Inference 83
6. for Two-Way Tables This chapter introduces a method for analyzing the relationship between two categorical variables. The data are usually summarized with a two-way table (see Section 3.2). Inferential procedures are applied to test whether the two categorical variables are independent. Example 3-2: Age and Political Interest, continued, from page 56. This data set was summarized in Table 3-6, page 59, and it appeared that there was an association between age and interest in politics, with interest appearing to increase with age. The 1265 respondents to this survey are a random sample (or close to it) of all American adults so we might ask if we can conclude that these data provide evidence of an association between age and interest for all American adults or whether the observed association in the sample could be due to sampling variability. A chi-square test of independence between two categorical variables can be used to assess this (independence means no association). The steps for carrying out a chi-square test of independence follow. The steps are very similar to those for creating a two-way table as described in Creating a two-way table, page 55. Remember that in this example, the data were entered as counts, so the Weight Cases option under Data was activated (see Entering count data, page 56). 1. Click Analyze, click Descriptive Statistics, and click Crosstabs. The Crosstabs window in Figure 6-1 appears. 2. Click on age, then click to move age to the Row(s) box. to move interest to the Column(s) box. 3. Click on interest, then click 4. Click the Cells button at the bottom of the Crosstabs window. The Crosstabs: Cell Display window in Figure 6-2 appears. 5. If you want the expected count in each cell, click Expected in the Counts box so that a check appears in the box in front of Expected. If you want the row percents, click Row in the Percentages box so that a check appears in the box in front of Row. 6. Click Continue. 7. Click the Statistics button at the bottom of the Crosstabs window (Figure 6-1). The Crosstabs: Statistics window in Figure 6-3 appears. 8. Click Chi-square so that a check appears in the box in front of it. 9. Continue. Click 10. Click OK. The resulting SPSS output appears in Table 6-1 and Table 6-2. The row percentages were previously computed in Table 3-6, page 59, and show that the percentage of people who are not much interested in politics decreases with age while the percentage who are very much interested increases with age. Comparing the expected counts to the observed counts also shows this pattern. We can also see that the expected counts are all at least 5 so the expected cell frequency condition for using the chi-square test is met (page 547 of the text). The test statistic and p-value are obtained from the first line (Pearson Chi-square) of Table 6-2; ignore the rest of the table. This line gives us a test statistic of X 2 = 41.446 with df = 4 and p-value (Asymp. Sig.) of less than .0005 (its .000 when rounded to 3 decimal places so must be less than .0005; its not exactly 0). The p-value is very small indicating that the data provide very strong evidence that there is an association between age and political interest among all U.S. adults and that, in fact, older people tend to be more interested in politics than younger people. Remember that a significance test does not tell us how strong the association is, so we need to look at the row percents in Table 6-1 to see how big a differen...