11.5 Chi-Square Homogeneity Test

The chi-square homogeneity test is the same as the chi-square independence test, except for the wording of the hypotheses. The hypotheses for a chi-square homogeneity test are:

\begin{align*} H_0 &: \text{The population proportions are homogeneous.}\\H_a &: \text{The population proportions are nonhomogeneous.} \end{align*}

For example, recall the chi-square independence test for “cancer status” and “smoking status”. The hypotheses for the corresponding chi-square homogeneity test are:

\begin{align*} H_0 &: \text{The proportions of females with & without cancer status are homogeneous}\\&\quad \text{   between smokers and nonsmokers.}\\H_a &: \text{The proportions of females with & without cancer status are nonhomogeneous}\\&\quad \text{   between smokers and nonsmokers.}\end{align*}

That is,

[latex]H_0: p_{\text{cancer|smoker}} = p_{\text{cancer|nonsmoker}}; p_{\text{cancer free|smoker}} = p_{\text{cancer free|non smoker}}.[/latex]
[latex]H_a: p_{\text{cancer|smoker}} \neq p_{\text{cancer|nonsmoker}}; p_{\text{cancer free|smoker}} \neq p_{\text{cancer free|non smoker}}.[/latex]

Note: In the case where the variable has more than 2 levels, lack of homogeneity does not imply that all pairs of proportions are different between the populations but that at least one pair of proportions is different.

The remaining steps for the chi-square homogeneity test are identical to the chi-square independence test.