HLTH 300 Biostatistics for Public Health Practice, Raul Cruz-Cano, Ph.D.

1. Fox/Levin/Forde, Elementary Statistics in Social Research, 12e • Chapter 9: Nonparametric Tests of Significance HLTH 300 Biostatistics for Public Health Practice,Raul Cruz-Cano, Ph.D. 4/28/2014, Spring 2014

2. CHAPTER OBJECTIVES 9.1 • Understand the logic of nonparametric tests 9.2 • Conduct one-way and two-way chi-square tests 9.3 • Perform the median test 9.4 • Perform the Mann-Whitney U and Kruskal-Wallis tests

3. Learning Objectives • After this lecture, you should be able to complete the following Learning Outcomes • 9.1 Understand the logic of nonparametric tests

4. 9.1 Nonparametric Tests t tests and F ratios require: • Normality (or especially large samples) • Interval level data What if these requirements cannot be met? • We must use nonparametric tests • Chi-square • The median test • Mann-Whitney U test • Kruskal-Wallis test Nonparametric tests are less powerful than parametric • Power = the probability of rejecting the null hypothesis when it is actually false and should be rejected

5. Learning Objectives • After this lecture, you should be able to complete the following Learning Outcomes • 9.2 Conduct one-way and two-way chi-square tests

6. 9.2 The One-Way Chi-Square Test Observed frequency: the set of frequencies obtained in an actual frequency distribution Expected frequency: the frequencies that are expected to occur under the terms of the null hypothesis • In general, this is found by dividing N by the number of categories Chi-square allows us to test the significance of differences between observed and expected frequencies

7. Examples Box 9.1, page 324 Problem 13

8. 9.2 The Two-Way Chi-Square Test How can we compare observed and expected frequencies for more than one variable? • Two-way chi-square test • This involves cross-tabulations The methods for calculating one-way and two-way chi-squares are very similar • In fact, the same formula is used • The only major difference is in how we calculate expected frequencies For each cell: df=(# of rows -1 )(# of columns -1)

9. 9.2 Table 9.2

10. Examples Box 9.2, page 331 Problem 15 (2 x 2) Problem 22 (more than 2 groups)

11. 9.2 Correcting for Small Expected Frequencies One of the few demands on the chi-square test is that the sample size should not be too small • Be wary of expected frequencies that are less than 5 • In this case, it might be best to collapse categories • When expected frequencies are greater than 5 but less than 10, use Yate’s correction • Reduces the size of the chi-square value • Only used for 2 X 2 tables, hence df= 1

12. Example Page 329

13. 9.2 Requirements for the Use of Two-Way Chi-Square • A Comparison between Two or More Samples • Nominal Data • Random Sampling • The Expected Cell Frequencies Should Not Be Too Small

14. Learning Objectives • After this lecture, you should be able to complete the following Learning Outcomes • 9.3 Perform the median test

15. 9.3 The Median Test Used when dealing with ordinal data • Determines the likelihood that two or more random samples have been taken from populations with the same median First, determine the median of the two groups combined Then, create a cross-tabulation with the two categories and the scores that fall above the median and the scores that do not fall above the median Finally, conduct a chi-square test • Using Yate’s corrections if there are any expected frequencies that are less than 10

16. Example Box 9.4, page 341 Problem 36

17. 9.3 Requirements for the Use of the Median Test • A Comparison between Two or More Medians • Ordinal Data • Random Sampling

18. Learning Objectives • After this lecture, you should be able to complete the following Learning Outcomes • 9.4 Perform the Mann-Whitney U Test and the Kruskal-Wallis Test

19. 9.4 The Mann-Whitney U Test The median test ignores the specific rank-order of cases This test examines the rank-ordering of all cases • It determines whether the rank values for a variable are equally distributed throughout two samples The smaller of the two U values is used for testing the differences between groups • This value is compared against the critical U value found in Table G in Appendix C We won’t study but be aware of its existence when comparing your work vs. answers in the back of the book

20. 9.4 The Kruskal-Wallis Test Can be used to compare several independent samples • Requires only ordinal-level data The H statistic is compared to the critical values of chi-square found in Table F in Appendix C We won’t study but be aware of its existence when comparing your work vs. answers in the back of the book

21. Homework Problem 14, 19, 28, 35

22. CHAPTER SUMMARY • Nonparametric tests of significance can be used to analyze data that are not normally distributed or are not measured at the interval level 9.1 • One-way and two-way chi-square statistics can be calculated for variables measured at the nominal level 9.2 • The median test can be used to examine data measured at the ordinal level 9.3 • The Mann-Whitney U and Kruskal Wallis tests are more powerful than the median test and can also be used to examine ordinal data 9.4