678 CHAPTER 13 Nonparametric Tests Requirements 1. The paired data are a simple random sample. 2. The data are ranks or can be converted to ranks. Note: Unlike the parametric methods of Section 10-1, there is no requirement that the sample pairs of data have a bivariate normal distribution (as described in Section 10-1 ). There is no requirement of a normal distribution for any population. Test Statistic Within each sample, first convert the data to ranks, then find the exact value of the rank correlation coefficient rs by using Formula 10-1: FORMULA 10-1 rs = n1Σxy2 - 1Σx21Σy2 2 n1Σx22 - 1Σx222n1Σy22 - 1Σy22 Simpler Test Statistic if There Are No Ties: After converting the data in each sample to ranks, if there are no ties among ranks for the first variable and there are no ties among ranks for the second variable, the exact value of the test statistic can be calculated using Formula 10-1 or with the following relatively simple formula, but it is probably easier to use Formula 10-1 with technology: rs = 1 - 6Σd2 n1n2 - 12 P-Values P-values are sometimes provided by technology, but use them only if they result from Spearman’s rank correlation. (Caution: Do not use P-values from linear correlation for methods of rank correlation. When working with data having ties among ranks, the rank correlation coefficient rs can be calculated using Formula 10-1. Technology can be used instead of manual calculations with Formula 10-1, but the displayed P-values from linear correlation do not apply to the methods of rank correlation.) Critical Values 1. If n … 30, critical values are found in Table A-9. 2. If n 7 30, critical values of rs are found using Formula 13-1. FORMULA 13-1 rs = {z 2 n - 1 1critical values for n 7 302 where the value of z corresponds to the significance level. (For example, if a = 0.05, z = 1.96.) Advantages of Rank Correlation: Rank correlation has these advantages over the parametric methods discussed in Chapter 10: 1. Rank correlation can be used with paired data that are ranks or can be converted to ranks. Unlike the parametric methods of Chapter 10, the method of rank correlation does not require a normal distribution for any population. 2. Rank correlation can be used to detect some (not all) relationships that are not linear. Disadvantage of Rank Correlation: Efficiency A minor disadvantage of rank correlation is its efficiency rating of 0.91, as described in Section 13-1. This efficiency rating shows that with all other circumstances being equal, the nonparametric approach of rank correlation requires 100 pairs of sample data to achieve the same results as only 91 pairs of sample observations analyzed through the parametric approach, assuming that the stricter requirements of the parametric approach are met.
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