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I. IntroductionII. Wilcoxon Rank Sum Test for Independent SamplesScoreLegendRankingAspirin RatingRankingIII. Sign Test For Matched PairsTable 5: Rating Scheme For Car ComfortLegendTable 6: Ratings For Comfort: European Car Vs. American CarRespondentEuropean Car RatingDifferenceIV. The Wilcoxon Signed rank Sum Test For Matched PairsTable 8: Differences in Travel Times, Absolute Differences RankedNov. 28, 2006 LEC # 16 ECON 240A-1 L. PHILLIPSNonparametric StatisticsI. IntroductionA principal use of nonparametric methods is for samples whose frequency distribution is not normal. This can be ascertained visually by looking at the histogram of the data series and seeing whether the histogram is bell shaped or not. A test for normalitycould be conducted using the Chi-Square or another approach.For example, suppose you would like to conduct a test for the difference between two means but the two independent samples are not normal. In this case, you could use the Wilcoxon rank sum test for independent samples. Another example arises from experimental design where you have matched pairs. If the data are not normally distributed then you could use the sign test if the data arise from a rating or ranking scheme. If the matched pairs result from quantitative data that is not normally distributed, then you can resort to the Wilcoxon signed rank sum test for matched pairs.II. Wilcoxon Rank Sum Test for Independent Samples This test is applied to problems of testing the difference between the means of twopopulations when they are non-normal. The text uses an example of rating a new pain- Table 1: Rating Scheme for New Painkilling DrugScore Legend5 Extremely effective4 Quite effective3 Somewhat effective2 Slightly effective1 Not at all effectiveNov. 28, 2006 LEC # 16 ECON 240A-2 L. PHILLIPSNonparametric Statisticskilling drug, compared to aspirin as a control. The rating scheme is displayed in Table 1. The data file is xm17-02. Thirty people were randomly selected and fifteen were given the new drug to rate and fifteen were given aspirin.There are fifteen ratings for the new drug and fifteen for aspirin, as displayed in Table 2.-------------------------------------------------------------------------------------------Table 2: Ratings For the New Painkiller and For AspirinNew Drug Aspirin3 45 14 33 22 45 11 34 45 23 23 25 45 35 44 5--------------------------------------------------------------------------------------The procedure is to take the thirty ratings and to rank them starting with the smallest number. There are three ones, ranked 1, 2, 3, and since they are tied they receive the average rank of two. There are five twos, ranked 4, 5, 6, 7, and 8 and since they are tied they receive the average rank of 6. The process of ranking proceeds in this manner. The ratings, sorted in ascending order, and the raw ranks, not accounting for ties, are displayed in Table 3, along with the ranks where ties have been taken into account.Table 3: Ratings of the Painkiller and Aspirin, Sorted in Ascending Order and RankedNov. 28, 2006 LEC # 16 ECON 240A-3 L. PHILLIPSNonparametric StatisticsRating Raw Rank Rank/TiesNov. 28, 2006 LEC # 16 ECON 240A-4 L. PHILLIPSNonparametric Statistics1 1 21 2 21 3 22 4 62 5 62 6 62 7 62 8 63 9 123 10 123 11 123 12 123 13 123 14 123 15 124 16 19.54 17 19.54 18 19.54 19 19.54 20 19.54 21 19.54 22 19.54 23 19.55 24 275 25 275 26 275 27 275 28 275 29 275 30 27-----------------------------------------------------------------------------------------------------The next step is to use Table 3 to modify Table 2, incorporating the rankings where ties are accounted for. This is displayed in Table 4. The entity used for testing is the rank sum of the new drug, 276.5, denoted T. Note that this rank sum is higher than therank sum for aspirin, indicating higher ratings for the new painkiller. The question is whether these ratings are significantly higher.Table 4: Ratings and Corresponding Rankings For the New Painkiller and Aspirin New Drug Rating Ranking Aspirin Rating Ranking3 12 4 19.55 27 1 2Nov. 28, 2006 LEC # 16 ECON 240A-5 L. PHILLIPSNonparametric Statistics4 19.5 3 123 12 2 62 6 4 19.55 27 1 21 2 3 124 19.5 4 19.55 27 2 63 12 2 63 12 2 65 27 4 19.55 27 3 125 27 4 19.54 19.5 5 27Rank Sum 276.5 188.5------------------------------------------------------------------------------------------------For sample sizes greater than ten, T is approximately normally distributed. The expected value of T is:E(T) = n1 (n1 + n2 + 1)/2 = 15(31)/2 =232.5, (1)Where the subscript 1 refers to sample one, the new drug, and 2 refers to sample two, aspirin. The standard deviation of T is:T = [n1 n2 (n1 + n2 + 1)/12]1/2 = 12/)31)(15(15 = 24.1 (2)The z statistic is:z = (T – E[T])/T = (276.5-232.5)/24.1 = 1.83 (3)A one-tailed test is used since the null hypothesis is that the central tendency or location for the new drug is the same as the central tendency for aspirin, i.e. there is no difference in locations between these two populations. The alternative hypothesis is that the central tendency or location for the new drug is greater than the locations between these two populations. The critical value for the normal distribution at a significance level of 5% is 1.645, as illustrated in Figure 1.---------------------------------------------------------------------------------------0.00.10.20.30.40.5-4 -2 0 2 4ZFREQUENCYFigure 1: One-Tailed Test, 5% Level, Normal DistributionNov. 28, 2006 LEC # 16 ECON 240A-6 L. PHILLIPSNonparametric StatisticsThe authors use their macro, STATS, to calculate this test. The data are in adjacentcolumns. The Wilcoxon Rank Sum Test is found under the Tools menu, data Analysis Plus.In the text, as another example, the authors use the data file workers in Ch. 17 to examine whether there is any difference in the duration of employment for 25 business graduates versus 20 non-business graduates.III. Sign Test For Matched PairsWe have used matched pairs as an experimental design to diminish unexplained variance. Once again, we resort to nonparametric methods if the data are not normally distributed. The text uses the data file xm17-03 which includes comfort ratings by 25 respondents who compare a European car to an American car. The rating scheme is listed in Table 5.1.6455%Nov. 28, 2006 LEC # 16 ECON 240A-7 L. PHILLIPSNonparametric Statistics------------------------------------------------------------------------------Table 5: Rating Scheme For Car ComfortScore Legend1 Ride


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