MY FATHER published a number of papers on blood analysis. In the proofs of one of them the following sentence, or something very like it, occurred: "Unless the blood is very thoroughly faked, it will be found that duplicate determinations rarely agree." Every biochemist will sympathise with this opinion. I may add that the verb "to lake," when applied to blood, means to break up the corpuscles so that it becomes transparent.

In genetical work also, duplicates rarely agree unless they are faked. Thus I may mate two brother black mice, both sons of a black father and a white mother, with two white sisters, and one will beget 10 black and 15 white young; the other 15 black and 10 white. To the ingenuous biologist this appears to be a bad agreement. A mathematician will tell him that where the same ratio of black to white is expected in each family, so large a discrepancy (though how best to compare discrepancies is not obvious) will occur in about 26 per cent. of all cases. If the mathematician is a rigorist he will say the same thing a little more accurately in a great many more words.

A biologist who has no mathematical knowledge, and, what is vastly more serious, no scientific honour, will be tempted to fake his results, and say that he got 12 black and 13 white in one family, and 13 black and 12 white in the other. The temptation is generally more subtle. In one of a number of families where equality is expected he gets 19 black and 6 white mice. It looks much more like a ratio of 3 black to 1 white. How is he to explain it? Wasn't that the cage whose door once seemed to be insecurely fastened? Perhaps the female got out for a while or some other mouse got in. Anyway he had better reject the family. The total gives a better fit to expectation if he does so, by the way. Our poor friend has forgotten the binomial theorem. A study of the expansion of would have shown him that as bad a fit or worse would be obtained with a probability of , or .0146. There is nothing at all surprising in getting one family as aberrant as this in a set of 20. But he is now on a slippery slope.

He gets his Ph.D. He wants a fellowship, and time is short. But
he has been reading *Nature* and noticed two letters* to that journal of which I was joint author, in
which I might appear to have hinted at faking by my genetical
colleagues. Thoroughly alarmed, he goes to a venal mathematician.
Cambridge is full of mathematicians who have been so corrupted by
quantum mechanics that they use series which are clearly divergent,
and not even proved to be summable. Interrupting such a one in the
midst of an orgy or Bhabha and benzedrine, our villain asks for a
treatise on faking. "I am trying to reconcile Milne, Born and Dirac,
not to mention some facts which don't seem to agree with any of them,
or with Eddington," replies the debauchee, "and I feel discontinuous
in every interval; but here goes.

* U. Philip and J. B. S. Haldane (1939).
*Nature*, **143**, p. 334.

Hans Grüneberg and J. B. S. Haldane (1940). *Nature*,
**145**, p. 704.

"I suppose you know the hypothesis you want to prove. It wouldn't
be a bad thing to grow a few mice or flies or parrots or cucumbers or
whatever you're supposed to be working on, to see if your hypothesis
is anywhere near the facts. Suppose in a given series of families you
expect to get four classes of hedgehogs or whatnot with frequencies
*p*_{1}, *p*_{2}, *p*_{3},
*p*_{4}, and your total is *S*, I shouldn't
advise you to say you got just *Sp*_{1},
*Sp*_{2}, *Sp*_{3} and *Sp*_{4}, or
even the nearest whole number. Here is what you'd better do. Say you
got *A*_{1}, *A*_{2}, *A*_{3} and
*A*_{4}, and evaluate

Your has three degrees
of freedom. That is to say you can say you got *A*_{1}
red, *A*_{2} green and *A*_{3} blue hedgehogs.
But you will then have to say you got *S* - *A*_{1} - *A*_{2} -
*A*_{3} purple ones. Hence the expected value of is 3, and its standard error is ; so choose your
*A*'s so as to give a
anywhere between 1 and 6. This is called faking of the first order.
It isn't really necessary. You might have , , and *A*_{1} = 9,
*A*_{2} = *A*_{3} = 3, *A*_{4} = 1. The probability
of getting this is , which is only just under .04. However, it looks
better not to get the exact numbers expected, and if you do it on a
population of hundreds or thousands you may be caught out.

"Your second order faking is the same sort of thing. Supposing
your total is made up of *n* families, and you say the
*r*th consisted of *a*_{r1},
*a*_{r2}, *a*_{r3}, *a*_{r4}
members of the four classes, *s*_{r} in all, you take

and sum for all values of *r*. Your total ought to be
somewhere near 3*n*. The standard error is , and it's better to be too
high than too low. A chap called Moewus in Berlin who counted
different types of algae (or so he said), got such a magnificent
agreement between observed and theoretical results, that if every
member of the human race had repeated his work once a month for
10^{12} years, they might expect as good a fit on one
occasion (though not with great confidence). So Moewus certainly
hadn't done any second order faking. Of course I don't suggest that
he did any faking at all. He may have run into one of those
theoretically possible miracles, like the monkey typing out the text
of Hamlet by mere luck. But I shouldn't have a miracle like that in
your fellowship dissertation.

"There is also third order faking. The 4*n* different
components of should be distributed round
their mean in the proper way. That is to say, not merely their mean,
but their mean square, cube and so on, should be near the expected
values (but not too near). But I shouldn't worry too much about the
higher orders. The only examiner who is likely to spot that you
haven't done them is Haldane, and he'll probably be interned as a Red
before you send your thesis in. Of course you might get R. A. Fisher,
which would be quite as bad. So if you are worried about it you'd
better come back and see me later."

Man is an orderly animal. He finds it very hard to imitate the
disorder of nature. In fact the situation is the exact opposite of
what the reader of Paley's *Evidences* might expect. But the
problem is an interesting one, because it raises in a sharp and
concrete way the question of what is meant by randomness, a question
which, I believe, has not been fully worked out. The number of
independent numerical criteria of randomness which can be applied
increases with the number of observations, but much more slowly,
perhaps as its logarithm. The criteria now in use have been developed
to search for excessive irregularity, that is to say, unduly bad fit
between observation and hypothesis. It does not follow that they are
so well adapted to a search for an unduly good fit. Here, I believe,
is a real problem for students of probability. Its solution might
lead to a better set of axioms for that very far from rigorous but
none the less fascinating branch of mathematics.

*Eureka*,
**6**.

Reproduced from Eureka 27 pages 21-24.

HTML conversion Copyright © 2002-4 The Archimedeans.

Errata: "122073" should read "122753", "or Bhabha and benzedrine" should read "of Bhabha and benzedrine", "" should read "".

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