168 
On Theories of Association 
cannot do it without confessing himself hopelessly in error!* — If or 77^ is the 
right thing to use for a fourfold table as Mr Yule now suggests, then his co and his 
Q are hopelessly wrong, for all the selections which do not alter his Q — and the fact 
that it is not altered by selection is according to him one of its great merits— do 
alter, and this to any extent we please, his " theoretical value of the correlation." 
This very point was emphasised by one of the present writers in a paper recently 
published in Biometrikaf but Mr Yule appears wholly to have missed the essential 
features of that criticism. The points were that (i) there was a wide range of 
values of Q — Mr Yule's coefficient of association — for a surface of the same correla- 
tion ; if it is impossible to compare the values of Q for the same surface divided at 
different places with any intelligible result, what possible comparison can be made 
of Q from one system to a second ?, and (ii) the values of Q and r/ llc or (f> are wholly 
different and tend in opposite directions as we change our divisions. Under 
" wholly different " we include liability to be " wholly differently interpreted." 
Both Q and range numerically between 0 and 1. Therefore in estimating the 
meaning of Q and or we have to consider where they stand on this range. 
Mr Yule examining the associations between developmental defects, nerve signs, 
low nutrition and mental dulness finds values ranging from 75 to "95 for his 
coefficient Q. He comments : " The associations are, however, all high (very high 
compared with most coefficients of organic correlation with which one has to dealj), 
ranging from '784 (? '750) to "952 §." Elsewhere in the same paper Mr Yule 
speaks of "174 as a "very small association," and a - 8 to - 9 association "as very 
high indeed ||." We know accordingly what Mr Yule understands by high and low 
association. Indeed if a scale of values is to lie between 0 and 1, those approaching 
0 must be very low and those approaching 1 must be very high. Now Heron 
applied Mr Yule's or Dr Boas' " theoretical coefficient " to precisely the same data 
as those for which Mr Yule had calculated his association Q, and found that it was 
very high. Heron found that for Q = - 921 and "753, the " theoretical value of r " 
= "011 and '006 respectively. If both these ways of investigating relationship are 
valid, then 'Oil and "006 must on a correlation scale represent a "very high degree 
of association." It would be interesting to know how Mr Yule would describe 
<f> = "95 or what represents a low association, if "01 corresponds to a high degree of 
association ! — But any one who is familiar with coefficients of correlation — and 
or is a real coefficient of correlation, — knows that values of "01 and under are 
extremely low values and, whatever their probable errors may be, are of no 
significance for purposes of prediction. All Mr Yule can say in reply to Heron's 
statement that one of Mr Yule's methods gives very high relationship and the 
* See also pp. 172-4 below, where this point is touched on again. 
t "The Danger of Certain Formulae suggested as substitutes for the Correlation Coefficient," 
by David Heron, Vol. vin. p. 109. 
% Here and at other points of his earlier papers, Mr Yule apparently considers that Q is really 
comparable with the true correlation. 
§ Phil. Trans. Vol. 194 A, p. 300. See also "high degree of association," Theory of Statistics, 
p. 34. 
|| Ibid. pp. 289 and 296. 
