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CD-ROM Chapter 15 Introduction to Nonparametric Statistics
Chapter 15 - Chapter Outcomes After studying the material in this chapter, you should be able to: Recognize when and how to use the runs test and testing for randomness. Know when and how to perform a Mann-Whitney U test. Recognize the situations for which the Wilcoxon signed rank test applies and be able to use it in a decision-making context. Perform nonparametric analysis of variance using the Kruskal-Wallis one-way ANOVA.
Nonparametric Statistics Nonparametric statistical procedures Nonparametric statistical procedures are those statistical methods that do not concern themselves with population distributions and/or parameters.
The Runs Test runs test The runs test is a statistical procedure used to determine whether the pattern of occurrences of two types of observations is determined by a random process.
The Runs Test run A run is a succession of occurrences of a certain type preceded and followed by occurrences of the alternate type or by no occurrences at all.
The Runs Test (Table 15-1)
The Runs Test (Small Sample Example) H 0 : Computer-generated numbers are random between 0.0 and 1.0. H A : Computer-generated numbers are not random. --- + - ++ -- ++ --- ++ -- ++ Runs: 1 2 3 4 5 6 7 8 9 10 There are r = 10 runs From runs table (Appendix K) with n 1 = 9 and n 2 = 11, the critical value of r is 6
The Runs Test (Small Sample Example) Test Statistic: r = 10 runs Critical Values from Runs Table: Possible Runs: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Reject H 0 Do not reject H 0 Decision: Since r = 10, we do not reject the null hypothesis.
Large Sample Runs Test MEAN AND STANDARD DEVIATION FOR r where: n 1 = Number of occurrences of first type n 2 = Number of occurrences of second type
Large Sample Runs Test TEST STATISTIC FOR LARGE SAMPLE RUNS TEST
Large Sample Runs Test (Example 15-2) OOOUOOUOUUOOUUOOOOUUOUUOOO UUUOOOOUUOOUUUOUUOOUUUUU OOOUOUUOOOUOOOOUUUOUUOOOU OOUUOUOOUUUOUUOOOOUUUOOO Table 15-2 n 1 = 53 “O’s”n 2 = 47 “U’s” r = 45 runs
Large Sample Runs Test (Example 15-2) Rejection Region /2 = 0.025 Since z= -1.174 > -1.96 and < 1.96, we do not reject H 0, Rejection Region /2 = 0.025 H 0 : Yogurt fill amounts are randomly distributed above and below 24-ounce level. H 1 : Yogurt fill amounts are not randomly distributed above and below 24-ounce level. = 0.05
Mann-Whitney U Test The Mann Whitney U test can be used to compare two samples from two populations if the following assumptions are satisfied: The two samples are independent and random. The value measured is a continuous variable. The measurement scale used is at least ordinal. If they differ, the distributions of the two populations will differ only with respect to the central location.
Mann-Whitney U Test U-STATISTICS where: n 1 and n 2 are the two sample sizes R 1 and R 2 = Sum of ranks for samples 1 and 2
Mann-Whitney U Test - Large Samples - MEAN AND STANDARD DEVIATION FOR THE U -STATISTIC where: n 1 and n 2 = Sample sizes from populations 1 and 2
Mann-Whitney U Test - Large Samples - MANN-WHITNEY U-TEST STATISTIC
Mann-Whitney U Test (Example 15-4) Since z= -1.027 > -1.645, we do not reject H 0, Rejection Region = 0.05
Wilcoxon Matched-Pairs Test The Wilcoxon matched pairs signed rank test can be used in those cases where the following assumptions are satisfied: The differences are measured on a continuous variable. The measurement scale used is at least interval. The distribution of the population differences is symmetric about their median.
Wilcoxon Matched-Pairs Test WILCOXON MEAN AND STANDARD DEVIATION where: n = Number of paired values
Wilcoxon Matched-Pairs Test WILCOXON TEST STATISTIC
Kruskal-Wallis One-Way Analysis of Variance Kruskal-Wallis one-way analysis of variance can be used in one-way analysis of variance if the variables satisfy the following: They have a continuous distribution. The data are at least ordinal. The samples are independent. The samples come from populations whose only possible difference is that at least one may have a different central location than the others.
Kruskal-Wallis One-Way Analysis of Variance H-STATISTIC where: N = Sum of sample sizes in all samples k = Number of samples R i = Sum of ranks in the i th sample n i = Size of the i th sample
Kruskal-Wallis One-Way Analysis of Variance CORRECTION FOR TIED RANKINGS where: g = Number of different groups of ties t i = Number of tied observations in the i th tied group of scores N = Total number of observations
Kruskal-Wallis One-Way Analysis of Variance H-STATISTIC CORRECTED FOR TIED RANKINGS
Key Terms Kruskal-Wallis One- Way Analysis of Variance Mann-Whitney U Test Nonparametric Statistical Procedure Run Runs Test Wilcoxon Test