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Worksheet 24a - ph6450 Understanding Wilcoxon Sign-Rank Test [example from: Rosner, Fundamentals of Biostatistics, page 351] An instrument that is in fairly common use in blood-pressure epidemiology is the random-zero device, whereby the zero point of the instrument is randomly set with each use and the observer is not aware of the actual level of blood pressure at the time of measurement. This instrument is intended to reduce observer bias. Before using such an instrument, it is important to check that the readings are, on average, comparable to those of a standard cuff. Two measurements were made on 20 children with each of the standard cuff and the random-zero instrument. Researchers are reluctant to assume that the distribution of the blood pressure in this group is normal. Person 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Mean SBP – Cuff 79 112 103 104 94 106 103 97 88 113 98 103 105 117 94 88 101 98 91 105 Mean SBP – Random Zero 84 99 92 103 94 106 97 108 77 94 97 103 107 120 94 87 97 93 87 104 Difference 5 -13 -11 -1 0 0 -6 11 -11 -19 -1 0 2 3 0 -1 -4 -5 -4 -1 Rank Sign Conduct a non-parametric test to see if the two measurement instruments are comparable. A. What are the null and alternative hypotheses for this test? B. What is the value of the test statistic? C. What is the expected value under the null? D. What is the standard error of the test statistic under the null? E. What is the p-value for this test? F. Draw a conclusion in the scientific context of this problem. > cuff = c(79,112,103,104,94,106,103,97,88,113,98,103,105,117,94,88,101,98,91,105) > rz = c(84,99,92,103,94,106,97,108,77,94,97,103,107,120,94,87,97,93,87,104) > wilcox.test(cuff,rz,paired=TRUE) Wilcoxon signed rank test with continuity correction data: cuff and rz V = 102.5, p-value = 0.07793 alternative hypothesis: true location shift is not equal to 0 Warning messages: 1: In wilcox.test.default(cuff, rz, cannot compute exact p-value with 2: In wilcox.test.default(cuff, rz, cannot compute exact p-value with paired = TRUE) : ties paired = TRUE) : zeroes > wilcox.test(cuff,rz,paired=TRUE,correct=FALSE) Wilcoxon signed rank test data: cuff and rz V = 102.5, p-value = 0.07366 alternative hypothesis: true location shift is not equal to 0 Warning messages: 1: In wilcox.test.default(cuff, rz, cannot compute exact p-value with 2: In wilcox.test.default(cuff, rz, cannot compute exact p-value with paired = TRUE, correct = FALSE) : ties paired = TRUE, correct = FALSE) : zeroes data bp; input obs cuff rz; datalines; 1 79 84 2 112 99 3 103 92 4 104 103 5 94 94 6 106 106 7 103 97 8 97 108 9 88 77 10 113 94 11 98 97 12 103 103 13 105 107 14 117 120 15 94 94 16 88 87 17 101 97 18 98 93 19 91 87 20 105 104 ; data bp; set bp; x = cuff - rz; run; proc univariate data=bp; var x; run; Relevant output: Tests for Location: Mu0=0 Test -Statistic- -----p Value------ Student's t Sign Signed Rank t M S Pr > |t| Pr >= |M| Pr >= |S| 1.84923 4 34.5 0.0800 0.0768 0.0756 Notice the p-value is not very different from the student’s T. This is because there are 20 data points and the normality assumption is not greatly violated. Still, the nonparametric test is more conservative because no specific underlying distribution is assumed.