The Path That Misses the Origin

I.The First Known Thing — the rock that is cleaned in steps

palaeomagnetism · the components of a natural remanence

A rock taken from an outcrop carries a magnetisation, and that magnetisation is seldom one thing. Part of it may have been acquired when the rock formed, as a lava cooled or a sediment settled, and records the field of that time. Part may have been added since, by burial, by chemical change, by lightning, or slowly and without any event at all from the field in which the rock has sat for the last few hundred thousand years, which is called a viscous magnetisation. What the magnetometer reports is the vector sum of these. Its strength and direction as found are the property of no single episode in the rock’s history, and a pole computed from it belongs to no age.

removal by steps, and the diagram that reads it

The components can be told apart because they are not equally stable. Each is carried by a population of grains with its own resistance to an alternating field or to heat, and a specimen exposed to successively stronger alternating fields, or heated to successively higher temperatures and cooled in zero field, loses the least stable first. The remaining vector is measured after every step. In 1967 J. D. A. Zijderveld gave the method the diagram it has kept: the tip of the vector is projected on two orthogonal planes and followed from step to step. While one component alone is being removed the tip moves along a straight line, and the direction of that line is the direction of the component being removed. Where another takes over, the line turns. The last segment, belonging to the most stable component, runs to the origin, since when it is gone nothing is left. A straight segment that is not aimed at the origin therefore says something definite before the cleaning is finished: there is a component beneath, more stable than the one now leaving, and the amount by which the line passes the origin by is its part across the line.

the fit, and the anchor

J. L. Kirschvink in 1980 gave the reading its standard arithmetic: a principal component analysis of the points in a segment, yielding a best line and a figure, the maximum angular deviation, for how well the points keep to it. He set out three ways of taking the fit, free, anchored to the origin, or with the origin counted as one further point, and the choice among them is a claim about the rock. To anchor is to assert that the component being fitted is the last. Heslop and Roberts examined the practice in 2016 by Bayesian model comparison and found that anchoring can rarely be justified statistically and returns uncertainties that are artificially small. The field’s considered position is that whether a path goes to the origin is a thing to be tested and not a thing to be assumed.

what the method cannot do, as its users state it

Its limits are as well known as its use. Two components whose stabilities overlap are removed together, and the path between them is a curve with no straight part to fit; the points then lie in a plane, and the most that one specimen yields is that plane, the stable direction being recovered only where the planes of several specimens intersect. Two components that share a direction are not separable at all, since removing one moves the tip along the other and the path is a single line to the origin through both. A viscous overprint is usually soft but is not always removed by the mildest steps. And a direction is credited only when specimens from one site, cleaned independently, agree on it. What the method abolished was an older habit, of measuring the magnetisation as found or after one cleaning step chosen in advance and treating what remained as the ancient part.

II.The Second Known Thing — the drug that is taken away at the end

drug discovery · disease definition, indication choice, endpoint selection

A treatment for a progressive disease may relieve its signs while it is being taken, or alter the course of the disease underneath, or both, and the second is a different claim, differently valued and far harder to support. The difficulty is one of endpoint. On any scale of severity, a patient who has been treated for a year and is better than an untreated one may be so because the disease has advanced less or because the drug is still acting, and the scale records the sum. Two designs were proposed in the 1990s, chiefly by Paul Leber, to take the sum apart: withdraw the drug at the end and see what remains, or start the comparison group late and see whether it catches up.

the trial that withdrew

The withdrawal design had its fullest trial in Parkinson’s disease. In ELLDOPA, reported by the Parkinson Study Group in 2004, 361 patients with early disease were given carbidopa and levodopa at 150, 300 or 600 milligrams of levodopa a day, or placebo, for forty weeks, after which treatment was withdrawn for two. The primary outcome was the change in the total score of the Unified Parkinson’s Disease Rating Scale between baseline and week 42. The placebo group had worsened by 7.8 units; the three levodopa groups by 1.9, by 1.9, and by −1.4. A remainder of some six to nine units in favour of the drug had survived its removal. An imaging substudy pointed the other way, the loss of striatal uptake of a dopamine-transporter tracer being greater on levodopa than on placebo.

what was made of the remainder

The remainder could not be read. Levodopa has, beside the response that follows each dose, a long-duration response that builds over weeks and fades over days to weeks, and the investigators stated that the clinical result was consistent either with a slowing of the disease or with a symptomatic effect outlasting the washout. Chan, Nutt and Holford in 2007 reproduced the trial by simulation under a model in which levodopa both slowed progression and washed out slowly, and concluded that two weeks had not been enough to eliminate the symptomatic benefit. Cilia and colleagues in 2020, studying patients who had come to levodopa only after years of untreated disease, attributed 60 to 65 per cent of the whole motor benefit to the long-duration response, found it still plainly present after a week without the drug, and saw a return to the untreated state only with withdrawals of more than fifteen days. The question was put again by the other design. In LEAP, reported in 2019, 445 patients received levodopa for eighty weeks or placebo for forty followed by levodopa for forty; the change in the total score at week 80 was −1.0 against −2.0, a difference of one unit, and the authors concluded that levodopa had no disease-modifying effect over the period.

what the field knows about the parts of its scale

Through all of this the object analysed was a single number. The scale is not single. The total used in these trials is the sum of three parts, one of mentation, behaviour and mood, one of activities of daily living and one a motor examination, and the motor examination is itself made of items for tremor, rigidity, slowness, gait and posture, which do not respond alike to levodopa and do not advance alike without it. The field knows this well. The delayed-start trial of rasagiline, ADAGIO, which in 2009 met its three hierarchical endpoints at one milligram and not at two, was followed by analyses of the parts of the scale; and longitudinal models at the level of the item, built since about 2017, treat tremor as a dimension apart from the rest and estimate progression domain by domain. The pharmacometric models that separate a symptomatic effect, as an offset, from a protective one, as a change of slope, are fitted to the total. What has not been taken from a withdrawal is the thing the palaeomagnetist takes first: the direction, across the parts of the scale, in which the benefit moved as it was lost.

III.The Bridge

no paper on the stepwise demagnetisation of rocks cites a paper on the design of withdrawal trials; both halls remove part of a composite to learn what it is made of, and only one of them reads the path

The halls do not meet. Progressive demagnetisation belongs to rock magnetism and is argued over coercivity spectra, overprints and the fitting of lines; the withdrawal design belongs to clinical neurology and regulatory statistics and is argued over washout lengths, missing data and the meaning of a slope. Neither cites the other and no review spans them. Three neighbours must be set aside before the proximity is worth anything. That several outcomes are better than one is not the bridge; the field has multivariate endpoints and global tests, and uses them to gain power, which is another purpose. That a symptomatic effect may fade slowly is not the bridge either; it is the known difficulty, stated by the trialists themselves. And the question whether a pause between two treatments restores a patient to the starting state concerns what a washout does to a hidden state, where the matter here is what a remainder is composed of. The bridge is a statement about where the information lies when removal is incomplete.

X: where a measured vector is a sum of contributions differing in stability, partial removal leaves a remainder whose size apportions nothing, while the path taken gives the direction of what is leaving and, by whether it is aimed at the origin, the part across that direction of whatever is not

Call it path directedness. Where a measured quantity has several dimensions and is the sum of contributions that differ in their stability against some removing agent — a field, heat, time without the drug — progressive removal moves it along a path, and the path is straight for as long as one contribution alone is being removed. The direction of the path is then the direction of that contribution. Whether the path is aimed at the origin is a separate fact, and it is the informative one: a path aimed at the origin leaves nothing behind it that differs in direction from what is leaving, and a path that passes the origin by leaves a more stable contribution whose part across the path equals the miss. That part is fixed by any two points on the path and does not wait for the removal to be completed. The part of the stable contribution lying along the path is the opposite case, and cannot be had from any removal short of a complete one. The size of the remainder at a single step, which is all a summed scale reports, does not distinguish the two.

three consequences, not one said three times

Three things follow. The first is that a truncated washout is not a spoiled one. Suppose, for illustration only, that the difference between drug and placebo on two domains scored in comparable units stands at −0.60 and −0.10 on the last day of treatment and at −0.30 and −0.10 two weeks later. All the loss has been in the first domain; the path runs parallel to that axis and passes the origin at a distance of 0.10; a tenth of a unit on the second domain is not among the things being removed, however much longer the washout runs. Had the second reading been −0.30 and −0.05, the path would be aimed at the origin and the remainder would be, on this evidence, the same effect half gone. On a summed scale the two remainders are −0.40 and −0.35, and nothing in either figure says which case it came from. The second consequence is that the discriminating quantity is an angle that the field has not reported. A slowing of the disease leaves a remainder lying along the direction in which untreated patients move over the same months; a symptomatic effect leaves one lying along the direction in which the drug acts on first being given. The withdrawal can separate them only in proportion to the sine of the angle between those two directions. Where a drug relieves exactly the features that advance, the angle is nothing and no washout short of a complete one will serve; where the disease advances chiefly in features the drug does not touch, a short one may. The third is that to read the remainder at the end of a fixed washout as lasting is to anchor the fit: it asserts that the removal was complete, which is the assertion the geophysicists found could rarely be defended and which the simulation of ELLDOPA found false.

IV.The Question That Becomes Possible

neither discipline would ask this alone

The palaeomagnetist has no reason to ask it. His vector has three dimensions given by space, his removing agent is under his hand and can be raised until nothing is left, and he has no patient who may not be kept from treatment a week longer. The trialist has no reason either, because his endpoint is declared in advance as a total, the parts of the scale enter afterwards as secondary analyses each tested for a difference of its own, and a direction among them is not an object his methods name. Held together, the halls pose a question that is neither’s: when a drug is withdrawn and a benefit survives, does the benefit retreat across the domains of the scale along a line aimed at zero, or along one that passes zero by.

the origin-directed washout hypothesis

Concretely. Take a withdrawal trial in which a scale of several domains was recorded at the last visit on treatment and at one or more visits after withdrawal. Fix the domains in advance — for the scale used in ELLDOPA, mentation, daily living, tremor, rigidity, slowness, and gait with posture — and put them on a common footing, each in units of its standard deviation at baseline. Form three vectors from the same trial. The first is the difference between drug and placebo at an early visit on treatment, before the disease has had time to move: the direction of the symptomatic effect. The second is the change in the placebo group from baseline to the end: the direction of progression. The third is the path, the difference between drug and placebo at the last treated visit and at each visit after withdrawal. The claim has three parts. First, that the directions of symptomatic effect and of progression are separated by an angle larger than its sampling error. Second, that the path lies along the symptomatic direction, as it must if what is being lost is the symptomatic effect. Third, that the line of the path, fitted free and not anchored, either meets the origin within its error or passes it by, and that where it passes it by the miss agrees in direction with that part of progression which lies across the path. The first part decides whether the instrument can see anything. The third is the finding.

why it would matter

Because the claim of disease modification is the most valuable a sponsor in these diseases can seek, and the means of supporting it are long, costly and frequently inconclusive. If the path can be read, a washout too short to be complete, which for ethical reasons is the only kind there will be, yields a lower bound on the lasting part and not an unreadable remainder. And the angle becomes a property to be consulted when the indication and the instrument are chosen: a disease stage and a scale in which the drug’s action and the disease’s advance point the same way cannot support the claim by withdrawal whatever the drug does, and it would be as well to know that before the trial and not after.

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. It needs the item-level records of withdrawal trials already run, of which ELLDOPA is the cleanest, having three doses and a placebo under one protocol, and it has in LEAP an answer obtained by another design against which its verdict can be held.

the honest status

Note the narrowness. The claim concerns trials in which a drug is withdrawn and a scale of several domains is recorded on both sides of the withdrawal. It does not say that levodopa modifies Parkinson’s disease; the delayed-start evidence is that it does not, and the claim is offered as a means by which the withdrawal trial might have said so itself. A path that passes the origin by shows a contribution more stable than the one leaving, not a permanent one: a slow symptomatic response with a pattern of its own would produce the same miss, and only the comparison with the direction of progression, or a longer washout, tells them apart. A path aimed at the origin does not exclude a lasting effect that shares the direction of the symptomatic one, which is the collinear overprint and is invisible to this method as to its original. The worked figures above are invented to show the geometry and carry no evidential weight. A direction in a space of unlike scales depends on how the scales are set, which is why the setting must precede the data. And where the scale has in effect one dimension the proposal has nothing to say, which is how it ought to behave.