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A Note on the Wilcoxon-Mann-Whitney Test for 2 x k Ordered Tables Author(s): John D. Emerson and Lincoln E. Moses Source: Biometrics, Vol. 41, No. 1 (Mar., 1985), pp. 303-309 Published by: International Biometric Society Stable URL: http://www.jstor.org/stable/2530667 . Accessed: 20/02/2011 10:09 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at . http://www.jstor.org/action/showPublisher?publisherCode=ibs. . Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. International Biometric Society is collaborating with JSTOR to digitize, preserve and extend access to Biometrics. http://www.jstor.org BIOMETRICS41, 303-309 March 1985 A Noteon theWilcoxon-Mann-Whitney for2 x k Test OrderedTables D. John Emerson Department Mathematics, of Middlebury College,Middlebury,Vermont05753,U.S.A. and E. Lincoln Moses Division Biostatistics, of Stanford SchoolofMedicine, University Stanford,California94305,U.S.A. SUMMARY and Biological medical investigations use ordered often data. are categorical Whentwogroups to be compared thedatafor groups in three moreordered and the fall or the categories, Wilcoxon-Mann- Whitney (WMW)test in usesinformation theordering givea test to is that usuallypowerfulagainst shift However, applications WMWoften alternatives. such of involvedistributions which for extensive ties an role. play important Newly availablecomputer for programs performing tests deeper exact give insights and of into the characteristics the exact WMW distributions the suitability normal of approximations. offer We practical advice, with basedon experience published datasets biomedical and on numerical studies hypothetical of orderedtables,forthe use of WMW and its normal approximations. 1. Introduction Ordered categoricalvariables in arisefrequently biomedical In research. some instances thesevariables providequalitative that information is intrinsically in imprecise; other situations arisefrom they classifying into of measurements ranges measured values. Moses, Emerson, 47 of and Hosseini(1984) identified instances ordered categoricalvariables in in appearing 32 ofthe 168 articles Volume306 (1982) oftheNewEngland Journalof Medicine. also These investigators reported thatthe statistical analyses rarelyused the inherent ordering;instead, they test typically a chi-square of homogeneity. used Further- the more, articles often indicated collapsing ordered that of was categories employed prior a to calculating chi-square An statistic. example these illustrates points. on Santoro al. (1982) reported a randomized et controlled trial compare clinical to two for combination treatments advanced drug Hodgkin's disease.Patient to response treatment was assessed and classified tumor by remission complete, as or partial, none.The results for seventy-five the patients as were follows: Response Complete Partial None Totals Treatment MOPpa with 27 2 9 38 Treatment MOPP and ABVDb with 34 3 0 37 a MOPP is Mechlorethamine, and Procarbazine, Prednisone. Vincristine, b ABVD is Doxorubicin, and Vinblastine, Dacarbazine. Bleomycin, Keywords: Exacttestin contingency tables; Mann-Whitney methods; Nonparametric procedure; Ordered data; categorical Wilcoxon ranksumtest. 303 304 March1985 Biometrics, of In reporting results their the of comparisons thetwotreatments, statistical Santoro et al. indicated thatthey clearly tests obtainP-values had used chi-square to and thatthey had made a continuity for In correction a smallsubgroup. theseanalyses they collapsed thetablein twoways: 27 11 and 29 9 34 3 37 0 Theirpaperpresented P-values .02 (without continuity of a for correction) thefirst table a and .005 (with continuity for correction) thesecondtable.Notethatthefirst collapsed to tablecorresponds categorizing as or to patients showing failing showcomplete remission and the secondcategorizes or themas having not having The any remission. choiceof collapsing a nonnegligible on theP-value, this true has effect and is of irrespectivewhether a one employs continuity to correction chi-square. The Wilcoxon-Mann-Whitney (WMW) procedure an provides alternate of analysis the tableofSantoro al. that et the incorporates natural ordering oftheresponse categoriesand It a of in avoidscollapsing. gives test thehypothesis no difference response treatment of to a against shift that alternative one treatment affords better The response. exacttwo-sided P-valueusing tiedconfiguration the is .010,ifwe agree measure extent departures to the of from nullhypothesis usingthedistance either the by (in direction) theWMW statistic of from the assumedmean. A normalapproximation givesa two-sided P-valueof .012 a (without continuity The correction). statistical analysis indicates an alternating that drug combination MOPP and ABVD is better of thanMOPP alone,at significance .01.level of An algorithm Mehta, and Patel, Tsiatis (1984) makes calculation exact the of P-values for WMW statistic the readilyavailable evenforfairly large tables. Thispapermakesuse oftheir in algorithm an examination theapplication WMW to 2 x k ordered of of tables. Other to approaches ordered in categorical havealso beenconsidered theliterature, data including those basedon oddsratios. refer Armitage We to (1955),Simon(1974),Andrich (1979),Patefield (1982),Agresti(1983),and to thereferences therein. 2. The ExactDistribution Its Approximations and One can describe WMW testusing the a either Wilcoxonranksumor Mann-WhitneyU We statistic. choosethe secondformulation adoptthe notation Armitage and of (1971, Chapter 13). Let x,, . .. ., xn denote a random sample froma random variableX, and let denotea random sample fromY. Then the Mann-Whitneystatistic .. ., yn2 is UXY= Rxi, y*) xi < A> I + '21 {(Xi, Yj Xi Ai 1, where S I is thenumber elements a setS. | of in Underthehypothesis X and Y haveidentical that distributions, has meann1 Uxy n2/2; whenties are absent,Uxycan assume an integer value from0 through n2 and is n, symmetrically aboutitsmean.Whentiesoccuramongthexi and theyj,the distributed attainable valuesof Uxyare integer half-integer need notbe equallyspaced.The or and exactdistribution on of depends theconfigurationtiesand is no longer symmetricaround It n1n2/2. typically moreerratic exhibits behavior whenthereare ties (Lehman,1961; Klotz,1966;and Klotzand Teng,1977).Calculation theexactnulldistribution Uxy, of of well in although understood principle Lehmann, (see 1975;? 1.4),hasonly become recently for feasible anybutthesmallest data sets.A computer of algorithm Mehtaet al. (1984) provides exactcalculations most the for data practical sets. Normalapproximations thenulldistribution Uxy wellknown[see Armitage for of are Whitney for2 x k Ordered Wilcoxon-Mann- Test Tables 305 (1971)and Lehmann are (1975)].The approximations very whenn1and n2areat accurate least10and no tiesarepresent. causethevariance Uxy shrink; then = nI + n2 Ties of to if observations on c distinct take valuesand tjis thenumber observations at thejth of tied the value, variance is - + nin2(n 1) 12 - E (t]- _ ~~j=1 fl ti)1 nl Normal can approximations be madeeither or a for with without correction continuity. The continuity is correction ordinarily the made by shifting value of the statistic being approximated closer theorigin one-half to by unit.In herstudy smalltableswith of ties, Lehman that (1961) concluded thecorrection a (using half-unit usually shift) improves the normal and to approximation tends giveconservative Her P-values. findings supported the of assertion Kruskal Wallis(1952) thata continuity and correction the improves approxi- mation whena P-valueis greater it than.02,butmayworsen otherwise. Lehmann (1975) recommends use ofthecontinuity the in He correction general. does notuse it whenties a in between andy's produce lackofequal spacing thevaluesof Uxy; x's notethat shifting thevalueby one-half no longer unit movestheteststatistic to halfway thenextpossible value. The normal to of is approximation thedistribution Uxy supported a limit by theorem: (Uxy- n1 n2/2)/[var(Uxy)]"/2 to be a standard tends normal as variable nj and n2getlarge, provided tj/n boundedaway from1 as n approaches is (see infinity Lehmann,1975, for Appendix, a proof).The resultmay suggest that,forfinite sampleswitha large proportion tiesin a particular of category, normal the need approximation notbe good evenfor fairlylargevaluesofn1and n2. A number authors, of including Lehman (1961),Klotz(1966),Lehmann (1975,Chapter 1),and Klotzand Teng(1977),haverecognized normal that approximations WMW are to ofteninaccurate whenthedataareheavily tied.Untilnewalgorithms madecalculation of theexactdistribution feasible, wasdifficult study it to these approximations in except very smalldata sets.Algorithms, including of Mehtaet al. (1984), now makeit easierto that compare exactWMW distributions their with and approximations to recommend when are exactcalculations needed. of Biomedical 3. Analyses Published Tables In their of in survey all 168articles Volume306 (January-June oftheNewEngland 1982) Journal Medicine,Moses et al. (1984) identified twenty-seven x k ordered of all 2 contingency tablesthatwereexplicitly presented or could be easilyreconstructed from information in thearticles.Among they these, six for identified tables which theresultsof statistical analyses were and wellreported, five other tables whose analyseswere onlypartly but reported which offered usefulillustrations ordered of for tables which WMW couldbe used.We present eleven the tables the of hereto illustrate exactcalculations P-values and their normalapproximations, because we believethatthesetablesare representative of those which for WMW is useful. Thepapers these containing tables almostalways reported or two-sided omnibus P-values for analyses the presented (usuallychi-square);herewegiveone-sided in P-values order to circumvent technical difficulties arisein defining two-sided that a P-valuefora nonsym- metric distribution Gibbonsand Pratt, (see 1975).Table 1 presents elevenpublished the tables,theirexactone-sided P-values from WMW analysis, normal a and approximations for theseP-values and a without with continuity correction. Theexamples suggest for tables that, the with more in thantencounts eachgroup, either normal does approximation wellfor mostpurposes; theagreement still, the between exact 306 March 1985 Biometrics, Table 1 Eleven x k ordered 2 306 contingency published Volume of The NewEngland tables in Journalof Medicine.One-sided are P-values providedforWilcoxon-Mann- Whitney usingexactcalculations, and usingnormal and a correction without with continuity approximations (c.c.). 2 x ktable P-value percent One-sided as Frequencies Totals with Exact Normal Normal c.c. A 9 0 1 1 11 .044 .10 .11 1 3 6 4 14 B 21 0 2 0 23 7.4 5.0 5.2 15 3 1 2 21 C 27 2 9 38 .41 .62 .64 34 3 0 37 D 5 2 12 19 .050 .042 .044 20 5 4 29 E 4 3 11 18 .048 .036 .040 21 4 5 30 F 14 33 47 3.4 1.9 3.0 48 235 283 G 18 4 14 36 20.8 17.9 18.1 7 3 2 12 H 44 62 57 163 .0015 .0015 .0015 19 53 89 161 I 2 6 5 5 18 5.9 5.7 5.8 2 10 6 0 18 J 1 7 10 18 .51 .38 .40 8 6 4 18 K 26 3 3 6 90 128 47.4 47.4 47.5 16 2 1 17 76 112 testsand theirapproximations substantially is worse than that forsimilarcomparisonsof groupsfreeof ties. The use of a continuity correction for makes verylittledifference these tables;the last two columns of Table 1 are in close agreement. Example B illustrates thatthe adoption of a normal approximation insteadof an exact test can change the outcome of a level-.05 significance test. Note in Example F the to discrepancy P-values despitethe largefrequencies. is in partattributable the strong in It of asymmetry the exact hypergeometric distribution this(collapsed) 2 x 2 table. for We stressthe importanceof the ties correctionfactorin the variance forthe normal approximation;we have used it for all normal approximationspresentedin this paper. Withoutit,the normalapproximation WMW forExample C givesa one-sidedP-value to (not shownin the table) of 4.6%, more than 10 timesthe correct value of .41%. Tables 4. Analysisof Hypothetical We consideredmany hypothetical x k tables in attempting enlargethe insights 2 to we gained fromexploringthe published data sets. Table 2 presentsresultsforten of these tables;we selectedsome of theseexamplesto illustratethat,whenmanyties occur,normal approximations WMW can do somewhatworse than the data sets of Table 1 seem to to seven examples, no single categorycontains more than half of the suggest.For the first totalfrequencies.One categorycontainsmostofthedata in each ofthelastthreeexamples. Examples A, B, and C illustratetables with nj and n2 at 12 or less; in these tables,the approximation P-values highbya factor approximatelyor too low normal gives too of 2 Wilcoxon-Mann-Whitney for2 x k OrderedTables Test 307 Table 2 Ten2 x k ordered to tables contingency selected compare exactone-sided P-valueswiththeir normal The approximations. examples illustrate large between relatively differences and exactP-values those obtainedfrom normal approximations. 2 x k table One-sided as P-value percent Frequencies Totals Exact with Normal Normal c.c. A 1 8 0 9 .17 .31 .36 8 0 1 9 B 2 6 1 9 2.2 1.0 1.1 7 2 0 9 C 3 8 1 12 1.5 .7 .8 9 3 0 12 D 4 10 1 15 2.4 1.3 1.4 10 5 0 15 E 3 9 3 0 15 .49 .63 .67 13 3 0 2 18 F 11 17 1 29 7.0 4.9 5.1 17 12 0 29 G 10 10 10 10 20 60 1.7 1.5 1.5 30 30 30 30 20 140 H 18 1 1 0 20 5.3 7.6 7.8 15 0 0 5 20 I 50 1 1 1 0 0 53 4.5 7.0 7.1 45 0 0 1 2 4 52 J 1 50 2 1 0 0 54 4.8 7.1 7.2 0 45 0 1 2 3 51 of bya factor approximatelyExamples 2. D-G showthat, evenformuchlarger tables,the normal approximation theWMWP-value differ to can from exactvaluebya nonneg- the amountin either ligible for the direction; theseexamples normal approximation usually produces P-values are that within 50% oftheexactvalue.The lastthree show illustrations that,whenmostcounts in a single fall the category, normal can approximation lead to a differentconclusionfrom exactvalueevenwhen50 or moreobservations in each the are group. Thus,investigators want use exactWMWP-values somerelatively may to for large datasets,whenmostobservations in a single fall category. 5. Recommendations Ourexamination published of tablesfrom biomedical the of consideration many literature, hypothetical tables, theoretical and results, an informal review relevant of lead literature us to offer following the for recommendations theuse ofWMW for analyzing 2 ordered x k contingency tables: (i) We recommend exactcalculations the unlessthe groupsizes,nj and n2,are 10 or more. makethisrecommendation We of of regardless theextent ties, and we referto one Mehtaetal. (1984) for suitable algorithm. (ii) We recommend whentiesappearin thedata,thetiescorrection thevariance that, to always usedwhenapplying normal be the approximation. (iii) Forordered tables withlargergroupsizesandwith many a ties, normal approximation usually a provides P-valuethatis within 50% oftheexactvalue.Whena P-valueis closeto a traditional level significance (for example, may 5%), investigators prefer to the report exactvalue. 308 Biometrics, March1985 fall (iv) Whenoverhalfof all observations in a single an category, approximate P-value maybe somewhat even unreliable, when bothn1and n2arelarger than10. (v) A continuity for correction a normal approximation makes difference both little when n1and n2are 10 or more.We recommend its against use forWMW withordered of becauseoftheunequalspacing valuesoftheWMW statistic. tables We believe the that Wilcoxon-Mann-Whitney for procedure investigating alterna- shift is well tives often suited analyzing 2 x k ordered to a table.Our recommendations should aid investigatorsthepractical in of aspects thisanalysis. ACKNOWLEDGEMENTS We thank of members theHarvard Study in Groupon Statistics theBiomedical Sciences who offered valuablecomments; theyincludeJohnBailar,GrahamColditz,Hossein Hosseini,KatherineGodfrey,Robert Lew,PhilipLavori,ThomasLouis,Frederick Mos- Katherine teller, and Taylor-Halvorsen, JohnWilliamson. also benefited We from the adviceofDavid Tritchler, CyrusMehta, and David Hoaglin.Dr Mehtamadeavailable to of us theprogram Mehta, and Patel, Tsiatis (1984)whichgivesexact of calculations WMW. We thank MarySchaefer Marilyn and for this Thorpe preparing manuscript. of Preparation thismanuscript facilitated Grant was by the RF-79026from Rockefeller Foundation. RE'SUME souvent lesinvestigations Onutilise dans et des biologiquesmedicales donnees qualitatives ordonnees. on Quand doit comparer groupes quelesdonnees deux et dans tombent trois classes ou ordonnees le plus, test Wilcoxon-Mann-Whitney tient de (WMW) de et compte l'ordre esthabituellement contre alternatives translation. puissant des detype de Cependant,telles applicationsWMW du font souvent des les intervenor distributions lesquelles ex-aequo pour un jouent roleimportant. De nouveauxprogrammes sur permettantfaire tests disponibles ordinateur de des exactsdonnent une comprehension meilleure desdistributions duWMW la pertinence descaracteristiques exactes et des approximations un base normales. apportons avispratique, surl'experience Nous de d'ensembles donnees publiees surdesetudes biomedicales et de numeriques tables dans ordonnees, l'utilisation duWMW desesapproximations et normales. REFERENCES A. Agresti, (1983). Testing marginal homogeneity ordinal for variables. categorical Biometrics 39, 505-510. Andrich, (1979). A model forcontingency D. tableshavingan ordered response classification. 35, Biometrics 403-415. P. Armitage, (1955).Testsfor linear in and trends proportions frequencies. 11, Biometrics 375-386. P. Armitage, (1971).Statistical in Methods MedicalResearch. NewYork:Wiley. Gibbons, D. and Pratt, W. (1975). P-values: J. J. and The Interpretation methodology. American Statistician 20-25. 29, Klotz,J. H. (1966). The Wilcoxon, ties,and the computer. of Journal theAmerican Statistical 61, Association 772-787. Klotz,J. H. and Teng,J. (1977). One-way layoutforcountsand the exactenumeration theof H Kruskal-Wallis distribution ties.Journal theAmerican with of Association Statistical 72, 165-169. W. in Kruskal, H. and Wallis, A. (1952).Use ofranks one-criterion W. analysis. variance Journal of theAmerican Association 583-621. Statistical 47, Lehman,S. Y. (1961). Exactand approximate for distributions the Wilcoxon statisticwithties. of American Journal the Association 293-298. Statistical 56, Lehmann, L. (1975).Nonparametrics: E. Methods Statistical BasedonRanks. Francisco: San Holden- Day. Mehta,C. R., Patel,N. R., and Tsiatis, A. (1984). Exact significance A. to testing establish the on equivalence twotreatments compared thebasisofordered of being data. categorical Biometrics 40, 819-825. Whitney for2 x k Ordered Wilcoxon-Mann- Test Tables 309 data Moses,L. E., Emerson, D., and Hosseini, (1984).Analyzing from J. H. ordered New categories. England of 311, Journal Medicine, 442-448. W. for in Patefield, M. (1982). Exacttests trends ordered contingency tables. AppliedStatistics 31, 32-43. P. Santoro, Bonadonna, Bonfante, and Valagussa, (1982).Alternating combinations A., G., V., drug in thetreatment advanced of Hodgkin's disease.The New England of Journal Medicine 306, 770-775. Simon,G. (1974). Alternative analyses the singly-ordered for contingency table.Journal the of American Association 971-976. Statistical 69, June1983;revised Received 1984. January

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