A displaced census leaves a hole, not a pile

A floor no better fit can lift ended on the finding with the most consequence. A shell’s place in Raup’s space is read from a section, by estimating its expansion $W$ and its distance from the axis $D$, and both come from one guessed centre. Swing that guess round and the pair traces a closed curve, one that crosses the contact line on the reference shell, with a mean product ten per cent low. So a mislocated centre does not blur a shell’s position. It moves it, and it moves it in a preferred direction.

From that it was a short step to a census. A scattered error is invisible in one specimen and obvious in a hundred, because the spread is the evidence. A systematic one is invisible in both, unless the shape of the cloud gives it away. And there was an argument that it might: the axis distance is read twenty-two times better far from the boundary than on it. If shells close to the line are moved a lot and shells far from it hardly move, a real population ought to be compressed towards the line, and so should show a density rising towards it. Density rising towards the contact line is exactly a claim people have made about real shells, that nature prefers to hold whorls just touching. The essay proposed a synthetic census to see whether the artefact was large enough to matter.

The synthetic census has been run, and its answer is that the prediction had the wrong sign. There is no pile. There is a hole. The test that follows from the hole is better than the one proposed.

A census with nothing to hide #

The population is as neutral as it can be made. Shells are drawn uniformly in the logarithm of the expansion and in the distance from the axis, $W$ from 1.1 to 6 and $D$ from 0.04 to 0.95, and every one whose whorls would touch is rejected, because a section whose inner wall is buried cannot be read at all. Twenty thousand remain. None prefers any place.

Each is then read once, in the same way as the reference shell’s locus. A centre is guessed, off by a stated share of the innermost radius, in a direction of its own, and from it the expansion is fitted to the outline and the axis distance averaged over a dozen azimuths. Four errors are run: 0.02, 0.05, a tenth and a quarter of the innermost radius. Those are sizeable. Earlier readings priced a good hand-found centre at the smaller end and a quick one at the larger.

The map looks much as a person might expect a map to look. With no error the cloud sits cleanly above the line, and a dense edge runs along it. With an error it loses that edge, and it has acquired something new: a thin scatter of points read below the line, where no section with free whorls has any business being. That small scatter is the most useful thing in the figure, and it comes back at the end.

How far a shell moves #

The first part of the earlier account holds. The product is biased low, and shells are carried on average towards the line. Its size depends on the shell. The ten per cent quoted before was at one shell, and over the whole population the mean change in the logarithm of $W \cdot D$ at a quarter of a radius is $-0.044$, about 4.4 per cent. It is $-0.016$ at a tenth. For shells of small expansion, $W$ under 2, it hardly exists until the error is a quarter, at $-0.016$. For $W$ from 3.5 to 6, it is $-0.025$ at a tenth, and $-0.054$ at a quarter.

So the direction was right, and the bias is real across most of the space. It is about half the size the reference shell suggested, and much stronger for tightly coiled shells than loosely coiled ones, since the shift is measured against the innermost radius and a large expansion makes that radius small.

No pile #

If the census had been compressed against the line, the ratio of read density to true density would rise above one near it. It does the opposite. In the bin nearest the line, within 0.05 in the logarithm of $W \cdot D$, it is 0.86 at an error of 0.05, 0.83 at a tenth and 0.81 at a quarter. In the next bin up it is 0.99, 0.94, 0.94. A little further out it runs near one, with the scatter of about two or three per cent that a bin of this size carries. Nowhere in the plot does the density rise above what it started with by more than that scatter.

Why it goes that way is plain once the map is in mind. The shells nearest the line are not stuck there. A shell read a little low, near the line, crosses it, and so leaves the region above the line altogether. A shell near the line read a little high moves up and away. Both of those empty the bin nearest the line. What would fill it are shells from further up, moved down, and there are not enough of those, because the bias is small and the spread is large: a shell’s position changes by much more in a random direction than it does in the biased one. Spread beats bias. A band next to an edge loses its members over the edge at a rate nothing refills.

And the argument about the axis distance being read twenty-two times better far from the line does not come into it as it seemed to. That was a statement about the size of the error on $D$ at a fixed $W$. Across a census, the error that decides most of what happens is the one on $W$, which a guessed centre moves a long way, and much more at large expansion. It moves the product up about as often as down, even if the mean is a little low.

A preference hidden rather than forged #

The result reverses the concern that led to the measurement. The fear was that a claimed preference for the line might be manufactured by the reading. If the map here is right, a real preference would be understated instead: if real shells do crowd against the line, a census read with uncertain centres would report them as crowding less than they do, by a sixth at the nearest bin at the errors here, and it would move some of them to the other side.

So a published occupancy map that shows density rising towards the line has shown it despite its centres rather than because of them, as far as this one error is concerned. This doesn’t make such a claim right. The band nobody can be placed in and its joint form both say individual specimens near the line cannot be placed there at all. And there are other errors than this one, especially a real census’s choice of which shells to section, which nothing here touches. But one specific way of getting that pattern falsely has been closed, and from an unexpected direction.

Shells that give themselves away #

The earlier essay hoped for a signature of the displacement in the census itself, and the map has one, of a kind that has nothing to do with shape. Every shell in this census was seen to run free: its inner wall could be followed, which is why it could be read at all. A reading of that same shell that gives a product below one is a contradiction of the section by its own numbers. It is not an error a person could overlook, since the section’s whorls are right there, and it is exactly the kind of thing that gets rounded away or put down to scatter.

Those contradictions are common. At an error of 0.02 of the innermost radius, 0.8 per cent of shells read below the line. At 0.05, 1.6 per cent. At a tenth, 2.9. At a quarter, 6.3. All four are from a population with not a single shell actually below it. With shells drawn from a narrower range of $W$ the rate differs, and it rises with the expansion at the lower errors: at a tenth, 1.7 per cent for $W$ under 2, 2.6 for the middle range and 3.7 per cent for $W$ from 3.5 to 6.

At a quarter of a radius the order changes, with the middle range at 8.1 per cent above the widest at 4.9. Those widest shells are moved so far and so unevenly by then that many more go upward as well, with nothing to do with the line, and fewer start close to it.

What each signature costs #

The two signatures have very different prices. A census could look for the hole, the shortfall of shells near the line. To tell that shortfall from what the census would have had with no error, at the usual 5 per cent and 90 per cent power, takes about 6,500 specimens at an error of a tenth, and 5,500 at a quarter. Nobody has that, and it would also require knowing what the density should have been, which is the whole of what such a survey is trying to find out. So the hole, though real, is of no practical use.

The contradictions are another matter. To have a nine-in-ten chance of finding at least one shell read below the line takes 143 specimens at an error of 0.05, 79 at a tenth and 36 at a quarter. That is the size of an ordinary published morphometric sample. It also has a precision that a density has not: the share of contradictions can be converted back into an estimate of the centre error, since it runs from 0.8 per cent at 0.02 to 6.3 at a quarter, if the population’s range of $W$ is at all known. A census carrying its own estimate of how badly it knows its centres is more than most published morphospaces carry.

The test requires one thing that a survey may have thrown away. It requires that a reading below the line is not silently removed, and that its section is checked. A worker who sees a product just below one on a shell plainly free of contact might well write it down as about one, or try again. Those would be the only readings that can tell anyone that the centres were off. Discarding them, or re-measuring until they were on the right side, would remove the check and leave the hole.

How deep a contradiction goes #

A reading below the line would be easy to explain away if it were always within a whisker of the line, and that is worth measuring, because a worker would treat a product of 0.99 very differently from one of 0.8. At a centre error of a tenth, half of the shells read below the line are read more than 0.061 below it in the logarithm of $W \cdot D$, about six per cent, and one in ten is more than 0.25 below it, a product under about 0.78. At a quarter of a radius those are 0.082 and 0.33. Even at 0.05 of a radius, the median is 0.033 and one in ten is 0.116 below. So most of these readings are a good way beyond any rounding, and they are not a matter of a shell sitting on the line and tipping over.

The shells they come from are close to the line but not all that close. At a tenth, half of them truly lie within 0.056 of it and one in ten lies more than 0.21 away. That is a further signature, since a shell far above the line in truth, read well below it, is a very large disagreement with what is plainly visible on the section. A reader who takes one of those as evidence of nothing is giving up the best check the data has on itself.

Across the space #

The hole is not even across the space. Within a tenth of the line at an error of a tenth, the shells with $W$ under 2 go from 25.3 per cent to 24.3, a loss of about 4 per cent of their members. Those with $W$ from 2 to 3.5 go from 6.9 to 6.3, about 9 per cent. Those with $W$ from 3.5 to 6 go from 3.4 to 2.1, a loss of 38 per cent. So the widest expansions have most of what is missing, and the band of shells close to the line at large expansion would be most under-counted by a real survey.

That turns the expected signature on its head in a further way. If a real census were to show relatively few shells near the line at large expansions and many at small ones, that would be exactly the shape this error predicts, with no biology involved at all. A real preference, if it did not depend on expansion, ought not to have that shape. A comparison of nearness to the line by expansion is therefore a second way of looking for this error, which should be read before anyone interprets that pattern as a fact about what shells do.

What this does not establish #

The population is uniform and arbitrary, and a real one is not. The ratio of densities at each distance is less sensitive to that than a density itself would be, though a population concentrated in one part of the space would change which errors matter. Only a guessed centre has been tried. The other parts of a real reading, tilt and a partial section above all, have their own systematic errors and are not in this map. Each shell has been read with $T$ at nothing, on a planispiral form, with no third number to complicate things. The error in each shell’s centre is independent, whereas a single worker’s habit in locating centres might be shared across a whole sample and that would behave as one. And shells with buried inner walls are excluded by construction, as any real census does, and it would be interesting to know how many of them read as free.

Measurements that would overturn it #

A density that rose above its true value near the line in a census read with a stated centre error would bring back the earlier prediction. A real census containing no reading below the line over several hundred specimens would mean that its centres were much better than a quarter of a radius, or that such readings were being removed, and a worker’s own records could tell which. A population with a strong real gradient that nonetheless piled up against the line under this reading would mean the hole belonged to this population rather than to the error.

Still open: a census whose centres share one habit #

Each shell here has a direction of its own, and that is what spreads the census out. A worker who habitually sets a centre slightly too far in, along the shell’s axis of largest whorl, has an error of one direction relative to every shell. Then a census’s displacements would all be alike, and the spread that made a hole would be gone. The next measurement runs the same census with centre errors drawn with a stated bias in direction relative to the aperture, from fully independent to fully shared, and asks at what degree of sharing the bin nearest the line stops losing members and starts gaining them, whether the contradictions still appear then, and what a real worker’s measured centres would have to look like for the pile that was predicted here to appear after all.