THE DERIVATION AND PARTITION OF χ2 IN CERTAIN DISCRETE DISTRIBUTIONS

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Abstract
1. (1) It is shown how the general term of any multinomial can be reduced to a series of binomial terms, to each of which corresponds a value of χ2 for one degree or freedom. In an accompanying paper (pp. 130-4 below), Dr J. O. Irwin (1949) has shown that corresponding to this reduction there is an exact partition of χ2 and a certain Helmert matrix. This partition can be formally related to regression analysis by showing that it is equivalent to the selection of variables of the form xi.j.k…/σi.jk…. (2)The expression for the probability of an r×8 contingency table can be partitioned into the product of the probability of (r-1)(8-1) fourfold tables, the χ of each of which is uncorrelated with that of any of the others. Asymptotically, when the expected frequencies are large, all the χ2 are normally distributed so that we have (r-1) (8-1) normal and uncorrelated deviates. Any difficulties as to degrees of freedom are avoided in this proof. (3) It is further shown that corresponding to the method of treating the r×8 table set out in (2), there is an exact partition of χ2 which can be obtained by pre- and post-multiplication of the (r×8) matrix of standardized variables by certain Helmert matrices. This operation makes the variables in the first row and in the first column all zero, leaving a matrix with (r-1) (8-1) standardized and uncorrelated variables. Each of these variables is the χ of one of the component fourfold tables, when calculated by the use of expectations obtained from the original margins and not from the component table itself. (4) Numerical examples are given for the case of a multinomial distribution, for the fourfold table and for a 3×3 table. In each case the partition of χ2 is illustrated.