1. Copyright © Cengage Learning. All rights reserved. 14 Elements of Nonparametric Statistics

2. Copyright © Cengage Learning. All rights reserved. 14.1 Nonparametric Statistics

3. 3 How Teenagers See Things

4. 4 Survey data are interesting but often do not follow the assumptions that the inferential statistics learned thus far require. In fact, most of the statistical procedures we have studied in this book are known as parametric methods. For a statistical procedure to be parametric, either we assume that the parent population is at least approximately normally distributed or we rely on the central limit theorem to give us a normal approximation.

5. 5 How Teenagers See Things The nonparametric methods, or distribution-free methods as they are also known, do not depend on the distribution of the population being sampled. The nonparametric statistics are usually subject to much less confining restrictions than their parametric counterparts. Some, for example, require only that the parent population be continuous.

6. 6 How Teenagers See Things The recent popularity of nonparametric statistics can be attributed to the following characteristics: 1. Nonparametric methods require few assumptions about the parent population. 2. Nonparametric methods are generally easier to apply than their parametric counterparts. 3. Nonparametric methods are relatively easy to understand.

7. 7 How Teenagers See Things 4. Nonparametric methods can be used in situations in which the normality assumptions cannot be made. 5. Nonparametric methods are generally only slightly less efficient than their parametric counterparts.

8. 8 Comparing Statistical Tests

9. 9 This chapter presents only a very small sampling of the many different nonparametric tests that exist. The selections presented demonstrate their ease of application and variety of technique. Many of the nonparametric tests can be used in place of certain parametric tests. The question is, then: Which statistical test do we use, the parametric or the nonparametric?

10. 10 Comparing Statistical Tests Sometimes there is also more than one nonparametric test to choose from. The decision about which test to use must be based on the answer to the question: Which test will do the job best? First, let’s agree that when we compare two or more tests, they must be equally qualified for use. That is, each test has a set of assumptions that must be satisfied before it can be applied.

11. 11 Comparing Statistical Tests From this starting point we will attempt to define “best” to mean the test that is best able to control the risks of error and at the same time keep the size of the sample to a number that is reasonable to work with. (Sample size means cost—cost to you or your employer.)

12. 12 Power and Efficiency Criteria

13. 13 Power and Efficiency Criteria Let’s look first at the ability to control the risk of error. The risk associated with a type I error is controlled directly by the level of significance . We know that P (type I error) =  and P (type II error) = . Therefore, it is  that we must control. Statisticians like to talk about power (as do others), and the power of a statistical test is defined to be 1 – .

14. 14 Power and Efficiency Criteria Thus, the power of a test, 1 – , is the probability that we reject the null hypothesis when we should have rejected it. If two tests with the same  are equal candidates for use, then the one with the greater power is the one you would want to choose. The other factor is the sample size required to do a job. Suppose that you set the levels of risk you can tolerate,  and , and then you are able to determine the sample size it would take to meet your specified challenge.

15. 15 Power and Efficiency Criteria The test that required the smaller sample size would seem to have the edge. Statisticians usually use the term efficiency to talk about this concept. Efficiency is the ratio of the sample size of the best parametric test to the sample size of the best nonparametric test when compared under a fixed set of risk values. For example, the efficiency rating for the sign test is approximately 0.63.

16. 16 Power and Efficiency Criteria This means that a sample of size 63 with a parametric test will do the same job as a sample of size 100 will do with the sign test. The power and the efficiency of a test cannot be used alone to determine the choice of test. Sometimes you will be forced to use a certain test because of the data you are given. When there is a decision to be made, the final decision rests in a trade-off of three factors: (1)the power of the test, (2) the efficiency of the test, and (3) the data (and the sample size) available.

17. 17 Power and Efficiency Criteria Table 14.1 shows how the nonparametric tests discussed in this chapter compare with the parametric tests covered in previous chapters. Comparison of Parametric and Nonparametric Tests Table 14.1