The Journal of Heredity 77:138.1986.
More on the too-good-to-be~ true paradox and Gregor Mendel
A. W. F. Edwards
In querying Fisher's(2) conclusion that Mendel's data were falsified in some respects, Pilgrim(3) produces some superficial arguments against statistical inference, all of which can be countered. As is well-known, Fisher's analysis led to a succession of Chi-square values each of which was improbably small and which together gave Chi-square84 = 41.61 (P~0.99993). Weiling(4) pointed out that adjusting Fisher's calculations so as to allow for the ascertainment bias to which Fisher had himself drawn attention increases this to Chi-square84 = 48.91 (P~O.999), and that a further increase occurs if it is assumed that a proportion of seed does not germinate. However, the impact of Fisher's analysis is not much lessened by these adjustments, and Pilgrim uses more general arguments, as follows.
First, he argues that it is simply not possible to conclude from a perfect or near-perfect fit of hypothesis and data that the fit is suspiciously good, especially if the perfect fit corresponds to the most probable outcome (as with data of 500:500 in a genetic segregation of 1:1). There is much to be said for this argument when viewing an isolated case, but it fails on Mendel's data because as many as 84 degrees of freedom are involved. The plain fact is that the expectation of Chi-square1 is, for a binomial random variable, exactly 1, so 84 of them have an expectation of 84; if, then, 84 of them can only muster a total of 48.91, something is awry. One can applaud the lucky gambler; but when he is lucky again tomorrow, and the next day, and the following day, one is entitled to become a little suspicious.
Second, Pilgrim asserts that Fisher multiplied together the P values from the several experiments, an unjustified procedure. But Fisher did not do that; he added the independent Chi-square values instead, in accordance with accepted procedures.
Third, Pilgrim raises the specter of the invalidity of probabilities viewed after the determination of the event (Fisher himself used to stress this point to his students by comparing the probability that he would have a son who had a son, etc., for a hundred generations hence, with the probability that he had a father who had a father, etc., for a hundred generations back). It is true that the probability of realizing Mendel's precise data (on the hypothesis of simple Mendelian segregations) is vanishingly small, but that was not the basis of Fisher's argument. His grounds for concluding that Mendel's data were falsified were not that it was exceedingly improbable that they would recur exactly on a repeat of the experiments (which it is) but that it was very improbable that any results so close to expectation would recur on a repeat. To put the point another way, there are millions of possible results that might arise from a repeat of Mendel's experiments, and even the most probable has an infinitesimal probability, but there are only two results pertaining to the value of Chi-square84 on a repeat : larger than or smaller than the observed value of 48.91.
I do not wish to imply that the logic of tests of significance
is beyond criticism(1), but the criticism would have to be much
deeper than Pilgrim's before it would have any effect on Fisher's
careful analysis of Mendel's data.
1.Edwards, A.W.F. Likelihood. Cambridge University Press, 1972. (Reissued in paperback 1984).
2.Fisher, R.A. Has Mendel's work been rediscovered? Ann. Sc. 1:115-137. 1936.
3.Pilgrim, I. The too-good-to-be-true paradox and Gregor Mendel. J. Hered. 75: 501-502. 1984.
4.Weiling, F. Hat J. G. Mendel bei seinen Versuchen "zu genau"
gearbeitet? -Der Chi-square Test und seine Bedeutung fur die Beurteilung
genetischer Spaltungsverhaltnisse. Der Zuchter 36: 359-365.
The author's address is: Department of Community Medicine, University of Cambridge. School of Clinical Medicine, Cambridge CR 2ES England.
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