Chapter 18 Statistical Inference for Ranked Data I Introduction to Assumption-Freer Tests

1. Chapter 18 Statistical Inference for Ranked Data I Introduction to Assumption-Freer Tests A. Two Types of Assumption-Freer Tests 1. Nonparametric tests 2. Distribution-free tests

2. B. Comparison of Assumptions 1. Parametric tests Population elements are randomly sampled or the elements are randomly assigned to the experimental conditions. Population is normally or binomially distributed. Null hypothesis is true. Other assumptions are sometimes required such as homogeneity of population variances.

3. 2. Assumption-freer tests Population elements are randomly sampled or the elements are randomly assigned to the experimental conditions. Population is continuous, which implies that no two population elements have the same value (no tied values). Null hypothesis is true. The assumption-freer test in this chapter also assume an ordered qualitative variable.

4. II Mann-Whitney U Test for Independent Samples A. Statistical Hypotheses H0: Population distributions for the experimental and control groups are identical. H1: Population distributions are not identical.

5. B. Computational Example 1. A random sample of 10 men and 10 women who are facing major surgery complete a check list that measures their apprehensiveness prior to surgery. Scores range from 10 (very high) to 0 (very low). Assume that the men and women are representative of people facing major surgery.

6. Table 1. Apprehensiveness of Patients Facing Major Surgery Men Women Men Women R1 R2 5 5 15.5 15.5 3 7 10.5 18.5 0 1 2.0 5.0 4 0 13.5 2.0 1 3 5.0 10.5 2 1 7.5 5.0 6 0 17.0 2.0 10 4 20.0 13.5 3 2 10.5 7.5 7 3 18.5 10.5 R1 = 120.0 R1 = 90.0

7. 2. Computational check: 3. Computation of U

8. 4. Critical value of U, U/2; n1, n2, is U.05/2; 10, 10 = 23. 5. To be significant, U ≤ U/2; n1, n2 6. The null hypothesis can not be rejected because U = 35 > U.05/2; 10, 10 = 23.

9. C. Computational Procedure When One or Both n’s Exceed 20 1. Approximate procedure using the normal distribution 2. c is a correction for continuity and is equal to 0.5

10. z.05 = 1.96 3. If there are tied scores, the denominator can be corrected for ties as follows:

12. D. One-Tailed Test With the Mann-Whitney U E. Relative Efficiency of the Mann-Whitney U Test 1. Power efficiency, PE, is where nL is the sample size required by test L to equal the power of the more efficient test S, based on nS observations

13. 2. Power efficiency of the Mann-Whitney U test relative to the t test for independent samples is 3. The value of PE is affected by , 1 – , H0, H1, and the sample size of the more efficient comparison statistic. Never the less, PE is a useful index of relative efficiency.

14. III Wilcoxon T Test for Dependent Samples A. Statistical Hypotheses H0: Population distributions for the experimental and control groups are identical. H1: Population distributions are not identical.

15. B. Computational Example 1. The number of times that seven study-abroad students called home during the first and second months abroad was determined. It was hypothesized that the students would place more calls during their second month abroad. 2. Assume that the seven students are representative of a population of students who participate in study-abroad programs.

16. 3. Statistical hypotheses H0: Population distributions of phone calls for the first and second months are identical. H1: Population distribution for the second month is displaced (shifted) above that for the first month.

17. Table 2. Distribution of Phone Calls for Study-Abroad Students 1st 2nd R+R– StudentMo. Mo. Diff. |Rank| Diff. Diff. 1 3 7 –4 4 –4 2 5 4 1 1 1 3 2 6 –4 4 –4 4 1 5 –4 4 –4 5 7 2 5 6 6 6 2 4 –2 2 –2 7 10 18 –8 7 –7 R+ = 7 |R–| = 21

18. 4. Computational check: R+ + |R–|=n(n +1)/2 7 + 21 = 7(7 + 1)/2 = 28 5. Test statistic, T, is the smaller of R+ and |R–|. 6. Critical value of T, T; n, is T.05, 7 = 2. 7. To be significant, T ≤ T.05, 7 8. The null hypothesis can not be rejected because T = 7 > T.05, 7 = 2.

19. C. Computational Procedure When n Exceeds 50 1. Approximate procedure using the normal distribution 2. c is a correction for continuity and is equal to 0.5.

20. z.05 = 1.645 3. If there are tied scores, the denominator can be corrected for ties as follows:

21. z.05 = 1.64

22. D. Relative Efficiency of the Wilcoxon T Test 1. Power efficiency of the Wilcoxon T test relative to the t test for dependent samples is 95.5% IV Comparison of Parametric Tests and Assumption-Freer Tests for Ranked Data A. Assumptions

23. Both kinds of tests assume that the population elements are randomly sampled or the elements are randomly assigned to the experimental conditions. Population is normally distributed versus the population is continuous. Homogeneity of population variances versus no assumptions about dispersion.

24. Quantitative variable versus an ordered quantitative variable 2. Level of mathematics necessary to understand their rational 3. Computational simplicity 4. Nature of the statistical hypothesis tested 5. Relative power

25. The End