13-1 Basics of Nonparametric Tests 645 Efficiency of Nonparametric Tests When the requirements of population distributions are satisfied, nonparametric tests are generally less efficient than their corresponding parametric tests. For example, Section 13-6 presents the concept of rank correlation, which has an efficiency rating of 0.91 when compared to linear correlation in Section 10-1. This means that with all other things being equal, the nonparametric rank correlation method in Section 13-6 requires 100 sample observations to achieve the same results as 91 sample observations analyzed through the parametric linear correlation in Section 10-1, assuming the stricter requirements for using the parametric test are met. Table 13-2 lists nonparametric tests along with the corresponding parametric test and efficiency rating. Table 13-2 shows that several nonparametric tests have efficiency ratings above 0.90, so the lower efficiency might not be an important factor in choosing between parametric and nonparametric tests. However, because parametric tests do have higher efficiency ratings than their nonparametric counterparts, it’s generally better to use the parametric tests when their required assumptions are satisfied. TABLE 13-2 Efficiency: Comparison of Parametric and Nonparametric Tests Application Parametric Test Nonparametric Test Efficiency Rating of Nonparametric Test with Normal Populations Matched pairs of sample data t test Sign test or 0.63 Wilcoxon signed-ranks test 0.95 Two independent samples t test Wilcoxon rank-sum test 0.95 Three or more independent samples Analysis of variance (F test) Kruskal-Wallis test 0.95 Correlation Linear correlation Rank correlation test 0.91 Randomness No parametric test Runs test No basis for comparison Ranks Sections 13-2 through 13-5 use methods based on ranks, defined as follows. DEFINITION Data are sorted when they are arranged according to some criterion, such as smallest to largest or best to worst. A rank is a number assigned to an individual sample item according to its order in the sorted list. The first item is assigned a rank of 1, the second item is assigned a rank of 2, and so on. Handling Ties Among Ranks If a tie in ranks occurs, one very common procedure is to find the mean of the ranks involved in the tie and then assign this mean rank to each of the tied items, as in the following example.
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