Nonparametric 14 Statistics OVERVIEW 14.1 Introduction to Nonparametric Statistics 14.2 Sign Test 14.3 Wilcoxon Signed Rank Test for Matched-Pair Data 14.4 Wilcoxon Rank Sum Test for Two Independent Samples 14.5 Kruskal-Wallis Test 14.6 Rank Correlation Test 14.7 Runs Test for Randomness Chapter 14 Formulas and Vocabulary Chapter 14 Review Exercises Chapter 14 Quiz Wavebreakmedia/Shutterstock Larose_3e_ch14.indd 1 10/30/15 11:03 AM Has Median Gas Mileage Increased? CASE STUDY More than ever, the increasing price of gasoline has made consumers aware of the gas mileage of their cars, trucks, and SUVs. Has the population median gas mileage improved from 2007 to 2014? We attack this problem using the sign test in Section 14.2. Then, in Section 14.6, we test whether a rank correlation exists between the miles per gallon of the vehicles in 2007 and 2014. THE BIG PICTURE Where we are coming from and where we are headed . ● In earlier chapters, we learned how to perform hypothesis tests for population parameters, such as the population mean m or the population proportion p. ● Here, in Chapter 14, we learn about a family of hypothesis tests known as nonparametric hypothesis tests, whose conditions are similar to those in earlier chapters but less stringent. Congratulations on getting this far in your discovery of the field of statistics! Your data analytic skills will enhance your marketability in the twenty-first century workplace. Best of luck in the future! 14-2 Larose_3e_ch14.indd 2 10/30/15 11:03 AM 14-3 Chapter 14 Nonparametric Statistics 14.1 Introduction to Nonparametric Statistics OBJECTIVES By the end of this section, I will be able to . 1 Explain what a nonparametric hypothesis test is, and why we use it. 2 Describe what is meant by the efficiency of a nonparametric test. 1 What Is a Nonparametric Hypothesis Test? In Chapters 9, 10, 12, and 13, we learned how to perform hypothesis tests for popula- tion parameters, such as the population mean m or the population proportion p. To perform each of these parametric hypothesis tests, certain conditions need to be sat- isfied. For example, Section 9.4 showed that the required condition of a t test for the population mean m, when we have a small sample size, is that the population be nor- mally distributed. However, what if we need to perform a t test with a small sample and the population is not normal? We turn to one of the nonparametric hypothesis tests, the subject of this chapter. Parametric hypothesis tests are used to test claims about a population parameter, such as the population mean m or the population proportion p. Often, parametric tests require that the population follow a particular distribution, such as the normal distribution. Nonparametric hypothesis tests, also called distribution-free hypothesis tests, generally have fewer required conditions. In particular, nonparametric tests do not require the population to follow a particular distribution, such as the normal distribution. Recall that we should not perform a parametric hypothesis test (such as the t test for the population mean m) if the conditions are not met. Why, then, would a data ana- lyst take a chance and use a parametric test when the conditions may not be satisfied? The answer is that there are advantages and disadvantages to each method. Advantages of Nonparametric Hypothesis Tests 1. Nonparametric methods may be used on a greater variety of data because they require fewer conditions than their parametric counterparts. For this reason, it is less likely that nonparametric hypothesis tests will be performed inappropriately. 2. Nonparametric methods can be applied to categorical (qualitative) data, such as class standing (freshman, sophomore, junior, or senior). 3. For certain nonparametric procedures, the manual computations tend to be easier than their parametric counterparts. (However, see Disadvantage 3 below.) Disadvantages of Nonparametric Hypothesis Tests 1. Nonparametric hypothesis tests are less efficient than parametric tests. This means that, for a given level of significance a, nonparametric tests require a larger sample size to reject a null hypothesis (more on efficiency below). 2. Nonparametric tests replace the actual data values with either signs (positive or negative) or ranks. Thus, the exact data values are wasted. For example, in the nonparametric sign test performed in Section 14.2, the actual data values are discarded and replaced with positive or negative signs. 3. Because the use of nonparametric hypothesis tests is less widespread, graphing calculators and statistical software often do not have dedicated procedures for these tests. Larose_3e_ch14.indd 3 10/30/15 11:03 AM 14.1 Introduction to Nonparametric Statistics 14-4 2 The Efficiency of a Nonparametric Hypothesis Test In general, parametric tests are more efficient than corresponding nonparametric tests. The efficiency of a nonparametric test is used to compare it with its corresponding parametric test. The efficiency of a nonparametric hypothesis test is defined as the ratio of the sample size required for the corresponding parametric test to the sample size required for the nonparametric test, in order to achieve the same result (such as correctly rejecting the null hypothesis). The efficiency ratings are reported on the assumption that required conditions for both the parametric and the nonparametric tests have been met. For example, in Section 14.3 we will learn about the Wilcoxon signed rank test for matched-pair data. The corresponding parametric test is the t test for the difference in means for dependent samples that we learned about in Section 10.1. If a certain result is achieved by using the Wilcoxon signed rank test with a sample size of 100, an equivalent result may be obtained using the dependent-samples t test with a sample size of 95. Thus, the eff iciency of the Wilcoxon signed rank test (assuming that the conditions have been met for both tests) is 95 efficiency 5 5 0.95 100 Thus, the Wilcoxon signed rank test is fairly eff icient compared with the dependent- samples t test. On the other hand, the sign test that we will learn about in Section 14.2 has an efficiency of only 0.63, meaning that the corresponding dependent-samples t test requires a sample size of only 63 to achieve the same result that the sign test achieves with a sample size of 100. Thus, the sign test is less efficient than the Wilcoxon signed rank test. However, as we shall see, the conditions for performing the Wilcoxon signed rank test are stricter than for performing the sign test. As is often the case, there is a trade- off between the efficiency of a test and the conditions required for performing the test. Table 1 contains the efficiency ratings of the nonparametric (distribution-free) hypothesis tests that we will learn about in this chapter. The efficiency ratings are cal- culated under the assumption that the conditions for both the parametric and the non- parametric tests have been met. Table 1 Efficiency of nonparametric tests compared with parametric tests Nonparametric Section Situation Parametric test test Efficiency 14.2 Matched pairs t test or Z test Sign test 0.63 (dependent samples) 14.3 Matched pairs t test or Z test Wilcoxon signed 0.95 (dependent samples) rank test 14.4 Two independent t test or Z test Wilcoxon rank 0.95 samples sum test 14.5 Several independent Analysis of variance Kruskal-Wallis 0.95 samples (F test) test 14.6 Correlation Linear correlation Rank correlation 0.91 Note: A data analyst could perform test both the parametric test and the 14.7 Randomness No parametric test Runs test — nonparametric test and leave it up to the client or the end user of the data to determine whether the In each case, the parametric test is more efficient than its nonparametric counterpart, greater efficiency of the parametric though, of course, this greater efficiency comes at the cost of more stringent required test is worth the cost of the more conditions for the parametric tests. Thus, when the conditions for the parametric test are stringent required conditions. met, it is preferable to perform the parametric test as opposed to the nonparametric test. Larose_3e_ch14.indd 4 10/30/15 11:03 AM 14-5 Chapter 14 Nonparametric Statistics Section 14.1 Summary 1. Nonparametric tests do not require the population to 2. The efficiency of a nonparametric hypothesis test is follow a particular distribution, such as the normal defined as the ratio of the sample size required for the distribution. Because of this, nonparametric hypothesis tests corresponding parametric test to the sample size are often called distribution-free hypothesis tests. There are required for the nonparametric test, in order to achieve advantages and disadvantages to using nonparametric tests the same result (such as correctly rejecting the null instead of parametric tests. hypothesis). Section 14.1 Exercises CLARIFYING THE CONCEPTS 5. What are the advantages to using nonparametric 1. What is a parameter? Explain why the hypothesis tests hypothesis tests as opposed to using parametric hypothesis from Chapters 9, 10, 12, and 13 are called parametric tests? (p. 14-3) hypothesis tests. (p. 14-3) 6. What are the disadvantages to using nonparametric 2. What is another term for nonparametric hypothesis tests? hypothesis tests? (p. 14-3) (p. 14-3) 7. Explain what is meant by “efficiency.” (p. 14-4) 3. Explain the difference between nonparametric tests and 8. True or false: There is a trade-off between the parametric tests. (p. 14-3) efficiency of a test and the conditions required for 4. Which types of tests have more stringent conditions, performing the test. (p.