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
