A dimmer session hides a lens band in the rods

A band in the lens reads as age or as macula gave a sixty-year-old observer’s lens a band of extra absorption — a tenth of an optical density, fifteen nanometres wide, the kind of shoulder the yellow pigments of an ageing lens add — and fitted the observer from colour matches between pairs of lamps, with the rod weight, the lens age and the macular pigment free. Centred at 410 to 430 nm the band was read as three to six years of lens age; centred at 440 to 460, as macular pigment. Either way the fitted rod weight — how much of the rods’ signal reaches the blue–yellow pathway, the quantity those matches exist to measure — moved by up to one and a half of its standard errors.

The essay ended on the one signal that could tell a lens band from the macula. The rods sit outside the fovea, behind the lens and not behind the macular pigment, so a band in the lens changes what they catch and the macula does not. In those matches the rod signal was a tenth of each cone’s peak, a small addition. A match made at a lower light level, where the rods carry more of the signal, would weight the lens more heavily against the macula. The prediction was that with the rod signal at a third — the mesopic end of the range — the macula and a band at 450 nm separate, and that the rod weight’s shift from the band falls below half a standard error; if so, an experiment on older observers should include a dimmer session, to measure the lens through the rods.

More rods hide the band, and a fourth number finds it #

The prediction fails in the direction that matters. With the rods at a third instead of a tenth and the band left out of the fit, one session’s rod-weight shift grows from 1.73 of its standard errors to 2.53, and the macula’s from 4.6 to 5.7; the misfit the session is left with, the only warning it gets, falls from 0.94 to 0.65 of its noise. The rods do see the band, and the rod weight is the parameter the fit uses to explain what the rods see, so more rods give the band more of the rod weight to hide in. What separates the band from the macula is fitting the band’s amplitude at 450 nm as a fourth number: it recovers the tenth of a density to within a hundredth at every rod share, from a tenth to a half, and keeps one session’s rod weight within half a standard error. More rods make even that slightly worse.

• A dimmer session is not the instrument for the lens. It moves the band’s effect into the rod weight, where it biases the thing being measured.
• The band needs a parameter of its own, and with one, a daylight session finds it.
• More rods tie the macula to the band more tightly, not less: their correlation in one session strengthens from 0.77 to 0.87.
• A band parameter at 450 nm catches bands near 450 and nothing at 410 or 420, where lens age still takes them.

The figure carries the result. The two solid lines, the fit the earlier essay used, rise across the whole range: the more of the signal the rods carry, the further a lens band pulls the rod weight in units of its own precision. The two dashed lines, with the band fitted, stay low — one session under half a standard error everywhere, two sessions under it at a tenth and climbing past it by a third.

The same observer with a louder rod signal #

The observer is the earlier essays’ sixty-year-old, built the same way: the three cone classes’ absorptances at the population’s median peaks, seen through the lens and, for the cones, through the macular pigment; the rod signal, seen through the lens alone, added to each cone channel at a stated share of that channel’s peak — the fourth curve the rods are a fourth curve showed no gain or white point can absorb — the short-wave channel’s at the rod weight, one half. At a tenth that is the earlier essay’s banded observer to a part in a trillion. The only change is the share, swept from a tenth to a half: a tenth is a bright photopic match with a little rod intrusion, a third the dim end of the mesopic range, a half beyond it.

The figure shows where the band acts. The short-wave channel is the S cones’ absorptance plus a rod signal, and a band at 450 nm cuts both, because both look through the lens. The macula is a separate filter in front of the cones alone, centred at 460 nm — the macular is a band, not a filter is the essay about its narrowness — and for the cones a band at 450 and a little more macula look nearly alike. What differs between the band and the macula, then, is only in the rod part of the channel, and the rod part is exactly what grows as the session dims.

The fit is the earlier essay’s: the matches between two lamps across the same family of surfaces, each setting with its own noise; the shift each parameter takes is the least-squares projection of the band’s effect on the matches onto the model’s sensitivities to each parameter, and each shift is divided by that parameter’s standard error, so that a shift of one is as large as the session’s own uncertainty. One session matches daylight against a phosphor LED; two add the LED against a three-emitter lamp.

The rod weight is what absorbs what the rods see #

Left out of the fit, the band pulls the rod weight further the more the rods carry: 1.73 standard errors at a tenth, 1.90 at 0.15, 2.08 at a fifth, 2.25 at a quarter, 2.53 at a third, 3.00 at a half, from one session. Two sessions, which measure the weight more precisely, are pulled further still in those units, from 2.7 to 4.9 at a third. The macula’s shift grows too, from 4.6 standard errors to 5.7, and the lens age’s changes sign.

The mechanism is in the model’s own parameters. The band at 450 nm reduces the short-wave channel’s rod part by roughly a tenth of a density’s worth at the wavelengths the lamps differ in. The fit has three ways to explain a smaller short-wave signal: an older lens, which darkens all short wavelengths; more macular pigment, which darkens the band round 460 nm in the cone part only; or a smaller rod weight, which scales the rod part alone. The last is the one that matches what the band does to the rods, and as the rods’ share rises, the rod part is a larger share of what the band changes, so more of the band’s effect on the signal is explained by moving the weight: the change in rod signal the fitted weight accounts for — the rods’ share times the weight’s shift — doubles from a tenth to a third. The rods do see the lens and not the macula, as the earlier essay said. The trouble is that the model already has a parameter for “the rods see less”, and it is the one being measured.

That is why the rods’ route is priced by the lamp found the rod weight uncertain in the first place, and a rod signal has no natural size found its scale set by the model rather than by the eye: it is the rods’ one free number, and anything that changes the rods’ signal and nothing else is, to a fit, a change in that number.

A dimmer session notices less #

The fit’s misfit is what an experimenter would see: how much of the matches the fitted model fails to explain, measured against what the session’s noise alone would leave. With the band at 450 nm, one session’s misfit falls from 0.94 of that noise at a tenth to 0.65 at a third. The earlier essay found no centre of the band at which a session’s misfit reached the two standard deviations an experimenter would stop at; a dimmer session is further from that than a brighter one.

So the dim session is worse on both counts. It biases the rod weight more, and it gives less sign of doing so, because the rod weight’s extra freedom — larger in effect the more the rods carry — is what makes the band fit so well. A lens of the wrong shape lands in the macula found a lens edge in the wrong place hiding in the same way, with a misfit the session could not see; a band at 450 nm, in a dim session, hides in the rod weight instead.

A fourth number finds the band in daylight #

The earlier essay tried one fourth parameter, the lens edge’s position, and it took none of the band. The parameter the band needs is itself: an amplitude for a band of absorption at 450 nm, fitted alongside the weight, the lens age and the macula. With it, one session recovers the band — 0.101 of a density at a tenth, against the true 0.1 — and moves the rod weight by 0.10 of its standard error. At a third, 0.098 and 0.23. At every share from a tenth to a half, one session finds the band to within a hundredth of a density and holds the rod weight’s shift under half a standard error; the band’s own amplitude is determined to three or four standard errors, clearly present.

Two sessions keep the weight’s shift under half a standard error at a tenth, 0.25, and not at a third, 0.75: more rods shrink the weight’s standard error faster than they shrink its shift, even with the band fitted. So the rods do not help the four-parameter fit either; they only fail to hurt it much in one session.

More rods trade the macula against the band more tightly #

The prediction’s own measure of separation is how independently the fit can place the macula and the band. In one session their correlation strengthens from −0.77 at a tenth to −0.87 at a third, and the band’s standard error grows from 0.026 of a density to 0.031: an error in the macula is traded for an opposite error in the band more readily, not less. In two sessions it stays between −0.80 and −0.85 at every share, easing slightly as the rods rise. The band and the macula overlap in wavelength by construction — one centred at 450, the other at 460 — and what distinguishes them in the matches is where the lamps’ spectra differ between 420 and 480 nm, which is set by the lamps and not by the light level. The rod share only decides how much of that difference reaches the fit through a channel the rod weight can absorb.

A band parameter catches only its own band #

A parameter for a band at 450 nm describes a band at 450 nm. Given a band at 440 it finds two thirds of it, 0.065 of a density; given one at 430, a fifth; at 410 and 420, almost nothing — and the lens age takes the band as 6.0 and 5.7 years, as it did without the fourth parameter. The earlier essay’s division holds: bands below about 430 nm look like an older lens, and a fit reads them as one; bands near 450 look like macula, and need their own parameter to be told apart from it.

An experiment on older observers therefore needs its lens model to carry the band where older lenses carry it, and the matches carry their own lens is the essay that says a lens age is already in the matches to be fitted. The pigments that accumulate in an ageing lens absorb most strongly below 440 nm, which is the region where a band is read as age anyway: the case this census found hardest, a band at 450 read as macula, is the one a real older lens least often presents.

What an experiment on older observers should do #

Run the matches at photopic levels, not dimmer ones. A dimmer session was proposed to measure the lens through the rods; the rods’ contribution is absorbed by the rod weight, which is the quantity the experiment is measuring, so a dimmer session biases the answer more and hides the bias better.

Fit a band where the lens might carry one. A fourth parameter for the band’s amplitude at 450 nm finds a band there to within a hundredth of a density in a single daylight session and leaves the rod weight within half a standard error. Where a lens band would sit lower, the lens age takes it already.

Treat a shift in the rod weight as a question about the lens first. Age can size the rod signal if the lens is known made the rod weight’s measurement depend on the lens model; this census adds that any error in the lens model that reaches the rods reaches the weight directly, and does so more the more the rods carry.

How the fits were made #

The observer is the sixty-year-old of the earlier essays: cone absorptances at the population’s median peaks and the standard optical density, the lens density of the model’s age function at sixty, the standard macular pigment as a Gaussian at 460 nm of width 42, and the rod signal — rhodopsin through the lens, scaled to its peak — added to each cone channel at a share of that channel’s peak, the short-wave channel’s at a weight of one half. The band is a Gaussian of density 0.1 and standard deviation 15 nm added to the lens. A session is the colour differences between two lamps’ renderings of the same family of surfaces through the observer, with each setting’s noise the earlier essays’ σ. The shifts are the Gauss–Newton step from the model’s observer, using forward-difference sensitivities with steps of 0.1 in weight, five years of lens age, 0.05 of macular density and 0.02 of band density; standard errors are from the inverse of the same normal matrix; the misfit is the chi-square of what the step leaves, in standard deviations of a chi-square with the matches’ degrees of freedom.

What this leaves out #

The rod share is a single number. A real mesopic match changes more than the rods’ share: the cones’ own sensitivities shift, the rods intrude into the red–green pathway differently from the blue–yellow, and the eccentricity at which a field is viewed changes the macula’s density. The census raises the rod share alone, which is the lever the prediction named.

The rod weight’s standard error falls with the rod share, from 0.059 at a tenth to 0.026 at a third in one session, and that is part of why every shift in its units grows. Measured on the weight’s own scale, the band’s pull falls, from 0.10 to 0.07: a larger rod share needs a smaller change of weight to produce the same change of signal. An experiment cares about the ratio, because a bias smaller than the noise does no harm and one larger than it is a wrong answer reported with confidence, and the dimmer session shrinks the noise faster than the bias.

Still open: whether a band in the lens is a second age #

A fourth parameter at a fixed centre catches a band near that centre. A real older lens adds absorption in a broad region below about 440 nm whose centre varies from one eye to the next, and a fit would need the band’s centre as well as its amplitude — a fifth number, nonlinear in the matches.

The calculation is the fit with both the band’s amplitude and its centre free, on observers whose bands are centred anywhere from 410 to 460 nm, from one session and two, with the question whether the centre is identified at all or trades with the lens age as the amplitude traded with the macula. The prediction is that one session cannot place a band’s centre below 430 nm to better than about fifteen nanometres, because there a band and an older lens change the matches almost identically, while two sessions with a lamp whose blue line sits near 440 place it within five. If that holds, the lamp an experiment on older observers most needs is not a dimmer one but one with a narrow line where the lens’s pigments and the macula overlap.

A proposed instrument has to be tested against the fit that will read it #

The habit is about testing a measurement proposal inside the model that will be used to interpret it.

The argument for a dimmer session was physical and correct as far as it went: the rods see the lens and not the macula, so more rod signal carries more information about the lens. What it did not ask was which of the fit’s parameters that information would be attributed to. The model already had a parameter for the rods — the one the experiment was built to measure — and the extra information arrived there, as bias. More signal about a quantity is worth nothing to a fit that has a nearer parameter to put it in.

The failure mode is to judge a design by what the data contain rather than by what the analysis will do with them. The test is to run the proposed design through the analysis before running it on people.