Introduction to Nonparametric Statistics: Distribution-Free Tests and Their Applications

1. Chapter 14 Nonparametric Statistics

2. Introduction: Distribution-Free Tests • Distribution-free tests – statistical tests that don’t rely on assumptions about the probability distribution of the sampled population • Nonparametrics – branch of inferential statistics devoted to distribution-free tests • Rank statistics (Rank tests) – nonparametric statistics based on the ranks of measurements

3. Single Population Inferences • The Sign test is used to make inferences about the central tendency of a single population • Test is based on the median η • Test involves hypothesizing a value for the population median, then testing to see if the distribution of sample values around the hypothesized median value reaches significance

4. Single Population Inferences • Sign Test for a Population Median η Conditions required for sign test – sample must be randomly selected from a continuous probability distribution

5. Single Population Inferences • Large-Sample Sign Test for a Population Median η Conditions required for sign test – sample must be randomly selected from a continuous probability distribution

6. Comparing Two Populations: Independent Samples • The Wilcoxon Rank Sum Test is used when two independent random samples are being used to compare two populations, and the t-test is not appropriate • It tests the hypothesis that the probability distributions associated with the two populations are equivalent

7. Comparing Two Populations: Independent Samples • Rank Data from both samples from smallest to largest • If populations are the same, ranks should be randomly mixed between the samples • Test statistic is based on the rank sums – the totals of the ranks for each of the samples. T1 is the sum for sample 1, T2 is the sum for sample 2

8. Comparing Two Populations: Independent Samples • Wilcoxon Rank Sum Test: Independent Samples • Required Conditions: • Random, independent samples • Probability distributions samples drawn from are continuous

9. Comparing Two Populations: Independent Samples • Wilcoxon Rank Sum Test for Large Samples(n1 and n2 ≥ 10)

10. Comparing Two Populations: Paired Differences Experiment • Wilcoxon Signed Rank Test: An alternative test to the paired difference of means procedure • Analysis is of the differences between ranks • Any differences of 0 are eliminated, and n is reduced accordingly

11. Comparing Two Populations: Paired Differences Experiment • Wilcoxon Signed Rank Test for a Paired Difference Experiment • Let D1 and D2 represent the probability distributions for populations 1 and 2, respectively Required Conditions Sample of differences is randomly selected Probability distribution from which sample is drawn is continuous

12. Comparing Three or More Populations: Completely Randomized Design • Kruskal-Wallis H-Test • An alternative to the completely randomized ANOVA • Based on comparison of rank sums

13. Comparing Three or More Populations: Completely Randomized Design • Kruskal-Wallis H-Test for Comparing k Probability Distributions • Required Conditions: • The k samples are random and independent • 5 or more measurements per sample • Probability distributions samples drawn from are continuous

14. Comparing Three or More Populations: Randomized Block Design • The Friedman Fr Test • A nonparametric method for the randomized block design • Based on comparison of rank sums

15. Comparing Three or More Populations: Randomized Block Design • The Friedman Fr-test • Required Conditions: • Random assignment of treatments to units within blocks • Measurements can be ranked within blocks • Probability distributions samples within each block drawn from are continuous

16. Rank Correlation • Spearman’s rank correlation coefficient Rsprovides a measure of correlation between ranks

17. Rank Correlation • Conditions Required: • Sample of experimental units is randomly selected • Probability distributions of two variables are continuous