Analysis of Contingency Tables

Abstract

A table showing the distribution of n units according to two or more criteria, each with a finite number of categories,is called a contingency table. If there are two criteria with I and J categories respectively, the contingency table can be presented in the form of a matrix

$$\left[ {\matrix{{{{\rm{x}}_{11}} \ldots } \hfill & {{{\rm{x}}_{1{\rm{j}}}} \ldots } \hfill & {{{\rm{x}}_{1{\rm{J}}}}} \hfill \cr \cdot \hfill & \cdot \hfill & \cdot \hfill \cr \cdot \hfill & \cdot \hfill & \cdot \hfill \cr \cdot \hfill & \cdot \hfill & \cdot \hfill \cr {{{\rm{x}}_{{\rm{i}}1}} \ldots } \hfill & {{{\rm{x}}_{{\rm{ij}}}} \ldots } \hfill & {{{\rm{x}}_{{\rm{iJ}}}}} \hfill \cr \cdot \hfill & \cdot \hfill & \cdot \hfill \cr \cdot \hfill & \cdot \hfill & \cdot \hfill \cr \cdot \hfill & \cdot \hfill & \cdot \hfill \cr {{{\rm{x}}_{{\rm{I}}1}} \ldots } \hfill & {{{\rm{x}}_{{\rm{Ij}}}} \ldots } \hfill & {{{\rm{x}}_{{\rm{IJ}}}}} \hfill \cr }} \right]$$

(19.1)

where x_ij is the number of units belonging to both category i of criterion 1 and category j of criterion 2. With reference to the matrix representation, the two criteria are called respectively the row criterion and the column criterion. If a unit belongs to row category i and column category j, it is said to be in cell (i,j) of the table.