130 Oll the Theonj of Association 



Replaciiifj C by N - {-y) aiul regrouping in similar pairs of terms containing (i)) 

 and (8) this will become 



{Aß^Z ...H-^) = jß, {i\^»-' + (ß) (7) (8) . . . (^) («') 



-i\r'(Z)) (£)... (J/)(iyr) 

 -i\r»(8)(^)...(j/)(ir) 



— etc.) 



and contimiing the sanie process until .all the frequencies (D) (E) . . . (^f) (X) are 

 eliminated, i.e. n — 1 times altogether, 



(A)(ß)(y)(8)...{t.){v) 



(AßyB ... fiv) = 



iV- 



That is to say the theorcm niust be triie quite generally : " A serie.s of n attributes 

 ABC ... MN are completely independent if the relation.s of iiidependence are 

 proved to hold for (2"— n+l) of the 2" ultimate frequeucies; such relations 

 niust then hold for the remaiuing n+\ frequencies also." If the ultimate 

 frequencies are only given by the relations of independence in n cases or less, 

 independence may ex ist for certain pairs of attributes in certain universes but 

 not in general. The mere fact of the relation hoiding for one cla.ss, e.g. 



iABGD . . . MiV) = (^)(^)(^ 'K^^- • •_ WW , 



implies nothing — in striking contrast to the simple case of two attributes, where 

 2" — jt+ 1 = 1 and ouly the one class-frequency need be tested in order to see if 

 independence exists. In the case of three attributes the number of third-order 

 classes is eight, of which four must be tested in order to be certain that complete 

 independence exists. In the case of four attributes there are sixteen fourth-order 

 classes of which eleven must be tested, and so on. 



I have dealt with the problem hitherto on the assumption that only the first- 

 order and the 9(tii order frequencies were given, and that the fiequencies of 

 intermediate onlcis were unknown— or at least uncalcuiated, for of course the 

 frequencies of all lower Orders may be expressed in terms of those of the »ith 

 Order. If however the fre((uencies of all onlers may be supposed known, the above 

 result may be thrown iuto a somewhat interesting form. It will be remembered 

 that the frequency of any class of any order may be expressed in terms of the 

 frequencies of the positive classes [(A) {AB) (AG) (ABC) etc.] of its own and 

 lower Orders. Tlien complete independence exists for a series of attributes if the 

 criterion of independence hold for all the positive-class frequencies up to that of the 

 nth Order. If \ve have for instance 



{ABCD ... MN) = ^^_^ {iA){B){C)iI)) ... W(iV)}, 



