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

2. (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.

3. 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

4. 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

5. (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.

6. (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

7. 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

8. (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.

9. (S) Statement
Same Conclusion!!!!

10. 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.

11. 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) =
E =
(899)(704)
1232

12. Find the Expected Counts:
Row
Totals
Black
White
Yellow/Orange
Controls (not injured)
Expected
491
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) =
E =
(899)(704)
1232

13. 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)
Expected
Cases (injured or killed)
Column Totals
28.459
10.541
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 if crash related injuries are independent of helmet color.

14. 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.

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

16. 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 =
The test statistic chi-square values need to be compared with the chi-square critical value found in Table A-4.

17. 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.

18. 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

19. 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.

20. 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

21. 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

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

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

24. 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

25. 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.