3.14 Setting Up a Chi-Square Test for Homogeneity or Independence

Describe chi-square distributions.

• The chi-square statistic measures the distance between observed and expected counts relative to expected counts.
• Chi-square distributions have positive values and are skewed right. Within this family of density curves, the skew becomes less pronounced with increasing degrees of freedom.

Identify an appropriate testing method for comparing distributions in two-way tables of categorical data including the populations and variables.

• To determine whether the distributions of a categorical variable for two or more populations are different, the appropriate test is the chi-square test for homogeneity.
• A chi-square test for homogeneity should reference the categorical variable and the populations in context.
• To determine whether row and column variables in a two-way table of categorical data might be associated in the single population from which the data were sampled, the appropriate test is the chi-square test for independence.
• A chi-square test for independence should reference the categorical variables and the population in context.

Identify the null and alternative hypotheses for a chi-square test for homogeneity or independence.

• The appropriate null hypothesis for a chisquare test for homogeneity is H0: there is no difference in the distributions of the categorical v ariable across populations or treatments. The appropriate alternative hypothesis for a chisquare test for homogeneity is Ha: there is a difference in the distributions of the categorical v ariable across populations or treatments.
• The appropriate null hypothesis for a chisquare test for independence is H0 : there is no association between two categorical variables in a given population or the two categorical variables in a given population are independent of each other. The appropriate alternative hypothesis for a chi-square test for independence is Ha: there is an association between two categorical variables in a given population or the two categorical variables in a given population are not independent of each other.

Justify the appropriateness of a chi-square test for independence or homogeneity by verifying conditions.

• A chi-square test for homogeneity or independence requires that three conditions must be met:
  • i. The randomization condition—the test of independence states that the data should be collected using a random sample. The test for homogeneity states that the data should be collected using independent random samples or a randomized experiment.
  • ii. The 10% condition—when sampling without replacement, check that n 10%N, where N is the size of the population and n is the sample size. (Note: This condition is unnecessary when the data are from a randomized experiment.)
  • iii. The expected counts condition—all expected counts should be greater than 5. Inference for Categorical Data: Proportions UNIT 3 112