MenuttestStatisticsSummariestablesandtestsClassicaltestsofhypothesesttestmeancomparisontestttestiStatisticsSummariestablesandtestsClassicaltestsofhypothesesttestcalculatorDescriptionttest ID: 451925
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2ttest—ttests(meancomparisontests) MenuttestStatistics�Summaries,tables,andtests�Classicaltestsofhypotheses�ttest(mean-comparisontest)ttestiStatistics�Summaries,tables,andtests�Classicaltestsofhypotheses�ttestcalculatorDescriptionttestperformsttestsontheequalityofmeans.Intherstform,ttestteststhatvarnamehasameanof#.Inthesecondform,ttestteststhatvarnamehasthesamemeanwithinthetwogroupsdenedbygroupvar.Inthethirdform,ttestteststhatvarname1andvarname2havethesamemean,assumingunpaireddata.Inthefourthform,ttestteststhatvarname1andvarname2havethesamemean,assumingpaireddata.ttestiistheimmediateformofttest;see[U]19Immediatecommands.Fortheequivalentofatwo-samplettestwithsamplingweights(pweights),usethesvy:meancommandwiththeover()option,andthenuselincom;see[R]meanand[SVY]svypostestimation.Options � Main �by(groupvar)speciesthegroupvarthatdenesthetwogroupsthatttestwillusetotestthehypothesisthattheirmeansareequal.Specifyingby(groupvar)impliesanunpaired(twosample)ttest.Donotconfusetheby()optionwiththebyprex;youcanspecifyboth.unpairedspeciesthatthedatabetreatedasunpaired.Theunpairedoptionisusedwhenthetwosetsofvaluestobecomparedareindifferentvariables.unequalspeciesthattheunpaireddatanotbeassumedtohaveequalvariances.welchspeciesthattheapproximatedegreesoffreedomforthetestbeobtainedfromWelch'sformula(1947)ratherthanfromSatterthwaite'sapproximationformula(1946),whichisthedefaultwhenunequalisspecied.Specifyingwelchimpliesunequal.level(#)speciesthecondencelevel,asapercentage,forcondenceintervals.Thedefaultislevel(95)orassetbysetlevel;see[U]20.7Specifyingthewidthofcondenceintervals.Remarksandexamplesstata.comRemarksarepresentedunderthefollowingheadings:One-samplettestTwo-samplettestPairedttestTwo-samplettestcomparedwithone-wayANOVAImmediateformVideoexamples 4ttest—ttests(meancomparisontests) Thetreatedvariableiscodedas1ifthecarreceivedthefueltreatmentand0otherwise.Wecantesttheequalityofmeansofthetreatedanduntreatedgroupbytyping.usehttp://www.stata-press.com/data/r13/fuel3.ttestmpg,by(treated)Two-samplettestwithequalvariances Group ObsMeanStd.Err.Std.Dev.[95%Conf.Interval] 0 1221.78817012.73030119.2652522.734751 1222.75.93844653.25087420.6844924.81551 combined 2421.875.62644763.06895420.5790923.17091 diff -1.751.225518-4.291568.7915684 diff=mean(0)-mean(1)t=-1.4280Ho:diff=0degreesoffreedom=22Ha:diff0Ha:diff!=0Ha:diff&#x-531;0Pr(Tt)=0.0837Pr(|T|&#x-531;|t|)=0.1673Pr(T&#x-531;t)=0.9163Wedonotndastatisticallysignicantdifferenceinthemeans.IfwewerenotwillingtoassumethatthevarianceswereequalandwantedtouseWelch'sformula,wecouldtype.ttestmpg,by(treated)welchTwo-samplettestwithunequalvariances Group ObsMeanStd.Err.Std.Dev.[95%Conf.Interval] 0 1221.78817012.73030119.2652522.734751 1222.75.93844653.25087420.6844924.81551 combined 2421.875.62644763.06895420.5790923.17091 diff -1.751.225518-4.28369.7836902 diff=mean(0)-mean(1)t=-1.4280Ho:diff=0Welch'sdegreesoffreedom=23.2465Ha:diff0Ha:diff!=0Ha:diff&#x-531;0Pr(Tt)=0.0833Pr(|T|&#x-531;|t|)=0.1666Pr(T&#x-531;t)=0.9167 TechnicalnoteIntwo-sampleusinggroupsrandomizeddesigns,subjectswillsometimesrefusetheassignedtreatmentbutstillbemeasuredforanoutcome.Inthiscase,takecaretospecifythegroupproperly.Youmightbetemptedtoletvarnamecontainmissingwherethesubjectrefusedandthusletttestdropsuchobservationsfromtheanalysis.Zelen(1979)arguesthatitwouldbebettertospecifythatthesubjectbelongstothegroupinwhichheorshewasrandomized,eventhoughsuchinclusionwilldilutethemeasuredeffect. ttest—ttests(meancomparisontests)5 Example3:Two-samplettestusingvariablesThereisasecond,inferiorwaytoorganizethedataintheprecedingexample.Weranateston24cars,12withouttheadditiveand12with.Wenowcreatetwonewvariables,mpg1andmpg2.mpg1mpg2 202423252121252218231718181724282024242723211923Thismethodisinferiorbecauseitsuggestsaconnectionthatisnotthere.Thereisnolinkbetweenthecarwith20mpgandthecarwith24mpgintherstrowofthedata.Eachcolumnofdatacouldbearrangedinanyorder.Nevertheless,ifourdataareorganizedlikethis,ttestcanaccommodateus..usehttp://www.stata-press.com/data/r13/fuel.ttestmpg1==mpg2,unpairedTwo-samplettestwithequalvariances Variable ObsMeanStd.Err.Std.Dev.[95%Conf.Interval] mpg1 1221.78817012.73030119.2652522.73475mpg2 1222.75.93844653.25087420.6844924.81551 combined 2421.875.62644763.06895420.5790923.17091 diff -1.751.225518-4.291568.7915684 diff=mean(mpg1)-mean(mpg2)t=-1.4280Ho:diff=0degreesoffreedom=22Ha:diff0Ha:diff!=0Ha:diff&#x-531;0Pr(Tt)=0.0837Pr(|T|&#x-531;|t|)=0.1673Pr(T&#x-531;t)=0.9163 Pairedttest Example4Supposethattheprecedingdatawereactuallycollectedbyrunningateston12cars.Eachcarwasrunoncewiththefueladditiveandoncewithout.Ourdataarestoredinthesamemannerasinexample3,butthistime,thereismostcertainlyaconnectionbetweenthempgvaluesthatappearinthesamerow.Thesecomefromthesamecar.Thevariablesmpg1andmpg2representmileagewithoutandwiththetreatment,respectively. ttest—ttests(meancomparisontests)7 Wecouldalsousettest:.ttestmedageifregion==1|region==4,by(region)Two-samplettestwithequalvariances Group ObsMeanStd.Err.Std.Dev.[95%Conf.Interval] NE 931.23333.34115811.02347430.4466232.02005West 1328.28462.49235771.77522127.2118629.35737 combined 2229.49091.44730592.09805128.5606930.42113 diff 2.948718.65903721.573994.323445 diff=mean(NE)-mean(West)t=4.4743Ho:diff=0degreesoffreedom=20Ha:diff0Ha:diff!=0Ha:diff&#x-531;0Pr(Tt)=0.9999Pr(|T|&#x-531;|t|)=0.0002Pr(T&#x-531;t)=0.0001Thesignicancelevelsofbothtestsarethesame. Immediateform Example6ttestiislikettest,exceptthatwespecifysummarystatisticsratherthanvariablesasarguments.Forinstance,wearereadinganarticlethatreportsthemeannumberofsunspotspermonthas62.6withastandarddeviationof15.8.Thereare24monthsofdata.Wewishtotestwhetherthemeanis75:.ttesti2462.615.875One-samplettest ObsMeanStd.Err.Std.Dev.[95%Conf.Interval] x 2462.63.22516115.855.9282569.27175 mean=mean(x)t=-3.8448Ho:mean=75degreesoffreedom=23Ha:mean75Ha:mean!=75Ha:mean&#x-531;75Pr(Tt)=0.0004Pr(|T|&#x-531;|t|)=0.0008Pr(T&#x-531;t)=0.9996 Example7Thereisnoimmediateformofttestwithpaireddatabecausethetestisalsoafunctionofthecovariance,anumberunlikelytobereportedinanypublishedsource.Fornonpaireddata,however,wemighttype 10ttest—ttests(meancomparisontests) Youcanalsousettestwithouttheunpairedoptioninaregressionsettingbecauseapairedcomparisonincludestheassumptionofconstantvariance.Thettestwithanunequalvarianceassumptiondoesnotlenditselftoaneasyrepresentationinregressionsettingsandisnotdiscussedhere.(xj�yj)=0+j. � WilliamSealyGosset(1876–1937)wasborninCanterbury,England.HestudiedchemistryandmathematicsatOxfordandworkedasachemistwiththebrewersGuinnessinDublin.Gossetbecameinterestedinstatisticalproblems,whichhediscussedwithKarlPearsonandlaterwithFisherandNeyman.Hepublishedseveralimportantpapersunderthepseudonym“Student”,andhelentthatnametothettestheinvented. ReferencesAcock,A.C.2014.AGentleIntroductiontoStata.4thed.CollegeStation,TX:StataPress.Boland,P.J.2000.WilliamSealyGosset—alias`Student'1876–1937.InCreatorsofMathematics:TheIrishConnection,ed.K.Houston,105–112.Dublin:UniversityCollegeDublinPress.Dixon,W.J.,andF.J.Massey,Jr.1983.IntroductiontoStatisticalAnalysis.4thed.NewYork:McGraw–Hill.Gleason,J.R.1999.sg101:Pairwisecomparisonsofmeans,includingtheTukeywsdmethod.StataTechnicalBulletin47:31–37.ReprintedinStataTechnicalBulletinReprints,vol.8,pp.225–233.CollegeStation,TX:StataPress.Gosset,W.S.1943.“Student's”CollectedPapers.London:BiometrikaOfce,UniversityCollege.Gosset[Student,pseud.],W.S.1908.Theprobableerrorofamean.Biometrika6:1–25.Hamilton,L.C.2013.StatisticswithStata:UpdatedforVersion12.8thed.Boston:Brooks/Cole.Hoel,P.G.1984.IntroductiontoMathematicalStatistics.5thed.NewYork:Wiley.Pearson,E.S.,R.L.Plackett,andG.A.Barnard.1990.`Student':AStatisticalBiographyofWilliamSealyGosset.Oxford:OxfordUniversityPress.Preece,D.A.1982.tisfortrouble(andtextbooks):Acritiqueofsomeexamplesofthepaired-samplest-test.Statistician31:169–195.Satterthwaite,F.E.1946.Anapproximatedistributionofestimatesofvariancecomponents.BiometricsBulletin2:110–114.Senn,S.J.,andW.Richardson.1994.Therstt-test.StatisticsinMedicine13:785–803.Welch,B.L.1947.Thegeneralizationof`student's'problemwhenseveraldifferentpopulationvariancesareinvolved.Biometrika34:28–35.Zelen,M.1979.Anewdesignforrandomizedclinicaltrials.NewEnglandJournalofMedicine300:1242–1245.Alsosee[R]bitest—Binomialprobabilitytest[R]ci—Condenceintervalsformeans,proportions,andcounts[R]esize—Effectsizebasedonmeancomparison[R]mean—Estimatemeans[R]oneway—One-wayanalysisofvariance[R]prtest—Testsofproportions[R]sdtest—Variance-comparisontests[MV]hotelling—Hotelling'sT-squaredgeneralizedmeanstest