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### Chapter 14

Nonparametric Statistics

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

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

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

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

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

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

Comparing Two Populations: Independent Samples

- Wilcoxon Rank Sum Test: Independent Samples
- Required Conditions:
- Random, independent samples
- Probability distributions samples drawn from are continuous

Comparing Two Populations: Independent Samples

- Wilcoxon Rank Sum Test for Large Samples(n1 and n2 ≥ 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

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

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

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

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

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

Rank Correlation Design

- Spearman’s rank correlation coefficient Rsprovides a measure of correlation between ranks

Rank Correlation Design

- Conditions Required:
- Sample of experimental units is randomly selected
- Probability distributions of two variables are continuous

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