Measures of Association for Nominal and Ordinal Variables

Abstract

Throughout this book, we have covered various ways of measuring the relationships among variables. We have already discussed tests of statistical significance and how they help us infer differences in a population based on a sample from the population. However, tests of statistical significance do not tell us about the strength of associations among variables. In criminal justice research, we often want to detect not only whether a relationship exists among variables but the size of this relationship as well. Determining the size of the relationship among variables makes the interpretation of our results much more meaningful and useful in real-life applications. In this chapter, we focus on how to use various measures of association for nominal- and ordinal-level variables in R by relying on data from the Seattle Neighborhoods and Crime Survey, which aimed to test multilevel theories of neighborhood social organization and crime using telephone surveys of 2,220 Seattle, WA, residents.

Key Terms

Concordant pairs of observations

Pairs of observations that have consistent rankings on two ordinal variables.

Cramer’s V

A measure of association for two nominal variables that adjusts the chi-square statistic by the sample size. V is appropriate when at least one of the nominal variables has more than two categories.

Discordant pairs of observations

Pairs of observations that have inconsistent rankings on two ordinal variables.

Gamma (γ)

PRE measure of association for two ordinal variables that uses information about concordant and discordant pairs of observations within a table. Gamma has a standardized scale ranging from −1.0 to 1.0.

Goodman and Kruskal’s lambda (λ)

PRE measure of association for two nominal variables that uses information about the modal category of the dependent variable for each category of the independent variable. Lambda has a standardized scale ranging from 0 to 1.0.

Goodman and Kruskal’s tau (τ)

PRE measure of association for two nominal variables that uses information about the proportional distribution of cases within a table. Tau has a standardized scale ranging from 0 to 1.0. For this measure, the researcher must define the independent and dependent variables.

Kendall’s τ _b

PRE measure of association for two ordinal variables that uses information about concordant pairs, discordant pairs, and pairs of observations tied on both variables examined. τb has a standardized scale ranging from −1.0 to 1.0 and is appropriate only when the number of rows equals the number of columns in a table.

Kendall’s τ _c

A measure of association for two ordinal variables that uses information about concordant pairs, discordant pairs, and pairs of observations tied on both variables examined. τc has a standardized scale ranging from −1.0 to 1.0 and is appropriate when the number of rows is not equal to the number of columns in a table.

Phi (φ)

A measure of association for two nominal variables that adjusts the chi-square statistic by the sample size. Phi is appropriate only for nominal variables that each have two categories.

Proportional reduction in error (PRE)

The proportional reduction in errors made when the value of one measure is predicted using information about the second measure.

Somers’ D

PRE measure of association for two ordinal variables that uses information about concordant pairs, discordant pairs, and pairs of observations tied on the independent variable. Somers’ D has a standardized scale ranging from −1.0 to 1.0.

Tied pairs of observations

Pair of observations that have the same ranking on two ordinal variables.