Karl Pearson and David Heron 
261 
But there are two other points about tetrachoric r t which also in our opinion 
weigh much on its side when comparison is made with Q: 
(i) It is modified by every form of selection and thus corresponds to our 
experience of every true measure of relationship. If association is to be of profit, 
it should pass into correlation as our detailed knowledge of the material becomes 
greater. All Mr Yule's association achieves is the passage with increasing 
knowledge into manifold and contradictory diversity. Every selection he has 
made on the basis of lesser knowledge must become an increased source of 
contradictory values, as he reaches more detailed knowledge of his material. 
There is a systematic variation of Q ; it rises continuously from median to 
extreme divisions in every distribution on which we have tested it. That is 
to say, there is a wide divergence of practical statistical data from the surface 
of constant Q, and this in a definite direction. Nor is this to be wondered at, 
for the surface of constant Q has marked heteroscedasticity, and markedly curved 
regression : see Appendix III. 
(ii) A true coefficient of association should not necessarily become perfect, 
when one of the four quadrants of the fourfold table becomes zero*. Whatever 
the degree of dependence of two variates may be, the frequency surface in practice 
is limited in extent, therefore by taking small percentages of one or both variates 
in the marginal totals we cau always in practice make one quadrant zero. We 
hold indeed that a fitting coefficient of association need not necessarily be perfect, 
even if two opposite quadrants have zero frequencies - ]". 
In both of these respects tetrachoric r t is superior to the coefficients of 
association or colligation. The one aspect in which it is inferior is the labour 
of calculation, but the tables published by Everitt and an additional table which 
we hope shortly will be published render the labour by no means severe. 
In order to establish our position it remains to show how even in the ex- 
ceptional cases selected by Mr Yule — which do not represent the run of ordinary 
practice — the tetrachoric r t is much more stable than his Q. 
* Consider the following Table for true ?-=-52 : 
Father's Stature. 
Under 71 S 
Over 71' '5 
Totals 
Under 65 -.5 
127-5 
0 
127-5 
Over 65 - .5 
8(53-5 
87 
950-5 
Totals 
991 
87 
1078 
What real knowledge do we gain by saying that association is perfect between "tall" sons and 
"tall" fathers, when it arises solely from the extreme position of the father-division not giving any 
content to a second quadrant on a limited surface of imperfect correlation ? 
t See p. 177 and later Appendix I. 
