One session has a second valley

An older observer’s colour matches are read by fitting six numbers — rod weight, lens age, macular density, a lens band’s amplitude and centre, and the macula’s peak — to twenty or more pairs of matched colours. A second session keeps a band and the macula apart built the census every later essay on the subject has read: nine simulated observers, each with a band at 415, 430 or 445 nm and a macula moved three nanometres short, unmoved, or three long; one, two and three sessions of matches; twenty experiments each with the matches’ own setting noise. Each experiment was fitted by a damped Gauss–Newton descent from two starts, the band centred at 410 and at 450 nm, and the better of the two kept.

A wrong density shifts every experiment alike near the macula refitted forty of those experiments from the truth for another purpose, and two of them came out better than the census had found — one with a band the census had fitted at nothing, at a $\chi^2$ lower by 7.6. A fit that stops in a worse valley reports a worse answer with full confidence, and nothing in a single fit’s output says so.

Its closing section predicted that the census’s starts miss the best valley in about one experiment in twenty, almost all where the band is small or the macula moved towards it, and that the misses change no correlation or count the earlier essays quoted by more than a few hundredths, because a miss is one experiment in twenty. If the misses clustered in one design or one band, the essays that read the census there would owe a correction.

One fit in eight, and most of them under one session #

Refitted from the truth and from the census’s start with the band at 430 and at 470 nm, 66 of the census’s 540 experiments have a fit with a lower $\chi^2$ than the census’s — 45 of 180 under one session, 9 under two and 12 under three. In 60 of the 66 the better fit puts the macula’s peak more than two nanometres further short than the census’s did, usually with a smaller band; in 31 it lies lower by under one unit of $\chi^2$, and in a few by more than five, the largest 14.5. With the best fits, two sessions find the band in the same number of experiments as the census for eight observers of nine and one more for the ninth, and a moved macula within two either way. One session’s counts change: an unmoved macula is reported moved in eight experiments of twenty for the band at 415 nm and six at 430, where the census reported one and two, and the band is found in up to fifteen experiments of twenty where the census found at most twelve.

• The prediction fails on the rate and on where. One in eight, not one in twenty; and the misses cluster in one design, which is the prediction’s own condition for a correction.
• Its consequence holds for two and three sessions. The counts the later essays leaned on stand.
• The earlier verdict on one session was too kind. One session is worse than the census said, not better.

The seven missed experiments make the pattern plain. Each census fit sits near the truth — the macula within a few nanometres of where the model has it, the band between 0.09 and 0.22 — and each better fit has jumped: the macula seven to nineteen nanometres short, the band smaller or about the same. Both are fits to the same matches, and in every one of the seven the matches prefer the jump by under two units of $\chi^2$. The census’s two starts both put the macula where the model has it, and a descent from there never went looking ten nanometres away.

Where the misses fall #

Under one session every observer has misses, between one and eight of twenty, and the observers with a band at 415 or 430 nm have the most. Under two sessions three observers have none and the rest one to three; under three sessions, four have none and two have four. The second session does not just shrink the error bars, it removes most of the second valley, which is what the earlier essay’s account of the second session implies: a second lamp pair whose lines fall elsewhere on the macula’s flanks gives the fit a reason to prefer the macula’s true position over a far-moved one.

The bands nearest the macula, at 445 nm, are missed least under every design. That is the opposite of where the prediction put them, and the next figure shows why: the second valley is a macula moved far short, and a band at 445 nm already sits on the macula’s short flank, where moving the macula further away from it changes the matches most and buys the fit least.

The second valley is the trade, far out #

The better fits gather in one quadrant: the macula further short, by five to twenty nanometres, and the band smaller, by up to a tenth of a density. This is the band–macula trade the census was built to measure, the one a macula three nanometres short is a lens band found under one session — a macula moved short puts absorption on its short flank as a band does, so the two can stand in for each other. Near the truth the trade appears as a correlation in the error bars. Far out it appears as a second minimum: a macula moved so far short that its long flank no longer covers the band’s wavelengths, the band shrunk to suit, and a fit about as good as the one near the truth.

The handful of misses in the other quadrants — a macula moved long, a band grown — are the exceptions, and none of them under two sessions moves the macula by more than a few nanometres.

Two valleys nearly level #

How much better the better fit is says what kind of miss it was. In 31 of the 66 it is lower by under one unit of $\chi^2$, two fits the matches can barely choose between; a few are lower by four to fifteen units, misses that would matter to anyone comparing the fits. Under one session the improvements run smoothly from a hundredth to fourteen and a half, and the few misses under two and three sessions spread over the same range: the second session makes second valleys rarer without making the ones that remain any less level.

A near-level second valley is a fact about the experiment, not a failure of the descent. It says that with one session’s matches, an observer with a band and an unmoved macula is almost as well described by a smaller band and a macula moved fifteen nanometres short, and the fit’s answer depends on where it started looking. The census’s standard errors, read from the Fisher information at the truth, say nothing about that: they describe the curvature of one valley.

What the counts become #

The earlier essay’s verdict on one session was that the band is found in at most twelve experiments of twenty and a moved macula in at most seven. With the best fits, the band is found in up to fifteen and a moved macula in up to twelve — but for the wrong reason. The extra shifts are the second valley’s far-moved maculas, and they appear for observers whose macula is not moved at all: eight of twenty at 415 nm and six at 430, against one and two in the census. One session does not merely fail to tell a band from a moved macula; with a thorough fit, it reports a moved macula a third of the time when there is none.

Under two sessions the bars pair up. The band counts are identical for eight observers and one higher for the ninth, at 445 nm with an unmoved macula, where the census’s one experiment with a band fitted at nothing turns out to have a band; the shift counts differ by at most two. Every count the later essays drew from two sessions stands, and so does every correlation and standard error, because those were read from the information at the truth rather than from the fits.

Why two starts missed it #

The census’s two starts differed in one number: the band’s centre, 410 or 450 nm. Both started the macula where the model puts it, with no shift. That choice made sense for the question the census was built to ask — where is the band? — and a second lamp pair finds a lens band’s centre had freed the band’s centre as a number of its own, and a fitted centre is the number most obviously able to settle on the wrong side of a line; two centres were tried for that reason. Nothing suggested that the macula’s peak had a far valley of its own, and a descent that starts at no shift moves the shift only as far as the local slope carries it.

A damped Gauss–Newton descent is a local method: it walks downhill from where it starts and stops in the first valley it reaches. Two starts make it less likely to stop in a shallow pit near one of them; they make it no more likely to find a valley in a direction neither start explored. Of the three extra starts here, the truth explores nothing new — it sits in the census’s own valley — and the two new band centres differ from the old ones by twenty nanometres. What found the second valleys was not a start placed near them but more descents, each taking a slightly different path; the census does not map the landscape between the valleys, and how gently its floor tilts towards a far-moved macula is not computed.

That suggests the audit’s count is a floor. A start with the macula placed ten nanometres short would walk into the second valley directly, and might find it in experiments where none of the five descents strayed that far. The figure that matters for a reader is not the exact count but its ratio across designs: a fifth or more under one session, a twentieth under two or three.

What this means for a real study #

A study of older observers’ matches does not have the truth to start from, and has one set of matches per observer rather than twenty. The audit’s lesson for it is concrete. Under one session, a fitted macula moved ten or more nanometres short is the second valley, not a finding: no real macula sits that far from the model’s, and the matches cannot tell it from a smaller band with the macula in place. Under two sessions the second valley is rare, and a fit that lands in it is the one experiment in twenty a reader should expect.

The habits the earlier essays recommended survive and gain a reason. The matches carry their own lens recommended fitting the lens rather than assuming it, and a band in the lens reads as age or as macula warned that a band the model lacks goes somewhere. Both assumed the fit finds the best description of the matches; the audit shows that under one session there are two good descriptions and a fit reports whichever it reached first. A dimmer session hides a lens band in the rods found a choice of session hiding a band; the choice of start can hide or invent a moved macula in the same way.

What the earlier essays owe #

The correction falls on one session, and on one essay’s description of it. A second session keeps a band and the macula apart reported one session finding the band in seven to twelve experiments and a moved macula in at most seven; with the best fits, one session finds both more often and finds the macula moved where it is not. Its conclusion — that one session cannot separate the two and two sessions can — is stronger for the audit, not weaker: the one-session fits it described were the better-behaved half of what one session’s matches allow.

A lost lens band goes to age or the macula and the essays on single sessions before the six-number census did not read this census and are not touched. The essays after it read two and three sessions, whose counts stand.

How the census was refitted #

Each of the 540 experiments is the census’s own: the same simulated observer, the same pairs of lights and surfaces, the same setting noise from the census’s seeds. Each was fitted three more times with the census’s own damped Gauss–Newton descent and its bounds — from the truth, and from the census’s start (rod weight and density at their standard values, age 32, half the band’s amplitude, no shift) with the band centred at 430 and at 470 nm instead of 410 and 450. The best of the census’s fit and these three, by $\chi^2$, is the best fit; the census’s counts as missed when another is lower by more than a hundredth of a unit. A band or a shift counts as found when it exceeds twice its standard error from the fit’s own covariance at its solution.

What this does not reach #

Five starts are not every start. The best fit here is the best of five; a sixth start might find a third valley. The audit’s counts of misses are therefore lower bounds, and the corrected one-session counts may move further.

The second valley’s macula is far from any real one. A macula moved fifteen nanometres from the model’s peak is outside what carotenoid mixes produce. A fit with bounds on the shift — a few nanometres either way — would rule the second valley out and might restore one session’s counts to something like the census’s; but the bound would then be doing the work the matches cannot.

$\chi^2$ alone decides. Two valleys within a unit of $\chi^2$ are, statistically, a tie; reporting either alone overstates what the matches know.

Still open: whether a bound on the macula’s shift rescues one session #

The second valley sits ten to twenty nanometres from the model’s macula, where no real macula is found. A fit constrained to shifts within, say, six nanometres either way — the range real lutein-to-zeaxanthin mixtures span — cannot reach it.

The calculation is the census’s one-session experiments refitted from all five starts with the macula’s shift bounded at ±6 nm, the counts of found bands and shifts read again, and the bound’s own effect read as how often a fit ends pressed against it. The prediction is that the bound restores the census’s one-session counts to within two of each observer’s, and that a bounded fit ends on its bound in about the experiments that had a second valley — one in four — so that the bound replaces a wrong answer with a visibly constrained one. If the counts stay inflated, the matches prefer a far-moved macula even within the bound, and the second valley has a near neighbour that a bound does not exclude. A bound on the macula trades false moves for hidden ones makes that calculation.

A fit reports the valley it found #

The habit here is about what a single fit cannot tell its reader.

The census’s fits were careful by any ordinary standard: a damped descent, two starts, the better kept. Each returned a solution, a $\chi^2$ and a covariance, and nothing in those three said that another solution as good or better lay ten nanometres away. A covariance describes the valley the fit is in.

The failure mode is to read a fitted model’s precision as its uniqueness. The census’s standard errors were honest about the valley’s width and silent about the other valley. The test is to start the fit from places it would not have gone — here, three more starts on every experiment — and count how often the answer changes; where it changes often, the experiment’s matches do not determine the answer, however small the error bars.