18(3 
On Theories of Association 
or 40 years since their last vaccination, for this " is dependent to a large degree 
on a purely arbitrary circumstance," the question of whether the population has 
been recently alarmed and revaccinated. The efficiency of vaccination depends 
both on size of cicatrix and on interval since vaccination. Yet if Mr Yule makes 
these equalisations he will wholly change his association coefficient. That coefficient 
admits not of a natural selection, but only of a wholly artificial one, namely, one 
which alters all sub-classes on one side of an arbitrary dividing line in the same 
ratio and not in their " natural " ratios. Every word Mr Yule has uttered with 
regard to equalisation and its " naturalness " applies equally to classifications in 
3, 4 or more groups. Why should the relationship of father to son in eye-colour 
depend upon the number of light blue eyed fathers taken? It is "natural" 
to equalise them with the dark blue. But this is only possible if the multiplying 
of a row or column will not influence the result, and the reader has only to test 
on the above table how Q is changed when he starts such a selection. A few of 
the values we have obtained are Q = '47, - 44, "36, '17 and 04 ! The fact is that 
Mr Yule's is not a general selection, but a perfectly artificial one, which is 
not in the least " natural " when we analyse its effect on the constituents of a 
given class. 
It is as well to stop and inquire what iufluence this " natural " equalising 
has on contingency tables. Below (Table II) is given the table for eye-colour in 
Father and Son reduced to five classes on account of the labour involved. The 
equalising factors are 
For the Bows : For the Columns : 
y 1+2 = 3-73512, x 1+2 = -61949, 
y 3 =3-92073, x 3 = -81174, 
y, =7-11764, x 4 = 1-00000, 
y s+6 = 7-47842, oc SH = 2-38871, 
y 7+8 = 6-35625, * v+8 = 1-34437, 
and the resulting table is given as Table III. 
TABLE II. 
Actual Eye-Colour Table for Father and Son in Five Glasses. 
Eye-Colour of Father*. 
o 
CO 
JO 
o 
o 
1+2 
3 
4 
5 + 6 
7 + S 
Totals 
1 + 2 
194 
70 
41 
9 
21 
335 
3 
83 
124 
41 
13 
23 
284 
25 
34 
55 
11 
12 
137 
5+6 
27 
12 
19 
24 
23 
105 
7+8 
29 
24 
24 
12 
50 
139 
Totals 
358 
264 
180 
09 
129 
1000 
* See Phil, Trans. Vol. 195 A, p. 138. 
