A lost lens band goes to age or the macula

An older lens absorbs more blue light than a young one, and not always in the same shape. The model these essays use ages the lens along one standard curve — the lens that the eye stops at, whose density grows fastest at the shortest wavelengths — and a band in the lens reads as age or macula asked what happens when a real lens adds absorption in a band of its own: a colour-matching experiment fitted with the standard curve reads the band as extra age when it sits low in the blue and as extra macular pigment when it sits near the macula’s own peak at 460 nm. A dimmer session hides a lens band in the rods gave the fit a fourth number, the band’s size, and a second lamp pair finds a lens band’s centre a fifth, its centre — and found from the matches’ sensitivities that one daylight-against-LED session places a band’s centre to within eleven nanometres below 430 and six above it, and that a second session, the LED against a three-emitter lamp, places every centre from 410 to 460 within five and a half.

A standard error is a statement about a fit that has found the truth. It says how far the answer would scatter if the fit started there and the noise pushed it about. A centre is a nonlinear parameter — the matches do not change in proportion to it — and a fit over one has to find the right valley before its standard error means anything. The closing section of the lamp-pair essay asked whether it does: the full fit, from starts spread across 400 to 470 nm, on simulated sessions with noise, for bands at 415, 430 and 445 nm. The prediction was that two sessions find a band at 430 or 445 from any start, and a band at 415 only from starts below 440 — above that, the fit would reach the macula’s flank first, settle there, and trade the band for extra macular pigment.

Two sessions find the band from anywhere, and a lost band is a worse fit #

With two sessions, every one of the eight starts finds each band within twice its standard error in fifteen to eighteen of twenty simulated experiments, where a fit started at the true observer finds it in eighteen. The start that the prediction said would fail at 415 nm — anywhere above 440 — finds it sixteen to eighteen times. A fit that misses either lands where the truth-started fit also lands, missed by the noise, or drops the band altogether: its size falls to nothing and its absorption is handed to another number — to lens age, three to five years of it, for a band at 415 or 430 nm, and to macular pigment for a band at 445. Every dropped band has a $\chi^2$ at least six above the best of its session’s eight starts, so the best of eight finds the band as often as the truth does. With one session the start decides: from 400 or 410 nm a band at 415 is found in eighteen or twenty experiments, from above 420 in five to fourteen.

• The prediction was wrong about where the fit fails. With two sessions the start hardly matters, and the band at 415 nm is as easy to find from 470 as from 400.
• It was half right about how. A lost band is traded, but to the macula only when the band sits beside it; a low band is traded to lens age, as the earlier essay found for an unmodelled band.
• The failures are escapable. Dropping the band is always a worse fit, so a few starts and the lowest $\chi^2$ are enough.
• One session is where the start matters, and there too the best of eight starts finds the band as often as the truth.

The three lines sit between fifteen and eighteen at every start, just under the ticks. The dips are at the starts farthest from the band at 445 nm — 400 and 470 — and, for the band at 415, at 430 and 440, just above it. None of them is a start that cannot find the band.

How the fits were made #

The observer is the one every essay in this line of work has used since age can size the rod signal if the lens is known: a sixty-year-old with the standard lens curve for that age, rods contributing a tenth of each cone’s peak, the median macula, and a band of extra lens absorption a tenth of a density high and fifteen nanometres wide. The two sessions are the lamp-pair essay’s: daylight against a phosphor LED, and that LED against the three-emitter lamp. Each session is forty-two surface matches, and each match is three numbers, so an experiment is 252 numbers; a simulated experiment is the true observer’s matches plus the site’s setting noise, drawn from a fixed seed so that the twenty are the same every time.

The fit is Levenberg–Marquardt on all five numbers at once — rod weight, lens age, macular density, the band’s size and its centre — with derivatives by forward differences and each number held inside its physical range: the centre between 390 and 500 nm, the size above nothing, the age between fifteen and ninety. Each fit starts from what an experimenter would guess before the matches: the standard observer’s lens at thirty-two, its median macula, the rods at their usual weight, a band of half the true size, and a centre somewhere between 400 and 470. A fit finds the band if its centre lands within twice the lamp-pair essay’s standard error of the truth — ten nanometres at 415, eight at 430 and seven at 445 — and drops it if the band’s fitted size is under a fifth of the truth’s.

As a check on the whole arrangement the same fit is started at the true observer. It finds the band in eighteen of twenty experiments at each of the three centres, which is what a standard error of that size allows: about one experiment in twenty lands outside two standard errors by noise alone, and with a few percent of slack in the linear approximation, two.

Where each fit finishes #

For the band at 415 nm the picture is a horizontal stripe. Wherever the fit starts — 400, 440, 470 — it finishes near 415 in most experiments, and the spread of the stripe is the spread of the truth-started fit. The fits that miss fall into two kinds that look nothing alike. Most finish close to 415 but just outside ten nanometres of it — at 426 or 428, say; they finish there from every start, because that is where the experiment’s own minimum is — the noise has moved the best answer, and a fit started at the truth lands there too. Forty-five of the seventy-three misses across the three bands are of this kind: the session’s best fit, missed by noise.

The other kind finishes at the edge of the range, at 390 or 500 nm, with the band’s size near nothing. Once the fit has shrunk the band away its centre no longer changes the matches, and the fit lets it wander to wherever its last step pushed it. Twenty-seven fits end this way, none to three from any given start for the bands at 415 and 430, and up to five from the far starts for the band at 445. They are not concentrated above 440 for the band at 415, as the prediction had them; the start at 430, just above the band, drops it three times, and the start at 470 twice.

Where a dropped band’s absorption goes #

A fit that drops the band has still to explain the matches the band produced, and it does it with the other three numbers. For the bands at 415 and 430 nm it makes the lens older — by five years on average for the band at 415 and three for the band at 430 — and makes the macula a little thinner, because an older lens absorbs at 440 to 460 as well and the macula has to give some of that back. For the band at 445 nm it adds macular pigment, about six hundredths of a density, and leaves the lens’s age where it was. The macular is a band, not a filter: it absorbs forty nanometres either side of 460, and a lens band at 445 sits inside that stretch, where the two are nearly the same change to a match.

That is exactly the confusion a band in the lens reads as age or macula found when the band was not in the model at all, and a lens of the wrong shape lands in the macula found for a lens whose edge was misplaced: a band low in the blue changes the matches almost as an older lens does, and one near 450 almost as more macular pigment does. The fit that drops the band is the four-number fit that essay was about, reached from inside the five-number one. The prediction had the confusion the wrong way round — the low band trading with the macula from high starts — because it reasoned about which feature the fit would reach first, when what decides it is which feature the band most resembles.

A dropped band is a worse fit #

Every one of the twenty-seven dropped bands has a $\chi^2$ at least six above the best of its session’s eight starts, and most far more than that. A $\chi^2$ difference of six on five fitted numbers is not a close call: the data disfavour the dropped band as strongly as a fit ever disfavours anything in a problem this size. Dropping the band is a local minimum — a valley the fit can fall into and cannot climb out of by small steps, because leaving it would mean regrowing the band at a centre it has already left — and not an equally good account of the matches.

So the practical remedy is the ordinary one. Start the fit several times and keep the lowest $\chi^2$. The best of the eight starts finds the band in eighteen, seventeen and eighteen experiments of twenty at 415, 430 and 445 nm, against eighteen for each from the truth.

One session is where the start decides #

With one session the matches constrain the band far less — the lamp-pair essay’s standard errors are eleven nanometres at 415 and six at 430 — and the valleys are shallower and more numerous. From a start at 400 or 410 nm, one session finds a band at 415 in eighteen or twenty experiments of twenty. From every start above 420 it finds it in five to fourteen, and most of the rest drop the band: fifteen of twenty from the starts at 440, 460 and 470. The band at 430 behaves the same way less sharply, fourteen or fifteen from the low starts and five to ten from the high. The band at 445 is found nine to fourteen times from anywhere.

This is the prediction’s shape, arriving one session early. The low starts reach a low band before anything else, and the high starts shrink it away before they arrive. And even here the remedy holds: the best of eight starts finds the three bands in twenty, fifteen and eighteen experiments, exactly as often as a fit started at the truth.

What an experiment on older observers should do #

Fit the band’s centre from several starts and keep the lowest $\chi^2$. With two sessions eight starts are more than enough; with one they are necessary, and a single start above 420 nm would lose a low band in most experiments.

Two starts are nearly as good as eight, if they straddle the blue. Keeping the better of two fits, one started with the band at 410 nm and one at 450, finds the band in exactly as many experiments as a fit started at the truth with two sessions — eighteen of twenty at each of the three centres — and with one session in twenty, fifteen and sixteen of twenty at 415, 430 and 445, against twenty, fifteen and eighteen. The reason is the shape of the failure. A start below the band reaches it before the macula can take its absorption; a start above it may shrink it away, but then its $\chi^2$ is far worse than the low start’s, and the comparison discards it. What the pair of starts buys is not two chances at the same valley but one start on each side of the confusion that makes the valleys, which is why it is the pair that works and not either start alone. A single start at 410 with one session finds a band at 445 only twelve times.

Treat a fit whose band has vanished as a failed fit, not a finding. A fitted band size near nothing, with its centre at the edge of the range, is the signature of the local minimum here. If a lower-$\chi^2$ fit exists with a band in it, the band is there; if none does, the observer may genuinely have no band — and then the fitted age and macula mean what they meant before bands were in the model.

Expect a lost band to show up as age or macula. The matches carry their own lens showed a session could fit the lens rather than measure it; a lost band is the case where what it fits is the wrong lens. An observer whose fitted lens is several years older than their real age, from a single start, may be an observer with a low band in the lens; one whose macula is fitted denser than typical may have a band near 445. The observer has no age is the warning that a fitted age is a property of the model’s lens curve rather than of the eye; this is one specific way the two come apart.

What this leaves out #

The noise is the model’s. Each match number is perturbed independently by the site’s setting noise. Real sessions carry correlated errors — an observer’s criterion drifting over a session, a lamp’s output drifting — and those could make some valleys deeper.

The lamps are two pairs. The lens is worst under tungsten found the lamp deciding how much a lens departure costs, and a pair built to find bands — with emitters on either side of 430 nm — would change which starts fail.

The band has one shape. A Gaussian fifteen nanometres wide and a tenth of a density high. A wider band is more like an older lens everywhere and would be dropped more often below 440; a narrower one would be easier to see and harder to find from far away.

The starts are one grid. Eight centres, every other number at a standard guess. A start with the lens already much older, or the macula already dense, would begin inside the very valleys the dropped fits end in.

Still open: whether a real older lens has a band to find #

Five essays here have now been spent on a band nobody has shown to be there. Lens absorption in older eyes is measured in vivo and ex vivo, and the measurements scatter about the standard age curve by more than the curve’s own change per decade; whether that scatter is a band, a steeper curve, or a scaled one is the question the fits here are built to answer.

The test is to fit the five numbers to real matches from observers over sixty, from several starts, and ask how many have a band whose fitted size is more than twice its standard error. The prediction worth recording is that a minority do — perhaps one in five — with centres below 430 nm, where the absorption of the lens’s own yellow chromophores lies, and that for the rest the band’s size is consistent with nothing. If most older observers carry a band, the standard age curve is the wrong shape for them and every fitted age in these essays has been partly a band; if none does, the band was a hypothesis the matches could have tested and did not need.

A local minimum is a statement about the start #

The habit is about what a failed fit is evidence of.

A fit that returns a band of no size is easy to read as a finding — no band — and the fits here that did so were one in eight to one in twenty, from starts nobody would call unreasonable. Every one of them was a worse fit than one the same data admitted with the band in it.

The failure mode is to report the answer a fit returned without asking whether it was the best answer the data allowed. The test costs a few more fits: start again from elsewhere and compare $\chi^2$. Here that turned every lost band back into a found one.