284 MR. G. UDNY YULE ON THE ASSOCIATION 



38. CASE 2. Non-ultimate Groups. Let A and B be the groups, to take the 

 simplest case only, which is all we at present require. Then 



(A) + (B) - N = (AB) - (a/?), 

 or, dividing by N, say 



<i 4- fa 1 = TI ""a 



8$, + Bfa = STT-! STT 3 ........ (5). 



Squaring 



<r.? + <r>; + 2 ^o;,^ = <?* + <,,? -2^*^, . . (6). 



2<r #l <r,, R Mi = jj { (TJ-! + 7T 3 ) (B-! - TT 3 )~ + fa fa* + fa - < a 3 }. 



Substitute for (TT, 7r s ) in terms of the first order frequencies, and also for it z in 

 the first bracket. Then we have 



^ 1 <r^R Wj = ^(7T 1 ^ 2 ) ........ (7), 



or for R^ i4j when these are non-ultimate- groups 



T> _ TTj - (ft^g > 



*'*' -- ' 



Now if the attributes AB are independent TT, = <^ x <^. 2) so that if the attributes 

 are unassociated errors in their frequencies are uncorrelated. On the other hand, 

 errors in the frequencies will be perfectly correlated only if 



(AB) = (0) = 

 or else 



(aB) = (A/3) = 0, 



which is more than is necessary for complete association (Q = 1). If the groups are 

 ultimate we see from equation (3) that errors in their frequencies are always 

 correlated, unless, indeed, the frequency of one of the groups be vanishingly 

 small. 



39. We may now proceed to find the probable errors or standard errors of K and 

 Q. Let < 1( fa, < 3 , fa be the values of <f> for any four groups forming a tetrad, e.g., 

 let 



fa = (ABCDE)/N fa = (ACDE)/N 

 fa = (a/3CDE)/N fa = (aBCDE)/N. 



Then for this tetrad K (equation (2), p. 273) is given by 



K = ^* 

 fa fa 



OK _ &fa ,Sfa_ fyl 

 K fa $i fa 



