1. © Copyright McGraw-Hill 2000 13-1 CHAPTER 13 Nonparametric Statistics

2. © Copyright McGraw-Hill 2000 13-2 Objectives State the advantages and disadvantages of nonparametric methods. Test hypotheses using the sign test. Test hypotheses using the Wilcoxon rank sum test. Test hypotheses using the signed-rank test.

3. © Copyright McGraw-Hill 2000 13-3 Objectives (cont’d.) Test hypotheses using the Kruskal-Wallis test. Compute the Spearman Rank correlation coefficient. Test hypotheses using the runs test.

4. © Copyright McGraw-Hill 2000 13-4 Introduction Nonparametric statistics or distribution-free statistics are used when the population from which the samples are selected is not normally distributed.

5. © Copyright McGraw-Hill 2000 13-5 Advantages of Nonparametric Methods They can be used to test population parameters when the variable is not normally distributed. They can be used when the data are nominal or ordinal. They can be used to test hypotheses that do not involve population parameters.

6. © Copyright McGraw-Hill 2000 13-6 Advantages of Nonparametric Methods In most cases, the computations are easier than those for the parametric counterparts. They are easier to understand.

7. © Copyright McGraw-Hill 2000 13-7 Disadvantages of Nonparametric Methods They are less sensitive than their parametric counterparts when the assumptions of the parametric methods are met. Therefore, larger differences are needed before the null hypothesis can be rejected. They tend to use less information than the parametric tests. For example, the sign test requires the researcher to determine only whether the data values are above or below the median, not how much above or below the median each value is.

8. © Copyright McGraw-Hill 2000 13-8 Disadvantages of Nonparametric Methods They are less efficient than their parametric counterparts when the assumptions of the parametric methods are met; that is, larger sample sizes are needed to overcome the loss of information. For example, the nonparametric sign test is about 60% as efficient as its parametric counterpart, the z test. Thus, a sample size of 100 is needed for use of the sign test, compared with a sample size of 60 for use of the z test to obtain the same results.

9. © Copyright McGraw-Hill 2000 13-9 Ranking the Data Many nonparametric tests involve the ranking of data — that is, the positioning of a data value in a data array according to some rating scale.

10. © Copyright McGraw-Hill 2000 13-10 Nonparametric Methods Sign test Wilcoxon rank sum test Wilcoxon signed-rank test Kruskal-Wallis test Spearman rank coefficient Runs test

11. © Copyright McGraw-Hill 2000 13-11 Single-Sample Sign Test The sign test is the simplest of the nonparametric tests and is used to test the value of a median for a specific sample. When using the sign test, the researcher hypothesizes the specific value for the median of a population; then he or she selects a sample of data and compares each value with the conjectured median.

12. © Copyright McGraw-Hill 2000 13-12 Single-Sample Sign Test (cont’d.) If the data value is above the conjectured median, it is assigned a “+” sign. If it is below the conjectured median, it is assigned a “ –” sign. If it is exactly the same as the conjectured median, it is assigned a “0”.

13. © Copyright McGraw-Hill 2000 13-13 Single-Sample Sign Test (cont’d.) If the null hypothesis is true, the number of + and – signs should be approximately equal. If the null hypothesis is not true, there will be a disproportionate number of + or – signs. The test value is the smaller number of + or – signs.

14. © Copyright McGraw-Hill 2000 13-14 z Test Value in the Sign Test when n  26 where  X = smaller number of + or – signs  n = sample size

15. © Copyright McGraw-Hill 2000 13-15 Wilcoxon Rank Sum Test The Wilcoxon rank sum test is used for independent samples. Both sample sizes must be  10.

16. © Copyright McGraw-Hill 2000 13-16 Formula for Wilcoxon Rank Sum Test where   R = sum of the ranks for the smaller sample size ( n 1 )  n 1 = smaller of the sample sizes, n 1  10  n 2 = larger of the sample sizes, n 2  10

17. © Copyright McGraw-Hill 2000 13-17 The Wilcoxon Signed-Rank Test When the samples are dependent, as they would be in a before-and-after test using the same subjects, the Wilcoxon signed-rank test can be used in place of the t test for dependent samples. This test does not require the condition of normality. When n  30, the normal distribution can be used to approximate the Wilcoxon distribution.

18. © Copyright McGraw-Hill 2000 13-18 The Wilcoxon Signed-Rank Test (cont’d.) The formula for the Wilcoxon signed-rank test is: where  n = number of pairs where the difference is not 0  w s = smaller sum in absolute value of the signed ranks

19. © Copyright McGraw-Hill 2000 13-19 The Kruskal-Wallis Test The Kruskal-Wallis test, also called the H test, is used to compare three or more means. Data values are grouped and then are ranked.

20. © Copyright McGraw-Hill 2000 13-20 Formula for the Kruskal-Wallis Test Where  R 1 = sum of the ranks of sample 1  n 1 = size of sample 1  R 2 = sum of the ranks of sample 2  n 2 = size of sample 2  R k = sum of the ranks of sample k  n k = size of sample k  N = n 1 + n 2 + … + n k  k = number of samples

21. © Copyright McGraw-Hill 2000 13-21 The Spearman Rank Correlation Coefficient where  d = difference in the ranks  n = number of data pairs

22. © Copyright McGraw-Hill 2000 13-22 Run Test A run is a succession of identical letters preceded or followed by a different letter or no letter at all, such as the beginning or end of the succession.

23. © Copyright McGraw-Hill 2000 13-23 Summary In many research situations, the assumptions for the use of parametric statistics cannot be met. Some statistical studies do not involve parameters such as means, variances, and proportions. For both situations, statisticians have developed nonparametric statistical methods, also called distribution-free methods.

24. © Copyright McGraw-Hill 2000 13-24 Summary (cont’d.) There are several advantages to the use of nonparametric methods — the most important one is that no knowledge of the population distribution is required. The major disadvantage is that they are less efficient than their parametric counterparts when the assumptions for the parametric methods are met.

25. © Copyright McGraw-Hill 2000 13-25 Summary (cont’d.)

26. © Copyright McGraw-Hill 2000 13-26 Conclusions Nonparametric or distribution-free tests are used when situations are not normally distributed.  A sportswriter may wish to know whether there is a relationship between the rankings of two Olympic swimming judges.  A sociologist may wish to determine whether men and women enroll at random for a specific rehabilitation program.