# notes3 - Nonparametric tests Timothy Hanson Department of...

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Nonparametric tests Timothy Hanson Department of Statistics, University of South Carolina Stat 704: Data Analysis I 1/16

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Nonparametric one and two-sample tests If data do not come from a normal population (and if the sample is not large), we cannot use a t-test. One useful approach to creating test statistics is through the use of rank statistics . Resampling methods provide alternative approaches for testing simple hypotheses and obtaining conﬁdence intervals. For example, the t approach can be used with a permutation test to test H 0 : μ 1 = μ 2 versus any of the alternatives, regardless of whether the data are normal or not . This is covered in Section 16.9 (pp. 712–716) and available in proc multtest . 2/16
Sign test for one population The sign test assumes the data come from from a continuous distribution with model Y i = η + ± i , i = 1 , . . . , n . η is the population median and ± i follow an unknown, continuous distribution. Want to test H 0 : η = η 0 where η 0 is known versus one of the three common alternatives: H 0 : η < η 0 , H 0 : η 6 = η 0 , or H 0 : η > η 0 . 3/16

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Sign test Test statistic is B * = n i = 1 I { y i > η 0 } , the number of y 1 , . . . , y n larger than η 0 . Under H 0 , B * bin ( n , 0 . 5 ) . Reject H 0 if B * is “unusual” relative to this binomial distribution. Question : How would you form a “large sample” test statistic from B * ? You would not need to do that here, but this is common with more complex test statistics with non-standard distributions. 4/16
Example: eye relief data Data are time in minutes that a drug takes to relieve n = 20 irritated eyes, measured redness. Rao (1998) page 178. proc univariate gives the sign test (and the Wilcoxin signed-rank test), but for a two-sided alternative. How do we get the p-value for a one-sided alternative?

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notes3 - Nonparametric tests Timothy Hanson Department of...

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