The Chi-Square Test and Accompanying Effect Size Indices

Abstract

In a previous column, we described statistical methods used to test the significance of a relationship between two variables that were either continuous or had many ordered categories or levels. The purpose of the present column is to describe a statistical test, the Pearson chi-square (χ 2 ) test for independence, that examines the relationship between two variables that are scaled at the nominal level. The variables can be dichotomous or have a few nonordered categories. Nominal scale data provide less information than interval or ordinal scale data. The finer the gradations on the measurement scale, the more information is transmitted, as long as there is evidence for reliability and validity. We recommend that you not divide your data into a few categories if the data are continuous or have a number of ordered levels, unless the measure to be divided has been validated against an external criterion that justifies “cut points.” When there are more than two categories of at least one of the variables and these categories are ordered (i.e., the levels or categories differ quantitatively), such as education level, you would lose power if you used a χ 2 test to analyze the data. It is recommended that these data be analyzed by using nonparametric statistics for ordinal data. Examples of these statistics (discussed in previous columns) are a Spearman rank order correlation, if both variables have more than two ordered levels, or a Mann-Whitney U test, if one of the two variables has only two levels and the other is ordered. In this column, the data to be considered are frequencies. Specifically, our interest is in how many people (the frequency count) fall into a particular category, relative to a different category. There are two major requirements of the χ 2 test. The