Short-Cut Formulas for the Exact Partition of χ 2 in Contingency Tables

Abstract

Recently, Lancaster (1949, 1950) and Irwin (1949) showed how x2 may be partitioned exactly into single degrees of freedom when expected frequencies are estimated from the data. In particular they considered the case of an r X s (r rows, s columns) contingency table in which the expected frequencies are computed from the marginal totals. Their work suggests several different methods of carrying out the necessary computations, some of which they have illustrated. For tables of any order, their contingency table formulas may be reduced to surprisingly simple and useful forms which effect a considerable saving of computational labor. The short-cut formulas are given here without proof since the reduction involves only straightforward algebra. Since this paper is intended primarily as a simplification of the Lancaster-Irwin formulas, their notation has been changed slightly in the hope that computers will find the formulas easier to read. The method is applied to a radiation experiment with grasshopper neuroblasts.