Wilcoxon Rank Sum Test

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  Nonparametric Statistical Methods

  • Myles Hollander, Douglas A. Wolfe, Eric Chicken(Authors)
  • 2013(Publication Date)
  • Wiley

    (Publisher)

  With this modified value of W ∗ , approxima-tions (4.10), (4.11), and (4.12) can be applied. 4.1 A Distribution-Free Rank Sum Test (Wilcoxon, Mann and Whitney) 119 The Mann–Whitney Statistic Procedures (4.4), (4.5), and (4.6) based on the rank sum statistic can also be performed using the Mann–Whitney statistic. Let U = m i = 1 n j = 1 φ( X i , Y j ) , (4.15) where φ( X i , Y j ) = 1 if X i < Y j , 0 otherwise . The statistic U counts the number of “ X before Y ” predecessors. It is easy to show (see Comment 7) that W = U + n ( n + 1 ) 2 . (4.16) Thus tests based on W and U are equivalent. For example, the one-sided test given by (4.4) that rejects if W ≥ w α is equivalent to the one-sided test that rejects if U ≥ u α where u α is the upper α percentile point of the null distribution of U . From (4.16) it follows that w α = u α + ( n ( n + 1 )/ 2 ) . Some textbooks and some software find it more convenient to use U rather than W . For example, the R functions wilcox.test , pwilcox , and qwilcox , illustrated in Comment 3 and Example 4.1, utilize U = U − mn , (4.17) the number of Y before X predecessors. The possible values of U and U are 0, 1, . . . , mn . Furthermore, when H 0 is true, the mean and variance of U and U are, respectively, E 0 ( U ) = E 0 ( U ) = mn / 2 (4.18) Var 0 ( U ) = Var 0 ( U ) = mn ( m + n + 1 )/ 12 . (4.19) The null distributions of U and U are symmetric about the mean mn / 2. EXAMPLE 4.1 Water Transfer in Placental Membrane. The data in Table 4.1 are a portion of the data obtained by Lloyd et al. (1969). Among other things, these authors investigated whether there is a difference in the transfer of tritiated water (water containing tritium, a radioactive isotope of hydrogen) across the tissue layers in the term human chorioamnion (a placental membrane) and in the human chorioamnion between 3-and 6-months’ gestational age. The objective measure used was the permeability constant Pd of the human chorioamnion to water.

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  Introduction to Probability and Statistics Metric Edition

  • William Mendenhall, Robert Beaver, Barbara Beaver, , William Mendenhall, Robert Beaver, Barbara Beaver(Authors)
  • 2019(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  What answers are obtained if Wilcoxon’s signed-rank test is used in analyzing the data? Compare with your earlier answers. 5. Precision Instruments Assume (as in the case of measurements produced by two well-calibrated measur-ing instruments) the means of two populations are equal. Use the Wilcoxon rank sum statistic for testing hypoth-eses concerning the population variances as follows: a. Rank the combined sample. b. Number the ranked observations “from the outside in”; that is, number the smallest observation 1 , the largest 2 , the next-to-smallest 3 , the next-to-largest 4, and so on. This sequence of numbers induces an ordering on the symbols A (population A items) and B (population B items). If 2 2 A B s s . , one would expect most of the A’s near the beginning of the ranking sequence, and thus a relatively small “sum of ranks” for the A observations. c. Given the measurements in the table produced by well-calibrated precision instruments A and B, test at near the .05 a 5 level to determine whether the more expensive instrument B is more precise than A. (Note that this implies a one-tailed test.) Use the Wilcoxon Rank Sum Test statistic. Instrument A Instrument B 1060.21 1060.24 1060.34 1060.28 1060.27 1060.32 1060.36 1060.30 1060.40 d. Test using the equality of variance -test F . 6. Interviewing Job Prospects A large corpora-tion selects college graduates for employment using both interviews and a written test. Interviews conducted at the company site are far more expensive than the written test conducted on campus. Consequently, the personnel office was interested in determining whether the test scores could be substituted for inter-views. The idea was not to eliminate interviews but to reduce their number. Ten prospects were ranked during interviews and tested. The paired scores are as listed here: Subject Interview Rank Test Score 1 8 74 2 5 81 3 10 66 4 3 83 5 6 66 6 1 94 7 4 96 8 7 70 9 9 61 10 2 86 DATA SET DS1538 a.

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  Discovering Statistics Using SAS

  • Andy Field, Jeremy Miles(Authors)
  • 2010(Publication Date)
  • SAGE Publications Ltd

    (Publisher)

  This process results in high scores being represented by large ranks, and low scores being represented by small ranks. The analysis is then carried out on the ranks rather than the actual data. This process is an ingenious way around the problem of using data that break the parametric assumptions. Some people believe that non-parametric tests have less power than their parametric counterparts, but as we will see in Jane Superbrain Box 15.1 this is not always true. In this chapter we’ll look at four of the most common non-parametric procedures: the Mann–Whitney test, the Wilcoxon signed-rank test, Friedman’s test and the Kruskal–Wallis test. For each of these we’ll discover how to carry out the analysis on SAS and how to interpret and report the results. 15.3. Comparing two independent conditions: the Wilcoxon rank-sum test and Mann–Whitney test When you want to test differences between two conditions and different participants have been used in each condition then you have two choices: the Mann-Whitney test (Mann & Whitney, 1947) and Wilcoxon’s rank-sum test (Wilcoxon, 1945; Figure 15.2). These tests are the non-parametric equivalent of the independent t -test. In fact both tests are equivalent, and there’s another, more famous, Wilcoxon test, so it gets extremely confusing for most of us. To make life slightly more confusing, there are two Wilcoxon tests – the Wilcoxon rank-sum test, and the Wilcoxon signed-rank test. As you can imagine, these similar names mean that it’s very easy to confuse things – many people prefer to never mention the rank-sum test, always calling it the Mann–Whitney test. However SAS calls them both Wilcoxon tests. To make it easier to tell which test I am talking about, I’m going to refer to the Wilcoxon two-sample test when we have independent groups (i.e. two samples), and the Wilcoxon signed-rank test when we have repeated measures (i.e

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  Introductory Statistics

  • Prem S. Mann(Author)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  • Determine a rejection region for the Wilcoxon Rank Sum Test for both a small-sample and a large-sample case. • Apply the Wilcoxon Rank Sum Test procedure for both a small-sample and a large-sample case. In Chapter 10 we discussed tests of hypotheses about the difference between the means of two independent populations using the normal and t distributions. In this section we com- pare two independent populations using the results obtained from samples drawn from these populations. In a Wilcoxon Rank Sum Test, we assume that the two populations have iden- tical shapes but differ only in location, which is measured by the median. Note that identical shapes do not mean that they have to have a normal distribution. To apply this test, we must be able to rank the given data. Note that the Wilcoxon Rank Sum Test is almost identical to the Mann‐Whitney test. In the hypothesis tests discussed in this section, the null hypothesis is usually that the two population distributions are identical. The alternative hypothesis can be that the two popula- tion distributions are not identical or that one distribution is centered to the right of the other or that one distribution is centered to the left of the other. Assuming that the null hypothesis is true and that the two populations are identical, we rank all the (combined) data values of the two samples as if they were drawn from the same population. Any tied data values are 15.3 The Wilcoxon Rank Sum Test for Two Independent Samples 643 assigned the ranks in the same manner as in the preceding section. Then we sum the ranks for the data values of each sample separately. If the two populations are identical, the ranks should be spread randomly (and evenly) between the two samples. In this case, the sums of the ranks for the two samples should be almost equal, given that the sizes of the two samples are almost the same.

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  Sensory Evaluation of Food

  Statistical Methods and Procedures

  • Michael O'Mahony(Author)
  • 2017(Publication Date)
  • Routledge

    (Publisher)

  An experimenter may decide that the scores from a rating scale cannot be realistically considered as having equal intervals due to the judges’ lack of skill in generating numbers. He may decide that a score of 10 does not signify an intensity twice as great as one denoted by 5 and that the psychological distance between 4 and 5 is not the same as that between 1 and 2. However, a high score does at least indicate a higher intensity than a low value; the data may at least be ranked. After assigning ranks to his data, a nonparametric test will be needed for its analysis. The experimenter may even decide that rather than force judges to attempt the complex skill of rating, the simpler skill of ranking might be more appropriate; again a nonparametric analysis is required. Ranking would seem especially appropriate for consumer testing where little training is given and where there is not much time available for testing. Ranking has the advantage of simplicity and the fact that, in our culture, we are well practiced in this skill.

  The nonparametric tests considered in this chapter will be ones suitable for analyzing ranked (ordinal) data. We will examine the Wilcoxon and Mann-Whitney U two-sample difference tests, as well as the rank sums test confusingly attributed to Wilcoxon, Mann and Whitney. For many-sample difference tests we will consider the Freidman, Kramer and Kruskal-Wallis tests as well as other tests which test for a ranking trend between treatments: the Page test and the Jonckheere test. We will also consider Spearman’s rank correlation. These are nonparametric versions of t , F , and r ; consult the Key to Statistical Tables to see the relationship between these and other tests. Also note their relationship to each other and to other nonparametric tests already considered: the binomial test, the McNemar test, the sign test, chi-square, the Cochran Q test, and the contingency coefficient.

  We begin with two-sample difference tests.

  16.2 The Wilcoxon Test: A Two-Sample, Related-Samples Difference Test

  Computation of the Wilcoxon Test and Comparison with the Sign Test

  This test is sometimes called the Wilcoxon signed rank test . The test is a related-samples difference test, the nonparametric equivalent of the related-samples t test. However, where the t test indicates whether means are significantly different, the Wilcoxon test indicates whether medians are significantly different. It can best be considered by first remembering the sign test. Consider plant growth rates (in cm per 2 weeks) before and after treatment by a gardener who claimed that talking to the plants made them grow more quickly. The rates are given on the opposite page. The sign test can be applied as discussed earlier (see Section 5.8

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  Fundamentals of Biostatistics

  • Bernard Rosner, , , (Authors)
  • 2015(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Appendix Table 11 was constructed under the assumption of no tied values. If there are ties, and min ( n 1 , n 2 ) < 10, then an exact-permutation test must be used to assess statistical significance (see Lehmann [1] or section 9.6 for more details). The Wilcoxon rank-sum test is sometimes referred to in the literature as the Mann-Whitney U test . The test statistic for the Mann-Whitney U test is based on the number of pairs of observations ( x i , y j ), one from each sample, such that x i < y j ; in addition, 0.5 is added to the test statistic for each ( x i , y j ) pair such that x i = y j . The Mann-Whitney U test and the Wilcoxon rank-sum test are completely equivalent. The relationship between them is as follows; U R m m ( 1 ) 2 , 1 = -+ where m is the size of the x sample It can be shown that E ( U ) = E ( R 1 ) -m ( m + 1)/2 = mn /2, Var ( U ) = Var ( R 1 ) and the p -values from the Wilcoxon rank-sum test and the Mann-Whitney U test are the same. Therefore, the choice of which test to use is a matter of convenience (or more specifically, the choice of the software being used). Using the Computer to Perform the Wilcoxon Rank Sum Test The R command wilcox.test can also perform the Wilcoxon rank-sum test both for the large sample method and the exact method. The scores for the two groups need to be stored in a single variable (e.g., z ). The grouping variable, which identi-fies the groups for individual observations, is stored in a separate variable (e.g., g ) and should have two possible values.

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  Nonparametric Statistics

  A Step-by-Step Approach

  • Gregory W. Corder, Dale I. Foreman(Authors)
  • 2014(Publication Date)
  • Wiley

    (Publisher)

  In this chapter, we will describe how to perform and interpret a Wilcoxon signed rank test and a sign test, using both small samples and large samples. In addition, we demonstrate the procedures for performing both tests using SPSS. Finally, we offer varied examples of these nonparametric statistics from the literature.

  3.3    Computing the Wilcoxon Signed Rank Test Statistic

  The formula for computing the Wilcoxon T for small samples is shown in Formula 3.1 . The signed ranks are the values that are used to compute the positive and negative values in the formula:

  (3.1)

  where ΣR+ is the sum of the ranks with positive differences and ΣR− is the sum of the ranks with negative differences.

  After the T statistic is computed, it must be examined for significance. We may use a table of critical values (see Table B.3 in Appendix B ). However, if the numbers of pairs n exceeds those available from the table, then a large sample approximation may be performed. For large samples, compute a z-score and use a table with the normal distribution (see Table B.1 in Appendix B ) to obtain a critical region of z-scores. Formula 3.2 , Formula 3.3 , and Formula 3.4 are used to find the z-score of a Wilcoxon signed rank test for large samples:

  (3.2)

  where is the mean and n is the number of matched pairs included in the analysis,

  (3.3)

  where sT is the standard deviation,

  (3.4)

  where z* is the z-score for an approximation of the data to the normal distribution and T is the T statistic.

  At this point, the analysis is limited to identifying the presence or absence of a significant difference between the groups and does not describe the strength of the treatment. We can consider the effect size (ES) to determine the degree of association between the groups. We use Formula 3.5 to calculate the ES:

  (3.5)

  where |z| is the absolute value of the z-score and n is the number of matched pairs included in the analysis.

  The ES ranges from 0 to 1. Cohen (1988) defined the conventions for ES as small = 0.10, medium = 0.30, and large = 0.50. (Correlation coefficient and ES are both measures of association. See Chapter 7

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  Biostatistics

  Basic Concepts and Methodology for the Health Sciences, 10th Edition International Student Version

  • Wayne W. Daniel, Chad L. Cross(Authors)
  • 2014(Publication Date)
  • Wiley

    (Publisher)

  Ranks y N Mean Rank Sum of Rank x 1.000000 15 9.67 146.00 2.000000 10 18.00 180.00 Total 25 Test Statistic b x 0 0 0 . 5 2 U y e n t i h W - n n a M 0 0 0 . 5 4 1 W n o x o c l i W Z 2.775 6 0 0 . ) d e l i a t - 2 ( . g i S . p m y s A Exact Sig. [2*(1-tailed Sig.)] .004 a a. Not corrected for ties b. Grouping Variable: y FIGURE 13.6.3 SPSS output for Example 13.6.1. 13.6 THE MANN–WHITNEY TEST 695 Mann–Whitney-Wilcoxon test. Indeed, many computer packages give the test value of both the Mann–Whitney test (U) and the Wilcoxon test (W). These two tests are algebraically equivalent tests, and are related by the following equality when there are no ties in the data: U þ W ¼ m m þ 2n þ 1 ð Þ 2 (13.6.3) Computer Analysis Many statistics software packages will perform the Mann– Whitney test. With the data of two samples stored in Columns 1 and 2, for example, MINITAB will perform a one-sided or two-sided test. The MINITAB procedure and output for Example 13.6.1 are shown in Figure 13.6.2. The SPSS output for Example 13.6.1 is shown in Figure 13.6.3. As we see this output provides the Mann–Whitney test, the Wilcoxon test, and large-sample z approximation. EXERCISES 13.6.1 Cranor and Christensen (A-4) studied diabetics insured by two employers. Group 1 subjects were employed by the City of Asheville, North Carolina, and group 2 subjects were employed by Mission– St. Joseph’s Health System. At the start of the study, the researchers performed the Mann–Whitney test to determine if a significant difference in weight existed between the two study groups. The data are displayed in the following table. Weight (Pounds) Group 1 Group 2 252 215 240 185 195 220 240 190 302 310 210 295 205 270 312 212 190 202 200 159 126 238 172 268 170 204 268 184 190 220 170 215 215 136 140 311 320 254 183 200 280 164 148 164 287 270 264 206 214 288 210 200 270 170 270 138 225 212 210 190 265 240 258 182 192 203 217 221 225 126 Source: Data provided courtesy of Carole W.

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  Introductory Probability and Statistics

  Applications for Forestry and Natural Sciences (Revised Edition)

  • Robert Kozak, Antal Kozak, Christina Staudhammer, Susan Watts(Authors)
  • 2019(Publication Date)
  • CAB International

    (Publisher)

  Section 14.1 , uses only plus and minus signs to identify differences between the observations and their median. In the 1940s, Frank Wilcoxon created a similar but more sophisticated test that uses both the direction and the magnitude of the differences between the observations and their median.

  The null and alternative hypotheses for the so-called Wilcoxon signed rank test are the same as those in Section 14.1 . As with the sign test, the Wilcoxon signed rank test can be used to test the null hypothesis of = c in a one-sample test and the null hypothesis of in a paired difference two-sample test. Also, if the samples (either one-sample or paired) are taken from a continuous symmetric population, the signed rank test is applicable for testing unknown population means, as well as medians. In the one-sample case, the absolute values of the differences between the observations and the unknown hypothetical population median (or mean) are ranked. In the paired sample cases, the absolute values of the paired differences (

  di

  ) are ranked. In both cases, zero differences are discarded in the process of ranking. If there are ties, we assign the average of the ranks that would have been assigned if the differences were distinguishable. This concept of ‘ties’ is perhaps best illustrated with an example. In the following problem, the lowest ranked differences are –2, –2 and 2. Since we are looking at absolute values only, this is considered a three-way tie. If the differences between these values were distinguishable, they would be ranked 1, 2 and 3. However, since they are tied, each receives an average rank of (1 + 2 + 3)/3 = 2.

  If the null hypothesis is true, the total of the ranks corresponding to the positive differences (w + ) should be approximately equal to the total of the ranks of the negative differences (w – ). When repeated samples are taken from a population, the w + and w – are considered individual values of W + and W –

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  Statistics for Psychologists

  An Intermediate Course

  • Brian S. Everitt(Author)
  • 2001(Publication Date)
  • Taylor & Francis

    (Publisher)

  not normal. A number of commonly used distribution-free procedures will be described in Sections 8.2–8.5.

  A further class of procedures that do not require the normality assumption are those often referred to as computationally intensive for reasons that will become apparent in Sections 8.6 and 8.7. The methods to be described in these two sections use repeated permutations of the data, or repeated sampling from the data to generate an appropriate distribution for a test statistic under some null hypothesis of interest.

  Although most of the work on distribution-free methods has concentrated on developing hypothesis testing facilities, it is also possible to construct confidence intervals for particular quantities of interest, as is demonstrated at particular points throughout this chapter.

  8.2.  The Wilcoxon–Mann–Whitney Test and Wilcoxon’s Signed Ranks Test

  First, let us deal with the question of what’s in a name. The statistical literature refers to two equivalent tests formulated in different ways as the Wilcoxon Rank Sum Test and the Mann–Whitney test. The two names arise because of the independent development of the two equivalent tests by Wilcoxon (1945) and by Mann and Whitney (1947). In both cases, the authors’ aim was to come up with a distribution-free alternative to the independent samples t test.

  The main points to remember about the Wilcoxon–Mann–Whitney test are as follows.

  1. The null hypothesis to be tested is that the two populations being compared have identical distributions. (For two normally distributed populations with common variance, this would be equivalent to the hypothesis that the means of the two populations are the same.)

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