# Contingency Tables

A contingency table is a table of counts or frequencies. It lists the number of times that each of 2 or more variables falls into a variety of different categories. For example, if you were examining differences in the frequency of hair colour between the sexes, you would have a table with 2 variables (sex and hair colour) and a number of categories (eg male and female for sex and black, brown, red, blonde etc for hair colour.). The table would then record the frequency with which each combination of categories was observed e.g. how many blonde haired males, how many black haired females etc. A contingency test simply examines the null hypothesis that the frequencies of observations found for one variable are independent of the frequencies of observations in the other. For the above example this could be stated as “The frequency of hair colour is the same for each sex”. statistiXL will accept this data in either a cross-tab form (for contingency tables with 2 variables) or a list form (for tables with 2 or more variables).

statistiXL provides a flexible module for analysis of contingency tables. Two-way and multi-way contingency tables can be analysed. The statistics available for the frequency test are Chi^{2} and log-likelihood, the latter being a good alternative approach to Chi^{2} if the expected frequencies are small. Yates’s and Cochran’s corrections for continuity are provided for 2×2 contingency where such an adjustment is recommended because of the low degrees of freedom.

There are no post hoc tests for contingency table analysis, but a divided Chi^{2} analysis can be used if the null hypothesis is rejected (i.e. if it is concluded that the observed distribution differs significantly from the expected distribution) to explore which particular category or categories is contributing to the difference. statistiXL explains how to subdivide a contingency table and warns of the limited statistical value of this approach. A Heterogeneity Chi^{2} can be used in a contingency table analysis to determine if a number of sets of observed frequency data can be combined into a single set. statistiXL explains how to analyse contingency tables for heterogeneity.

Results are presented in tabulated form, starting with an optional table of a summary of the observed and expected frequencies. The test results are then presented, with the Chi^{2} and Log-likelihood values, along with their degrees of Freedom and P values.

The help file included with statistiXL provides an overview of contingency table analysis and three examples, a 2 x 2 contingency table with heterogeneity Chi^{2} analysis, a 2 x 4 contingency table with subdivided Chi^{2} analysis, and a 2 x 2 x 2 multiway contingency table analysis.