The Comparison of Treatment With Control Group Means in Toxicological Studies

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The Comparison of Treatment with Control Group Means in Toxicological Studies

Author(s): Eryl A. C. ShirleyReviewed work(s):Source: Journal of the Royal Statistical Society. Series C (Applied Statistics), Vol. 28, No. 2(1979), pp. 144-151Published by: Blackwell Publishing for the Royal Statistical SocietyStable URL: http://www.jstor.org/stable/2346731 .

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Appl. tatist. 1979),28, No. 2, pp. 144-151

TheComparisonfTreatment ith ontrolGroupMeans n ToxicologicaltudiesBy ERYL A. C. SHIRLEY

HuntingdonResearch Centre,Huntingdon, ngland[Received anuary978.Final evision ovember978]SUMMARYThreemethods fcomparing roupmeans nthe tandard our roup oxicologicalstudy,where hree roups f animals eceivingncreasingoses of a substance recontrasted ith zerodose control roup, re compared yestimatingowers singMonteCarlomethods. t is found hat,ngeneral, artholomew'sest or rdered

means s themost owerful,utthatWilliams's est as virtuallyhe amepower tdetectingffectst the ll-importantow dose evel.Keywords: MULTIPLE COMPARISONS; TESTS FOR TREND; MONTE CARLO METHODS; TOXICOLOGICALSTUDIES

1. INTRODUCTIONTHERE re many significanceests vailableformaking omparisons mongstgroup meansobtainedfrom xperimentalata. Some tests re to be used when ll possiblecomparisonsamongstmeans re of nterest,therswhen nly omecomparisonsre of nterest,or nstance,when everal reatmentsr increasingoses of thesametreatmentre tobe comparedwithcontrol.The purposeof thispaper s to examine hree pecialtests n thecontext fthestandardtoxicological tudywherethree qually sized groupsof animals,each grouptreatedwithdifferentosesofthesame substancereferredo as the ow,medium nd highdosegroups),are to be comparedwith n untreatedcontrol) roup. Similar xperimentalesigns rise nother ituations, .g. three evelsof a fertilizer,esticide r fungiciden agricultural rials,threeevels fsome ngredientna manufacturingrocess, rthree evels ffurtherducationin a sociological tudy. In thetypical oxicological tudy, he ow dose groupare given namount fthe ubstancewhichsthe ameas theexpected quivalent umanntakemultipliedbya safety actorwhich ariesfrom to 5 fordrugs o 100forfoodadditives nd pesticides.The highdose groupare given n amount of the substance onsidered ufficiento inducetoxicsymptomswithoutncreasinghemortalityate. The mediumdose group generally

receive n amountofthe substancewhich iesequallybetween he amountsgiven o the owand highdose groupson a logarithmiccale. The comparisons re usuallymade on body-weightgains during he course of the experiment, rgan weights t sacrifice nd varioushaematological nd biochemicalparametersmeasured t intervals uring he experiment.Othercomparisons re made on qualitative bservations,uch as presenceor absence oftumours r other bnormalities,utthesewillnotbe considerednthispaper.In practice, o singlemeasure s consideredn isolation nd thetoxicologist illbase hisconclusions n the totalpicture resented ytheexperimentalata. Nevertheless,t s usefulfor he toxicologisto have availableefficient ethods fcomparinghe treated roupswiththecontrolgroup foreach parameternd to knowthe chancesof either indingreatmenteffects herenone existor offailingo detect reatmentffects here hey o exist.Thispaper s primarilyntendeds a guideto the choiceof a testofsignificancen datawhichfulfil he conditions ecessary or nalysisofvariancemethods, amely pproximate

144


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MULTIPLE COMPARISONS 145normalityf he nderlyingistributionndhomogeoeityfvariance. owever,istribution-freenaloguesxist f ll the estsxaminednd arealso discussed.2. THETHREETESTSExAMINEDAll three ests reprimarilyesigned o test henullhypothesisI = P = P3= L4against

the lternativeypothesis1


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146 APPLIED STATISTICSin these ircumstances,he izeof the est ould xceed henominalizeof the 2 test y asmallmargin. etanothermethods not o amalgamatehegroups nd to use theoriginalmaximumikelihoodstimatorshroughout.owever,his est snot s powerfuls Marcus'sin the pecial ituationustdescribed.2.2. Williams's est or Comparing rderedMeanswith ControlFor previous escriptionfthis est eeWilliams1971, 972). The maximumikelihoodestimatorsfthegroupmeans re obtainedn the ameway s forBartholomew'sest.Thetest tatisticor omparinghehighestosegroupwith he ontrols

tk = k *-x9l)/(S. . D .),where l is the amplemeanof thecontrolroup nd S.E.D. is the tandardrror fthedifferenceetweenhe wogroupmeans.If 1k is foundo be significantlyifferentrom he ontrolhenhenext owest osegroupis compared ithhe ontrolnthe ameway ndso on.Tables fsignificanceointsreavailable or qual group izes nd for ncreasedontrolgroup eplicationnthe eferencesbove.In this tudy,hemaximumikelihoodstimatesfthegroupmeans asedonthe otaldata etwere sed t each tage f he rocedure.eeWilliams1972) or discussionf headvantage, henhemediumosegroupssmallerhan hehigh osegroup,frecalculatingthemaximumikelihoodstimatorst each tage.

2.3. A ModificationftheMethod fLeast SignificantifferencesL.S.D.)whichssumes heMeans are OrderedThe procedures as follows. feitherhedifferenceetweenhehighdose group ndcontrol ean s greaterhan heL.S.D.,orthedifferencesetweenhe wo ower osemeansandthe ontrol ean reboth reaterhan heL.S.D. and nthe amedirection,henheressaid o be a treatmentffectt thehighestose evel.Given hat hehigh osegroup asbeenfoundo besignificantlyifferentromhe ontrol,hemediumosegroups assumedobesignificantlyifferentromhe ontrolf itherhe ifferenceetweentsmean ndthe ontrolmean r thedifferenceetweenhe owdosemean ndthe ontrolmean sgreaterhan heassociated.S.D. and nthe ame irectionsthehigh osedifference.fboth hehigherosegroups avebeen oundobesignificantlyifferentromhe ontrol,hen he owdosegroupmean s declared obe significantlyifferentromhecontrolroupmean fthedifferencebetweenhemsgreaterhan he ssociated .S.D. and nthe amedirections thehighergroupmeandifferences.The nvestigationsfBartholonew1961) uggesthathistests more owerfulhan heF testndtests ased n contrastsetweenroupmeans.Williams1971) nd Marcus1976)have hown hat artholomew'sest sgenerallyore owerfulhanWilliams'sestwhen ourormore roupsrecompared. his,however,efersnly o thedetectionfeffectst thehighestose evel.The observationshatWilliams'sestmproveselativeo Bartholomew'stest s the ffectt the owest ose evel ncreases,ndthatWilliams'sests generallyhesuperior hen nly hree roupsrecompared,uggesthat ifferentonclusions ay risefromcomparisonf he owersf he estsodetectffectst the owest ose evelwhenheyareused na stepwise anner.Marcus1976)producedmodificationfWilliams'sestwherehemaximumikelihoodestimatorf he ontrolroupmeanwasused nthe est tatisticnsteadf he ontrolroupsamplemean. In thefour roup ituationheshowedhat hepower f this est enerallylaybetweenhat fWilliams'sndBartholomew'sests,o this est as notbeen tudiedere.Themodified.S.D. procedure as ncludeds a commonsensese of imple .S.D.'s inthemonotonicoseresponseituation,nvolvingospecial ables.


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MULTIPLE COMPARISONS 1473. METHODOLOGYThe experimentwiseype I errorsof Williams's and Bartholomew's estswere set at5 per ent two-tailedests).TheType error fthemodified.S.D. procedure,sing wo-tailed per ent .S.D.'s, s about5$6 er ent. Thisvaluewas kindly btained ormebyD. A.Williams.) his lightncreasen theType error ouldbe eliminatedy hanginghe

L.S.D.'s appropriately.owever, hiswouldnegate he mainreason ordevelopinghisprocedure, amelyts simplicity;onsequentlyhepowerof the testbased on standardL.S.D.'swasexamined.The powers fthetestswere stimatedyMonteCarlo methods. our groups f tennormallyistributedbservationsere eneratedsing heCamlibRNO1 andRNNORMroutines n theUniversityf CambridgeBM370/165 omputer. his is not the mostefficientayof generatingherequired ata for hese articularestswhich rebased ongroupmeans ndthepooledwithin-groupariances. our normalmeans nda x2valuewith 6 degrees ffreedomouldhavebeengenerated. owever,hese imulationsormedpartof a larger imulationxerciselso involvingistribution-freeethods asedon therankingf the ndividualbservationso this attermethod f generatinghedatawas notused.Typical atliverweight ata were hosen s the subject f thesimulationso thattoxicologistsould eehowvariousestsf ignificanceehaved n "realdata". Themean fthepopulationf iverweightsrom ntreatedats s approximately0g and the tandarddeviationg. However,nthe able fresultshemeans re caled ydividingy he tandarddeviation,hismethod fpresentinghe esults eing onsideredreferableor tatisticians.The configurationsf meanswere hosen ocover hedifferentinds f doseresponsesthatmighte encountered.hus, n the able fresults,hefirstonfigurationepresentstreatmentffecttthehigh ose evel nly; henext wo onfigurationsepresentreatmenteffectstthemediumndhigh ose evels nly,ndtherest fthe onfigurationsf meansrepresentifferentegreesf treatmentffectsoundtall dose evels.These rethe ametypesfconfigurationsfmeans s those tudiedyMarcus1976).Themagnitudefthe reatmentffectt thehigh ose evelwas chosen orepresenthesituation herehechance fpickingpthegiven ffects about80percent.This s theminimumevel f detectionoxicologistsould ike oachieve or differencefbiologicalimportance.In addition,twasof nterestoexamineherelativeowers f the ests pplied othemediumnd owdose evelsnsituationshere he ffectt thehigh osewas so large s tobe foundignificantnalmost 00per ent foccasionsndthere ere onsiderablereatmenteffectstthe ower ose evels.These ituationsrepresentedn Table2.Thetests ere llcomparednthe ame ets f imulatedata ndwere arried ut nthestepwise anner escribedreviously.ivethousandimulations ere arriedut oneachconfigurationfmeans. 4. RESULTSThe estimatedowers or achtreatmentroup-controlroup omparisonredisplayedin Tables1 and 2. (Themarginaltandardrrors f these stimatesreall 0-7percent rsmaller.The estimatesre correlated ithinowsof the table o thestandardrrorsfdifferencesetweenhese ower stimatesillbe smaller han hemarginaltandardrrorsmultipliedy 2.) Thus, able1showshat, hen herereno owdose ffects,artholomew'stestdetects ighdose effectsn about10 percentmoreoccasions hanWilliams's est.Williams'sest, owever,s overall s powerfulr more owerfulhanBartholomew'sest tdetectingigh ose effectsn theoccasionswhenow doseeffectsxist.Onthewhole,hemodified.S.D. procedureomparesavourablyithheotherwo ests, ut heres a lossofpower fabout7 percentwhen ompared ith artholomew'sest nd 5 percentwhen


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148 APPLIED STATISTICSTABLE1

Estimatedt owers f thethree ests or three osage evels(Percentagesftimes reatmentffectst the5 percent evel ofsignificancearefoundfor hehigh,medium nd low dose levels)Dose ModifiedL.S.D. Williams's Bartholomew's(Control,L, M and H) means level procedure test test

5 00, 5 00, 5 00, 6-25 H 78-7 75-7 85-2M _L _5 00, 5 00, 5-63, 6-25 H 77 3 75-8 83-3M 27-1 25-3 29-4L5 00, 5 00, 6-25,6-25 H 76-9 82-1 93 9M 66-0 70-8 82-9L5 00, 5-42,5-85, 6-25 H 77 0 76-7 77-5M 44 9 42-9 43-3L 14-5 12-3 12-85 00, 5-63,5-63,6-25 H 77-2 75-7 71-0M 37 8 29-6 29-3L 26-5 18-7 19-65 00, 5*63,6*25, 6*25 H 78-3 82-6 84-0M 68.4 71.6 72-3L 27-4 26-3 26-65 00, 6-25,6-25,6-25 H 85-4 85-8 84.2

M 81d1 78-6 78-7L 73.4 69-5 70-2f Estimates based on 5,000 simulations.

TABLE2Estimatedt owers fthe hree estswhen here s astrong reatmentffectt thehigh ose evel(Percentage ftimes reatmentffectst the5 percent evelofsignificancearefoundfor hehigh,medium nd lowdose levels)

Dose ModifiedL.S. D. Williams's Bartholomew's(Control,L, M and H) means level procedure test test5 00, 5 00, 6&25, 7 50 H 100 0 1000 1000M 76-8 74-5 81-8L - _500, 592, 6-83,7-75 H 1000 1000 1000M 98-0 97-6 96-8L 52-2 52-0 52-15 00, 6-25, 6&25, 50 H 99 9 99 9 99 9M 89-7 83-3 83-3L 77 9 72-0 73-1

t Estimates based on 5,000 simulations.


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MULTIPLE COMPARISONS 149comparedwithWilliams's est n the situationwhere here s no effectt the ow dose and a"plateau" effectt the medium nd highdose levels.Bartholomew's est s also more powerfulhan Williams's est t detectingmediumdoseleveleffects hereno low dose leveleffectsxist in one case there s an increasenpowerof12 per cent) nd has about the same power nmostof the other ases. Overall, he modifiedL.S.D. procedure etectsmedium ose effects oreoften hanWilliams's estbut not as oftenas Bartholomew's est.At thelow dose level,Williams's nd Bartholomew's estsfind ignificantffectslmostequally often Bartholomew's est has the edgeby 0f6 per cent),and the modified .S.D.proceduremore often, his ncrease n powerbeing ess than 5 per centexceptfor one con-figuration here t s about 8 per cent.From Table 2 it can be seenthat,where reatmentffectsre large, herankings fthepowersofthethree ests emain he sameas for he smaller reatmentffects.In addition, he conditional ype I errorswere estimated orthemedium nd low dosegroups nthe case of a largetreatmentffectn thehighdose group nd found o be 4f8 nd2f6per cent espectivelyor hemodified .S.D. procedure; f6 nd 07 percentforWilliams'stest;and 2f7 nd 0f8percentforBartholomew's est. These lowType errors, articularlyforWilliams's nd Bartholomew's ests, risebecausethe testsof significancen the lowerdose groups re onlycarried ut in thedirection f the effectound t thehighdose level.The good performancefthe modified .S.D. procedure t detectingreatmentffectst themediumand low dose levels should be balanced againstits higherType I errorswhencomparedwithWilliams's nd Bartholomew's ests.

5. CONCLUSIONIn general,because of its increasedpowerof up to 12 per cent,Bartholomew's est spreferableo Williams's estor itsmodification.However, t is morecomplicated o applyand only imited ablesofsignificanceoints re available. Also,because Williams's esthasvirtuallyhe samepoweras Bartholomew's est t detectinghe ow dose effects, illiams'stest s a good alternativeo Bartholomew's estwhentheexperimenters not too concernedat missing ighdose effectsrovided here re no low dose effects. his is often hecase intoxicologicaltudieswhere t is the ow dose which s related o theexpectedhumandosage,and thehigher oses are included o assesswhatmight ccur fa subjectwas supersensitivetothe ubstance eing ested.Also,as stated yWilliams1971) f here s a small ffectt thelow dose level ntheoppositedirectiono theeffectt the higher ose levels,Williams's estremains alid whereasBartholomew'sest nd themodifiedWilliams's estdo not. This isbecause whenapplyingWilliams's est hemaximum ikelihood stimates fthe dose groupmeans are comparedwiththe controlgroup samplemean not the maximum ikelihoodestimate nder hehypothesisf orderedmeans. Thisflexibilitys also usefulwhen ssessingsomesubstance or oxicity.Marcus's modificationfWilliams's estwouldalso be a goodalternativefmoretablesofsignificanceointsweremade available. Tables ofapproximatesignificanceoints re available n a paper byWilliams1977).The results erived rom hemodified .S.D. proceduremply hat n intelligentse oftheordinary .S.D.'s is a reasonablealternativeo a two-tailed artholomew's r Williams'stest. A strict omparisonbetween he modified .S.D. procedure nd the othertwo testscouldonlybe madeifdifferent.S.D.'s wereused such as to bring hepercentage f Typeerrors ownto 5 percent. However, hecomparisonmadehere s adequateformostpracticalpurposes nd such a modification ould counteracthemainreasonforusing hisprocedure,namelytssimplicity.Finally, omepointsof interestrise when one-sided estsof significance,ample sizes,non-parametricests nd studieswith woorfourdosagelevels re considered.One pointof interestrises when the conditional ype I errors t themedium nd lowdoses, given n effect t thehigh dose, are considered.The powerat the medium nd low


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150 APPLIED STATISTICSdoselevels fthetwo-sided ersions fBartholomew'sndWilliams's ests ouldbe improvedby using ignificanceointsforone-sided ests t thesedose levels, esting eingonly nthedirection fthe highdose level ffect. hemodified .S.D. procedure ouldnotbe improvedinthismannerwithouthe conditional ype error t the medium ose exceeding per centbya considerablemargin.

Significanceointshavebeencalculated yWilliams1972)and Chase 1974)forWilliams'sand Bartholomew's estsrespectivelyppliedto the situationwhere here re more obser-vations nthecontrol roup han n each of the treated roups. Williams 1972) showed hatthe most fficientllocationof a givennumber fanimals o groups n a toxicity xperimentwithk groups nall is such that here re about1f3 1(k- 1) as many nimals nthe controlgroup as in each of the treatment roups. Williams lso produced ables which howedtheeffectf nequalitiesnthenumbers fobservationsnthedosegroups nthe ignificanceevelsof i statistic. He foundthat, provided he differencesetweenreplicationsre moderate,andcertainlyf he ratios i/rk where i sthereplicationnthedosegroup ,i< k) lie between0O8 nd 1 25, there s a negligible ffect n thevalues of the significance oints. No suchinvestigationas yet been made on Bartholomew's 2 test,but one would not expectthesignificanceoints o be upsetbymoderate ifferencesngroup izes. Whatever est s beingapplied,where here s a highmortality iguren thehighdosage group comparedwith heotherdosegroups, tmaybe advisable o consider he secondhighest ose levelregardlessfwhether r notthehighdose groupmean is significantlyifferentrom hecontrolgroupmean Williams, 972).Non-parametricnalogues exist for the parametric estsevaluatedhere. These non-parametric estsare constructed y replacing roupmeansby groupmean ranksand theresidual ariance yN(N + 1)/12minus correctionor iedvalues,whereN isthe otalnumberofobservationsntheexperiment. or details eeKruskal nd Wallis 1952),Shorack 1967),Hollander nd Wolfe 1973,pp. 114-133) ndShirley1977). Theasymptoticelativefficiencyof these tests to theirnormaltheory quivalents s 12u2{f+if2(x) x}2,where (x) is theassociateddensity unction. t follows hat heconclusions n relative owers btainedfrompower omparisonsmade nthenormal heoryasealso apply symptoticallyo the quivalentrank tests.The conclusions fthisstudy pply only o the standard ourgroupexperiment,hevastmajority ftoxicitytudiesbeingof thistype. However, hreegroupand fivegroupexperi-mentsare sometimes onducted. Both Williams 1971) and Marcus (1976) foundthatWilliams'stest was generallymorepowerful han Bartholomew's est n the threegroupexperiment.n the fivegroup experiment,Williams 1971) foundthat Bartholomew's estwas generallymorepowerfulhan histest, ncludinghe case where herewas a linear rendwithdose. This latter indingmplies hat Bartholomew's estmaybe morepowerfulhanWilliams's est t detectingow dose effectsn fivegroupexperimentsutmoresimulationsneed to be performed efore nyfirmonclusions an be made.

REFERENCESBARLOW, . E., BARTHOLOMEW,. J., BREMNER,. M. and BRUNK, . D. (1972). Statisticalnferenceunder Order Restrictions.New York: Wiley.BARTHOLOMEW,. J. 1959a). A test f homogeneityor rdered lternatives. iometrika,6, 36-48.(1959b). A test orhomogeneityor rdered lternativesI. Biometrika,6, 328-335.(1961). A test fhomogeneityfmeans nder estrictedlternatives..R. Statist. oc.B, 23, 239-281.CHACKO,V. J. 1963). Testinghomogeneity gainst ordered alternatives. Ann. Math. Statist., 34, 945-956.CHASE, G. R. (1974). On testing for ordered alternatives with increased sample size for a control.Biometrika,61, 569-578.HOLLANDER,M. and WOLFE,D. A. (1973). Non-parametric tatisticalMethods. New York: Wiley.KRUSKAL,W. H. and WALLIS, W. A. (1952). Use of ranks in one-criterionvariance analysis. J. Amer.Statist.Ass., 47, 583-621.MARCUS, . (1976). The powers f ome ests f he quality fnormalmeans gainst nordered lternative.Biometrika, 3, 177-183.


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MULTIPLE COMPARISONS 151MARCUS, R., PERITZ, E. andGABRIEL, K. R. (1976). On closed esting rocedures ith pecial eferenceoordered nalysis f variance.Biometrika,3, 655-660.NELSON, L. S. (1977). Tables for estingrdered lternativesn theanalysis f variance. Biometrika,4,335-338.SHIRLEY, E. (1977). A non-parametricquivalentfWilliam's est or ontrastingncreasingose evels fa treatment.iometrics,3,386-389.SHORACK, G. R. (1967). Testing gainst rderedlternativesn model1analysis fvariance; ormal heoryandnon-parametric.nn.Math.Statist., 8,1740-1752.WILLIAMS, D. A. (1971). A test ordifferencesetweenreatment eanswhen everal ose evels recom-paredwith zero dose control.Biometrics,7,103-177.- (1972). Thecomparisonf several ose evelswith zero dosecontrol.Biometrics,8,519-531.(1977). Some nferenceroceduresormonotonicallyrdered ormalmeans. Biometrika,4,9-14.

The Comparison of Treatment With Control Group Means in Toxicological Studies

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