A density a tenth wrong costs the band more than fitting it

An older eye’s colour matches carry more than the standard observer has numbers for — the matches carry their own lens 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, a lens band’s amplitude and centre, and the macula’s peak. A second session keeps a band and the macula apart found two sessions of matches carrying all six, at a price paid by a band near the macula: at 445 nm the band’s amplitude and the macula’s shift trade, correlated at 0.82. A line on the macula’s flank buys less than a repeated session found no third session breaking that trade.

Knowing the macula’s density frees two sessions, not the line on its peak took the density out of the fit, as a measurement by heterochromatic flicker photometry would allow, and found two sessions’ trade falling from 0.83 to 0.62 — the density was the road the band and the peak traded along. It also found that a measurement only helps in proportion to how much better it is than the matches’ own estimate, 0.027 for two sessions, and that a realistic flicker measurement, good to 0.03 or 0.05, buys a few hundredths of correlation.

Its closing section raised a second problem with an outside measurement. Flicker photometry and a colour match need not see the same amount of pigment. The pigment is densest at the centre of the fovea and falls off over a couple of degrees, so a flicker stimulus of one size and a matching field of another average it differently, and the density fixed in the fit can be wrong by a predictable amount — commonly a tenth to a quarter. The prediction was that a density fixed a tenth too high takes about a third of a standard error off the band at 445 nm and moves the macula’s peak by under half a nanometre, because the density’s roads to the band and the shift run with the same sign; and that at 415 nm the bias goes wholly into the band.

A tenth’s error moves the band by more than its own error #

With the density fixed a tenth above its true 0.35, the five-number fit to two sessions lowers a band at 445 nm by 0.032 of its 0.1 amplitude — 1.4 of the fixed fit’s own standard errors — and moves the macula’s fitted peak 1.25 nm short; with the line on the macula’s peak, by 1.7 standard errors and 1.1 nm. A band at 415 nm moves the other way, up by 0.052, 2.4 standard errors, its fitted centre 5.6 nm shorter and the lens’s age 2.9 years younger. Bias and error together, a density fixed a tenth wrong gives the band at 445 nm under two sessions a larger error than the free six-number fit; the break-even is about eight per cent. A density supplied as a measurement with a standard error of 0.03, wrong by the same tenth, biases the band by under half its standard error.

• The prediction is too small by a factor of four. A tenth’s error costs the band more than a standard error, not a third of one.
• The far band is not spared; it is moved more. At 415 nm the bias goes into the band, as predicted, and into its centre and the lens’s age, which the prediction did not mention.
• Fixing is the mistake, not measuring. The same wrong number, entered with its error, is mostly overruled by the matches.

The two bars at each scale point the same way, and both are large. Fixed a fifth too high, the density lowers the band by 2.8 standard errors and moves the macula’s peak 2.5 nm short, 2.9 of its standard errors; a fifth too low, the reverse. The first-order reading is linear in the density’s error, so a tenth gives half of a fifth: 1.4 and 1.5 standard errors. Full fits from the truth, with the density held at its wrong value, move the band within a twentieth of what the first-order reading says and the peak within a tenth, so at 445 nm the straight line is the fit’s own; at 415 nm, as the next section shows, the full fit goes further than the line.

Why a wrong density moves both #

A wrong density is a wrong amount of light removed between about 420 and 500 nm, and the fit has to put that light back somewhere. With the density fixed too high, the matches look as though less light got through at those wavelengths than the fit’s density allows for; the fit compensates by taking absorption away from the other things that absorb there — the lens band, shortened, and the macula’s peak, moved short so that more of its absorption falls where it can be spared. That is the road the density was found to share with the band and the shift, run the other way — the same road on which a band in the lens reads as age or macula found an unmodelled band being read as macular pigment, and a macula three nanometres short is a lens band found a shifted macula being read as a band. The free fit traded the three among each other; the fixed fit has one of them pinned at the wrong place and spends the other two to make up for it.

The prediction reasoned from the same road and got its size wrong. It took the free fit’s correlations — about −0.7 between the density and each of the band and the shift — as the measure of how much a wrong density would move them. But a correlation measures how two uncertain numbers move together, not how far one moves when another is forced. A forced error of a tenth, 0.035 of density, is more than a full standard error of the matches’ own density estimate, and the fit absorbs a full standard error’s worth of density error by moving each partner by about a standard error of its own.

The far band moves the other way, and further #

At 415 nm the band rises by 0.052 — 2.4 standard errors, and 0.059 in a full fit — its centre moves 5.6 nm shorter, and the lens’s fitted age falls by 2.9 years; the macula’s peak barely moves, a fifth of a nanometre. The sign is the reverse of 445 nm’s. A band at 415 nm sits below the macula’s short flank, and the fit makes up for too much macular absorption there by rebuilding the whole short-wave absorption from three numbers at once: a younger lens absorbing less across the blue, and a taller band, sitting shorter, putting some of it back below 420 nm. The band gets taller and moves; the age gives ground.

At 430 nm the bias is smaller and splits, the band down by 0.7 standard errors and the macula’s peak 1.2 nm short. The three bands bracket the macula’s short flank, and a wrong density moves each one according to which side of that flank it sits on. No band is spared, and the one the prediction expected to take the bias quietly takes it in three numbers at once.

Fixing a wrong density is worse than fitting it #

What a reader of a fitted band carries is not its standard error alone but its bias as well, and the two combine as the root of their squares. With the density fixed exactly, the band at 445 nm has an error of 0.023 under two sessions, against 0.034 when the density is fitted. Fixed a tenth too high or low, its error is 0.039 — larger than the free fit’s. The break-even, where a fixed density’s bias and error together equal the free fit’s error, is about eight per cent.

So the earlier essay’s gain is fragile in a precise way. Knowing the density exactly breaks the trade; knowing it to within eight per cent still helps; anything worse, and a flicker measurement taken at the wrong field size makes the band less reliable than the matches alone would have. Real flicker and matching fields differ by enough that a tenth is a common error, which puts many real analyses on the wrong side of the line.

More matches make a wrong density worse #

The two-session fit is the census’s weakest design, and the natural hope is that a richer one — a third session, a line on the macula’s peak — leaves less room for a wrong density to do harm. It leaves less room for the standard error and the same room for the bias. Under every design tried, a density a tenth too high lowers the band at 445 nm by between 1.2 and 1.9 of its own standard errors, and every design does worse with the wrong fixed density than with the density free: by 1.17 times its free error under two sessions, 1.26 with the second session repeated, 1.40 with the line at 465 nm and 1.51 with the cyan line at 487.

The ordering is the point. A richer design measures the band better, so its free fit’s error is smaller; the bias a wrong density puts into the band shrinks less, because it comes from the trade between the band, the peak and the density, which every design shares to some degree. A smaller error with an undiminished bias is a larger bias in units of the error, and every third session, each of which the earlier essays found better for the band than two sessions, is hurt more than two sessions are. An analysis that has gone to the trouble of a third session is the one that can least afford to fix the density from a flicker measurement taken at another field.

The matches barely notice #

A wrong density that moves the band by more than its error might at least leave a trace in how well the fit fits. It leaves very little. With the density fixed a tenth too high, the five-number fit to two sessions’ 252 matches misses them by a chi-squared of 1.7 in units of the setting noise; fixed a fifth too high, by 6.7. A real experiment’s matches carry their own noise, which adds about 247 to the chi-squared, give or take 22, so a misfit of 1.7 is invisible and one of 6.7 is a third of the noise’s own scatter. The fit with the wrong density fits the matches almost exactly as well as the truth does.

That has a direct reading. The rate at which the misfit grows with the density’s error is the information the matches carry about the density, and a chi-squared of 1.7 at a tenth’s error means the matches alone pin the density’s scale to about eight per cent — the same eight per cent at which a fixed density stops paying for itself. That is not a coincidence: fixing the density beats fitting it exactly when the outside value is closer to the truth than the matches could have put it, and the matches put it within eight per cent. A flicker measurement taken at the wrong field is, in effect, a second estimate competing with one about as good as itself, and fixing it throws the better-calibrated of the two away.

Entered as a measurement, the wrong density does little harm #

The fix is the one the earlier essay already recommended for a different reason: enter the outside density as a measurement with its own error, not as a constant. A measurement good to 0.03 and a tenth wrong biases the band at 445 nm by 0.014, half its standard error, because the matches’ own estimate of the density, good to 0.027, carries as much weight as a measurement no better than itself and pulls the fit halfway back towards the truth; good to 0.05, the bias is under a quarter of a standard error. A measurement good to 0.01 and a tenth wrong biases it by 0.028, more than a standard error, because the fit now trusts it — the price of believing a precise number that is wrong.

That gives an analysis of older observers a rule. State the outside density’s error honestly, including what the field mismatch could add, and the fit will weigh it correctly: a measurement that is precise but measured on the wrong field should carry the mismatch in its error, so that the matches can overrule it.

How the fits were read #

The observers are the band essays’: rods feeding the S pathway at half weight, a lens of the model’s shape scaled by age, a lens band a Gaussian fifteen nanometres wide of a tenth of a density at 415, 430 or 445 nm, and a macula of density 0.35 centred at 460 nm moved by −3, 0 or +3. The sessions and their matches, step and setting noise are the earlier fits’. The five-number fit with the density fixed at a wrong value is read to first order from the six-number fit’s information matrix: the bias in the other five numbers is the fixed fit’s covariance times their cross-information with the density, times the density’s error. For two sessions, a full Gauss–Newton fit from the truth on noise-free matches, the density held a tenth too high, moves the band at 445 nm within a twentieth of the first-order reading and the shift within a tenth; at 415 nm it moves the band 13 per cent further than first order, so the far band’s biases quoted here are, if anything, understated.

A density entered as a measurement adds one over its variance to the density’s diagonal in the information; its bias then propagates through the full covariance. The error a reader carries is the root of the bias squared plus the standard error squared.

What this does not reach #

A model of the field mismatch. The density’s error is applied as a scale; how much a given flicker field and matching field actually differ depends on the pigment’s spatial profile in the eye, which the model does not carry.

Noise-free bias. The biases are those of the expected matches; with noisy matches each fit scatters about its biased value by its standard error, which is the combination the error figure uses.

The model’s macula. A single Gaussian; the real pigment’s shoulder near 480 nm would give a wrong density a second place to go.

One band’s centre. The band sits at 445 or 415 nm as the earlier essays placed it; a second lamp pair finds a lens band’s centre found the matches placing a band above 430 nm far better than one below, and a wrong density may move a free centre as well as the amplitude.

Still open: whether the bias survives in noisy fits #

Every number here is read from the expected matches. A real experiment’s matches carry noise, and a fit with the density fixed wrongly has one fewer number to spend absorbing that noise, so its spread may grow where its bias does not, or the bias may wander from one experiment to the next.

The calculation is the earlier essays’ twenty simulated experiments for each observer, refitted with the density fixed at its true value and at 0.9 and 1.1 times it, under two sessions and with the line on the macula’s peak, and the band and the macula’s peak read from each experiment. The prediction is that the experiments scatter about the first-order biases found here 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 the band at 415 nm, whose full fit already moved further than the first-order reading, moves further still in some experiments. If the experiments scatter more widely, a fixed density costs the band more than its bias, and the case against fixing it is stronger than this essay makes it. A wrong density shifts every experiment alike near the macula makes that calculation.

A constant is a measurement with no error #

The habit here is about the difference between fixing a number and knowing it.

The earlier essay asked what happens when the density is known, and the answer was striking. Knowing and fixing are not the same thing, and the gap between them is the measurement’s error, which fixing sets to nought. A number fixed at the wrong value does more damage than a number left free, because the fit has no way to disagree with it. A lens of the wrong shape lands in the macula found the same with a shape rather than a number: an assumption the fit could not question sent its error into the parameter beside it.

The failure mode is to enter an outside estimate as a constant because it was measured, rather than as a measurement because it has an error. The test is to perturb the fixed value by the size of its plausible error and see whether the fit’s other numbers move by more than their own uncertainty. Here a tenth moved them by more than a standard error, and entering the same tenth with its error moved them by a quarter.