Abstract
PROF. FISHER1 is an apt controversialist, but he knows as well as I do that what I understand by graduation is not confined to curves; that I should term graduation the fitting of a binomial to a series of observations, or the determining whether a system of correlation coefficients could be reasonably supposed to have arisen from samples of material drawn from a population with a given correlation coefficient. The difference between Prof. Fisher and myself lies in the use (and abuse) of the acceptance and rejection of hypotheses. There is only one case in which an hypothesis can be definitely rejected, namely, when its probability is zero. He cites a case which I criticised in the paper he refers to, in which two recessives (say) had produced a dominant, and theory was absolutely contradicted. It did not require an application of the (P, 2) test to assert that either theory or observations must be rejected! I merely showed that the (P, 2) test did not fail in this case. But let us look into what actually happens, and I cannot do better than illustrate it on some statistics provided by Sir James Jeans in NATURE of September 14, 1935 (p. 432). He is comparing the eccentricities of visual binaries, 116 in number, against a theory of equipartition (not a curve, but frequencies are considered). His data expressed by a frequency series run as follows:— If the P, 2 test be applied to the total 116 binaries, we have P < 0.000,0005. On the other hand, if it be applied to the 83 stars of lowest eccentricity, P = 0.79. In neither case can you say the hypothesis is true or false. You reject it in the former case because it is a poor graduation, you say in the latter case that it is a reasonably good graduation because 79 per cent of random samples would, were the hypothesis true, give a worse result than the observations do. But in accepting it as a working graduation, you do not assert its truth any more than you assert the falsity of the hypothesis applied to the whole 116 stars; you merely say the latter is a bad graduation, and try for a better. Had Sir James Jeans taken all stars with eccentricity 0.07, instead of 0.06, he would have found P = 0.105, and if he had proceeded to e 0.08, the result would have been P = 0.00001, that is, he might have got a worse sample in 100,000 trials. Actually he gives his reasons for cutting off the higher eccentricities. With them I am not concerned, although the exact cutting off at e = 0.06 is not discussed; the difficulty of detecting high eccentricity binaries and of then determining their orbits may account for the irregularity of the last four frequency entries, as he holds, or there may be other reasons why the falling-off occurs at e = 0.06. Hypotheses non fingo!
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01 October 1935
In Prof. Karl Pearson's letter on “Statistical Tests” in NATURE of October 5, p. 550, the values of the eccentricities e should have been given in tenths, not hundredths; that is, e = 0.6, not 0.06. This has no bearing on the argument.
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NATURE, 136, 474, Sept. 21, 1935.
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PEARSON, K. Statistical Tests. Nature 136, 550 (1935). https://doi.org/10.1038/136550a0
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DOI: https://doi.org/10.1038/136550a0
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