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McGraw-Hill/Irwin Copyright © 2013 by The McGraw-Hill Companies, Inc. All rights reserved. A PowerPoint Presentation Package to Accompany Applied Statistics in Business & Economics, 4 th edition David P. Doane and Lori E. Seward Prepared by Lloyd R. Jaisingh

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16-2 Nonparametric Tests Chapter Contents 16.1 Why Use Nonparametric Tests? 16.2 One-Sample Runs Test 16.3 Wilcoxon Signed-Rank Test 16.4 Wilcoxon Rank Sum 16.5 Kruskal-Wallis Test for Independent Samples 16.6 Friedman Test for Related Samples 16.7 Spearman Rank Correlation Test Chapter 16

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16-3 Chapter Learning Objectives LO16-1: Define nonparametric tests and explain when they may be desirable. LO16-2: Use the one-sample runs test. LO16-3: Use the Wilcoxon signed-rank test. LO16-4: Use the Wilcoxon rank sum test for two samples. LO16-5: Use the Kruskal-Wallis test for c independent samples. LO16-6: Use the Friedman test for related samples. LO16-7: Use the Spearman rank correlation test. LO16-8: Use computer software to perform the tests and obtain p-values. Chapter 16 Nonparametric Tests

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Why Use Nonparametric Tests? Parametric hypothesis tests require the estimation of one or more unknown parameters (e.g., population mean or variance).Parametric hypothesis tests require the estimation of one or more unknown parameters (e.g., population mean or variance). Often, unrealistic assumptions are made about the normality of the underlying population.Often, unrealistic assumptions are made about the normality of the underlying population. Large sample sizes are often required to invoke the Central Limit Theorem.Large sample sizes are often required to invoke the Central Limit Theorem. Parametric Tests Parametric Tests Chapter 16 Nonparametric Tests Nonparametric Tests Nonparametric or distribution-free testsNonparametric or distribution-free tests - usually focus on the sign or rank of the data rather than the exact numerical value. - do not specify the shape of the parent population. - can often be used in smaller samples. - can be used for ordinal data. LO16-1 LO16-1: Define nonparametric tests and explain when they may be desirable.

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16-5 Advantages and Disadvantages of Nonparametric Tests Advantages and Disadvantages of Nonparametric Tests Chapter 16LO Why Use Nonparametric Tests?

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16-6 Some Common Nonparametric Tests Some Common Nonparametric Tests Chapter 16LO Why Use Nonparametric Tests?

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16-7 To test the hypothesis of a random pattern in a sequence of binary events, first count the number of outcomes of each type. n 1 = number of outcomes of the first type n 2 = number of outcomes of the second type n = total sample size = n 1 + n 2To test the hypothesis of a random pattern in a sequence of binary events, first count the number of outcomes of each type. n 1 = number of outcomes of the first type n 2 = number of outcomes of the second type n = total sample size = n 1 + n 2 A run is a series of consecutive outcomes of the same type, surrounded by a sequence of outcomes of the other type.A run is a series of consecutive outcomes of the same type, surrounded by a sequence of outcomes of the other type. Wald-Wolfowitz Runs Test Wald-Wolfowitz Runs Test Chapter One-Sample Runs Test When n 1 10 and n 2 10, then the number of runs R may be assumed to be normally distributed with mean R and standard deviation R.When n 1 10 and n 2 10, then the number of runs R may be assumed to be normally distributed with mean R and standard deviation R. For a given level of significance, find the critical value z for a two-tailed test. Reject the hypothesis of a random pattern if z +z. LO16-2

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16-8 To compare the sample median (M) with a benchmark median (M 0 ), the hypotheses are:To compare the sample median (M) with a benchmark median (M 0 ), the hypotheses are: When evaluating the difference between paired observations, use the median difference (M d ) and zero as the benchmark.When evaluating the difference between paired observations, use the median difference (M d ) and zero as the benchmark. Chapter Wilcoxon Signed-Rank Test Calculate the difference between the paired observations.Calculate the difference between the paired observations. Rank the differences from smallest to largest by absolute value.Rank the differences from smallest to largest by absolute value. Add the ranks of the positive differences to obtain the rank sum W.Add the ranks of the positive differences to obtain the rank sum W. LO16-3 LO16-3: Use the Wilcoxon signed-rank test.

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16-9 Chapter 16 For small samples, a special table is required to obtain critical values.For small samples, a special table is required to obtain critical values. For large samples (n 20), the test statistic is approximately normal.For large samples (n 20), the test statistic is approximately normal. Use Excel or Appendix C to get a p-value.Use Excel or Appendix C to get a p-value. Reject H 0 if p-value.Reject H 0 if p-value Wilcoxon Signed-Rank Test LO16-3

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16-10 The hypotheses are H 0 : M 1 = M 2 (no difference in medians) H 1 : M 1 M 2 (medians differ)The hypotheses are H 0 : M 1 = M 2 (no difference in medians) H 1 : M 1 M 2 (medians differ) Step 1: Sort the combined samples from lowest to highest.Step 1: Sort the combined samples from lowest to highest. Step 2: Assign a rank to each value.Step 2: Assign a rank to each value. If values are tied, the average of the ranks is assigned to each. Step 3: The ranks are summed for each column (e.g., T 1, T 2 ).Step 3: The ranks are summed for each column (e.g., T 1, T 2 ). Step 4: The sum of the ranks T 1 + T 2 must be equal to n(n + 1)/2, where n = n 1 + n 2.Step 4: The sum of the ranks T 1 + T 2 must be equal to n(n + 1)/2, where n = n 1 + n 2. Step 5: Calculate the mean rank sums – mean of T 1 and mean of T 2.Step 5: Calculate the mean rank sums – mean of T 1 and mean of T 2. Step 6: For large samples (n 1 10, n 2 10), use a z test.Step 6: For large samples (n 1 10, n 2 10), use a z test. Performing the Test Performing the Test Chapter 16LO Wilcoxon Rank Sum Test (Mann Whitney Test) LO16-4: Use the Wilcoxon rank sum test for two samples.

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16-11 Performing the Test Performing the Test Step 7: For a given, reject H 0 if z +zStep 7: For a given, reject H 0 if z +z Chapter 16LO Wilcoxon Rank Sum Test (Mann Whitney Test)

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16-12 The hypotheses to be tested are: H 0 : All c population medians are the same H 1 : Not all the population medians are the sameThe hypotheses to be tested are: H 0 : All c population medians are the same H 1 : Not all the population medians are the same For a completely randomized design with c groups, the tests statistic isFor a completely randomized design with c groups, the tests statistic is Performing the Test Performing the Test Chapter Kruskal-Wallis Test for Independent Samples The H test statistic follows a chi-square distribution with d.f. = c – 1 degrees of freedom.The H test statistic follows a chi-square distribution with d.f. = c – 1 degrees of freedom. This is a right-tailed test, so reject H 0 if H > 2 or if p-value.This is a right-tailed test, so reject H 0 if H > 2 or if p-value. LO16-5

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16-13 The groups must be of the same size.The groups must be of the same size. Treatments should be randomly assigned within blocks.Treatments should be randomly assigned within blocks. Data should be at least interval scale.Data should be at least interval scale Friedman Test for Related Samples Chapter 16LO16-6 LO16-6: Use the Friedman test for related samples.

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16-14 In addition to the c treatment levels that define the columns, the Friedman test also specifies r block factor levels to define each row of the observation matrix.In addition to the c treatment levels that define the columns, the Friedman test also specifies r block factor levels to define each row of the observation matrix. The hypotheses to be tested are: H 0 : All c populations have the same median H 1 : Not all the populations have the same medianThe hypotheses to be tested are: H 0 : All c populations have the same median H 1 : Not all the populations have the same median Unlike the Kruskal-Wallis test, the Friedman ranks are computed within each block rather than within a pooled sample.Unlike the Kruskal-Wallis test, the Friedman ranks are computed within each block rather than within a pooled sample. Chapter Friedman Test for Related Samples LO16-6

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16-15 Compute the test statistic:Compute the test statistic: Performing the Test Performing the Test Chapter 16 The Friedman test statistic F calc follows a chi-square distribution with = c – 1 degrees of freedom.The Friedman test statistic F calc follows a chi-square distribution with = c – 1 degrees of freedom. Reject H 0 if F > 2 or if p-value.Reject H 0 if F > 2 or if p-value Friedman Test for Related Samples LO16-6

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16-16 The sign of r s indicates whether the relationship is direct - ranks tend to vary in the same direction, or inverse - ranks tend to vary in opposite directions The magnitude of r s indicated the degree of relationship. If r s is near 0 - there is little or no agreement between rankings r s is near +1 - there is strong direct agreement r s is near 1 - there is strong inverse agreement Chapter Spearman Rank Correlation Test LO16-7 LO16-7: Use the Spearman rank correlation.

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16-17 The sums of ranks within each column must always be n(n+1)/2.The sums of ranks within each column must always be n(n+1)/2. Next, compute the difference in ranks d i for each observation.Next, compute the difference in ranks d i for each observation. The rank differences should sum to zero.The rank differences should sum to zero. Calculate the sample rank correlation coefficient r s.Calculate the sample rank correlation coefficient r s. Performing the Test Performing the Test Chapter Spearman Rank Correlation Test LO16-7

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16-18 For a right-tailed test for example, the hypotheses to be tested are H 0 : True rank correlation is zero ( s 0) H 1 : True rank correlation is positive ( s > 0)For a right-tailed test for example, the hypotheses to be tested are H 0 : True rank correlation is zero ( s 0) H 1 : True rank correlation is positive ( s > 0) If n is large (at least 20 observations), then r s may be assumed to follow the normal distribution using the test statisticIf n is large (at least 20 observations), then r s may be assumed to follow the normal distribution using the test statistic Performing the Test Performing the Test Chapter 16 Here we will reject H 0 if z calc > z or if p-value.Here we will reject H 0 if z calc > z or if p-value Spearman Rank Correlation Test LO16-7

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