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# Chapter 14 - PowerPoint PPT Presentation

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

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### Chapter 14

Nonparametric Statistics

• 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

• 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

• Sign Test for a Population Median η

Conditions required for sign test – sample must be randomly selected from a continuous probability distribution

• Large-Sample Sign Test for a Population Median η

Conditions required for sign test – sample must be randomly selected from a continuous probability distribution

• 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

• 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

• Wilcoxon Rank Sum Test: Independent Samples

• Required Conditions:

• Random, independent samples

• Probability distributions samples drawn from are continuous

• Wilcoxon Rank Sum Test for Large Samples(n1 and n2 ≥ 10)

• 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

• 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

• Kruskal-Wallis H-Test

• An alternative to the completely randomized ANOVA

• Based on comparison of rank sums

• 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

• The Friedman Fr Test

• A nonparametric method for the randomized block design

• Based on comparison of rank sums

• 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