The Table That Chose the Champion

I.The First Known Thing — ten events and one number

athletics · the decathlon scoring tables

The decathlon is ten events in two days, and the difficulty at its centre is not athletic but arithmetical. A time in the 100 metres, a height in the pole vault and a distance in the discus have no common unit, and the contest can be decided only by a table that converts each into points and adds them. The table is an instrument of the governing body rather than a fact of nature, and it has been rebuilt several times. The tables of 1912 were linear, with points set against the Olympic records of the day. The Finnish tables of 1934 introduced progression, with each further increment of performance worth more than the last. The tables of the 1950s were progressive throughout; those of the 1960s and 1970s mixed progressive running events with regressive throws; and the tables adopted in 1984 and in force since 1985 fix each event by a formula of the same shape, points equal to a constant times the margin over a zero-point performance raised to an exponent, with the exponent set separately for every event.

the exponent is the argument

In the tables now in use the exponents are near 1.8 and 1.9 for the running events, near 1.35 to 1.42 for the jumps, and between 1.05 and 1.1 for the throws. An exponent above one makes the schedule convex: a second of improvement is worth more to the fast man than to the slow one, and a centimetre in the long jump is worth more at seven metres than at six. The history of the tables is the history of what this curvature does to the competition. Each generation of tables has been criticised, with numbers, for the kind of athlete it selects. The strongly progressive tables of the 1950s were found to favour specialists, since an outstanding mark in one event bought more than solid marks in all ten, and the regressive throwing schedules of the following decades were found to make elite throws nearly worthless and to hand the event to the runners. The present tables are argued over on the same ground: the near-linear throws and the steeply convex sprints are said to make a fast decathlete with ordinary throws the expected champion, and proposals for revised tables are published every few years with the aim of restoring balance.

how a table is calibrated

The decisive feature is how the exchange rates between events are set. They are not set by any judgement of what a metre of javelin is worth against a tenth of a second in the hurdles, because no such judgement exists. They are set by looking at what decathletes actually do. The constants are chosen so that performances of comparable standing in the population of decathletes — the tenth-best mark of a season in each event, say, or the seventy-fifth — earn comparable points, so that the table prices each unit of each event by how rare it is among the people who have been competing. Two things follow and both are known to the people who make tables. First, because rarity climbs steeply in the tail of every event, a table calibrated on rarity is convex unless something is done to flatten it, and a convex table rewards the athlete who concentrates his excellence. Second, the table is only calibrated where the population is. A performance outside the range the table was built from is priced by continuing the formula past the data, and the further out it lies, the more the price is an artefact of the exponent rather than of anything the calibrating population ever did. Neither observation has any application the sport can see beyond the sport.

II.The Second Known Thing — the score that stands for the disease

drug discovery · disease definition, indication choice and endpoint selection

Many diseases have no single measurable quantity that is the disease, and where that is so the trial endpoint is a composite index. Rheumatoid arthritis is the discipline's most developed case. The Disease Activity Score was built in Nijmegen at the end of the 1980s by asking rheumatologists, in an ordinary clinic, which patients they judged to need a change of treatment, and then finding by discriminant analysis the combination of measurements that best reproduced that judgement. The result, in its 28-joint form of 1995, is a weighted sum: 0.56 times the square root of the tender joint count, 0.28 times the square root of the swollen joint count, 0.70 times the natural logarithm of the erythrocyte sedimentation rate, and 0.014 times the patient's global assessment. A score below 2.6 is remission. The square roots and the logarithm are the exchange rates; they were fitted, not chosen, and they were fitted on patients treated with the drugs of that decade, in whom the sedimentation rate rose and fell with the joints and rarely strayed far from the range the clinic saw.

the other tables

The field has several tables and knows they disagree. The American College of Rheumatology's response criteria of 1995 are a threshold composite: a patient responds at the twenty per cent level if tender and swollen joint counts both improve by a fifth and three of five further measures do too, so that improvement beyond the threshold buys nothing and a large gain on one measure cannot rescue a shortfall on another. The Simplified and Clinical Disease Activity Indices of the following decade are linear sums, the first including C-reactive protein in milligrams per decilitre and the second omitting the laboratory altogether. Each is validated against the others and against outcomes, and the choice among them for a trial's primary endpoint is a question sponsors and regulators take seriously, because a drug can reach remission by one table and not by the next.

the episode the field remembers

The disagreement became a controversy with interleukin-6 receptor blockade. Tocilizumab acts directly on the hepatic acute-phase response, so that under treatment the sedimentation rate and C-reactive protein fall to the floor whatever the joints are doing. In its trials, remission by the Disease Activity Score was reached considerably more often than remission by the indices without a laboratory term, and the discrepancy was traced, in a series of papers around 2010 and 2011, to the acute-phase component. The field's response was a rule of practice: for drugs acting on interleukin-6, prefer the clinical index. The episode is filed as a peculiarity of one mechanism and one term. What the field did not do, and had no vocabulary to do, was to state the general condition under which any index will misprice any mechanism, and to compute it before the trial rather than discover it in the readout.

III.The Bridge

no paper on scoring tables cites a disease index; no index paper has heard of the decathlon

The halls do not meet. One is a small technical literature on the scoring of a track-and-field event, written by statisticians of sport for a governing body; the other is clinical rheumatology and the methodology of outcome measures, written for regulators and trialists. Neither cites the other, no review spans them, and their shared words — score, table, points — are used on both sides without any sense that the same object is being described. Yet the object is the same. In each, several quantities with no common unit are combined into one number by exchange rates fitted to the distribution of a reference population, and in each, the number is then used to rank performers who may not resemble that population at all.

X: a fitted exchange rate is a price for the population it was fitted on, and an extrapolation for everyone else

Call it extrapolated pricing. Where several incommensurable components are combined into one score by exchange rates fitted to a reference population, the score prices a unit of each component by its standing in that population and not by its worth, and the pricing has a curvature: convex where the fit follows rarity into the tail, concave where a root or a logarithm has been applied to flatten it, and saturating where a threshold has been imposed. Three consequences follow. The curvature, not the total, decides whether concentration or breadth wins, so that the type of the champion is fixed by the table before any performer arrives. The exchange rate is meaningful only over the range the reference population occupied, and a performer who moves one component beyond that range is priced by extrapolation, in the direction and to the degree that the curvature there dictates. And because the reference population is by construction the performers who existed when the table was built, a genuinely new kind of performer — a new event technique, a new mechanism of action — is precisely the case the table was never calibrated on, so that the index will be found to misprice exactly the innovations it is asked to judge. The interleukin-6 episode is not a peculiarity. It is what a logarithm fitted on sedimentation rates between roughly ten and sixty does when handed a drug that puts the rate at two, and the size of the mispricing could have been read off the calibration range and the transform's slope at its edge before a single patient was dosed.

IV.The Question That Becomes Possible

neither discipline would ask this alone

The statistician of sport has no reason to ask it, because the population of decathletes changes slowly and the table is revised by committee when the complaints accumulate. The rheumatologist has no reason to ask it, because the field treats the disagreement between its indices as a matter of validation against outcomes, settled index by index, and has no notion that a mechanism could be scored against a table before its trial. Held together they produce a question that can be answered from documents and data that already exist.

the extrapolated pricing hypothesis

For a composite index whose exchange rates were fitted on a reference population, can the disagreement that a new mechanism will produce between that index and its alternatives be predicted before the trial, from two quantities available in advance — how far the mechanism's expected effect on each component lies outside the range the calibrating population occupied, and the curvature of the index's transform in that region — and does that predicted mispricing order the mechanisms as the trials then find them ordered? In rheumatoid arthritis the prediction is definite. Interleukin-6 blockade, which moves the acute-phase reactants furthest outside their calibration range, should show the largest discordance between the laboratory-bearing and laboratory-free indices; Janus kinase inhibition, which lowers C-reactive protein substantially through the same signalling but less completely, should show a smaller one; tumour necrosis factor blockade, under which the reactants fall in proportion to the joints, should sit within the calibration range and show little; and the ordering should be the same for the linear protein term of one index and the logarithmic sedimentation term of another, differing only in size as the curvatures differ. The stronger form of the question concerns the type of the champion: on item-level data, sweep the curvature of the composite from saturating through concave to convex while holding the components fixed, and ask whether the rank order of the indication's drugs changes. If it does, then the choice of primary endpoint in this indication has been choosing the winner's mechanism, and the field can say by how much.

why it would matter

Because it converts the choice of endpoint from a selection among validated instruments into a computation about a mechanism, made before the trial from the index's own fitting population and the drug's phase 2 component effects; because it turns the interleukin-6 rule of practice into a general one that names in advance which future mechanisms will need which index; and because the curvature sweep would tell an indication whether its composites have been quietly selecting breadth over depth, or the reverse, for thirty years. And if it fails — if the discordance is explained by nothing more than whether the index contains a laboratory term and the drug acts on the liver, which is the rule the field already has — then the tables have supplied a vocabulary and nothing else.

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 is unusually cheap, because the reference populations are described in the index papers, the component-level trial data sit in the shared data repositories, and the prediction can be written down before the data are opened.

the honest status

Note the narrowness. The claim concerns composite indices whose exchange rates were fitted to a reference population, and mechanisms that move at least one component outside that population's range, and it is made for one indication in which both the indices and the item-level data are unusually well documented. It says nothing about single-quantity endpoints, nothing about composites whose weights were set by consensus rather than fitted, and nothing about whether any index is wrong; a laboratory term that misprices a hepatic drug may be exactly right for the patients it was built for. The analogy is imperfect in a way worth stating. A scoring table is a convention adopted by a body that may change it at will, and it answers to no truth beyond fairness; a disease index answers to joint damage, disability and death, and is validated against them. What transfers is not the sport's politics but its calibration logic: that exchange rates set from a population are prices only within that population, that their curvature decides what kind of performer wins, and that the case lying outside the calibration is exactly the new case the instrument is asked to adjudicate. Until the calibration ranges have been recovered and the discordances compared against the field's own binary rule, this remains what its stamp says: a candidate. A structural echo between two halls of the library, waiting for someone with the tools of both to walk the corridor.