128 On (he Theory of Association 



If a series of attributes are complctely imlependent accordiiig to this definition 

 relations of thc form (9) must hold for the frequency of every class of every 

 possible Order. Take the class-frequency (ABCD) of the foiirth order for instauce. 

 A and B are, by the terms of the definition, independent witliiii thc universe CD. 

 Therefore 



(ABCD)J-^(^\ 

 {LI)) 



But A and 0, and also B and 0, are independent within the universe D. Therefore 

 the fraction ou the right is equal to 



1 (AD) (CD) (BD) (CD) _ {AD)(BlJ) {CD) 

 (CD)- {D) ■ (D) (Df 



But again AD, BD, CD are cach independent within the univeree at large; 



therefore finally 



(Anrm- ^ (A)(D) {B){D) {G)(D) {A){B){C){D ) 

 {ABtD)--^^.^^~ .^-^-.-^= W • 



Any other frequency can be reduced step by step in preci.sely the same way. 



Now consider the converse problem. The total freqni'ncy N is given and also 

 the n frequencies {A), (B), (0), etc. In how many of the ultimate frequencies 

 {ABCD. ..MX), {aBCD...MX), etc. must "relations of independence" of the fonn 



{A){B){C){D)...{M){X) 



{ABCD...MN). 



m- 



hold good, in order that complete independence of the attributes may be inferred ? 

 The answer is suggestcd at once by the following consideration. The number of 

 ultimate frequencies (frequencies of order n) is 2"; the number of frequencies 

 given is /i + 1. If thcn all but n + 1 of the ultimate frequencies are given in 

 terms of the equations of independence, the remaining frequencies are deter- 

 minate ; either thcse detcrminate values must be those that would be given 

 by equations of independence, or a State of complete independence must be impossible. 

 Supposc all the ultimate class-frequencies to have been tested and found to be 

 given by the e<)uati()ns of independence, with the exception of thc negative class 

 ( 5/378... /ii«) and the n classes with one positive attribute {AßyB . . . /jlv), (aByo . . . nv), 

 etc. Take any one of these untested class-frequencies, {AßyS ... nv), and we have 

 for example 



iAßyB...^iv) = {A)-{ABCD ... MN) 



-{ABCD...Mv) 



— other terms with one negative 

 -{ABCD ... fxv) 



— other terms with two negatives 



— {AByB ... fiv) 



— other terms with w — 2 negatives. 



