306 
On Theories of Association 
quite close to the value we obtained for Table I from which we originally deduced 
Table II. Hence the probable error of x = "67449 x 1000 x 0 o-„ = "2505. It will 
be clear therefore that with such a probable error for x, the frequency of the 
d quadrant might actually be anything from 0 to '5, and yet the Table of 
experience takes the form VI, when we can record only by units. 
Suppose d had been '5, x = — - 3579, then we have for the theoretical table : 
TABLE VII. 
971-2199 
22-6388 
993-8587 
5-6413 
•5000 
6-1413 
976-8612 
23-1388 
1000 
which solved by the tetrachoric process gives 
r t = + -224 + -187, 
i.e. the value of r t is not significant*. 
On the other hand if we put d = 0, i.e. deal with Table VI, tetrachoric r, is 
unity, because on the Gaussian hypothesis only complete association is compatible 
with absolute zero in this quadrant and then any sample will exhibit absolute 
correlation. But, as we have just seen, the zero in quadrant d could arise from 
samples of a population in which d was small but not zero, and that in a particular 
case 87 °/ 0 of samples of a material with zero association would show this zero 
quadrant. The accompanying diagram gives the values of tetrachoric r t for 
various hypotheses as to x. The dotted rectangle is bounded by the lines which 
give vertically twice the probable error of x for x = 0, and horizontally twice the 
probable error of r t for true r t = 0. We have also placed once the probable error of 
tetrachoric r t from the plotted r t on either side giving the broken curve. It will 
be seen that from the case of d = '5, r t = + '224 + '187 up to rf = '0025 and 
r t = — *400 + "779, there is no significance in the values of r t found by the tetra- 
choric process. Even after this we cannot assert that the values of r t obtained 
would be significant*]' ; for the proof of the formula of the probable error of r t 
depends upon Bd being small as compared to d. Now a d = \/d (1 — djN) = yd, 
very nearly when d is very small. Hence Bdjd is of the order \!djd— 1/VcZ which 
will be large if d be less than unity j. The probable error of r t — 1 is only zero if 
d is absolutely zero for the population which is being sampled and not if it is 
merely zero in the sample. But we only know that population through the 
sample, and we see that in such cases as we are considering the zero in the 
sampled population is only likely to occur in a small proportion of the cases dealt 
with. The tetrachoric process clearly fails in such cases, but we see that with 
* If we apply Q we have Q= "584 ± -332, pointing rather more in the direction of significance, but 
such application of Q is illegitimate, as the fractionising of the theoretical surface has no meaning for a 
coefficient which is based on complete neglect of the nature of the frequency-distribution. 
t In fact the tetrachoric r t series-equation rapidly becomes divergent and the formula for the 
probable error takes an indeterminate form. 
X The like failure occurs in Mr Yule's proof for the probable error of Q, although he has not 
warned his readers of this; it is accordingly not applicable, if one quadrant has zero frequency. 
