The Point That Does Not Move

I.The First Known Thing — the wavelength where nothing changes

physical chemistry · the isosbestic point in absorption spectroscopy

A solution that absorbs light is characterised by its spectrum, the absorbance plotted against wavelength. When something happens in the solution — an acid is titrated, a compound isomerises, a reaction runs — the chemist records the spectrum again and again as it proceeds and lays the curves over one another. Most of the picture moves. One band falls and another rises. But in a great many such families there is a wavelength at which every curve passes through the same point, where the absorbance does not change from the first spectrum to the last. It is called an isosbestic point, from the Greek for equal extinction, and the definition kept by the International Union of Pure and Applied Chemistry is simply that: a wavelength at which the total absorbance of a sample does not change during a chemical reaction or a physical change of the sample.

why the curves must meet

The reason is arithmetic. Absorbance is additive, each species contributing in proportion to its concentration. If one species is being converted into another and the total amount of the two is constant, then every spectrum in the family is a weighted average of the same two pure spectra, with weights that sum to one and differ only in how far the conversion has gone. At any wavelength where the two pure spectra happen to agree, the weighted average takes that common value whatever the weights. The curves therefore meet there, and they meet however many are recorded. Nothing about the rate of the reaction, its mechanism or the shapes of the two spectra enters. The point exists because only one thing is changing, and that thing is a proportion.

what is read from its presence, and from its loss

Chemists read the point in both directions. A sharp crossing held through the whole course of a reaction is taken as evidence of a single linearly independent process: a clean conversion, with no intermediate accumulating and no side reaction under way. A crossing that blurs, or drifts steadily along the wavelength axis as the reaction proceeds, is taken as evidence of the opposite — a third absorbing species, a second reaction, or a change in the medium that is moving the bands themselves and not the proportions between them. And where the point holds it is put to use, because at that wavelength the absorbance reports the total amount present and is indifferent to how the total is divided. Oximetry relies on it: oxygenated and deoxygenated haemoglobin share isosbestic points, one of them near 800 nanometres, at which total haemoglobin can be read whatever the saturation.

the cautions that came with it

The discipline has also learnt what the point does not show, and the cautions are written into the definition. Its existence does not prove that one species is converted into a single other, nor that only two are in equilibrium; it shows that the stoichiometry is unchanged and that no secondary reaction intervened while the spectra were taken. Three species that change in fixed proportion to one another will produce it as surely as two. It is necessary for a simple conversion and not sufficient. And it has a threshold of evidence that is seldom stated because it is obvious at the bench: two spectra that cross have shown nothing, since two curves may cross anywhere. It is the third curve passing through the same point, and the tenth, that carries the inference.

II.The Second Known Thing — the responder and the line that defines one

drug discovery · trial design, controls, and evidence standards

A parallel-group trial assigns patients at random to a drug or to a control and compares the outcome in the two arms, in the first instance as a difference between means. A mean difference does not say how the benefit is distributed, and those who must act on a trial want to know. The standard answer is a responder analysis. A line is drawn on the outcome scale — a halving of a depression score, a reduction in pain of thirty or of fifty per cent, a seventy-five per cent improvement in a psoriasis index, the loss of five per cent of body weight — and the proportion of patients beyond the line is reported for each arm. Regulators ask for it. The Food and Drug Administration’s 2009 guidance on patient-reported outcomes goes further and asks that the whole cumulative distribution of responses be examined in each treatment group, to characterise the effect and to examine the possibility that a mean improvement reflects different responses in subsets of patients.

the objection that has stood for decades

The objection to reading such proportions as counts of people who responded is old and has not been answered. Each patient in a parallel-group trial is observed under one treatment only, so no patient’s response to the drug is ever seen; what is seen is two distributions. The same pair of means is produced by every patient improving a little and by a few improving a great deal, and the proportion beyond a line is produced by either. Senn has pressed the point for many years: the line is a convention, dichotomising a continuous measure discards information, and a statement that forty per cent of patients respond is not licensed by the design that produced it.

the two instruments the field has

Two instruments are used to decide between the pictures. The first is the variance ratio. If patients differ in their response to a drug, the spread of outcomes in the treated arm should exceed the spread under placebo, and the ratio of the two variances can be pooled across trials. Meta-analyses of antipsychotic and of antidepressant trials published in 2019 found the ratio close to one and concluded that there was little evidence of individual differences in response. The weakness of the instrument is acknowledged: a ratio of one may conceal a distribution in two humps whose departures cancel, and the variance of many clinical measures scales with the mean for reasons unconnected with treatment. The second instrument is the finite mixture model, which supposes two latent classes of patient, each normally distributed, and estimates their proportions in each arm. Applied by Thase, Larsen and Kennedy in 2011 to 1,357 patients from five trials of escitalopram, it returned a benefit of about sixteen points on the Montgomery–Åsberg scale among those who benefited, and about a fifth of patients benefiting on the drug who would not have done so on placebo. Its weakness is that it assumes what is in question. Two normal curves can be fitted to any skewed distribution, and the fit does not show that the classes exist.

how the field holds it

It is worth noticing how the question is held. Both instruments compare one arm with another. The thresholds are fixed by convention or anchored to what patients report as a meaningful change, which is a question about the scale and not about whether a class of responders exists. And the trial on which approval rests is ordinarily a comparison of one dose with a control, the dose-ranging trial having done its work once the dose was chosen, and having been analysed for that purpose as a curve of mean response against dose. What is not held anywhere is that the number of arms decides whether the question can be put at all.

III.The Bridge

no paper on isosbestic points cites a paper on responder analysis; each hall lays a family of curves over one another, and only one of them looks for the point the family shares

The halls do not meet. The literature of the isosbestic point is physical and analytical chemistry; the literature of the responder is biostatistics and regulatory guidance. Neither cites the other and no review spans them. Five neighbours must be set aside, and the first is close enough that it must be named and not merely dismissed. Mathematical psychology has held since Falmagne’s note of 1968 a fixed-point property of binary mixtures: if response times arise from two processes mixed in varying proportion, their densities share a common point, and the property is used, with at least three experimental conditions, to ask whether a subject is switching between two strategies. That is the same theorem in a third hall. It has been applied to task switching, to decisions under risk and to recognition memory, and not to the arms of a trial; and it arrived without the chemists’ century of practice in reading the point’s drift, its loss and its false presence, which is what the present bridge draws on. The second neighbour is the variance ratio, which reads one moment of two curves where this reads the whole of three or more. The third is the mixture model, which assumes two classes of a stated shape where this assumes nothing about either. The fourth is bimodality, the search for two humps with a trough between them: a mixture may have one hump and still have a common point, so the test does not wait for the distribution to divide visibly. The fifth is the minimal important difference, which fixes a line by what patients value and is silent on whether anything changes at it.

X: curves that differ only in how much has passed from one fixed state to another must all pass through the point where the two states agree, and whether they do can be asked of three curves and of no fewer

Call it the common crossing. Where each member of a family of curves of constant total — spectra at fixed total concentration, outcome distributions each of which sums to one — is a weighted average of the same two fixed curves and the members differ only in the weights, every member passes through the points at which the two fixed curves agree, whatever the weights; so a point common to three or more members is evidence that what changed between them was occupancy alone and that the two states did not move, and the absence of such a point is evidence that the states themselves moved. Four things follow, and they are not one thing said four times. The first is that two curves are no test. Any two distinct distributions cross somewhere, so a trial of one dose against a control cannot put the question at any sample size, while a trial of two doses and a control can; the capacity to ask is a property of the number of distinct occupancies the design contains and not of the number of patients. The second is that the common point is a threshold with a derivation. The difference between two arms in the proportion of patients beyond a line is greatest at a point where their densities cross, and if all the arms cross at one point then one line is the most discriminating for every pair of arms at once and is given by the data. If every patient is instead shifted by an amount that grows with dose, the crossing between any two arms lies midway between their means and travels as the dose rises, and there is no such line. A responder threshold is therefore a property of the drug and the disease exactly where a common crossing exists, and a convention everywhere else. The third is that the chemists’ cautions transfer whole. The point shows one degree of freedom and not two classes, since three classes moving in fixed proportion would give it; its steady drift with dose is itself a reading, of states that move as occupancy changes, as when those who respond respond more deeply at a higher dose; and a control arm that misses a point the active doses share would indicate that improvement on placebo and improvement on the drug are different states, which the mixture model has to assume are the same. The fourth concerns what the test withstands and where it fails. Independent measurement error broadens both states alike, and a mixture of two broadened states is still a mixture, so the common point survives noise that would defeat any attempt to classify single patients. But where the differences between arms are small against the spread, a uniform shift also produces crossings that nearly coincide, close to the centre of the control distribution, separated only by half the difference between the means; so the test has force only where arms differ by a substantial fraction of a standard deviation, and below that it is silent, which is not the same as passing.

IV.The Question That Becomes Possible

neither discipline would ask this alone

The spectroscopist has no reason to ask it. A patient is not a solution, and the chemist’s interest in the point ends with the reaction in the cell. The trialist has no reason to ask it either. The question of whether responders exist is posed arm against arm, with instruments built for two curves, and the trial that has three or more arms at fixed doses is analysed as a curve of means and then set aside. Held together the two give a test that can be run on distributions already recorded.

the common crossing hypothesis

In fixed-dose trials with a control and at least two active doses, where the arms differ in mean by a substantial fraction of a standard deviation, do the outcome distributions of all the arms pass through a common point that lies away from where a uniform shift would put each pairwise crossing — and where they do, is that point reproduced at other visits and in other trials of the same drug, does it coincide with the threshold at which the difference in responder proportions is greatest for every pair of arms, and does its presence or absence, where the variance ratio is near one in both cases, separate the drugs for which a class of responders was later identified by a marker from those for which the search found nothing?

why it would matter

Because it would give the responder threshold either a derivation or an honest label. Where a common point exists, the line belongs to the drug and the disease and the proportion beyond it estimates an occupancy; where none exists, the proportion is an artefact of where the line was drawn and should be reported as such. Because it would say what a confirmatory programme gives up when it carries one dose forward: a second active arm retained at a fixed dose keeps the question open, and its cost can be weighed against that. And because it would tell a stratification effort, before any marker is sought, whether there is a class to be found, which the variance ratio cannot do when the departures cancel and the mixture model cannot do without assuming the answer.

V.How It Could Be Proven — or Broken

a candidate must be breakable, or it is merely poetry

An intercognix candidate earns nothing until it names its own test. This one needs no new trial. Fixed-dose trials with a control and two or more active doses exist in number, with patient-level data held by sponsors and by the data-sharing platforms: the fixed-dose trials of antidepressants and antipsychotics, the three-dose trials of the newer agents for weight loss in which the active arms stand far from placebo, the dose-ranging trials of biologics in psoriasis. What it needs is the observed outcomes of each arm, unimputed, at a fixed visit.

the honest status

Note the narrowness. The claim is about families of curves and not about patients. A common point, where it is found, shows that a population is divided between two distributions in a proportion the dose alters; it does not say which patient is in which, and it identifies no one. It shows one degree of freedom and cannot show that the classes number two. It does not say that responder analyses should be abandoned or required, that any threshold now in use is wrong, or that any label misstates a benefit. It does not apply to a trial of one dose against a control, to flexible-dose trials in which the dose follows the response, to outcomes counted as events over time, or to effects that are small against the spread, which is most of them. The whole of the claim is that a design with three or more fixed arms contains a test of whether dose changes how many patients occupy a state or changes the states, that the test needs no assumption about the shape of either, that it yields a threshold where it passes, and that the dose-ranging trial is not, so far as a search of both literatures shows, at present read for it.