Age can size the rod signal, if the lens is known

A rod signal has no natural size found an ambiguity in the collection’s own observer model. Rods leak a signal into the colour pathways, and the model adds it to each cone class as a tenth of that class’s peak absorptance. An older lens absorbs short wavelengths before either receptor sees them and takes more than half of the S cones’ peak, so a rod signal defined as a fraction of the S cones’ peak shrinks with age. Defined instead as a fixed amount, the same signal grows relative to the S cones as the lens yellows. Between twenty and seventy-five the rod signal’s share of the S cones’ catch changes by under half again under one reading and more than doubles under the other, and nothing in the model says which is right: its normalisation, not the retina, decides.

That essay called it a question rather than a permanent ambiguity. The reference lamp must not move had found the experiment that measures the rod signal’s weight in the S pathway — an asymmetric match between daylight and a phosphor LED, a surface under one lamp matched by a setting under the other — and the proposal was to run it on observers of widely separated ages, fitting the weight per observer. “The two normalisations make opposite predictions about how the fitted weight varies with age.” It named the quantity to report — the fitted weight’s standard error against age — and the obvious hazard: an older observer differs in the lens, and a difference in fitted weight is not the rod route’s unless the lens is measured.

Opposite trends, and a confound as large as the effect #

Under the model’s scaled reading, the daylight-against-LED match moves 0.69 ΔE00 across the full range of the rod signal’s S weight at twenty and 0.47 at seventy-five; under the fixed reading, 0.55 and 0.76. A weight fitted from a forty-setting session is 47 per cent less precise at seventy-five than at twenty under the first, and 28 per cent more precise under the second. If the fixed reading is true and the weight is fitted with the scaled one, the fitted weight looks 62 per cent larger at seventy-five than at thirty-two. And a lens mistaken by ten years moves the match as far as a third to a half of the whole weight.

• The two readings do predict opposite trends, and the trends differ by a factor of two, as the proposal estimated.
• The fitted weight’s drift is the quantity to watch, more than its standard error, because the standard error is the model’s own prediction and the drift is a measurement.
• The experiment is small if the weight is a property of the species: one or two observers at each end of the age range show the drift at two standard errors.
• The lens has to be known to two or three years’ worth of density. Anything looser and the lens error is as large as the effect being measured.

Two readings of one observer #

The observer is the collection’s cone model at a stated lens age, with a rod curve behind the same lens, and the rod signal entering the L and M channels at weight one and the S channel at a weight between nought and one — the weight the experiment is trying to measure. The two readings differ in one number: the size of the rod term in each channel. Scaled, it is a tenth of that channel’s peak absorptance at the observer’s own age, the model’s own definition. Fixed, it is a tenth of that channel’s peak at thirty-two, an absolute amount that does not shrink when the lens takes the cone’s peak. At thirty-two the two are one observer.

The whole difference is in the S channel, and it is the lens. A seventy-five-year lens takes more than half of the S cone’s peak absorptance and very little of the L and M cones’. The scaled reading shrinks the S channel’s rod term with it; the fixed reading does not. At seventy-five the fixed rod signal in the S channel is 1.9 times the scaled one, the factor a rod signal has no natural size found between the two readings’ shares.

What the match sees #

The match signal is the thing an experiment resolves: how far the asymmetric match moves, as a median over the collection’s surface family, when the S weight goes from nought to one. At thirty-two it is 0.58 ΔE00 under either reading. Under the scaled reading it falls with age — 0.69 at twenty, 0.52 at forty-five, 0.47 at seventy-five — because the rod term shrinks with the S cone and the weight has less to act on. Under the fixed reading it rises — 0.55 at twenty, 0.64 at forty-five, 0.76 at seventy-five — because the rod term keeps its size while the S cone’s own signal falls away beneath it, so the weight moves a larger share of what the S channel carries.

That is the prediction the proposal made, computed: “under the fixed-size reading an older observer’s matches move more per unit of weight.”

The standard error of a fitted weight is the setting noise over the signal, divided by the square root of the number of settings, so it inherits the signal’s trends upside down. With forty settings of 0.5 ΔE00 precision, the scaled reading’s standard error rises from 0.115 at twenty to 0.169 at seventy-five, 47 per cent; the fixed reading’s falls from 0.144 to 0.104, 28 per cent. The two trends differ by a factor of 2.03 across the range — the “factor of two the catch ratios differ by” that the proposal expected.

The drift is what an experiment can measure #

There is a difficulty with reporting a standard error as the result. A standard error computed from a fit is the model’s prediction of the precision, using the model’s own signal, and the model’s signal is what depends on the reading. An experimenter fitting with the scaled reading would compute the scaled standard errors whatever the retina does. The standard error tells an experimenter how precise each observer’s weight will be; it does not by itself say which reading is true.

What does is the fitted weight itself, across observers.

If the retina’s rod signal has a fixed size and the weight is fitted with the scaled model, the fitted weight is the true weight times the ratio of the two signals: ×0.80 at twenty, ×1 at thirty-two, ×1.24 at forty-five, ×1.38 at fifty-five, ×1.50 at sixty-five and ×1.62 at seventy-five. The other way about — scaled true, fixed fitted — the reciprocal. A cohort of observers who share one true weight would show the wrong reading as a trend of the fitted weight with age, and the right reading as no trend. That is a measurement, not a prediction, and it is the one the experiment should report.

How many observers #

If the true weight is one half and the fixed reading is true, a scaled fit differs between twenty-year-olds and seventy-five-year-olds by 0.41. Against per-observer standard errors of 0.115 and 0.169 from forty settings, showing that at two standard errors needs one observer in each group if everyone’s true weight is the same, two if it varies between people by a standard deviation of a tenth, three at two tenths, and six at three tenths.

How much the weight varies between people is not known, and that is the real limit on the design. The collection’s population essays treat the rod weight as one number; nothing in the colour-matching literature measures its spread. The figure therefore states the experiment’s size as a function of that unknown rather than at a guessed value. On any spread up to a third of the weight itself, a study of a dozen observers — six young, six old — would settle which reading is right, at forty settings each.

The lens is the confound, and it is large #

The proposal’s hazard is where the numbers are least comfortable.

A lens ten years older than assumed moves the daylight-against-LED match as far as a change of 0.55 in the S weight at twenty-five, 0.41 at forty-five and 0.34 at sixty-five. An experiment trying to detect a drift of 0.41 across the age range cannot afford lens errors of that size; holding the confusion under a tenth of the weight needs each observer’s lens known to within 1.8 years’ worth of density at twenty-five and 3.0 at sixty-five.

That is a demanding requirement. The lens’s density at a given age varies between people by far more than two or three years’ worth — the observer has no age found the collection’s age model to be a statement about an average eye, and individual lenses scatter widely around it. So an experiment that assigns each observer the average lens for their age will confound the rod weight with lens density at every age, and the drift it finds could be either. The lens has to be known on each observer — measured by a densitometric method that reads its optical density directly, to a fraction of what a decade of ageing does to it, or fitted from the matches alongside the weight, which is only possible if a lens error moves the matches in a different pattern from a change of weight. The comparison here is of sizes alone; whether the patterns differ is the question this essay leaves open.

This turns the proposal’s covariate from an age into a number, as it said it would. It also says how good the number has to be, however it is obtained, which the proposal did not. And it has an uncomfortable corollary: the young observers, whose lenses are clearest, are the ones for whom a lens error costs most — 0.55 of the weight per decade at twenty-five against 0.34 at sixty-five — because at twenty-five the S cones are seeing the most of the short wavelengths a lens takes.

And the macula is a second lens #

The lens is not the only thing in front of the S cones that varies between people. The macular pigment absorbs between about 400 and 520 nanometres, over the central few degrees where a matching field sits, and it sits in front of the cones and not the rods. The macular is a band, not a filter is where the collection priced its shape; what matters here is its density, which differs between people of the same age by a few tenths of an optical density and bears little relation to age at all.

In the same model and at the same weight, a macular pigment a tenth of an optical density denser than assumed moves the daylight-against-LED match as far as 0.30 of the S weight at twenty-five and 0.42 at fifty-five; two tenths, 0.61 and 0.85. A tenth of an optical density of macular pigment is therefore a confound the size of a decade of lens, and ordinary variation between people is several times that.

So the covariate the experiment needs is not one number but two, and the second is not predicted by age at all. Heterochromatic flicker photometry measures macular density on each observer in a few minutes, and a lens densitometer measures the lens; an experiment that neither measures nor fits them would find drift, or no drift, in whichever direction the two pigments’ spread happened to push its small sample. With both known to the precision computed here, the rod signal’s size is a small study. Taken from age and a population average, it is not a study at all.

What the experiment would establish #

A trend in fitted weight with age, measured on observers whose lenses have been measured, decides the rod signal’s size. No trend under the scaled reading: the model’s definition is right, and the rod signal in the S pathway shrinks with the S cone’s absorptance, as a signal routed through the same photoreceptor-to-bipolar gain would. A rising trend: the rod signal has a size of its own, and the S cone’s shrinking peak leaves it proportionally larger in older eyes. The rods are a fourth curve introduced the rod term as the model’s statement that rod intrusion is real; this would say what it is proportional to.

And it would settle something the lamp essays depend on. The rods’ route is priced by the lamp found that which lamps make the rod route matter depends on the rod signal’s share of the S catch, and the share’s trend with age is the one thing the two readings disagree about. Every statement about how the rod route ages is currently a statement about the normalisation chosen.

How the observers were built #

The cone absorptances are the collection’s template model, with the CIE lens-density function at the stated age and the standard macular pigment and photopigment densities; the rod curve is the same model’s scotopic sensitivity behind the same lens, normalised to its peak. The rod term added to each cone channel is a tenth of that channel’s peak absorptance — at the observer’s own age for the scaled reading and at thirty-two for the fixed — times the rod curve, times the channel’s weight: one for L and M, the S weight under test. The match signal is the one the earlier rod-signal essays used: for each surface of the collection’s family, the displacement of the observer at S weight nought and at one from a rod-free observer, under daylight minus under the phosphor LED, added to the rod-free observer’s colour under the LED, and the CIEDE2000 between the two results; the median over the family. The standard error of a weight fitted from n settings of precision σ is σ over the signal times the square root of n, with n = 40 and σ = 0.5 ΔE00 as declared in the lamp essay.

What this leaves out #

Age here is the lens only. Older eyes also lose cone photopigment density, change their macular pigment and pupil size, and may change the rod signal itself. Each would add a term to the drift; the model varies the one the proposal named.

The true weight is taken as one half and as shared. The drift is proportional to the true weight, so a smaller true weight needs proportionally more observers; a weight that itself changes with age is indistinguishable, in this design, from the fixed reading.

And the setting precision is the declared 0.5 ΔE00. Older observers set asymmetric matches less precisely, which would raise their standard errors under both readings alike and cost observers without changing the direction of either trend.

Still open: whether the matches can carry their own lens #

The lens confound is large because the daylight-against-LED match responds to lens density as well as to the S weight, and the comparison here was of how far each moves the match, not of how. Over a family of surfaces a lens error and a change of weight each move the matches in a pattern, and if the patterns differ a fit can take both. A second match, under a pair of lamps chosen so that it responds to lens density as strongly but to the S weight weakly, would sharpen the separation further and measure the lens from within the colour experiment itself.

The calculation is the census of lamp pairs from the reference lamp must not move, with each pair scored twice — its signal per unit of S weight and its signal per decade of lens — and the pair chosen that maximises the second relative to the first. The prediction is that a pair of lamps both rich in short wavelengths, differing in how far into the violet they reach, responds to the lens many times more than to the S weight, because the rod curve peaks at 500 nanometres and the lens acts below 450. If such a pair exists, two matches per observer would give both the lens and the weight without a densitometer; if every pair that sees the lens also sees the weight, the densitometric measurement is not optional.

A covariate has a required precision #

The habit is about asking how well a confound has to be measured, not only whether it has to be.

The proposal knew the lens was a hazard and said it should be measured. That is necessary and not sufficient. Measured to what precision was a computation, and the answer — two to three years’ worth of density, in a quantity that varies between people of the same age by much more than that — is what decides whether the experiment is feasible with an ordinary densitometric measurement or needs a better one.

The failure mode is to list a confound and stop. A confound that is measured but measured too loosely leaves an experiment exactly as ambiguous as one that ignores it, and only the arithmetic says which side of that line a given measurement falls.