304 
On Theories of Assoeiation 
absolutely compatible with Q being zero. It is true that the value of 0 <7q as 
estimated from the sample is nearly double what its value would be truly esti- 
mated from the population, but both values suffice to show that Q = 0 is as 
reasonable an hypothesis as to the constitution of the material as Q = — 1, and 
far more reasonable than Q = — 1 + 0. The method of mean square contingency 
gives us x s = -1455, leading by Palin Elderton's Tables to P = -97l, or if the 
material were truly independent only in three cases per hundred should we get 
a better fit than Table II to Table I in taking samples of 1000. 
The fact is that in drawing random samples of 1000 from Table I, the 
distribution of frequency in the cell d is given by 
(•999862 + -000138) 1000 = -8712 + "1202 + -0086 (for all terms beyond second). 
Hence in 100 samples of Table I, d would be zero in 87 cases, unity in 12 and 
greater than unity in about one case. In other words Mr Yule's Association 
Coefficient would for material with true zero association be — 1 in about 87 °/ 0 
of cases and of the order + "75 in 12 °/ o other cases. In all these cases, however, 
the probability method shows practical independence. It must accordingly be 
recognised that it is extremely dangerous, if zero frequency be found in one 
quadrant, to assert that Q = + 1 + 0. For, a population of zero association would 
give such a value in 87 °/ 0 of samples of 1000 in a case like that under consideration. 
The tetrachoric method fails also for this extreme case, for the simple reason 
that the very continuity of the method excludes thinking in isolated units. 
If the real population were that of Table I, then on the basis of continuity we 
should expect - 138 from the infinite skirt of the Gaussian surface in quadrant d. 
But we might extend this volume of the skirt up to something under "5 before we 
should anticipate a whole unit to appear in d. There is no suggestion of this 
kind about Q, for Mr Yule directly discards all conception of such continuous 
frequency surfaces. This point must be borne in mind in applying tetrachoric r t ; 
a quadrant of zero frequency does not necessarily signify that in the theoretical 
Gaussian distribution this quadrant would have zero frequency. We may equally 
reasonably assume it anything up to "5. In such a distribution as 
TABLE III. 
A 
Not-.4 
Totals 
B 
21 
450 
471 
Not-Z?... 
529 
0 
529 
Totals 
550 
450 
1000 
we shall alter the tetrachoric coefficient from negative unity to a high value 
somewhat less than negative unity, by inserting anything up to "5 in quadrant d, 
but we shall not swing the correlation from — 1 through 0 to a small positive 
value by the process. We shall get a reasonable minimum value for r t , and we 
shall be convinced that r t = — 1 ± 0 is not necessarily a true representation of the 
