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On Theories of Association 
the theoretical frequency is zero often occur*." In other words, observation 
records something as occurring which existing theory says cannot occur, and 
Dr Pearl asks for a criterion which shall make the impossible only mildly 
improbable. He must either remould his theory or explain away his observations. 
We see no alternative. 
Dr Pearl's demand for a criterion which shall not be crucial, but allow elastic 
records to fit a plastic theory, is well illustrated by the following paragraph from 
the work of another American Mendelian who is convinced that "feeble-mindedness" 
is a Mendelian unit character, but has found his " green ball " in the normal 
offspring of two feeble-minded parents: 
"These two are apparent exceptions to the law that two feeble-minded parents 
do not have anything but feeble-minded children^. We may account for these 
two exceptions in one of several ways. Either there is a mistake in calling them 
normal, or a mistake in calling the parents feeble-minded ; or else there was 
illegitimacy somewhere and these two children did not have the same father as 
the others of the family. Or %ve may turn to the Mendelian laiv and we discover 
* The remainder of Dr Pearl's paragraph runs : "To determine the probable error of the individual 
frequency in measuring the goodness of fit of Mendelian observation and theory, as was first practised 
by Weldon, and later by Johannsen and by Mendelian workers generally, does not appear to the writer 
to be an altogether sound procedure. It fails to take account of the correlations in errors amongst the 
several frequencies. Yet these are just as important and just as certainly existent in a Mendelian 
' category ' type of distribution as in the ordinary variation polygon of a continuously variable 
character.... Pearson's test covers this point, and were it not for the other difficulty noted above 
would be much more widely useful in Mendelian work than is actually the case" (loc. cit. p. 204). 
The " widely useful " test in Mendelian work is quite obviously one which will overlook negation of 
theory, or not drive the observer back to question the validity of his records or his categories. But 
there is a misstatement in the above sentence which needs correction. If there be only alternative 
categories, e.g. the total of (RTtys compared with the total of (Di?)'s + (2)Z))'s, then Pearson's test is 
absolutely identical with the probable-error test. This is of course well recognised ; for, if and be 
the observed frequencies and »i and iu the theoretical frequencies, 
2 = ("U^i) 2 (m* z ntf = _ )2 / 1 1 \ 
= -i— ■ a- =-!— ? si-r-, where N-ni+n 2 , 
*('-}) »( l -J) 
(Deviation of either category) 2 
(Standard Deviation) 2 ' 
and for this case P — . I e d%. This test therefore Weldon applied with perfect legitimacy 
to the consideration of Mendelian quarters. When Weldon came in tlte very paper cited by Dr Pearl to 
test more complex Mendelian results, he did not fail to take account of correlations in errors, and 
actually applied Pearson's criterion (Biometrika, Vol. i. p. 235). Dr Pearl's sentence therefore requires 
remodelling ; he has clearly failed to appreciate what Weldon was doing. 
t The pedigrees published by Weekes and Goddard show other exceptions to Mendelian law, which 
they appear to have overlooked. This is confirmed by similar pedigrees in our Laboratory. The very 
idea that the continuous and highly variable character "feeble-mindedness" is a "unit character" 
in the Mendelian sense will do much to check real research into the grave complexities and difficulties 
of this very vague and broad category. 
