A lens of the wrong shape lands in the macula

The matches carry their own lens took the experiment designed across the earlier essays — an asymmetric colour match, surfaces seen under daylight matched to the same surfaces under a phosphor LED, to measure how strongly the rods feed the blue–yellow pathway — and asked whether it needed a densitometer for every observer. Age can size the rod signal, if the lens is known had found that a lens ten years older than assumed moves the matches as far as four tenths of the signal being measured. Over a family of forty-two surfaces, the rod weight, the lens and the macula each move the matches in their own pattern, and a fit could take all three: from one session the lens came out to ±1.2 years, better than a densitometer had been asked to deliver.

That fit returns a lens age, and a lens age means something specific. In the collection’s model every lens is the same curve — an optical density falling exponentially with wavelength, with a decay length of 68 nanometres — scaled up as the lens yellows. A real lens is not only denser with age but differently shaped, and two people of the same age differ in both. The essay’s closing section asked where a shape error goes: give a synthetic observer a lens with a sixty-year-old’s density at 400 nanometres and its absorption edge moved by ten, fit it with the model, and see what the fitted weight does. The prediction was that the error would go mostly into the fitted lens age and the macula, both of which absorb short wavelengths, and that less than a tenth of the weight’s standard error would go astray.

Six times the tenth, and invisible #

From one daylight-against-LED session, a lens edge ten nanometres from the model’s moves the fitted rod weight by 0.59 to 0.63 of its standard error — six times the predicted tenth — the macula by 2.4 to 2.8 of its own, and the lens age by 0.4 to 0.7 of its own. The misfit it leaves is a third of a standard deviation of the session’s own noise. A second session, the LED against a three-emitter lamp, halves the weight’s standard error and moves the weight by 3.5 of the smaller ones. Three of the four lamp pairs using the three-emitter lamp move it by 2.8 to 5.3; the other six by under 0.65. Fitting the edge’s position as a fourth parameter moves the weight by under a tenth of its standard error, costs it 3 to 14 per cent of its precision, and recovers the edge to within 0.4 of a nanometre.

• The prediction half holds. The error goes mainly into the macula — not the lens — and a real share of it reaches the weight.
• A session cannot tell. What the fit leaves unexplained is lost in the noise of forty-two settings.
• Sharper is not safer. The two-session design that fits everything best is the most biased by a shape error, in its own units.
• The repair is a better lens model, not a densitometer. The matches carry the lens’s shape as well as its density.

What a moved edge is #

An exponential cannot be moved along the wavelength axis without becoming the same exponential. The model’s lens at sixty has a density of 1.3 at 380 nanometres falling by a factor of e every 68; slide it ten nanometres longer and it is the same curve times e to the 10/68, which with its density at 400 held fixed is exactly the curve it started as. So the proposal’s “edge moved by ten nanometres” has to mean something else, and the natural reading is where the lens passes half the light — a density of 0.30. The model’s sixty-year-old passes half at 479.5 nanometres. Holding the density at 400 and moving that point ten nanometres longer needs a decay length of 76.6; ten shorter, 59.4. The shaped lenses are shallower and steeper versions of the same curve, crossing it at 400.

That is a physically ordinary difference. A lens’s absorption comes from several chromophores — ultraviolet filters derived from tryptophan, and the yellow pigments that accumulate with age — whose proportions vary between people, and the model’s single exponential is an average over them. The observer has no age found the lens the one part of the eye that changes in one direction with age; its shape is the part that changes with the person.

Each shaped lens is put in front of the cones and the rods alike, in an observer built exactly as the earlier fit builds its own — rod weight one half, the standard macula — and with its edge where the model puts it, the shaped observer is the model’s sixty-year-old to a part in a trillion.

Where one session puts the error #

The fitter assumes the model’s lens shape and fits the three numbers it was built to fit — the rod weight, the lens age and the macular density — to the forty-two matches. The shape error’s effect on each is the least-squares projection of the difference in the matches onto the fit’s sensitivities, which for an error this small is the whole of the bias.

The macula takes most of it. A lens edge ten nanometres longer is read as 0.038 more optical density of macular pigment, 2.4 of its standard errors; ten shorter, 0.045 less, 2.8 of them. The macular is a band, not a filter is why: macular pigment absorbs within forty nanometres of 460, and a shallower lens edge adds density in exactly that band while taking some away below 400, which is a band-shaped change in the light reaching the cones — the macula’s shape, not the lens’s. The lens age moves too, by 0.8 years one way and 0.5 the other, under one of its standard errors each.

And the weight moves by 0.035 to 0.037, 0.59 and 0.63 of its standard error of ±0.059. The prediction said under a tenth. A shape error is not something a fit’s free parameters can simply absorb between themselves; whatever part of the difference in the matches lies outside the space the lens and macula span, the weight is the next parameter asked to explain.

Whether the session can tell #

It cannot. A lens edge ten nanometres off changes the forty-two matches by 0.13 to 0.14 CIELAB units on each axis, root mean square; the fit takes up four fifths of that in variance, and what it leaves, 0.06 on each axis, is well below the 0.29 on each axis that the experiment’s 0.5 ΔE a setting amounts to. Summed over the session the leftover is a chi-square of 5 to 6 on 123 degrees of freedom, against a spread in that statistic of 16 from the noise alone: a third of a standard deviation. An experimenter looking at the fit’s residuals would see nothing wrong.

That is the unwelcome half of the finding. The reference lamp must not move and its sequels designed the experiment around what it could measure precisely, and a precisely measured wrong answer is indistinguishable, from inside the experiment, from a precisely measured right one. The earlier essay’s ±0.059 on the weight is the spread of the fit’s answers over repeated sessions with this observer. It is not the distance from the right answer if the observer’s lens is not the model’s shape.

Which lamps a wrong lens misleads #

Every pair built on the three-emitter lamp is exposed; every other pair is not. Daylight against the three-emitter lamp moves the weight by 5.3 of its standard errors, tungsten against it by 3.3, the tube against it by 2.9, the LED against it by 1.7. The six pairs without it move the weight by 0.1 to 0.6.

The reason is how the lamps sample the lens. The three-emitter lamp’s light comes in three narrow bands, one of them in the blue, and a match under it depends on the lens’s density at a few wavelengths rather than averaged across the edge. A lens of the wrong shape through those few wavelengths looks like a lens of the right shape and a different age, or a different macula, or a different weight — and which it is taken for depends on where the lines fall. Broad lamps average the edge, and the average of a steeper and a shallower curve crossing at 400 is close to the model’s.

This matters because the earlier essay’s best design used the three-emitter lamp: the matches carry their own lens found that the LED against it, added to daylight against the LED, nearly freed the weight from the price of fitting the macula.

The sharper design, the larger error #

Adding the second session makes the weight twice as precise and the shape error seven times more consequential. With both sessions the weight’s standard error is ±0.028; the edge error moves it by 0.099 to 0.110, which is 3.5 to 3.9 of those. The macula moves by 8 to 10 of its now much smaller standard errors, the lens by 2.7 to 3.8.

The misfit rises too, to 1.8 to 2.2 standard deviations of the noise — a chi-square of 40 to 48 on 249 degrees of freedom, against a noise spread of 22. That is a warning an experimenter might notice across a group of observers and would not trust from one. The design’s precision is real; it is precision about the model’s lens, and it is spent on a narrow-line lamp that reads the model’s shape assumption at its most specific.

The design is not wrong to be precise. It is wrong to be precise about a quantity it cannot see, and a design that fits more parameters from fewer, sharper measurements is more exposed to every assumption behind the parameters it does not fit.

As the edge moves #

The weight’s error is a straight line in the edge’s. At five nanometres it is 0.30 standard errors in one session and 1.8 to 1.9 in two; at ten, 0.59 to 0.63 and 3.5 to 3.9. The line runs through nought at the model’s shape and changes sign with the edge’s direction, so a population whose lens edges scatter about the model’s by a few nanometres would scatter its fitted weights accordingly, with no bias on average and a spread added to the fit’s own.

Fitted with the edge as well, the line lies on the axis. The dashed lines in the figure are the same fits with a fourth parameter: the lens edge’s position, at the density at 400 already given by the lens age. Across the whole range the weight moves by under a tenth of its standard error.

Fitting the edge #

The matches carry the lens’s shape as well as its density. With the edge free, one session fits the weight to ±0.061 instead of ±0.059, a cost of 3 per cent, and two sessions to ±0.032 instead of ±0.028, 14 per cent. The ten-nanometre edge comes back as 9.65 ± 4.2 nanometres from one session and 9.59 ± 1.5 from two, and the weight is moved by 0.005 and 0.016 of its standard errors.

So the proposal’s conclusion was right, and its reasoning wrong. It predicted that the shape error would go harmlessly into the lens and macula; it does not. But the densitometer it worried about is still not needed, because the fix is inside the experiment: a lens model with one more number, fitted from the same settings. The collection’s claim that one match names the observer holds for a richer observer, and the precision lost to naming more of it is small.

How the lenses were fitted #

The model’s lens has optical density 0.5 + 0.02 × (age − 20) at 380 nanometres, falling as exp(−(λ − 380)/68); at sixty it is 0.968 at 400 and 0.301 at 479.5. A shaped lens keeps 0.968 at 400 and falls as exp(−(λ − 400)/τ), with τ chosen so that density 0.301 falls at 479.5 plus the edge’s shift. Cones and rods are built as in the earlier fit: the site’s pigment templates at their median peaks and density, through the lens and a macula of standard density; the rod signal, rhodopsin through the lens, added to each cone at 0.1 of its peak, the S cone’s at weight one half. Each match is the CIELAB difference of a surface under the two lamps, adapted through CAT16, over the forty-two surfaces of the collection’s family. Sensitivities are finite differences at the true observer’s lens age of sixty; each setting carries 0.5 ΔE split across three axes. The bias is the Gauss–Newton step from the model observer of age sixty towards the shaped one, and the standard errors are the Fisher information’s. The edge parameter’s sensitivity is the finite difference of the shaped observer’s matches in the edge’s position.

What this leaves out #

The lens’s shape is one number here. A real lens’s absorption is a sum of chromophores whose proportions vary independently, and the difference between two lenses of the same density at 400 nanometres need not be a change of slope; it could be a shoulder, or a second edge further into the blue. One extra parameter absorbs a change of slope; whether it absorbs a shoulder is untested.

The macula is fitted as the model’s shape too, a band of fixed width at 460 nanometres. Real macular pigment’s spectrum is well characterised and varies little in shape between people, which is why it can be modelled that way; the argument here would apply to it with a smaller effect.

The observer is one, a sixty-year-old with rod weight one half. The weight’s standard error and the biases scale with the observer; their ratios should move little across the range the earlier essays used.

Still open: whether a shoulder in the lens is fitted as an edge #

The fitted edge takes up a change of slope because a change of slope is what it was built from. A real older lens often has something else: extra absorption between 400 and 450 nanometres from the yellow pigments that accumulate with age, which is a shoulder on the curve rather than a steeper one.

The calculation is this fit on observers whose lens carries a shoulder — a Gaussian band of absorption near 420 nanometres added to the model’s exponential, sized to add a tenth of an optical density there — fitted with and without the edge parameter. The prediction is that the edge parameter takes up most of a shoulder near 420, because both change the lens’s density most in the band just past 400 and the lamps sample that band only coarsely, but that a shoulder near 450 is fitted partly as macula whatever the lens model, because there the two bands overlap. If it does, a matching experiment on older observers needs its lens model to carry the yellow pigments’ band explicitly, and the three-emitter lamp — whose blue line sits in that region — is where it would show first.

A standard error is a promise about repetition, not about truth #

The habit is about what a precise fit’s standard error describes.

The earlier fit’s standard errors were computed correctly, and they say how far the fitted weight would scatter if the same observer did the session again and again. They say nothing about whether the observer is the one the fit assumes. A lens of a slightly different shape moves every repetition’s answer to the same wrong place, and the scatter stays exactly as small.

The failure mode is to read a small standard error as a small error, when the model behind it was fixed rather than fitted. Every assumption a fit does not free is an error it cannot report, and the sharper the design, the more of its precision rests on them.