Measures of Association for Categorical Variables: Chi-square

Need for categorical analysis

• many questions deal with data on the nominal or ordinal scale

  • usually questions of proportions or frequencies

• such data are analyzed by determining if there is a difference between proportions observed within a set of categories and the proportions that would be expected

• the chi-square family of tests is used for these purposes

  • non-parametric statistics that determine if a distribution of observed frequencies differ from theoretical expected frequencies

Chi-square assumptions

• frequencies represent individual counts (as opposed to percentages, ranks, etc.)

• categories are exhaustive and mutually exclusive

Chi-square equation

x² = Σ (observed - expected)²/expected

Goodness of fit tests

• determine if a set of observed frequencies differs from a given set of theoretical frequencies that define a specific distribution

• testing if a sample is actually split 50:50 (or some other ratio) on some attribute

• comparing frequency counts with a known or theoretical distribution

Common models of goodness of fit tests

• uniform distributions

• known distributions

Tests of independence

• examine the degree of association between two independent classification variables

• if the categorical variables are dependent/paired/repeated, McNemar’s test is used

• examine the potential association between two categorical variables

Fisher’s Exact test

• can be used with a small sample size

• directly comparing counts statistically

Odds ratio

risk measure derived from categorical 2×2 tables

McNemar test for correlated samples

examine the potential association between two categorical variables in a repeated measures design