IV.—PARTIAL ASSOCIATION. 55 
11. 1t might appear, at first sight, that theoretical considera- 
tions would enable us to lessen it still further. As we saw in 
Chapter I., all class-frequencies can be expressed in terms of those 
of the positive classes, of which there are 2" in the case of n 
attributes. For given values of the n+ 1 frequencies &, (4), (B), 
(C), . . . of order lower than the second, assigned values of the 
positive class-frequencies of the second and higher orders must 
therefore correspond to determinate values of all the possible 
associations. But the number of these positive class-frequencies 
of the second and higher orders is only 2 —n +1 ; therefore the 
number of algebraically independent associations that can be 
derived from = attributes is only 2"-m+1. For successive 
values of n this gives— 
n 2" —m 1] 
; 
Hence if we give data, in any form, that determine four 
associations in the case of three attributes, eleven in the case of 
four attributes, and so on, in addition to V and the class-frequencies 
of the first order, we have done all that is theoretically necessary. 
The remaining associations can be deduced. 
12. Practically, however, the mere fact that they can be deduced 
is of little help unless such deduction can be effected simply, 
indeed almost directly, by mere mental arithmetic almost, and 
this is not the case. The relations that exist between the ratios 
or differences, such as (4B) — (4B),, that indicate the associations 
are, in fact, so complex that an unknown association cannot be 
determined from those that are given without more or less lengthy 
work ; it is not possible to infer even its sign by any simple 
process of inspection. We have, for instance, from (5), by the 
process used in obtaining (4) for the special case of § 6— 
| (427) - LC | (4B) - (4B) - (5 (140) - (4050) - BO 
- (40)(BC) 
| az iC | 
which gives us the difference of (4By) from the value it would 
have if 4 and B were independent in the universe of y’s in terms 
of the difference of (ABC) from the value it would have if 4 and