Chapter 8 Nonparametric statistics

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Chapter 8Non-parametric statistics
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Agenda

• Sign test
• Wilcoxon signed-rank test
• Wilcoxon rank-sum test
• Kruskal-Wallis test

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Introduction

• Most of the hypothesis-testing and confidence
  interval procedures discussed in previous
  chapters are based on the assumption that we are
  working with random samples from normal
  populations.
• These procedures are often called parametric
  methods
• In this chapter, nonparametric and distribution
  free methods will be discussed.
• We usually make no assumptions about the
  distribution of the underlying population.

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Sign Test Description of the Test

• The sign test is used to test hypotheses about
  the median of a continuous distribution.
• Let R represent the number of differences
• that are positive.

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Sign Test Description of the Test
If the following hypotheses are being tested
The appropriate P-value is
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Sign Test Description of the Test
If the following hypotheses are being tested
The appropriate P-value is
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Sign Test Description of the Test
If the following hypotheses are being tested
If r lt n/2, then the appropriate P-value is
If r gt n/2, then the appropriate P-value is
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Example 1
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Example 1
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Example 1
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Sign Test
The Normal Approximation
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Example 2
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Example 2
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Sign Test for Paired Samples
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Example 3
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Example 3
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Type II Error for the Sign Test
Calculation of ? for the sign test. (a) Normal
distributions. (b) Exponential distributions

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Wilcoxon Signed-Rank Test

• The Wilcoxon signed-rank test applies to the
  case of symmetric continuous distributions.
• Under this assumption, the mean equals the
  median.
• The null hypothesis is H0 ? ?0

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Example 4
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Example 4
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Wilcoxon Signed-Rank Test Large-Sample
Approximation
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Wilcoxon Signed-Rank Test Paired Observations
Example 5
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Wilcoxon Signed-Rank Test Paired Observations
Example 5
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Wilcoxon Signed-Rank Test Paired Observations
Example 5
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Wilcoxon Rank-Sum Test Description of the Test
We wish to test the hypotheses
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Wilcoxon Rank-Sum Test Description of the Test
Test procedure Arrange all n1 n2 observations
in ascending order of magnitude and assign
ranks. Let W1 be the sum of the ranks in the
smaller sample. Let W2 be the sum of the ranks
in the other sample. Then W2 (n1 n2)(n1
n2 1)/2 W1
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Example 6
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Example 6
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Example 6
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Example 6
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Nonparametric Methods in the Analysis of
Variance Kruskal-Wallis test
The single-factor analysis of variance model for
comparing a population means is
The hypothesis of interest is
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Nonparametric Methods in the Analysis of
Variance Kruskal-Wallis test
The test statistic is
Computational method
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Example 7
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Example 7