Presentation on theme: "Introduction to Nonparametric Statistics"— Presentation transcript:
1 Introduction to Nonparametric Statistics CD-ROM Chapter 15Introduction to Nonparametric Statistics
2 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.
3 Nonparametric Statistics Nonparametric statistical procedures are those statistical methods that do not concern themselves with population distributions and/or parameters.
4 The Runs TestThe 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.
5 The Runs TestA 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.
7 The Runs Test (Small Sample Example) H0: Computer-generated numbers are random between 0.0 and 1.0.HA: Computer-generated numbers are not random .Runs:There are r = 10 runsFrom runs table (Appendix K) with n1 = 9 and n2 = 11, the critical value of r is 6
8 The Runs Test (Small Sample Example) Test Statistic:r = 10 runsCritical Values from Runs Table:PossibleRuns:Reject H0Reject H0Do not reject H0Decision:Since r = 10, we do not reject the null hypothesis.
9 MEAN AND STANDARD DEVIATION FOR r Large Sample Runs TestMEAN AND STANDARD DEVIATION FOR rwhere:n1 = Number of occurrences of first typen2 = Number of occurrences of second type
11 Large Sample Runs Test (Example 15-2) Table 15-2OOOUOOUOUUOOUUOOOOUUOUUOOOUUUOOOOUUOOUUUOUUOOUUUUUOOOUOUUOOOUOOOOUUUOUUOOOUOOUUOUOOUUUOUUOOOOUUUOOOn1 = 53 “O’s”n2 = 47 “U’s”r = 45 runs
12 Large Sample Runs Test (Example 15-2) H0: Yogurt fill amounts are randomly distributed above and below 24-ounce level.H1: Yogurt fill amounts are not randomly distributed above and below 24-ounce level. = 0.05Rejection Region /2 = 0.025Rejection Region /2 = 0.025Since z= > and < 1.96, we do not reject H0,
13 Mann-Whitney U Test • The two samples are independent and random. 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.
14 Mann-Whitney U Test U-STATISTICS where: n1 and n2 are the two sample sizesR1 and R2 = Sum of ranks for samples 1 and 2
15 Mann-Whitney U Test - Large Samples - MEAN AND STANDARD DEVIATION FOR THE U-STATISTICwhere:n1 and n2 = Sample sizes from populations 1 and 2
16 Mann-Whitney U Test - Large Samples - MANN-WHITNEY U-TEST STATISTIC
17 Mann-Whitney U Test (Example 15-4) Rejection Region = 0.05Since z= > , we do not reject H0,
18 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.
19 Wilcoxon Matched-Pairs Test WILCOXON MEAN AND STANDARD DEVIATIONwhere:n = Number of paired values
20 Wilcoxon Matched-Pairs Test WILCOXON TEST STATISTIC
21 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.
22 Kruskal-Wallis One-Way Analysis of Variance H-STATISTICwhere:N = Sum of sample sizes in all samplesk = Number of samplesRi = Sum of ranks in the ith sampleni = Size of the ith sample
23 Kruskal-Wallis One-Way Analysis of Variance CORRECTION FOR TIED RANKINGSwhere:g = Number of different groups of tiesti = Number of tied observations in the ith tied group of scoresN = Total number of observations
24 Kruskal-Wallis One-Way Analysis of Variance H-STATISTIC CORRECTED FOR TIED RANKINGS
25 Key Terms • Kruskal-Wallis One-Way Analysis of Variance • Run • Mann-Whitney U Test• Nonparametric Statistical Procedure• Run• Runs Test• Wilcoxon Test