226 
On Theories of Association 
give the correlation of ranks and not that of true variates*. The fallacy of 
Mr Yule's arguments and the extreme inferiority of his method to that of con- 
tingency will be manifest in the following comparisons. We, however, draw the 
attention of the reader to this : that if the method of pseudo-ranks did approach 
a limit it would be the correlation of ranks uncorrected for huge brackets, i.e. we 
should still have to correct for passing from ranks to variates and for class-indices. 
Comparison of Method of Contingency with Method of Pseudo-Ranks. 
Coefficients deduced from Table XV, Stature of Father and Son. Product- 
Moment Correlation = "52. 
Order of 
Table 
Nature of Divisions 
Pearson's 
Contingency 
Yule's 
Pseudo-Banks 
7x7 
1 : 
2:3:4 
5+6 : 7 : 8 
•49 
•48 
6x6 
1 : 
2:3:4 
5+6 : 7+8 
•49 
•48 
1st 5x5 
1 : 
2 + 3 + 4 
5+6 : 7 : 8 
•52 
•37 
2nd 5 x 5 
1+2:3:4 
5+6 : 7+8 
•52 
•47 
1st 4x4 
1+2:3:4 
5+6+7+8 
•51 
•46 
2nd 4x4 
1 : 
2 + 3 + 4 
5+6 : 7+8 
•52 
•36 
3rd 4x4 
1+2:3:4 
5+6+7+8 
•51 
•46 
1st 3x3 
1+2+3 : 4 
5+6+7+8 
•52 
•37 
2nd 3 x 3 
1 + 
2 : 3 + 4 
5+6+7+8 
•51 
•44 
The inferiority of the method of pseudo-ranks will be obvious. The contingency 
gives as good results for a 3 x 3 table as for a 5 x 5 table f ; but for two different 
tables of the same order the method of pseudo-ranks will give results differing by 
as much as - 10, ten times the difference of the contingency method. 
Here is another Table j, to which Mr Yule has applied his method of reaching 
a limit to the actual correlation, namely that for eye-colour for pairs of brothers. 
Mr Yule having so to speak ingeniously " dressed the window " to show a 
falling correlation of pseudo-ranks with his increase of classes, then asserts that this 
pseudo-rank correlation approaches a limit below 0"28, and extending his fallacious 
reasoning holds that this limit of ranks is the limit to the true correlation of 
* The fog in Mr Yule's mind on this subject is well illustrated by his table on p. 619. He takes 
a table for a Gaussian distribution of correlation 0 - 3 and says that with an infinite number of classes 
the Yulean coefficient would become 0-3. There is not a trace of any knowledge on his part that 
the limit of a process by which unit-range is given to each individual is not the same as that of a 
process by which unit-area of the frequency curve is given to each individual. Here and elsewhere 
he makes no distinction between the correlation of variates and the correlation of ranks. In the actual 
case his limit would have been - 2876 and not "3000, but far greater differences will arise if the 
material be skew. We return to this point later. 
t Like all statistical methods, that of contingency must be used with due regard to the data to 
which it is applied and to the manner in which it is applied. Compound or heterogeneous material 
may give a contingency coefficient differing considerably from true correlation, and groupings of great 
inequality in the cells may render idle the corrective factor, e.g. if the great bulk of the material be 
placed in one or two cells. 
+ Phil. Trans. Vol. 195 A, p. 140. 
