Decomposing Chi-Square

Abstract

According to the statistical properties of the chi-square distribution, a variable which is distributed according to the chi-square distribution with p degrees of freedom can be written as the sum ofp chi-square variables, each with one degree of freedom. This means that a contingency table which gives rise to a chi-square with p degrees of freedom can be decomposed into as many as p smaller tables. The magnitudes of the chi-squares for these smaller tables will often give a better understanding of the relationship between the two variables than was possible to obtain from the overall chi-square for the original table. This paper gives explicit rules for how to decompose a table into smaller tables, and the rules are illustrated with an example from a content analysis of value orientations in different maga zines.