276 
On Theories of Association 
The weighted standard deviation test, which is probably a better one, shows 
that r t is superior to Q for the Brothers table and equal to it for the Father and 
Son table ; both are terribly bad and a slight change in the nature of the test 
may make one or other apparently slightly superior. In this eye-colour material 
in its present state we are quite clear that the tetrachoric method applied 
to extremely skew divisions will not give consistent results. But this had been 
stated years ago, in passages which Mr Yule refrains from quoting. We are 
equally clear that Q gives no better results, while for the bulk of tables of 
statistical practice, it certainly gives less stable results than r t . 
(G) Age of Husband and Wife Data. 
This is a case of extreme skewness* selected by Mr Yule to show how tetra- 
choric r t varies. It is also a case of heterogeneity, for second marriages are mixed 
up with first marriages. With his usual ingenuity Mr Yule has exhibited a curve 
and a table, in which he has increased the number of the terminal values, where 
tetrachoric r t gives too low values and has little weight, and shown few of the 
central values where r t is relatively steady and has great weight. Further, he 
tells us that : 
" From the standpoint of the calculator, however, the table presents the 
disadvantage that the correlation is high, viz. '91, and the approximation to the 
value of the normal coefficient correspondingly slow, eight to ten or twelve terms 
of the equation being necessary to give a value fairly trustworthy in the second 
place of decimals " (loc. cit. p. 624). Mr Yule may have got results " fairly trust- 
worthy in the second place of decimals," but as either the odd or even series 
of powers of r t in the equation rarely becomes convergent till much beyond the 
twelfth term — we have had in some cases to go to 18 or 20 terms — the confidence 
Mr Yule put in his values was quite unjustified. The case is an interesting one 
because the true correlation is very high, i.e. "9253 + '00004 f. Accordingly, 
Mr Yule's association coefficient whatever division is taken is constrained to lie 
between something like "952 and 1. It is therefore difficult to compare the range 
of instability of Q with that of tetrachoric r t . We are bound to consider both 
in relation to their possible ranges. 
That the curve given for tetrachoric r t indicates no defect in r t relatively 
to other coefficients, will be at once appreciated by comparing it with the curve 
for Mr Yule's coefficient of colligation w ! That coefficient varies just as much, only 
it is the other way round : see Diagram XXI, p. 278. As for the Boas-Yulean 
coefficient </>, we could only assume from it, that there was, when the dichotomies 
were at young or old ages, a low relation between the ages of Husband and Wife. 
* The actual skewnesses are: for Husbands Sk. = -71 and for Wives Sk.= - 76. These are among 
the highest values on record for skewness, and the surface is not " moderately skew " as Mr Yule 
without publishing any numbers (loc. cit. p. 624) states it to be. 
t The regression line is sensibly curved at the terminals, but this does not markedly influence r) 
which uncorrected is -9142, as against r='9136 without Sheppard. With Sheppard r rises to -9253. 
Mr Yule gives - 91 for this correlation, which appears to us an uncorrected value. 
