Chapter 18: Nonparametric Methods

Nonparametric MethodsChapter 18Copyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/IrwinNonparametric Methods18.1 The Sign Test: A Hypothesis Test about the Median18.2 The Wilcoxon Rank Sum Test18.3 The Wilcoxon Signed Ranks Test18.4 Comparing Several Populations Using the Kruskal-Wallis H Test18.5 Spearman’s Rank Correlation Coefficient18-*18.1 Sign Test: A Hypothesis Test about the MedianDefineS = the number of sample measurements (less/greater) than M0x to be a binomial random variable with p = 0.5We can reject H0: Md = M0 at the  level of significance (probability of Type I error equal to ) by using the appropriate p-valueLO18-1: Use the sign test to test a hypothesis about a population median.18-*Sign Test: A Hypothesis Test about the Median ContinuedAlternativeTest Statisticp-ValueHa: Md > MoS=number of measurements greater than MoThe probability that x is greater than or equal to SHa: Md n2 We can reject H0: D1 and D2 are identical probability distributions at the  level of significance if and only if the test statistic T satisfies the appropriate rejection conditionLO18-2: Compare the locations of two distributions using a rank sum test for independent samples.18-*The Wilcoxon Rank Sum Test ContinuedAlternativeReject H0 ifHa: D1 is shifted right of D2T ≥ Tu if n1 ≤ n2T ≤ Tu if n1 > n2Ha: D1 is shifted left of D2T ≤ TL if n1 ≤ n2T ≥ TL if n1 > n2Ha: D1 is shifted right or left of D2T ≤ Tu or T ≥ TuLO18-218-*18.3 The Wilcoxon Signed Rank TestGiven two matched pairs of n observations, selected at random from populations 1 and 2 with distributions D1 and D2 compute the n differences (D1 – D2)Rank the absolute value of the differences from smallest to largestDrop zero differences from sampleAssign average ranks for tiesT- = sum of ranks, negative differencesT+ = sum of ranks, positive differencesWe can reject H0: D1 and D2 are identical probability distributions at the  level of significance if and only if the appropriate test statistic satisfies the corresponding rejection point conditionLO18-3: Compare the locations of two distributions using a signed ranks test for paired samples.18-*The Wilcoxon Signed Rank Test ContinuedAlternativeTest StatisticReject H0 ifHa: D1 is shifted right of D2T-T- ≤ T0Ha: D1 is shifted left of D2T+T+ ≤ T0Ha: D1 is shifted right or left of D2T=smaller of T- or T+T ≤ T0LO18-318-*18.4 Comparing Several Populations Using The Kruskal-Wallis H TestGiven p independent samples(n1, , np  5) from p populationsRank the (n1+ + np) observations from smallest to largest (average ranks for ties)Let T1 equal sum of ranks, sample 1, continuing until Tp equals sum of ranks, sample pLO18-4: Compare the locations of three or more distributions using a Kruskal–Wallis test for independent samples.18-*The Kruskal-Wallis H Test ContinuedTo testH0: The p populations are identicalHa: At least two of the populations differ in locationTest statisticReject H0 if H > 2 or if p-value 0rs > rHa: Ha: ps r/2LO18-518-*