© 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey All Rights Reserved HLTH 300 Biostatistics for Public Health Practice, Raul Cruz-Cano, Ph.D. 4/28/2014, Spring 2014 Fox/Levin/Forde, Elementary Statistics in Social Research, 12e Chapter 9: Nonparametric Tests of Significance 1

© 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey All Rights Reserved Understand the logic of nonparametric tests Conduct one-way and two-way chi-square tests Perform the median test Perform the Mann-Whitney U and Kruskal-Wallis tests CHAPTER OBJECTIVES

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

4 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 Nonparametric Tests

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

6 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 The One-Way Chi-Square Test

7 Examples Box 9.1, page 324 Problem 13

9.2 8 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 The Two-Way Chi-Square Test For each cell: df=(# of rows -1 )(# of columns -1)

9.2 Table 9.2

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

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 Correcting for Small Expected Frequencies

12 Example Page 329

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

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

15 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 The Median Test

16 Example Box 9.4, page 341 Problem 36

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

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

19 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 The Mann-Whitney U Test We won’t study but be aware of its existence when comparing your work vs. answers in the back of the book

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 The Kruskal-Wallis Test 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

© 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey All Rights Reserved Nonparametric tests of significance can be used to analyze data that are not normally distributed or are not measured at the interval level One-way and two-way chi-square statistics can be calculated for variables measured at the nominal level The median test can be used to examine data measured at the ordinal level The Mann-Whitney U and Kruskal Wallis tests are more powerful than the median test and can also be used to examine ordinal data CHAPTER SUMMARY