Wilcoxon Signed Ranks TestLowe Ana Marie L. Ligad-TranceStudent, Ph. D. in Science Education - Biology

Frank was an outstanding chemist, whose interest in statistics first started while studying fungicides, when he and colleagues studied Fishers newly published Statistical Methods for Research Workers.In 1945, he published his paper setting out the rank-sum and signed-rank tests which are still named after him.FRANK WILCOXON (1892 1965)Lowe Ana Marie L. LigadTranceWILCOXON SIGNED RANKS TESTNovember 2014

WHEN TO USE WILCOXON?Use this when the same participants perform both conditions, i.e., it is appropriate for analyzing the data from a repeated-measures design with two conditions.Use it when the data do not meet the requirements for a parametric test (i.e. if the data are not normally distributed; if the variances for the two conditions are markedly different; or if the data are measurements on an ordinal scale.Lowe Ana Marie L. LigadTranceWILCOXON SIGNED RANKS TESTNovember 2014

WHEN TO USE WILCOXON?The Wilcoxon signed-ranks test can also be used for paired data.Where you have two unrelated samples, which you wish to compare, the Wilcoxon rank sum test is used; this is very similar to the Mann Whitney test.Lowe Ana Marie L. LigadTranceWILCOXON SIGNED RANKS TESTNovember 2014

WHEN TO USE WILCOXON?The assumption made for the Wilcoxon test is that the variable being tested is symmetrically distributed about the median, which would also be the mean. Remember too that it is still vitally important that the samples has been randomly chosen from the population.The Wilcoxon signed-rank test can be used for a single sample.Lowe Ana Marie L. LigadTranceWILCOXON SIGNED RANKS TESTNovember 2014

SAMPLE PROBLEMS USING WILCOXON SR TESTA survey collected information about how much students spend when buying presents for different people. Each student recorded the amount they would spend on a present for a partner and also when buying a present for a friend of the same sex. In evolutionary terms, we might expect students to spend more on the (potential) mothers/fathers of their children than on friends of the same sex.Lowe Ana Marie L. LigadTranceWILCOXON SIGNED RANKS TESTNovember 2014

SAMPLE PROBLEMS USING WILCOXON SR TESTSuppose we wanted to know if people's ability to report words accurately was affected by which ear they heard them in. To investigate this, we performed a dichotic listening task. Each participant heard a series of words, presented randomly to either their left or right ear, and reported the words if they could. Each participant thus provided two scores: the number of words that they reported correctly from their left ear, and the number reported correctly from their right ear. Do participants report more words from one ear than the other?Lowe Ana Marie L. LigadTranceWILCOXON SIGNED RANKS TESTNovember 2014

SAMPLE PROBLEMS USING WILCOXON SR TESTEight patients suffering from OCD were exposed to both a Placebo and an Experimental treatment (at different times). They were measured on severity of symptoms, on a 20-point scale were 20 = highest severity. Previous research indicates that severity of symptoms is not normally distributed in this population. Was there a significant difference in severity of symptoms between the two conditions? Lowe Ana Marie L. LigadTranceWILCOXON SIGNED RANKS TESTNovember 2014

STEPS/PROCEDURESFor each item in a sample of n items, compute a difference score, Di, between the two paired values.

Neglect the + and signs and list the set n absolute differences, |Di|.

Omit any absolute difference score of zero from further analysis, thereby yielding a set of n nonzero absolute difference scores, where n n. n becomes the actual sample size.Lowe Ana Marie L. LigadTranceWILCOXON SIGNED RANKS TESTNovember 2014

STEPS/PROCEDURESAssign ranks Ri from 1 to n to each of the |Di| such that the smallest absolute difference score gets rank 1 and the largest gets rank n. If two or more |Di| are equal, assign each of them the mean of the ranks they would have been assigned individually had ties in the data not occurred.

Reassign the symbol + and to each of the n ranks, Ri, depending on whether Di was originally positive or negative.Lowe Ana Marie L. LigadTranceWILCOXON SIGNED RANKS TESTNovember 2014

STEPS/PROCEDURESCompute the Wilcoxon test statistic, W, as the sum of the positive ranks.

For samples of n > 20, the test statistic W is approximately normally distributed with mean W and standard deviation W defined as

- may also be the negative ranks, which ever is lower.Lowe Ana Marie L. LigadTranceWILCOXON SIGNED RANKS TESTNovember 2014

STEPS/PROCEDURESLarge-Sample* Wilcoxon Signed Ranks Test

* - in some references , higher than 15 samples is already considered as large sample (as in Siegel, Castellan)

Lowe Ana Marie L. LigadTranceWILCOXON SIGNED RANKS TESTNovember 2014

HYPOTHESESThe null and alternative hypotheses for the Wilcoxon signed rank test areTwo-Tail TestOne-Tail TestOne-Tail TestHo: MD = 0Ho: MD 0Ho: MD 0H1: MD 0H1: MD < 0H1: MD > 0

Lowe Ana Marie L. LigadTranceWILCOXON SIGNED RANKS TESTNovember 2014

TABLE (CRITICAL VALUES FOR WILCOXON SR TEST) Lowe Ana Marie L. LigadTranceWILCOXON SIGNED RANKS TESTNovember 2014

TABLE (CRITICAL VALUES FOR WILCOXON SR TEST) Lowe Ana Marie L. LigadTranceWILCOXON SIGNED RANKS TESTNovember 2014

TABLE (NORMAL CURVE)Cumulative Probabilities of the Standard Normal DistributionLowe Ana Marie L. LigadTranceWILCOXON SIGNED RANKS TESTNovember 2014

EXAMPLETwelve adult males were put on a liquid diet in a weight-reducing plan. Weights were recorded before and after the diet. The data are shown in the table below. Determine if the plan was successful. Use = 0.05.

Hypotheses:Ho: MD 0,H1: MD < 0.Lowe Ana Marie L. LigadTranceWILCOXON SIGNED RANKS TESTNovember 2014

SOLUTIONLowe Ana Marie L. LigadTranceWILCOXON SIGNED RANKS TESTNovember 2014

AfterBeforeDi|Di|RiSignSigned RiRi+Ri-188186-224 - -44177171-667 - -77176177112 + 22169168-112 - -22196191-555.5 - -5.55.51721720ignore165177121210 + 1010190191112 + 22165170555.5 + 5.55.5180171-999 - -99181188778 + 88172187151511 + 11116638.527.5

SOLUTIONLowe Ana Marie L. LigadTranceWILCOXON SIGNED RANKS TESTNovember 2014n = 12 1 = 11. Based on Table for WSR Test,

the critical value at = 0.05 is 10 (2-tailed test) and 13 (1-tailed test).

Since lower(38.5, 27.5) = 27.5 and is higher than any critical value (p > 0.05), then it is concluded that there is no significant evidence to reject the null hypothesis.

USING SPSSEnter the After data values into one variable and the Before data values into another variable (see left figure, below). Note that the row/case numbers correspond to the subject numbers, as in the table above.Lowe Ana Marie L. LigadTranceWILCOXON SIGNED RANKS TESTNovember 2014

USING SPSSSelect Analyze Nonparametric Tests 2 Related Samples (see right figure, above).Select Weight After Diet and Weight Before Diet together, then click the arrow button to enter them into the Test Pair(s) List. Select Wilcoxon as the Test Type and then click OK. (See the figure, below.)Lowe Ana Marie L. LigadTranceWILCOXON SIGNED RANKS TESTNovember 2014

USING SPSSYour output should look like this.Lowe Ana Marie L. LigadTranceWILCOXON SIGNED RANKS TESTNovember 2014

USING SPSSLowe Ana Marie L. LigadTranceWILCOXON SIGNED RANKS TESTNovember 2014

USING SPSSLowe Ana Marie L. LigadTranceWILCOXON SIGNED RANKS TESTNovember 2014

SUMMARYThe Wilcoxon test is still valid for data from any distribution, whether Normal or not, and is much less sensitive to outliers than the two-sample t-test.If one is primarily interested in differences in location between the two distributions, the Wilcoxon test has the disadvantage of also reacting to other differences between the distributions such as differences in shape.Lowe Ana Marie L. LigadTranceWILCOXON SIGNED RANKS TESTNovember 2014

SUMMARYWhen the assumptions of the two-sample t-test hold, the Wilcoxon test is somewhat less likely to detect a location shift than is the two-sample t-test. However, the losses in this regard are usually quite small.The Mann-Whitney test is essentially identical to the Wilcoxon test, even though it uses a different test statistic.Lowe Ana Marie L. LigadTranceWILCOXON SIGNED RANKS TESTNovember 2014

REFERENCESAccessed Date: November 2014bbs12e_onlinetopic_ch12-8.pdfCh10.wilcoxon.pdfexample_wilcoxon.pdfM12_BERE8380_12_SE_C12.8.qxd. 2/21/11. 3:45 PM.SPSS.PairedWilcoxonTests.pdfSPSS_SignedRankTest.pdfSPSS.WilcoxonTests.pdfStatSheet10_WilcoxonT.pdfv2appendixc.pdfWilcoxonTable2005.pdf

Siegel, S. & Castellan, N. (1956, 1988, 1989). Nonparametric Statistics for Behavioral Sciences. Fong & Sons Printers Pte Ltd.

Lowe Ana Marie L. LigadTranceWILCOXON SIGNED RANKS TESTNovember 2014

Thank you!Lowe Ana Marie L. LigadTranceWILCOXON SIGNED RANKS TESTNovember 2014