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. T 1 is the sum for sample 1, T 2 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 F r 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 F r -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 R s provides 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