146 On the Theories of 3Iultlple and Partial Contingency 
gives a measure of the divergence from independence*. This is a multiple con- 
tingency coefficient. 
Another case not infrequently arises ; the population N has the characteristics 
A, B, C, ... not independent, but related; the cell uvw ... ifj contains i^„„w...<i' 
the question arises how far it is safe to consider the population M as a sample 
of this population. In this casef 
M 
= S 
.(ii). 
In both these cases we have the relation 
Suvw...^ {mu^w...^!,) = M (iii), 
and accordingly the number of cell frequencies is one more than the number of 
independent variates. Thus in using the tables J of "goodness of fit" the n' of 
the argument is ajSy ... A, but the value of P, the probability, has actually been 
determined from «' — 1. 
(2) Now there are a number of cases in which not only do the cell-contents of 
the sample obey the linear relation (iii), but also other linear relations are imposed 
on the cell-contents. In the most general case we can suppose q linear relations 
between the cell-contents ?»ut:w...i{'j and obtain the probability P corresponding to 
a value of limited by these q relations. The theory of sampling, when such 
conditions are introduced, I term the theory of partial contingency. The reason 
for this terminology will be clearer as we develop the theory. 
As far as we are concerned at present, it is of no importance whether we are 
dealing with one or other of our two cases, i.e. whether we are questioning the 
possibility of our material being a sample from a population with independent 
A, B, C, ... L characteristics, i.e. determining a coefficient of mean squared 
contingency, or are investigating the possibility of its being a probable sample 
from a population with any associations between these characteristics. We can 
accordingly write in the form 
X =^uvw...^] = \ (IV), 
L "''uvw...'p J 
or for convenience we may even drop the descriptive subscripts and, numbering 
the cells in some sequence 1, 2, 3, ... s, ... {a^y ... A), write 
.^g inis-rUsV (^). 
* If the characteristics may be assumed to be continuous variates, certain corrections for units of 
grouping can be made. There is also a correction due to the necessarily positive value of 4>'. These 
corrections, which have been for some time in use, will be considered elsewhere. 
I We must in this case of course actually know the value of n^^^yj ^, it cannot be judged from the 
sample. 
X For discussion of the deduction of P from x' '■ see Phil. Mag. Vol. L. p. 157 (July 1900) ; and for 
Tables : see Tables for Statisticians and Biomctricians, Table XII (Cambridge University Press). 
