A wrong density shifts every experiment alike near the macula

An older eye’s colour matches carry more than the standard observer has numbers for — the matches carry their own lens first found one session fitting rod signal, lens and macula together — and the fits that read them have grown to six: rod weight, lens age, macular density, and a lens band’s amplitude and centre beside the macula’s peak — the set a second session keeps a band and the macula apart found two sessions of matches carrying, at the price of a trade between a band near the macula and the macula’s position. Knowing the macula’s density frees two sessions, not the line on its peak found that taking the density out of the fit, as a flicker measurement would allow, breaks much of that trade.

A density a tenth wrong costs the band more than fitting it then asked what happens when the outside measurement is wrong — as it can be, since flicker photometry and a colour match average the pigment over different fields. Read from the expected matches, a density fixed a tenth too high lowers a band at 445 nm by 1.4 standard errors and raises a band at 415 nm by 2.4, and bias and error together leave the band worse off than if the density had been fitted.

Every number in that essay came from the expected matches — the matches the observer would make with no noise at all, read through the Fisher information and checked by full fits on noise-free data. A real experiment’s matches carry setting noise, and the fit has to absorb it with one number fewer when the density is held. Its closing section predicted that the experiments scatter about the first-order biases with the fixed fit’s standard error, so that a tenth’s error moves the band beyond twice its error in about a third of experiments at 445 nm and two thirds at 415, and that at 415 nm, where the full fit had already moved further than the first-order reading, some experiments move further still. If the experiments scattered more widely than that, the case against fixing the density would be stronger than the expected matches make it.

Rigid near the macula, lopsided far from it #

Twenty simulated experiments per case — the joint census’s own noise, under two sessions and with a third session’s line at 465 nm on the macula’s peak, the band at 445 or 415 nm and the macula unmoved — each fitted with the density free and held at 0.9, 1 and 1.1 of its truth. At 445 nm the density a tenth high moves every experiment’s band by −0.031 under two sessions, within a fortieth of the first-order bias, and the twenty experiments’ moves differ from one another by a standard deviation of 0.0024 against the band’s standard error of 0.023. A tenth high puts the band beyond twice its standard error in 5 of 20 experiments under two sessions and 8 with the line; at 415 nm in 16 and 18. At 415 nm the density a tenth high raises the band by 0.058 and a tenth low lowers it by 0.032, and the low side’s moves scatter three times as widely; in one experiment of twenty, under both designs, the density a tenth low fits no band at all. In every case a density a tenth wrong in either direction gives the band a root-mean-square error at least two fifths larger than fitting the density.

• The prediction’s counts bracket the result. A quarter at 445 nm where it said a third, four fifths at 415 where it said two thirds.
• Its mechanism holds near the macula exactly. The experiments do not scatter about the bias; the bias is added to each of them, rigidly.
• Far from the macula it does not. The bias depends on which way the density is wrong and, when it is too low, on the experiment.

The three traces are the same experiments fitted three ways, and they run parallel. Experiment nine fits a band of 0.16 at the true density, 0.20 with the density a tenth low and 0.13 with it a tenth high; experiment ten fits 0.08, 0.11 and 0.05. Whatever the noise did to an experiment, a wrong density adds the same displacement on top, about three hundredths of amplitude either way. A reader who could see all twenty experiments would see the band’s scatter unchanged and its centre moved — the error a systematic one, the kind no amount of repetition averages away.

That rigidity is what the first-order reading assumes and could not check. A bias read from the Fisher information is the derivative of the fit with respect to the density, taken at the truth; it holds in a noisy experiment only if the derivative is the same wherever the noise has put the fit. Near the macula it is. The fitted band at 445 nm sits in a part of parameter space where the matches respond nearly linearly to the band, the density and the macula’s shift together, and moving the density moves the solution along a straight road.

The bias, experiment by experiment #

Pairing the fits of each experiment removes the noise every fit shares and leaves only what the density did. At 445 nm the eight numbers on the right of each bar are a tenth of a standard error or less — 0.10 and 0.10 under two sessions, 0.15 and 0.07 with the line. The density’s error enters each experiment the same way, and the means, −1.35 and +1.33 standard errors under two sessions, are the first-order reading’s to within a twentieth.

At 415 nm the picture divides. A density a tenth high moves the band up by 2.59 standard errors under two sessions, a little further than the first-order 2.35 — the full fit’s overshoot the earlier essay found — and its spread stays modest at 0.27. A density a tenth low moves it down by only 1.46, and its spread is 0.82: one experiment’s band moves by almost nothing, another’s by more than the mean high-side shift. With the line on the macula’s peak the low side’s mean is −2.32 and its spread 0.73. A band far from the macula, read with a density that is too low, has no single bias to quote.

Why the far band is lopsided #

A density too low leaves part of the macula’s absorption unexplained, and at 415 nm the fit’s nearest lever is the lens band, which sits on the macula’s short flank and can absorb some of the same light. The band at 415 nm therefore shrinks — but it cannot shrink below nothing, and a band of 0.1 asked to give up 0.04 of its amplitude in an experiment whose noise had already pulled it down to 0.04 has nowhere to go. Experiment fourteen is that experiment. Its band fits at 0.03 with the density at its truth under two sessions, 0.04 with the line, and with the density a tenth low the fit takes the band to zero and moves the remaining misfit into the lens’s age, which comes out a year and a half to two and a half years older.

That destination is not new. A lost lens band goes to age or the macula found a band the fit could not hold being absorbed into the lens’s age, and a band in the lens reads as age or as macula found the same two numbers standing ready to take a band’s light when the fit’s model had no band in it. Here the model has a band and the fit still gives it up, because a wrong density has used the band’s road: the macula’s absorption, overstated or understated, reaches the short wavelengths through the same lever the band does, and when the density is too low the fit has nowhere left to put the band’s absorption but into the lens’s general yellowing. An analyst reading that experiment would see a lens two years older than its owner and no band — a normal ageing lens, by every number the fit reports.

A density too high is the opposite case: the extra macular absorption must be taken back, and the band at 415 nm grows to give the short wavelengths their light. Growing has no floor, and every experiment does it by nearly the same amount. The asymmetry is the boundary, and it bites only when the band is small and the density errs in the direction that asks the band to shrink. That is a statement the expected matches could not make, because the expected matches never put the band near its floor.

How often the band is out of bounds #

Counting how many experiments land beyond twice the band’s standard error from its truth is the reader’s view of the bias: a band beyond that line would be reported as different from a normal lens’s at the usual confidence. With the density a tenth high, five of twenty at 445 nm and sixteen of twenty at 415 nm cross it under two sessions; with the line, eight and eighteen. The line makes the count worse, not better, because it shrinks the band’s standard error — from 0.023 to 0.016 at 445 nm — without shrinking the bias, so the same displacement is more standard errors.

That is the uncomfortable half of the earlier essay’s case for designs that measure more. A better design tightens the error bar around the wrong answer when the density it was given is wrong, and the fraction of experiments that report a spurious band rises with the design’s precision. A line on the macula’s flank buys less than a repeated session ranked third sessions by what they bought a free fit; with a wrong constant in the fit, the ranking of what they cost runs the same way.

The fitted density and the true density stay near the expected one in twenty, as they should — two, one, one and four experiments — except where the line meets the far band, where the free fit’s eight of twenty reflect the free fit’s own larger error measured against the fixed fit’s smaller bar. The wrong densities fill the rest of the chart: 3, 8, 4 and 12 experiments a tenth low; 5, 16, 8 and 18 a tenth high.

The cost in error #

The root-mean-square error is the number a reader of fitted bands actually carries, bias and scatter together, and it ranks the four fits identically in every case. The density held at its truth beats the free fit everywhere, by a tenth at 445 nm and by up to forty per cent at 415 nm with the line; and the density held a tenth wrong in either direction loses to the free fit everywhere, by between half again and a factor of two. At 415 nm a density a tenth high gives 0.062 against the free fit’s 0.031 under two sessions — the band’s error doubled, nearly all of it bias.

So the earlier essay’s break-even — a fixed density pays only if it is within about eight per cent of the truth — survives the noise, and its conclusion that a flicker measurement should be entered as a measurement with its error rather than as a constant is, if anything, stronger. A constant that is wrong is wrong in every experiment, and at 415 nm wrong by more when it is too high and erratically when it is too low.

The matches barely notice #

One defence remains: a wrong density might show itself in the fit’s quality. It does, faintly. The mean worsening in $\chi^2$ with the density a tenth high is 1.9 under two sessions at 445 nm, 4.6 at 415, 3.3 and 6.6 with the line — against $\chi^2$ values near 245 and 375, the matches’ own count. Some experiments in every case fit better with the wrong density than with the right one, four of twenty at 445 nm under two sessions. A single experiment’s $\chi^2$ cannot tell an analyst the density was wrong; twenty experiments’ could, if they were all fitted at two densities and compared, which is to say if the density were fitted.

What a lens-band study should take from this #

Near the macula, a fixed density’s error is a calibration offset. Every experiment carries the same displacement, which can be quoted, corrected for if the density’s error is known, and cannot be averaged out by repetition. Far from the macula it is a hazard. A density too high inflates every band; a density too low shrinks most of them and erases the occasional one outright. A study that reports a band’s absence at 415 nm from a fit with a fixed density has reported, in part, the density.

More precise designs make a wrong constant worse. The line on the macula’s peak improves every free and every correctly fixed fit and raises the count of spurious bands. Whatever precision a design buys is precision about the fit’s own premises.

How the experiments were fitted #

Each case’s twenty experiments are the joint census’s: the observer at sixty with the standard macular density, 0.35, a lens band of amplitude 0.1 and width 15 nm at 445 or 415 nm, the macula unmoved, and matches with the census’s setting noise drawn from its own seeds, so that under two sessions every experiment is one the census already fitted. The design with the line adds a third session setting the phosphor LED against a lamp whose second emitter sits at 465 nm. Each experiment is fitted four times by damped Gauss–Newton descent from the truth: all six numbers free, and with the density held at 0.315, 0.35 and 0.385 and the other five free. The free fits reproduce the joint census’s bands to a thousandth in 38 of the 40 experiments under two sessions; in the other two, the fit from the truth reaches a lower $\chi^2$ than the census’s descent from its two starting points did. The standard errors quoted are the fixed fit’s at the truth, read from the Fisher information.

What this does not reach #

One observer per band. The macula is unmoved and the band’s amplitude one value. A macula three nanometres short is a lens band found a moved macula reading as a band under one session; with the density also held wrong, a moved macula adds a second road for the error, and that is not simulated. A band of half the amplitude would sit nearer its floor in more experiments, and the lopsidedness should grow.

Twenty experiments. Counts out of twenty carry their own uncertainty — five of twenty could be three or eight in another twenty — and the claims here are about which side of the prediction they fall and by how much, not their exact values.

A density wrong by a constant. The density’s error is the same in every experiment. A flicker measurement taken afresh for each experiment would be wrong by a different amount each time, and the rigid shift near the macula would become a scatter.

Still open: whether the joint census’s starts miss other valleys #

Two of the forty experiments fitted here found a better fit from the truth than the joint census found from its two starting points — at 445 nm a band the census fitted at nothing, at 415 nm one it fitted at 0.04 instead of 0.06, the census’s $\chi^2$ worse by 7.6 and 2.7. Every essay since the second session has read its fitted bands and shifts from that census.

The calculation is every experiment in the joint census — nine observers, three designs, twenty experiments each — refitted from the truth and from a grid of starting centres, and the census’s own fit compared with the best found. The prediction is that the census’s starts miss the best valley in about one experiment in twenty, almost all of them experiments where the band is small or the macula moved towards it, and that the misses change no correlation or standard error the earlier essays quoted by more than a few hundredths, because a miss is one experiment in twenty. If the misses cluster in one design or one band, the census’s starts were biased for that case, and the essays that read it there owe a correction. One session has a second valley makes that calculation.

A bias read at the truth #

The habit here is about where a derivative is taken.

The earlier essay’s bias was a derivative at the truth, and a derivative at the truth is what a fit does to the expected matches. A real fit starts wherever the noise has put it, and the derivative there need not be the same — near a boundary, or where the matches respond nonlinearly, it is not.

The failure mode is to quote a first-order effect as the effect. It is the effect on an experiment that sits at the truth, and no experiment does. The test is to simulate the experiments and apply the change to each — here, three densities on the same twenty sets of matches — and to read the effect as a distribution. Near the macula the distribution is a spike and the first-order reading is the whole story; at 415 nm the distribution has a tail, and the tail is the experiment that loses its band.