Karl Pearson and David Heron 
183 
tion data*. The recognition, however, that the fourfold table may give discordant 
results — a recognition made by the Biometric School within four years of the 
publication of the pigmentation investigations for eye-colour in man and coat- 
colour in horses — does not dismiss the fourfold table from practical statistics, but 
only from that portion of it where multiple contingency tables are available. 
Given a fourfold classification alone, how is it to be treated ? We reply unhesi- 
tatingly that in the great bulk of cases the use of tetrachoric r t is the best treat- 
ment. We base this on the experience that where nothing is known the Gaussian 
is far more likely to describe approximately the frequency than any other hypothesis. 
Even taken as a mere coefficient of association, tetrachoric r t is better than 
Mr Yule's Q or the Boas-Yulean <jf>, except for absolutely discrete units as in 
the purely theoretical Mendelian cases ; and in those cases the correlation of ranks 
is the correlation of variates, as Pearson indicated in his memoir of 1904f. 
(5) On the Surface of Constant Association and on "Natural" Equalisation. 
As we have indicated, Mr Yule never states adequately the conditions under 
which his coefficients of association and colligation are to be applied. He 
apparently considers the nature of the continuity of his frequency surface, if his 
variates are continuous, to be absolutely immaterial. Now in the case of every 
two continuous variates, whatever their nature, a frequency surface does exist for 
which Mr Yule's association or colligation is constant wherever the divisions may 
be taken upon which the fourfold table is based. Let n pg be all the first quadrant 
frequency corresponding to the total frequencies p and q of the two variates, where 
p and q are supposed to be absolutely known. If the association coefficient be Q 
and the notation 
a 
>> 
a+b 
c 
d 
c + d 
a + c 
b + d 
N 
we have (1 + Q)/(l - Q) = adjbc = x, say. 
* In the very same number of Biometrika (Vol. in. 1904) in which the Huxley Lecture appeared, there 
is a paper on the inheritance of pigmentation in the Greyhound ; it is the work of Pearson's Laboratory 
and started about the same time as the Huxley Lecture reductions. The following words occur : 
"When we first started work on the greyhounds, the method of contingency had not been developed, and 
accordingly we made tables for the inheritance of melanism and of red pigment and proceeded to find 
the correlations by the fourfold division process" (I. c. p. 252). And again " In order to compare the 
fourfold method with contingency methods, 16-fold tables and 25-fold tables were worked out to 
compare with the fourfold tables adopted for the inheritance of red and black pigment respectively" 
(p. 253). ' ' The results deduced by contingency D method are singularly uniform and steady as 
compared with those of the fourfold-table methods, and we believe, if it be adopted generally for 
such pigmentation problems, it will not only free us from any question of pigmentation scale, but 
afford a good result on a not excessive expenditure of calculating energy" (p. 253). It is clear that 
the Laboratory publicly admitted the difficulties of the fourfold-table method two years before Mr Yule 
started to criticise it as applied to pigmentation data ! Yet Mr Yule never mentions this fact. 
t "On a Generalised Theory of Alternative Inheritance with special reference to Mendel's Laws," 
Phil. Trans. Vol. 203, p. 53. 
