A line on the macula's flank buys less than a repeated session

An older observer’s colour matches carry more than the standard observer model has numbers for, and a run of essays has been adding them one at a time. The matches carry their own lens fitted the rod signal, lens age and macular density from one session. A band of extra absorption in the lens — the yellow chromophores an ageing lens accumulates — added two more numbers, its amplitude and centre, which a second lamp pair finds a lens band’s centre showed needed a second session to place. Then a macula three nanometres short is a lens band found that a macula peaking a few nanometres from the model’s 460 nm is read, by a fit without a number for it, as a lens band near 438 nm.

A second session keeps a band and the macula apart gave the fit all six numbers — rod weight, lens age, macular density, the band’s amplitude and centre, and the macula’s peak — and found two sessions enough to carry both departures: the band found in at least twelve of twenty simulated experiments for every observer, a moved macula in at least twelve. The price fell on the band nearest the macula. At 445 nm the band’s amplitude and the macula’s shift were correlated at 0.82, and the band’s standard error was 1.8 times what it is when the macula’s peak is not fitted at all.

The reason offered was overlap. A lens band at 445 nm and a macula moved a few nanometres both change absorption on the macula’s short flank. What tells them apart, the essay argued, is that a moved macula also changes its long flank, past 470 nm, where no lens band reaches — and neither session’s lamps put a narrow line there. The proposal was a third session, the phosphor LED against a lamp with a narrow line between 480 and 490 nm, and the prediction was that the band–macula correlation would fall from 0.82 to under 0.6 and the band’s standard error return to within a quarter of its band-only value.

The flank line buys less than repeating a session #

With the third session — the LED against a four-emitter lamp whose second emitter is a narrow line at 487 nm — the correlation between the band’s amplitude and the macula’s shift at 445 nm falls from 0.82 to 0.80, and the band’s standard error from 1.78 to 1.51 times its band-only value, for the unmoved macula. A third session that repeats the second session’s lamps, with as many matches and no new line, does better on both standard errors: the band’s to 1.41 times, the macula shift’s to 0.98 nm against the cyan session’s 1.04. Moving the line from 487 to 465 nm, onto the macula’s peak, brings the band’s error to 1.16 times its band-only value — inside the predicted quarter — while its correlation with the macula’s shift rises to 0.86. Across every third session tried, the correlation never falls below 0.78.

• The prediction fails on both numbers. The correlation barely moves and the band’s error stays half again its band-only value.
• The long flank is not where the information is. A line there buys less than a session with no line there at all.
• A line on the macula’s peak frees the band but not the trade. The band gets cheaper because everything does; the band and the macula stay just as entangled.

The three lines are the three macula positions, and they move together. At 487 nm, where the proposal put the line, the band’s error sits at 1.39 to 1.63 times its band-only value, above the dotted line for the repeated session. At 500 and 510 nm, further down the long flank, it is the same. Only at 465 and 475 nm does it fall clearly below the repeat, to 1.10 to 1.26 times. The band-only value is itself a two-session fit, so a three-session fit can in principle go below it; at 445 nm none does.

What the proposal got right, and what it missed #

The argument for the long flank was sound as far as it went. The macular pigment’s absorption in the observer model is a Gaussian centred at 460 nm, and moving it changes absorption fastest about thirty nanometres either side of its peak — near 430 nm on the short flank and near 490 nm on the long. A lens band at 445 nm overlaps the short flank and stops well short of the long one. A line at 487 nm sits almost exactly where a moved macula’s absorption changes fastest and a band changes nothing.

What the argument missed is that the fit has six numbers, not two, and a match is a single number that all six move together. A line at 487 nm changes a match when the macula moves, but it changes it even more when the macular density changes, because 487 nm is still well inside the pigment’s absorption. The fit uses the new session mostly to measure how much macular pigment there is, and the macula’s peak position, which the line was supposed to pin, gets only what is left. It is the same habit a lens of the wrong shape lands in the macula found from the lens’s side: the macular density is the number the matches reach for first.

At 487 nm the line sits on the steepest part of the macula’s long flank, as proposed; moved to 465 it sits almost on the peak, and moved to 510 it is nearly off the pigment altogether. The three numbers beside the lamp change less than the line’s position suggests they should: the correlation stays between 0.80 and 0.88 everywhere, and the band’s error is lowest only where the line sits on the peak.

The numbers show it. The macular density’s standard error falls from 0.027 with two sessions to 0.018 with the cyan line at 487 nm — the lowest of every design tried, lower than the repeated session’s 0.022 or the line at 465 nm’s 0.020. The density is the number the flank line reads best. And it reads another thing the proposal did not intend: with the line at 487 nm, the band’s amplitude and its centre become correlated at −0.63, where two sessions had −0.45 and the line at 465 nm −0.42. A line on the long flank, at the edge of where a band at 445 nm reaches, lets a slightly taller band sitting slightly shorter look like a slightly shorter band sitting slightly taller. The band pays for that in its own error.

Repeating the second session is the fair comparison #

A third session adds matches whatever its lamps are, and more matches lower every standard error. The question the proposal asked is not whether three sessions beat two — they always will — but whether a line on the long flank adds something a third session without it would not. The control is the second session repeated: the same LED against the same three-emitter lamp, as many new matches as the cyan session adds, and no line anywhere new.

On the number the line was supposed to read, the macula’s shift, the repeated session gives 0.98 nm for the unmoved macula and the cyan line at 487 nm gives 1.04 — a little worse, not better. The same holds for the macula moved either way: 1.01 against 1.10, 0.96 against 0.98. The cyan session is not useless — it improves on two sessions’ 1.29 nm — but everything it buys, a session that adds no new line buys too, and slightly more. Lines at 465 and 475 nm do beat the repeat, at 0.90 and 0.88 nm, and lines at 500 and 510 nm do worse.

The repeated session is a fair control because it holds the number of matches fixed. Both third sessions add one lamp pair’s worth of matches over the same surfaces; they differ only in what the second lamp of the pair emits. That the plain repeat does as well as the designed one says the cyan lamp’s spectrum carries no more about the macula’s peak, per match, than the three-emitter lamp it was designed to improve on.

No line breaks the trade #

The prediction’s headline number was the correlation, and it is the number that moves least. Two sessions give 0.82 to 0.83 at 445 nm. The cyan line at 487 nm gives 0.78 to 0.80; the repeated session 0.79 to 0.81; the line at 465 nm, which frees the band’s error most, 0.85 to 0.86; the lines at 500 and 510 nm 0.83 to 0.88. No third session tried brings it below 0.78, and the prediction’s 0.6 is far below anything on the chart.

A correlation and a standard error measure different things. The standard error says how well a number is known; the correlation says how much of its uncertainty it shares with another number. A third session that improves both the band and the macula’s shift together lowers both errors and leaves their correlation where it was — which is what the line at 465 nm does. For the band at 445 nm, every third session tried makes both numbers better known and none makes them more separable. That is the alternative the earlier essay named: the band and the macula’s peak are told apart by their shapes at the short flank, and the second session already took the separation the matches can give. A third adds precision, not distinction.

What each third session buys each band #

The band at 445 nm was the one the proposal was for, but the census has three. For the band at 415 nm, far from the macula, every third session brings the band’s error to within a tenth of its band-only value, and the cyan line takes it below, to 0.91. At 415 the cyan session is the best of the four: its line adds information where the band and the macula do not compete, and the extra density measurement helps the band directly. For the band at 430 nm the four designs land between 1.38 and 1.61 times, the line at 465 nm best and two sessions worst. For the band at 445 nm the spread is widest, from 1.78 with two sessions to 1.16 with the line at 465.

The pattern is that the line’s best position follows the band. A band far from the macula benefits from any extra session, and most from one that pins the macula’s density, so that nothing about the macula is left to borrow from the band. A band overlapping the macula’s short flank benefits most from a line near the macula’s peak, where the band and the macula absorb in different proportions from their overlap — not from a line on the far side of the macula, where only the macula absorbs but where the fit uses the line for density.

Three sessions find the band a little more often #

The fits so far are read from the information at the true observer. The census also runs twenty simulated experiments for every observer, each a noisy set of matches fitted from two starts, and those say how often the departures are actually found.

With the cyan session added, the band is found beyond twice its standard error in at least sixteen of twenty experiments for every observer, and in nineteen or twenty for the band at 445 nm — against eighteen or nineteen with two sessions. A macula moved three nanometres either way is found moved in ten to twenty experiments, and an unmoved one is found moved in at most one. The gains are real and small, about what a third session of any lamps would buy; they are the experiments’ version of the standard errors above. The rod weight’s standard error barely changes with the third session, from 0.032 to 0.029, as it barely changed with the second — the rods see the lens and not the macula, which is why a dimmer session hides a lens band in the rods found them no help in telling the two apart.

What an analysis of older observers should take from this #

A third session is worth running for the precision it adds, and a lamp designed for the macula’s long flank is not worth designing. A repeat of the second session gives the band at 445 nm a smaller error than the cyan lamp does, at no cost in new hardware. If a lamp is to be designed, its narrow line belongs near the macula’s peak, at 465 to 475 nm, where it brings a band near the macula within a fifth of its band-only error.

The band–macula correlation at 445 nm is a property of the observer, not the design. Above 0.78 under every design tried, it means a fit reporting a band near 445 nm and a macula moved short has found a combination it can only partly split. The honest report gives both numbers with their correlation. A lost lens band goes to age or the macula and a band in the lens reads as age or macula found the same habit of the matches from the other direction: what one number cannot carry goes to the nearest number that can.

How the observers were simulated and fitted #

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 of a tenth of a density at 415, 430 or 445 nm, and a macula of density 0.35 whose Gaussian absorption, 42 nm wide, is centred at 460 nm moved by −3, 0 or +3. A session is a set of asymmetric matches over the collection’s surface family between two lamps. The two-session design is daylight against the phosphor LED and the LED against a three-emitter lamp. The cyan session is the LED against a four-emitter lamp — a blue pump at 452 nm, a narrow line sixteen nanometres wide at 487, a broad green at 535 and a narrow red at 625 — and the moved-line variants change only the second emitter’s centre.

The six-number fit is Levenberg–Marquardt from two starts with the band at 410 and 450 nm. Standard errors and correlations are read from the Fisher information at the true observer, with the same setting noise as the earlier fits; for the two-session and cyan designs that information is the fits’ own, reproduced to a part in a billion, and for the repeated session and the moved lines it is computed the same way. Each observer’s twenty experiments add independent Gaussian noise to every match.

What the census does not reach #

One line width and one set of other emitters. The cyan lamp’s line is sixteen nanometres wide on a fixed pump, green and red. A narrower line, or a lamp whose other emitters avoid the macula, could read the peak with less density mixed in.

Information, not experiments, for the controls. The repeated session and the moved lines are compared by their Fisher information at the truth; their experiments were not simulated, so how often each would find the band is inferred from its standard error rather than counted.

Simulated observers. The macula’s shape is the model’s Gaussian; a real macula’s lutein and zeaxanthin give it shoulders, and a shift of its peak changes its shape as well.

Still open: whether a line on the macula’s peak, with density fixed, splits the trade #

The line at 465 nm frees the band at 445 nm by measuring everything better, and the correlation stays because macular density soaks up what the macula’s peak and the band would otherwise share. If the density were known independently — as it can be, from heterochromatic flicker photometry, which measures macular pigment directly — the fit would have five numbers, and the line near the peak would have only the band and the peak to tell apart.

The calculation is the six-number fit with the macular density fixed at its true value, under two sessions, the repeated session, the cyan line at 487 nm and the line at 465 nm, for the band at 445 nm and the three macula positions. The prediction is that fixing the density takes the correlation between the band and the macula’s shift below 0.6 with the line at 465 nm, and leaves it near 0.8 with two sessions, because the second session’s lamps never separated density from peak. If the correlation stays near 0.8 with density fixed, the trade is between the band and the peak themselves, at the short flank, and no measurement of the macula’s amount can buy it out.

A designed measurement needs a plain one beside it #

The habit here is about how to test whether a measurement designed for a purpose serves it.

The cyan lamp was designed with a reason: a line where one departure acts and the other does not. Run as a third session, it improved every number the fit reports, and a reader comparing three sessions with two would call the design a success. Only the plain repeat — the same number of matches, no new line — shows that the improvement belonged to the extra matches and not to the line.

The failure mode is to compare a designed addition with nothing, rather than with an undesigned addition of the same size. The test is to run the cheapest addition that changes nothing but the amount of data, and ask whether the design beats it. Here it did not, the reasoning behind it had a gap — the fit spent the line on density — and the position that did help was one the reasoning had not proposed.