Two-sample t-test - MATLAB ttest2 - MathWorks

This MATLAB function returns a test decision for the null hypothesis that the data in vectors x and y comes from independent random samples from normal ... Skiptocontent HelpCenterHelpCenter SearchHelpCenter HelpCenter MathWorks SearchMathWorks.com MathWorks Support CloseMobileSearch OpenMobileSearch Off-CanvasNavigationMenuToggle DocumentationHomeAI,DataScience,andStatistics StatisticsandMachineLearningToolbox ProbabilityDistributionsContinuousDistributionsStudent'stDistributionStatisticsandMachineLearningToolbox HypothesisTests ttest2 Onthispage SyntaxDescriptionExamplesTwo-Samplet-TestforEqualMeanst-TestforEqualMeansWithoutAssumingEqualVariancesOne-Sided,Two-Samplet-TestInputArgumentsxyName-ValueArgumentsAlphaDimTailVartypeOutputArgumentshpcistatsMoreAboutTwo-Samplet-testMultidimensionalArrayFirstNonsingletonDimensionTipsExtendedCapabilitiesVersionHistorySeeAlso DocumentationExamplesFunctionsBlocksAppsVideosAnswers TrialSoftware TrialSoftware ProductUpdates ProductUpdates Resources DocumentationExamplesFunctionsBlocksAppsVideosAnswers MainContent ttest2Two-samplet-testcollapseallinpage Syntaxh=ttest2(x,y)h=ttest2(x,y,Name,Value)[h,p]= ttest2(___)[h,p,ci,stats] =ttest2(___)Descriptionexampleh=ttest2(x,y)returns atestdecisionforthenullhypothesisthatthedatainvectorsxandycomes fromindependentrandomsamplesfromnormaldistributionswithequal meansandequalbutunknownvariances,usingthetwo-samplet-test. Thealternativehypothesisisthatthedatainxandycomes frompopulationswithunequalmeans.Theresulthis1if thetestrejectsthenullhypothesisatthe5%significancelevel, and0otherwise.exampleh=ttest2(x,y,Name,Value)returns atestdecisionforthetwo-samplet-testwith additionaloptionsspecifiedbyoneormorename-valuepairarguments. Forexample,youcanchangethesignificancelevelorconductthe testwithoutassumingequalvariances.example[h,p]= ttest2(___)alsoreturnsthep-value,p, ofthetest,usinganyoftheinputargumentsintheprevioussyntaxes.example[h,p,ci,stats] =ttest2(___)alsoreturnstheconfidenceinterval onthedifferenceofthepopulationmeans,ci, andthestructurestatscontaininginformation abouttheteststatistic.ExamplescollapseallTwo-Samplet-TestforEqualMeansOpenLiveScriptLoadthedataset.Createvectorscontainingthefirstandsecondcolumnsofthedatamatrixtorepresentstudents’gradesontwoexams.loadexamgrades x=grades(:,1); y=grades(:,2);Testthenullhypothesisthatthetwodatasamplesarefrompopulationswithequalmeans.[h,p,ci,stats]=ttest2(x,y)h=0 p=0.9867 ci=2×1 -1.9438 1.9771 stats=structwithfields: tstat:0.0167 df:238 sd:7.7084 Thereturnedvalueofh=0indicatesthatttest2doesnotrejectthenullhypothesisatthedefault5%significancelevel.t-TestforEqualMeansWithoutAssumingEqualVariancesOpenLiveScriptLoadthedataset.Createvectorscontainingthefirstandsecondcolumnsofthedatamatrixtorepresentstudents’gradesontwoexams.loadexamgrades x=grades(:,1); y=grades(:,2);Testthenullhypothesisthatthetwodatavectorsarefrompopulationswithequalmeans,withoutassumingthatthepopulationsalsohaveequalvariances.[h,p]=ttest2(x,y,'Vartype','unequal')h=0 p=0.9867 Thereturnedvalueofh=0indicatesthatttest2doesnotrejectthenullhypothesisatthedefault5%significancelevelevenifequalvariancesarenotassumed.One-Sided,Two-Samplet-TestOpenLiveScriptLoadthesampledata.Createacategoricalvectortolabelthevehiclemileagedataaccordingtothevehicleyear.loadcarbig.mat; decade=categorical(Model_Year<80,[true,false],["70s","80s"]);Plotboxchartsofthemileagedataforeachdecade.boxchart(decade,MPG) xlabel("Decade") ylabel("Mileage")Createvectorsfromthemileagedataforeachdecade.Usealeft-tailed,two-samplet-testtotestthenullhypothesisthatthedatacomesfrompopulationswithequalmeans.Usethealternativehypothesisthatthepopulationmeanforthemileageofcarsmadeinthe1970sislessthanthepopulationmeanforthemileageofcarsmadeinthe1980s.MPG70s=MPG(decade=="70s"); MPG80s=MPG(decade=="80s"); [h,~,~,stats]=ttest2(MPG70s,MPG80s,"Tail","left")h=1 stats=structwithfields: tstat:-14.0630 df:396 sd:6.3910 Thereturnedvalueofh=1indicatesthatttest2rejectsthenullhypothesisatthedefaultsignificancelevelof5%,infavorofthealternativehypothesisthatthepopulationmeanforthemileageofcarsmadeinthe1970sislessthanthepopulationmeanforthemileageofcarsmadeinthe1980s.PlotthecorrespondingStudent'st-distribution,thereturnedt-statistic,andthecriticalt-value.Calculatethecriticalt-valueatthedefaultconfidencelevelof95%byusingtinv.nu=stats.df; k=linspace(-15,15,300); tdistpdf=tpdf(k,nu); tval=stats.tstattval=-14.0630 tvalpdf=tpdf(tval,nu); tcrit=-tinv(0.95,nu)tcrit=-1.6487 plot(k,tdistpdf) holdon scatter(tval,tvalpdf,"filled") xline(tcrit,"--") legend(["Student'stpdf","t-statistic",... "CriticalCutoff"])Theorangedotrepresentsthet-statisticandislocatedtotheleftofthedashedblacklinethatrepresentsthecriticalt-value.InputArgumentscollapseallx—Sampledatavector|matrix|multidimensionalarray Sampledata,specifiedasavector,matrix,ormultidimensional array.ttest2treatsNaNvalues asmissingdataandignoresthem. Ifxandyare specifiedasvectors,theydonotneedtobethesamelength.Ifxandyare specifiedasmatrices,theymusthavethesamenumberofcolumns.ttest2performs aseparatet-testalongeachcolumnandreturns avectorofresults.Ifxandyare specifiedasmultidimensional arrays,theymusthavethesamesizealongallbutthefirstnonsingleton dimension. DataTypes:single|doubley—Sampledatavector|matrix|multidimensionalarray Sampledata,specifiedasavector,matrix,ormultidimensional array.ttest2treatsNaNvalues asmissingdataandignoresthem. Ifxandyare specifiedasvectors,theydonotneedtobethesamelength.Ifxandyare specifiedasmatrices,theymusthavethesamenumberofcolumns.ttest2performs aseparatet-testalongeachcolumnandreturns avectorofresults.Ifxandyare specifiedasmultidimensional arrays,theymusthavethesamesizealongallbutthefirstnonsingleton dimension.ttest2worksalongthefirst nonsingletondimension. DataTypes:single|doubleName-ValueArgumentsSpecifyoptionalpairsofargumentsas Name1=Value1,...,NameN=ValueN,whereNameis theargumentnameandValueisthecorrespondingvalue. Name-valueargumentsmustappearafterotherarguments,buttheorderofthe pairsdoesnotmatter. BeforeR2021a,usecommastoseparateeachnameandvalue,andenclose Nameinquotes. Example:'Tail','right','Alpha',0.01,'Vartype','unequal'specifies aright-tailedtestatthe1%significancelevel,anddoesnotassume thatxandyhaveequalpopulation variances.Alpha—Significancelevel0.05(default)|scalarvalueintherange(0,1) Significancelevelofthehypothesistest,specifiedasthe comma-separatedpairconsistingof'Alpha'and ascalarvalueintherange(0,1). Example:'Alpha',0.01 DataTypes:single|doubleDim—Dimensionfirstnonsingletondimension(default)|positiveintegervalue Dimensionoftheinputmatrixalongwhichtotestthemeans, specifiedasthecomma-separatedpairconsistingof'Dim'and apositiveintegervalue.Forexample,specifying'Dim',1tests thecolumnmeans,while'Dim',2teststherowmeans. Example:'Dim',2DataTypes:single|doubleTail—Typeofalternativehypothesis'both'(default)|'right'|'left' Typeofalternativehypothesistoevaluate,specifiedasthecomma-separatedpairconsisting of'Tail'andoneof: 'both'—Testagainstthe alternativehypothesisthatthepopulationmeansarenot equal.'right'—Testagainstthe alternativehypothesisthatthepopulationmeanof xisgreaterthanthepopulationmean ofy.'left'—Testagainstthe alternativehypothesisthatthepopulationmeanof xislessthanthepopulationmeanof y. ttest2teststhenullhypothesisthatthe populationmeansareequalagainstthespecifiedalternative hypothesis. Example:'Tail','right' Vartype—Variancetype'equal'(default)|'unequal' Variancetype,specifiedasthecomma-separatedpairconsisting of'Vartype'andoneofthefollowing. 'equal'Conducttestusingtheassumptionthatxandyare fromnormaldistributionswithunknownbutequalvariances.'unequal'Conducttestusingtheassumptionthatxandyare fromnormaldistributionswithunknownandunequalvariances.This iscalledtheBehrens-Fisherproblem.ttest2uses Satterthwaite’sapproximationfortheeffectivedegreesof freedom. Vartypemustbeasinglevariancetype,even whenxisamatrixoramultidimensionalarray. Example:'Vartype','unequal' OutputArgumentscollapseallh—Hypothesistestresult1|0 Hypothesistestresult,returnedas1or0. Ifh=1, thisindicatestherejectionofthenullhypothesisattheAlphasignificance level.Ifh=0, thisindicatesafailuretorejectthenullhypothesisattheAlphasignificance level. p—p-valuescalarvalueintherange[0,1] p-valueofthetest,returnedasascalar valueintherange[0,1].pistheprobability ofobservingateststatisticasextremeas,ormoreextremethan, theobservedvalueunderthenullhypothesis.Smallvaluesofpcast doubtonthevalidityofthenullhypothesis. ci—Confidenceintervalvector Confidenceintervalforthedifferenceinpopulationmeansofxandy, returnedasatwo-elementvectorcontainingthelowerandupperboundaries ofthe100×(1–Alpha)%confidence interval. stats—Teststatisticsstructure Teststatisticsforthetwo-samplet-test, returnedasastructurecontainingthefollowing: tstat—Valueofthetest statistic.df—Degreesoffreedom ofthetest.sd—Pooledestimateof thepopulationstandarddeviation(fortheequalvariancecase)or avectorcontainingtheunpooledestimatesofthepopulationstandard deviations(fortheunequalvariancecase). MoreAboutcollapseallTwo-Samplet-testThetwo-samplet-testis aparametrictestthatcomparesthelocationparameteroftwoindependent datasamples.Theteststatisticis t=x¯−y¯sx2n+sy2m,wherex¯andy¯arethesamplemeans,sxandsyare thesamplestandarddeviations,andnandmare thesamplesizes.Inthecasewhereitisassumedthatthetwodatasamplesare frompopulationswithequalvariances,theteststatisticunderthe nullhypothesishasStudent'stdistribution withn+m– 2degreesoffreedom,andthesamplestandard deviationsarereplacedbythepooledstandarddeviation s=(n−1)sx2+(m−1)sy2n+m−2.Inthecasewhereitisnotassumedthatthetwodatasamples arefrompopulationswithequalvariances,theteststatisticunder thenullhypothesishasanapproximateStudent'stdistribution withanumberofdegreesoffreedomgivenbySatterthwaite'sapproximation. ThistestissometimescalledWelch’st-test.MultidimensionalArrayAmultidimensionalarrayhasmorethantwo dimensions.Forexample,ifxisa1-by-3-by-4 array,thenxisathree-dimensionalarray.FirstNonsingletonDimensionThefirstnonsingletondimensionisthefirst dimensionofanarraywhosesizeisnotequalto1.Forexample,ifxis a1-by-2-by-3-by-4array,thentheseconddimensionisthefirstnonsingleton dimensionofx.TipsUsesampsizepwrto calculate:Thesamplesizethatcorrespondstospecifiedpower andparametervalues;Thepowerachievedforaparticularsamplesize,given thetrueparametervalue;Theparametervaluedetectablewiththespecified samplesizeandpower.ExtendedCapabilitiesGPUArraysAcceleratecodebyrunningonagraphicsprocessingunit(GPU)usingParallelComputingToolbox™.ThisfunctionfullysupportsGPUarrays.Formoreinformation,seeRunMATLABFunctionsonaGPU(ParallelComputingToolbox).VersionHistoryIntroducedbeforeR2006aSeeAlsottest|ztest|sampsizepwr × OpenExample Youhaveamodifiedversionofthisexample.Doyouwanttoopenthisexamplewithyouredits? No,overwritethemodifiedversion Yes × MATLABCommand YouclickedalinkthatcorrespondstothisMATLABcommand: RunthecommandbyenteringitintheMATLABCommandWindow. WebbrowsersdonotsupportMATLABcommands. Close × SelectaWebSite Chooseawebsitetogettranslatedcontentwhereavailableandseelocaleventsandoffers.Basedonyourlocation,werecommendthatyouselect:. Switzerland(English) Switzerland(Deutsch) Switzerland(Français) 中国(简体中文) 中国(English) Youcanalsoselectawebsitefromthefollowinglist: HowtoGetBestSitePerformance SelecttheChinasite(inChineseorEnglish)forbestsiteperformance.OtherMathWorkscountrysitesarenotoptimizedforvisitsfromyourlocation. 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