Statistical Inference: A Short Course

13.5 Testing Independence

In the Pearson chi-square tests covered in the preceding section, the random sample data were classified via a single criterion into a set of mutually exclusive and collectively exhaustive categories or cells. However, as will be demonstrated shortly, this approach can be applied to situations other than these resulting from multinomial experiments. For example, suppose our observations are classified according to two or more attributes. Then the question naturally arises: “Are these attributes or categories of classification statistically independent?” (“Do they lack statistical association?”) To answer this, we shall look to Pearson's chi-square test for independence of attributes or Pearson's chi-square test of association. As will be evidenced below, this technique can be applied to either quantitative or qualitative data.

Given a specific criterion of classification, suppose we cross-classify our data set according to a second criterion or attribute, where the population proportion within each resulting cell is unknown, for example, given a set of categories on the factor “educational attainment” (high school diploma, college degree, graduate school degree), we may cross-classify according to a second factor such as “sex” or “income level” (low, medium, high). We shall refer to the resulting two-way classification scheme as a two-way contingency table.

As stated above, we are interested in determining whether there is any dependency relationship or ...