A macula three nanometres short is a lens band

A colour match made by an older observer carries the observer’s lens in it. Light reaching the cones has passed through the lens first, the lens yellows with age, and a match between two lights of different spectra is set by what survives. The matches carry their own lens showed that one session of matches — daylight against a phosphor LED, forty-two settings — fits the lens’s age, the density of the macular pigment over the fovea, and the weight with which the rods feed the blue-yellow pathway, all at once — the weight that a rod signal has no natural size showed needs a lens to be read against, and that age can size the rod signal if the lens is known put a number on. Four essays since then have asked what happens when the lens is not the model’s shape. A band in the lens reads as age or as macula gave it a band of extra absorption, the kind an ageing lens’s yellow chromophores add; a second lamp pair finds a lens band’s centre freed the band’s centre as a fifth number; and a lost lens band goes to age or the macula found that two sessions fit all five numbers from almost any start. A dimmer session hides a lens band in the rods had already found the band trading with the macula more tightly than predicted.

That essay ended on the experiment the fits were built for: fit the five numbers to real older observers and count how many carry a band more than twice its standard error. It needs real older observers’ matches, which no simulation supplies. It also rests on an assumption that can be tested without any, which is that a significant band can only be a band — that nothing else the eye might do would show up in a fit as one.

The likeliest something else is the macula. The observer model’s macular pigment has one fixed shape, an absorption peaking at 460 nm. A real macula is two carotenoids, lutein and zeaxanthin, deposited in different proportions in different eyes, and the two absorb at wavelengths a few nanometres apart, so a macula’s peak moves with its mix. The macula overlaps exactly the region where the band essays looked for a band. The question is what the five-number fit makes of a macula whose peak sits a few nanometres from where the model puts it.

A short macula is fitted as a band #

An observer at sixty with no band in its lens and a macula peaking three nanometres short of the model’s, at 457 nm, is fitted from the exact matches of two sessions with a lens band of 0.061 at 438 nm — 3.1 of its standard errors — and twenty noisy experiments find a band more than twice its standard error in nineteen. Six nanometres short, the phantom is 0.121 at 437 nm, larger than the tenth of a density the band essays searched for, and every experiment finds it. A macula peaking long has no band to become: three nanometres long, the fit finds none, and six, it runs to the long edge of the range a band may take. A real band, fitted with the macula’s peak freed instead of a band, is read as a macula 2.4 to 3.0 nanometres short, significantly in sixteen to twenty experiments. Each confusion is found nearly every time by the fit that makes it. Comparing the two fits separates them: with two sessions, the band fit is better in every experiment on a real band, and the fit with the macula’s peak free is better in sixteen to nineteen of twenty on a short macula, though it has one number fewer. With one session it is better on the three-nanometre macula in only four.

• A band more than twice its standard error is not evidence of a band. A macula a few nanometres off produces one as reliably as a real band does.
• The confusion is one-sided in the macula’s direction. Only a macula peaking short of 460 mimics a band, because a band can only add absorption on the short side.
• The two are distinguishable, and only by comparison. The phantom leaves a misfit that a second session can see.
• The macula’s peak is measurable. Freed, it is recovered to under a nanometre from two sessions.

Each bar is one macula and one design, fitted to the matches with no noise at all — the fit’s answer to the question with nothing random in it. The macula where the model has it is fitted with no band. Each of the two maculas peaking short is fitted with a band in the blue, larger the further the peak moves and more significant with two sessions than with one: from two sessions the three-nanometre macula’s phantom is 3.1 standard errors and the six-nanometre macula’s 5.8. From one session the same phantoms are 2.1 and 3.9 standard errors, because one session’s standard errors are larger. Both maculas that peak long are different. The three-nanometre one is fitted with no band, the model’s three numbers absorbing what they can; the six-nanometre one is fitted, from two sessions, with a band at the 500 nm limit of the fit’s range, a sign the fit is reaching for something outside the lens’s blue rather than finding anything there.

Why the macula’s shift looks like a band #

A macula moved short adds absorption on its short flank and removes it on its long one. The model’s macula can only change its density, so the three-number fit cannot reproduce the move: it thins the macula to give back the long flank and leaves the short flank’s excess unaccounted for. That excess, a narrow ridge of absorption a little short of the macula’s peak, is what a lens band is. The five-number fit finds it at 437 to 438 nm, between the band essays’ bands at 430 and 445, and thins the macula from 0.35 to 0.31 at the same time.

A macula moved long does the reverse — removes absorption short of its peak and adds it beyond — and a band, which can only add absorption, has nothing to add where the excess is. The fit finds no band, or, at six nanometres, pushes one to the far edge of its range in an attempt to add absorption beyond the macula. The phantom exists only on one side, and it is the side a band in an older lens is looked for.

The size is the surprise. A three-nanometre move is a small fraction of the macula’s width — the model’s macula is a Gaussian 42 nm wide — and still produces a band of six hundredths, three fifths of the tenth of a density the band essays used as their standard band. The band essays found two sessions placing a band’s centre to about five nanometres; a macula’s peak moving three nanometres changes the matches as much as a band of half that standard size.

How often a fit finds what is not there #

The exact fits say what a perfect experiment would conclude. A real one carries setting noise, and the count that the proposed test would make is the number of experiments in which a band passes twice its standard error. For the macula three nanometres short, the five-number fit finds a band in nineteen of twenty two-session experiments and seven of twenty one-session ones; six nanometres short, in every experiment of either design. For an observer with nothing to find — no band, macula where the model has it — the same criterion fires in three of twenty two-session experiments and one of twenty one-session ones, the false-alarm rate the test carries whatever the eye is doing.

The reverse confusion is as strong. An observer with a real band of a tenth of a density at 415, 430 or 445 nm, fitted with the macula’s peak freed and no band, is found to have a macula shifted more than twice its standard error in sixteen, nineteen and twenty experiments of twenty. Each fit finds the departure it is built to find, whether or not that departure is the one present. A test that counts significant bands in a population of older observers would count every observer whose macula peaks a few nanometres short as one with a band, and a test that counted significant macula shifts would count every observer with a band as one with a shifted macula.

Where the phantom lands #

A phantom that landed somewhere a real band never does could be recognised by its centre. This one does not. For the macula three nanometres short, fourteen of the nineteen found bands lie between 430 and 450 nm — interleaved with the real bands at 430 and 445 — and three sit near the short edge of the range, at 403 to 408, where the fit has traded the band for extra lens age. The six-nanometre macula’s phantoms cluster tighter, sixteen of twenty between 432 and 447. Real bands are found where they are, 408 to 419 for the band at 415, 417 to 443 for 430 and 439 to 453 for 445, overlapping the phantoms throughout.

A second lamp pair finds a lens band’s centre proposed that a lamp with a narrow line near 440 nm, where the lens’s chromophores and the macula overlap, was the lamp an experiment on older observers most needed. That is also exactly the region in which a moved macula and a lens band are hardest to tell apart by where they sit. The centre of a found band says nothing about whether it is a band.

A real band, read as a short macula #

Run the other way, the confusion is as regular. A band at 415 nm is read as a macula 2.4 nanometres short, a band at 430 as 3.0 short, a band at 445 as 2.7 short, from the exact matches of two sessions; the noisy experiments scatter about those values by a nanometre either way, and almost none crosses zero. The macula-shift fit also has to do something about a band’s absorption that a moved macula cannot provide, and it does what the band essays found a dropped band doing: it ages the lens, by five years for the band at 415 and by under three for the band at 430, and adds macular density for the band at 445.

The symmetry is the point. A macula three nanometres short and a band of six hundredths at 438 nm are, to a fit with only one of the two parameters, nearly the same thing. They are two descriptions of one change in the matches, and a fit that can describe it only one way will describe it that way.

The comparison that separates them #

They are not the same thing to a fit that is offered both descriptions and asked which is better. The five-number band fit has one more parameter than the macula-shift fit, so on equal terms it should fit at least as well. For a real band it does, by a wide margin. For a short macula it does not: from the exact matches of two sessions the band fit leaves a misfit of $\chi^2$ 4.3 for the three-nanometre macula and 14.0 for the six-nanometre one, where the macula-shift fit leaves none. A phantom band reproduces most of what a moved macula does to the matches but not all of it, and what it leaves out is the long flank, where the band cannot remove absorption and the moved macula did.

With two sessions, the macula-shift fit is never better on a real band — in none of sixty experiments — and better on a short macula in sixteen experiments of twenty at three nanometres and nineteen at six, with one parameter fewer. With one session the misfit the phantom leaves is too small to see against the noise: $\chi^2$ 0.5 for the three-nanometre macula from the exact matches, and the macula-shift fit is the better in four experiments of twenty, near what chance gives for an unshifted macula, eight. One session can find a departure; it takes two to say which.

That makes the proposed test for older observers a comparison rather than a count. Fit each observer twice, once with a band and once with the macula’s peak free, and ask which is better; with two sessions, a real band is never mistaken for a moved macula by that comparison, and a moved macula is mistaken for a band in one experiment of twenty at six nanometres and four at three.

Where each fit takes the rod signal #

The sessions were designed to measure the rods’ weight in the blue-yellow pathway, and lens and macula were fitted alongside because they would otherwise bias it. A lens of the wrong shape lands in the macula found a lens edge ten nanometres off moving the rods’ weight by six tenths of its standard error from one session. A misplaced macula is the same kind of error, in the other medium.

For the macula six nanometres short, the model’s three numbers move the rods’ weight by 2.3 of its standard errors from two sessions, and the band fit by 1.0; the phantom band absorbs more than half the bias and keeps the rest. Three nanometres short, the three-number fit moves the weight 1.3 standard errors and the band fit 0.6. A macula peaking long moves the weight the other way, by 1.5 standard errors at three nanometres and 2.4 at six; the band fit finds no band to absorb the first, and absorbs a third of the second only by pushing a band to the edge of its range. The fit with the macula’s peak free returns the true weight in every case. A fit of the rod signal that ignores where the macula peaks is wrong by more than the quantity it was built to measure is resolved to, in either direction, for a shift of a few nanometres.

The peak is cheap to measure #

Freed, the macula’s peak shift is the best-determined number in the fit after the macula’s density. From two sessions its standard error is 0.71 to 0.77 nanometres for every simulated shift, and from one session 1.25 to 1.33; the exact matches return every shift exactly, and the other numbers with it. A field size is two changes showed how much of the difference between the two standard observers is macular pigment; this says that a matching session can measure not only how much an observer has but where its absorption peaks, to under a nanometre, at the cost of one more number. It is a cheaper number than the band’s centre, which two sessions determined to five.

How the observers were simulated and fitted #

The observer is the band essays’ at sixty: the collection’s lens at that age, rods at a tenth of each cone’s peak feeding the S pathway with weight one half, and a macula of density 0.35 whose absorption is a Gaussian 42 nm wide, moved here by −6, −3, 0, +3 and +6 nm from 460. The reverse observers keep the macula at 460 and add a band of a tenth of a density, a Gaussian 15 nm wide, at 415, 430 or 445 nm. The designs are the band essays’: one session of daylight against a phosphor LED, and the same with a session of the LED against a three-emitter lamp, forty-two settings a pair, the matches read as CIELAB differences over the collection’s surface family with its setting noise.

Each observer’s matches are fitted noise-free and in twenty experiments with seeded noise. The three fits are Levenberg–Marquardt with the parameters held to their physical ranges: the three numbers — weight, lens age and macular density; those with a band’s amplitude and centre, from starts at 410 and 450 nm, keeping the better, which the band-start census found as good as eight starts; and the three with the macula’s peak shift. Standard errors come from the fit’s own derivatives at its solution. A departure is found when its amplitude exceeds twice its standard error. The model’s macula shape, and its move as a whole, stand in for a change in its carotenoid mix — a template’s tails matter more than its peak, as a template is mostly its tail found for cone pigments, and moving the whole shape moves both tails; a real change in mix would change the shape a little as well as the peak, which this census does not model.

What the census does not reach #

One macular shape, moved. Lutein and zeaxanthin differ in more than peak position, and a macula’s shape also changes across the fovea, which a field of a different size samples differently. A shape change that added absorption short of the peak would mimic a band in the same way; one that only broadened it might not.

Simulated observers. The real question — how many older observers carry a band — is untouched. What has changed is what the count would mean.

A band and a moved macula together. Each simulated observer here has one departure or the other. An older eye may have both, and whether two sessions can carry a band and the macula’s peak in the same fit is a separate question.

Still open: whether two sessions carry a band and the macula’s peak together #

The comparison separates a band from a moved macula when only one is present. A real older eye may have a band in its lens and a macula whose peak is not the model’s, and then neither four-number nor five-number fit is right: the six-number fit, with the band’s amplitude and centre and the macula’s peak all free, is the one an analysis of older observers would need.

The calculation is this census with observers carrying both — a band of a tenth of a density at 415, 430 or 445 nm and a macula moved three nanometres either way — fitted with all six numbers from two sessions, with the standard errors of the band’s amplitude and the macula’s shift and the correlation between them. The prediction is that the band at 445 and a short macula trade almost completely, with a correlation beyond 0.95 and the band’s standard error tripled, while a band at 415 stays separable, because it lies beyond the macula’s short flank; so that two sessions can tell an older eye’s low bands from its macula and cannot tell its high ones. If all three bands stay separable, a six-number fit is the analysis to run on real older observers; if none does, the sessions need a third lamp whose lines fall on the macula’s long flank, where a band cannot reach.

A significant parameter tests the model, not the eye #

The habit here is about what a significance test on one parameter of a model can say.

A band more than twice its standard error says that the matches are better described with the band than without it, given everything else in the model. It says nothing about whether something outside the model would describe them better still, and the macula’s peak was outside the model. The test proposed for older observers would have counted phantoms as bands, every one of them significant.

The failure mode is to read a parameter’s significance as evidence for the thing the parameter is named after. The repair is to name the alternatives, give each a parameter, and compare the fits — which here costs one more fit per observer, and needs the second session that the band essays had already shown was worth running.