A second session keeps a band and the macula apart

An older observer’s colour matches carry more than the observer model has numbers for. The model fits a lens of one shape scaled by age, a macular pigment of one shape scaled by density, and the weight with which rods feed the blue-yellow pathway; the matches carry their own lens showed one session of matches fitting all three. A run of essays since then has added what an older eye actually has. A band of extra absorption in the lens — the yellow chromophores that accumulate with age — is fitted by two more numbers, its amplitude and centre, and a lost lens band goes to age or the macula found two sessions fitting all five from almost any start.

A macula three nanometres short is a lens band then found what that fit makes of a macula whose absorption peaks a few nanometres from the model’s 460 nm, as a real macula’s does when its mix of lutein and zeaxanthin differs. A short macula is fitted as a lens band near 438 nm, significantly in nineteen experiments of twenty, and a real band, fitted with the macula’s peak free instead, is read as a macula a few nanometres short. Each fit finds the departure it is built to find. With two sessions, comparing the two fits tells which departure is present — but only if one of them is.

An older eye may have both. The fit that would be run on real older observers is then the six-number one, band and macular peak together, and the question is whether the matches can carry six numbers at once or whether the band and the macula trade so tightly that neither is determined. The prediction was that it depends on where the band is: a band at 445 nm and a short macula, which overlap, would trade almost completely, with a correlation beyond 0.95 and the band’s standard error tripled; a band at 415 nm, beyond the macula’s short flank, would stay separable.

Two sessions carry both; one cannot #

With all six numbers fitted to two sessions of matches, the band is found beyond twice its standard error in twelve to twenty experiments of twenty for every one of the nine observers, and a macula moved three nanometres either way is found moved in twelve to nineteen; an unmoved macula is found moved in at most three. The band and the macula do trade — their correlation at the truth is 0.42 to 0.71 for the band at 415 nm and 0.73 to 0.83 at 430 and 445 — but never as tightly as the prediction said. With one session the predicted trade appears: correlations of 0.91 to 0.95 for the bands at 430 and 445 nm, a band found in at most twelve experiments for any observer, and a moved macula in at most seven. Carrying the macula’s peak raises the band’s standard error from two sessions by a quarter at 415 nm and by three quarters at 445; the rods’ weight loses under a sixth of its precision.

• The prediction is right about one session and wrong about two. The trade it described is real; the second session breaks it.
• Where the band sits decides the price, as predicted. The band nearest the macula pays most for the extra number, though it pays less than a factor of two.
• The analysis for real older observers can carry both. With two sessions, a six-number fit finds a band and a moved macula when they are there and rarely finds a moved macula when it is not.

Each group of four bars is one observer. The upper two are two sessions: the band found, the shift found. The band is found in seventeen to twenty experiments for every observer but the one with a band at 415 nm and an unmoved macula, where it is found in twelve — a band at 415 nm sits where a band and an older lens change the matches almost alike, and a tenth of a density there is under three standard errors. The shift is found in twelve to nineteen experiments where the macula is moved, and where it is not, in one, three and none for the bands at 415, 430 and 445 nm. The lower two bars are one session: the band found in three to twelve experiments, the shift in two to seven where the macula is moved. One session can neither find a band reliably nor tell a moved macula from an unmoved one when a band is also free.

How tightly they trade #

A band and a short-moved macula both put absorption on the macula’s short flank, so a fit that is free to use either can trade one for the other: a larger band with the macula moved further long, or a smaller band with it moved further short. The correlation between the two at the truth measures how freely. With one session it is 0.91 to 0.95 for the bands at 430 and 445 nm — nearly a single combination determined and its two parts not — and near nothing for the band at 415, which overlaps the macula least but is barely seen by one session at all. With two sessions the same correlations fall to 0.73 to 0.83, and the band at 415 nm to 0.42 to 0.71.

The second session is the LED against a three-emitter lamp, whose narrow lines fall at different places on the macula’s flanks than daylight’s or a phosphor LED’s broad spectra do. A second lamp pair finds a lens band’s centre found the same session placing a band’s centre where one session could not. A band and a moved macula differ in shape as well as position — the band is a Gaussian 15 nm wide, the macula 42 — and lines that sample the two shapes at different wavelengths see the difference that broad spectra average away.

The scatter shows the trade directly. For a band at 445 nm and a macula three nanometres short, one session’s twenty fits spread along a line from bands of almost nothing to a third of a density, and from shifts of nearly nine nanometres short to five long, the correlation across the twenty 0.82. Two sessions’ fits cluster about the truth: bands from 0.04 to 0.14, shifts from one to six nanometres short, the correlation across them 0.60. The same twenty experiments’ worth of noise, half as much again as data, turns a line into a cluster.

Each band has its own partner #

The band at 415 nm behaves differently from the other two on every measure here, and the reason is that it trades with a different number. At 415 nm the band’s amplitude correlates with the lens’s age at −0.95 from two sessions and −0.96 from one: a band that far into the violet changes the matches almost exactly as a slightly older lens does, since the lens’s own absorption rises steeply there. At 445 nm the band’s amplitude correlates with the lens’s age at only −0.12 from two sessions, and with the macula’s density at −0.73 and its shift at 0.83. At 430 nm it is shared, −0.50 with age and 0.78 with the shift.

That is why one session reads the band at 415 nm as nearly uncorrelated with the macula’s shift — 0.06 — and still finds it in only seven to twelve experiments: its partner is the lens, and the lens’s age is determined by one session to five years, against two from two sessions. A band in the lens reads as age or as macula found exactly this split for a band at a fixed centre, read as four to six years of extra lens age at 410 to 430 nm and as macular pigment at 440 to 460. Six numbers do not remove either partnership; a second session loosens both, by sampling the lens’s steep violet absorption and the macula’s flanks at wavelengths the first session’s broad spectra do not separate.

For an analysis of older observers this means the two kinds of confusion arrive at opposite ends of the band’s range. A band found near 415 nm with its amplitude uncertain is a statement about the lens’s shape near the violet, where the standard age curve is also least certain; a band found near 445 nm is a statement that has had to be distinguished from the macula, and the macula’s peak is the number that must be fitted beside it. A lens of the wrong shape lands in the macula found a lens edge moved ten nanometres being read as macular pigment; a band at 445 nm is the same overlap from the band’s side.

What the extra number costs #

An extra number always costs precision, and the question is whose. With two sessions the band’s standard error rises from 0.030 to 0.037 at 415 nm when the macula’s peak is fitted too, from 0.021 to 0.033 at 430, and from 0.019 to 0.034 at 445 — by a quarter, three fifths and three quarters. That is the prediction’s ordering, the band nearest the macula paying most, at about half the price it named. With one session the band’s standard error is 0.068 to 0.089, two to two and a half times the two-session figures, which is why one session finds the band so rarely.

The rods, which these sessions were designed to measure — age can size the rod signal if the lens is known is why the lens is fitted at all — barely notice. The rods’ weight is determined to 0.032 or 0.033 with all six numbers fitted from two sessions, against 0.029 to 0.031 with the band alone — under a sixth worse. The macula’s peak itself is determined to 0.9 to 1.3 nanometres, against 0.7 to 0.8 when it is the only departure fitted, and the band’s centre to four or five nanometres, about what a second lamp pair finds a lens band’s centre found for the band alone.

Where the found bands and shifts land #

A band found where it is not would be the phantom of the previous census again, now arriving with the macula’s peak carried. It does not arrive: the found bands sit about their own true centres for every observer, whether the macula is moved short, not moved, or moved long, with a few at the short edge for the band at 415 nm, where the fit trades the band for lens age as the band essays found. The phantom band that a three-nanometre-short macula produced at 438 nm, with the macula’s peak not in the fit, has no room to form once the peak is: the fit uses the shift for the macula’s change and the band for the lens’s.

The shifts land about their truths by about a nanometre and a quarter either way, slightly more for the band at 445 nm than at 415. No observer’s shifts are pulled systematically towards where the band would push them, which is what a fit in which the two had truly merged would show: with one session, the 445 nm band’s fits pulled their shifts nine nanometres short or five long along the trade line.

What an analysis of older observers should do #

The lead the band essays left was to fit five numbers to real older observers’ matches and count those with a band beyond twice its standard error; the previous census showed that count would include every observer whose macula peaks a few nanometres short. The repair this census supports is specific. Fit all six numbers, from two sessions with the second on a lamp with narrow lines, and count bands and macular shifts separately. Each is found in most experiments where it is present; a moved macula is falsely found in at most three experiments of twenty when it is not; and the rods’ weight, which was the point, keeps all but a sixth of its precision.

With one session the same analysis would find a band in at most three experiments of five even where one is present, and would report band and macula as one confounded quantity at the wavelengths where older lenses absorb most. A dimmer session hides a lens band in the rods found that making the session dimmer does not help; a second, different lamp pair does.

How the observers were simulated and fitted #

The observer is the band essays’ at sixty, rods at a tenth of each cone’s peak feeding the S pathway with weight one half, a lens band a Gaussian 15 nm 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. The designs are one session of daylight against a phosphor LED, and that with a second of the LED against a three-emitter lamp, forty-two settings a pair, read as CIELAB differences over the collection’s surface family with its setting noise. Each observer’s matches are simulated in twenty experiments with seeded noise and fitted by Levenberg–Marquardt with all six numbers held to their physical ranges, from two starts with the band’s centre at 410 and 450 nm and the macula unmoved, keeping the better; standard errors and correlations at the truth come from the matches’ derivatives there, and each experiment’s own from its fit’s. A departure is found when its amplitude, or the shift’s magnitude, exceeds twice its standard error.

What the census does not reach #

Fits from two starts. Every experiment here was fitted from two starting points. One session has a second valley refitted all of them from three more and found one session’s fits missing a better valley in a quarter of its experiments — a macula moved far short — so that one session, fitted thoroughly, reports a moved macula where there is none in up to eight experiments of twenty. The two-session counts here stand.

One macular shape, moved. A real change in carotenoid mix changes the macula’s shape a little as well as its peak; a shape change would need its own number, and seven numbers from two sessions is untested.

One band width. The band is fifteen nanometres wide in every case; a broad shoulder of absorption, which some older lenses show instead of a band, would overlap the macula’s flank more widely.

Simulated observers. Whether real older observers carry bands, shifted maculas or both is still the question the matches are for, and this says only that two sessions could answer it.

Still open: whether a third lamp’s line on the macula’s long flank frees the band at 445 nm #

The band at 445 nm pays most for the macula’s peak because the two overlap; what tells them apart is that a moved macula also changes its long flank, past 470 nm, where no lens band reaches. The second session’s three-emitter lamp has a line near 455 nm and its next at 530, and none on the long flank itself.

The calculation is this census with a third session added — the LED against a lamp with a narrow line at 480 to 490 nm — and the six-number fit repeated for the band at 445 nm and a macula moved −3, 0 and +3. The prediction is that the band–macula correlation falls from 0.82 to under 0.6 and the band’s standard error returns to within a quarter of its band-only value, because a line on the long flank measures the macula’s shift with no band in the way. If the third session buys little, the band at 445 nm and the macula’s peak are separated by their shapes at the short flank alone, and the second session has already taken what the matches can give.

A trade is a statement about the design, not the eye #

The habit here is about what a correlation between two fitted parameters says.

The prediction treated the band–macula trade as a property of the eye: two absorbers that overlap in wavelength cannot be told apart. With one session that is what the fit reports. With two, the same eye, the same absorbers and the same overlap give a correlation of 0.82 instead of 0.95, and the band is found nearly every time. The trade was in what the matches sampled, and a second lamp sampled it differently.

The failure mode is to read a parameter correlation as a limit of the thing measured when it is a limit of the measurement. The test is to change the measurement — here, add a session whose lines fall elsewhere — and see whether the correlation moves. If it moves, the design was the limit.