Contingency Tables: Independence and Homogeneity Section 11-3 Contingency Tables: Independence and Homogeneity

(or two-way frequency table) Contingency Table (or two-way frequency table) A contingency table is a table in which frequencies correspond to two variables. (One variable is used to categorize rows, and a second variable is used to categorize columns.) page 606 of Elementary Statistics, 10th Edition Contingency tables have at least two rows and at least two columns.

Case-Control Study of Motorcycle Drivers Is the color of the motorcycle helmet somehow related to the risk of crash related injuries? 491 213 704 377 112 489 31 8 39 899 333 1232 Black White Yellow/Orange Row Totals Controls (not injured) Cases (injured or killed) Column Totals page 606 of Elementary Statistics, 10th Edition

Test of Independence A test of independence tests the null hypothesis that there is no association between the row variable and the column variable in a contingency table. (For the null hypothesis, we will use the statement that “the row and column variables are independent.”) page 607 of Elementary Statistics, 10th Edition

(H) Hypothesis Statements The null hypothesis H0 is the statement that the row and column variables are independent; the alternative hypothesis H1 is the statement that the row and column variables are dependent.

(A) Assumptions/Requirements The sample data are randomly selected and are represented as frequency counts in a two-way table. For every cell in the contingency table, the expected frequency E is at least 5. (There is no requirement that every observed frequency must be at least 5. Also there is no requirement that the population must have a normal distribution or any other specific distribution.) page 607 of Elementary Statistics, 10th Edition

Total number of all observed frequencies in the table Expected Frequency (row total) (column total) (grand total) E = Total number of all observed frequencies in the table page 607 of Elementary Statistics, 10th Edition

(T) Test of Independence Test Statistic 2 =  (O – E)2 E Critical Values 1. Found in Table A-4 using degrees of freedom = (r – 1)(c – 1) r is the number of rows and c is the number of columns 2. Tests of Independence are always right-tailed. page 607 of Elementary Statistics, 10th Edition Same chi-square formula as for multinomial tables.

(S) Statement Same Conclusion!!!!

Test of Independence This procedure cannot be used to establish a direct cause-and-effect link between variables in question. Dependence means only there is a relationship between the two variables.

Find the Expected Counts: 491 213 704 377 112 489 31 8 39 899 333 1232 Black White Yellow/Orange Row Totals Controls (not injured) Cases (injured or killed) Column Totals 899 1232 704 For the upper left hand cell: (row total) (column total) E = (grand total) = 513.714 E = (899)(704) 1232

Find the Expected Counts: Row Totals Black White Yellow/Orange Controls (not injured) Expected 491 513.714 213 704 377 112 489 31 8 39 899 333 1232 Cases (injured or killed) Expected Column Totals (row total) (column total) E = (grand total) = 513.714 E = (899)(704) 1232

Find the Expected Counts: 491 513.714 213 704 377 112 489 31 8 39 899 333 1232 Black White Yellow/Orange Row Totals Controls (not injured) Expected Cases (injured or killed) Column Totals 356.827 132.173 28.459 10.541 190.286 Expected Calculate expected for all cells. To interpret this result for the upper left hand cell, we can say that although 491 riders with black helmets were not injured, we would have expected the number to be 513.714 if crash related injuries are independent of helmet color.

Case-Control Study of Motorcycle Drivers Using a 0.05 significance level, test the claim that group (control or case) is independent of the helmet color. H0: Whether a subject is in the control group or case group is independent of the helmet color. (Injuries are independent of helmet color.) H1: The group and helmet color are dependent.

Case-Control Study of Motorcycle Drivers Row Totals Black White Yellow/Orange Controls (not injured) Expected 491 513.714 213 704 356.827 132.173 377 112 489 28.459 10.541 31 8 39 899 333 1232 190.286 Cases (injured or killed) Expected Column Totals 190.286

Case-Control Study of Motorcycle Drivers H0: Row and column variables are independent. H1: Row and column variables are dependent. The test statistic is 2 = 8.775  = 0.05 The number of degrees of freedom are (r–1)(c–1) = (2–1)(3–1) = 2. The critical value (from Table A-4) is 2.05,2 = 5.991. The test statistic chi-square values need to be compared with the chi-square critical value found in Table A-4.

Case-Control Study of Motorcycle Drivers Figure 11-4 page 610 of Elementary Statistics, 10th Edition We reject the null hypothesis. It appears there is an association between helmet color and motorcycle safety.

Test of Homogeneity In a test of homogeneity, we test the claim that different populations have the same proportions of some characteristics. page 611 of Elementary Statistics, 10th Edition

How to Distinguish Between a Test of Homogeneity and a Test for Independence: Were predetermined sample sizes used for different populations (test of homogeneity), or was one big sample drawn so both row and column totals were determined randomly (test of independence)? The key to identifying it is a test of homogeneity is the predetermined sample sizes.

Example: Influence of Gender Using Table 11-6 with a 0.05 significance level, test the effect of pollster gender on survey responses by men. page 612 of Elementary Statistics, 10th Edition

Hypotheses H0: The proportions of agree/disagree responses are the same for the subjects interviewed by men and the subjects interviewed by women. H1: The proportions are different. page 612 of Elementary Statistics, 10th Edition

Calculations with Expected Values to use in Assumptions Chi-Square Test of Homogeneity Minitab page 613 of Elementary Statistics, 10th Edition

Assumptions We have expected counts greater than five in all categories. Assume two random samples of survey responses.

STatement Since the pvalue < .05 we reject the H0 , there is sufficient evidence to suggest there are differences in opinion for subjects interviewed by men and the subjects interviewed by women

Recap Contingency tables where categorical data is arranged in a table with a least two rows and at least two columns. Test of Independence tests the claim that the row and column variables are independent of each other. Test of Homogeneity tests the claim that different populations have the same proportion of some characteristics.