Chapter 11 Analyzing the Association Between Categorical Variables Section 11.4 Using Residuals to Reveal the Pattern of Association

Association Between Categorical Variables The chi-squared test and measures of association such as and are fundamental methods for analyzing contingency tables. The P-value for summarized the strength of evidence against independence.

Association Between Categorical Variables If the P-value is small, then we conclude that somewhere in the contingency table the population cell proportions differ from independence. The chi-squared test does not indicate whether all cells deviate greatly from independence or perhaps only some of them do so.

Residual Analysis A cell-by-cell comparison of the observed counts with the counts that are expected when is true reveals the nature of the evidence against . The difference between an observed and expected count in a particular cell is called a residual.

Residual Analysis The residual is negative when fewer subjects are in the cell than expected under . The residual is positive when more subjects are in the cell than expected under .

Residual Analysis To determine whether a residual is large enough to indicate strong evidence of a deviation from independence in that cell we use a adjusted form of the residual: the standardized residual.

Residual Analysis The standardized residual for a cell = (observed count – expected count)/se A standardized residual reports the number of standard errors that an observed count falls from its expected count. The se describes how much the difference would tend to vary in repeated sampling if the variables were independent. Its formula is complex, software can be used to find its value. A large standardized residual value provides evidence against independence in that cell.

Example: Religiosity and Gender “To what extent do you consider yourself a religious person?” Table 11.17 Religiosity by Gender, With Expected Counts and Standardized Residuals.

Example: Religiosity and Gender Interpret the standardized residuals in the table. The table exhibits large positive residuals for the cells for females who are very religious and for males who are not at all religious. In these cells, the observed count is much larger than the expected count. There is strong evidence that the population has more subjects in these cells than if the variables were independent.

Example: Religiosity and Gender The table exhibits large negative residuals for the cells for females who are not at all religious and for males who are very religious. In these cells, the observed count is much smaller than the expected count. There is strong evidence that the population has fewer subjects in these cells than if the variables were independent.