Karl Pearson and David Heron 
215 
Mr Yule's association coefficient is for the two cases 
Q = l and Q = -l. 
In the first case the tetrachoric r t is +10 and in the second tetrachoric 
r t = — 10, since A = 2"0 and k = 2'6*. As another illustration take 
5000 
0 
5000 
4772 
228 
5000 
9772 
228 
10000 
This table gives the least positive <p for the given marginal frequencies ; and 
with these frequencies </> can never rise above the value + "1527 which it has for 
this table. But both Q and tetrachoric r t = + TO. 
The minimum value of for the same total frequencies is given by the table 
4772 
228 
5000 
5000 
0 
5000 
9772 
228 
10000 
This has <j> = — '1527, while Q = — 1 and tetrachoric r t = — 1. In other words, 
<f> is restricted to lie between + '1527 and — T527, while Q and tetrachoric r t may 
pass through the whole range + l'O to — TO. Why has Mr Yule not pointed out 
these facts when recommending (j> as obviously suitable for all fourfold tables when 
we get rid of normal variation ? Clearly, if he had done so his criticism of 
uncorrected contingency would have been shown to apply with still greater force 
to his own coefficient. The standard deviations of the variates of the fourfold 
a | b . , V(a + c) (b + d) , V(a + b) (c + d) 
table — \—, are not given by — — A and ~ — A and the 
c | d & N N 
. . ad — be . . . , . . , . 
product moment by — — uA , unless we may concentrate each variate into 
points at distances A and A' which Mr Yule takes as units. But it is clear that 
when we have done this we (i) have fixed the standard deviations of the variates, 
(ii) can shift our dichotomic lines throughout the whole ranges of A and A' with- 
out influencing the result. If the variates are really not concentrated into points 
their standard deviations are wholly independent of the dichotomic lines, and 
every shifting of those lines will change the proportions of each variate falling into 
the two categories. The independence of the standard deviations of the dichotomic 
lines is the advantage of the tetrachoric r t over ; and in practical Mendelian 
statistics it is in most cases impossible to shift the dichotomic lines without 
* Actually /( = 1-999087 and k = 2-597180. As we have already indicated and shall further emphasise 
in Appendix I, the value of tetrachoric r t is indeterminable by the usual method in such cases. 
