisy matches, would show whether the fit actually finds the band as often as the smaller standard errors promise; the earlier essays found information and experiments agreeing for these observers, but a fit with one number fewer has fewer ways to wander.

A measurement of the right density. The outside measurement is unbiased here. Flicker photometry measures the density at the edge of a stimulus that need not match the field the colour matches use, and a biased density fixed in the fit would move the band and the peak to compensate — the same trade, now driven by an error.

The model’s macula. The pigment’s shape is a single Gaussian; real macular pigment, lutein and zeaxanthin together, has a shoulder near 480 nm, and a measured density says how much of it there is without saying where its shoulder falls.

Still open: whether a density measured at the wrong field biases the band #

A flicker measurement and a colour match can see different amounts of macular pigment. The pigment is densest at the centre of the fovea and falls off over a couple of degrees, so a flicker stimulus of one size and a matching field of another average it differently, and the density fixed in the fit can be wrong by a predictable amount — commonly a tenth to a quarter of its value.

The calculation is the five-number fit with the density fixed at its true value scaled by 0.8, 0.9, 1.1 and 1.2, for the band at 445 nm and at 415 nm under two sessions, the fitted band amplitude and macula shift read against their truth. The prediction is that a density fixed a tenth too high takes about a third of a standard error off the band at 445 nm and moves the macula’s peak by under half a nanometre, because the density’s roads to the band and the shift run with the same sign; and that at 415 nm the bias goes wholly into the band. If a tenth’s error moves the band by more than a standard error, a fixed density is worse than a fitted one unless the two fields are matched, and the right use of a flicker measurement is as a measurement with an error rather than as a constant. A density a tenth wrong costs the band more than fitting it makes that calculation.

An outside number is only as good as its margin over the inside one #

The habit here is about how to value a measurement brought in from outside a fit.

The proposal treated the macular density as an unknown that flicker photometry could supply, and asked what the fit would do without it. That is the right first question and it gave a striking answer: two sessions’ trade falls by a quarter. But a fit that estimates a number from its own data is already a measurement of that number, and the outside measurement competes with it rather than filling a hole.

The failure mode is to price an outside measurement by fixing the number it measures, rather than by adding it with its own error. Fixed, the density looked worth a correlation of two tenths; added at the precision a real instrument reaches, it is worth three hundredths, because the matches knew the density nearly as well. The test is to compare the outside measurement’s error with the inside estimate’s before planning around it.