Two sample ttest.ppt

comparisonoftwosamples summerprogrambrianhealy previousclasses hypothesistestingnullandalternativehypothesesteststatisticp valueconclusionconfidenceintervalscomparisonofcitohypothesistestpowerandsamplesize whatarewedoingtoday two samplet testpairedt testindependentsamplesequalvarianceunequalvariancesamplesizefortwosamples bigpicture uptothispoint wehaveonlyconcernedourselveswithonesample oftenwewanttocompareonegrouptoanother whathappenswhenwearecomparingtwosamples variabilityinbothsamples andpotentiallytwosamplesarerelatedmuchofthetheoryisthesame example oneofthefirststudiesianalyzedwasatumorsizestudy havinganaccuratemeasureoftumorsizeisextremelyimportantbecauseitallowsaphysiciantoaccuratelydetermineifatumorisgrowing shrinkingorremainingconstant theproblemisthatoftenthemeasurementsofthetumorsizevaryfromphysiciantophysician inthepast tumorsizewasmeasuredusingthelineardistanceacrossthetumor butthiswasfoundtobeveryvariablebecauseoftheirregularshapeofsometumors anewmethodcalledtherecistcriteriatracestheoutsideofthetumor therecistmethodwasbelievedtogivemoreconsistentmeasuresofthevolumeofthetumor availabledata foraportionofthestudy apairofdoctorswereshownthesamesetoftumorpictures thevolumeofthetumorwasmeasuredbytwoseparatephysiciansundersimilarconditions questionofinterest didthemeasurementsfromthetwophysicianssignificantlydiffer ifnot thentherewouldbenoevidencethatthevolumemeasurementschangebasedonphysician 20scansweremeasuredbyeachphysician 10areshownhere measurementsincm3whatcanyousayaboutthesesamples twomeasurementonthesamepersontheyarerelatedsowemustaccountforthismuchresearchinstatisticsdealswithhowtohandlecorrelateddata butinthiscaseitisprettyeasy dependentsample wecanmeasuretheeffectofthetreatmentineachpersonbytakingthedifferenceinsteadofhavingtwosamples wecanconsiderourdatasettobeonesampleofdifferencesjustliketheonesampleproblem differences volumefromdr 1populationmean samplemean volumefromdr 2populationmean samplemean differencepopulationmean samplemean distributionofdifferences assumingdi sarenormallydistributed canuset distributionwithn 1dofwherenisthenumberofdifferencesstandarddeviationofdifferencesteststatisticactsjustlikeonesample picture wecanseethattheassumptionofnormalityofthedifferencesisreasonableinthiscase pairedt test nullhypothesis nodifferencebetweenphysicianseffecttwodependentsamples alpha 0 05teststatistic t statisticwithdofp value 0 53failtorejectnullhypothesisconclusion thereisnoevidenceofadifferenceintumorvolumemeasurementbasedonphysician confidenceinterval confidenceintervalforpairedt testconstructedinthesamewayasone samplet testforourexample theconfidenceintervalis 1 010 54 notethattheconclusionfromthehypothesistestandtheconfidenceintervalarethesame pairedt testinr usingthehelpmenu determinehowtocompletethepairedt testinr pairedt testinr data read table p pairedscans dat header f dr1 data 1 dr2 data 2 t test dr1 dr2 paired t theoutputprovidesthep valueandtheconfidenceintervalpairedt testdata data 1 anddata 2 t 0 6456 df 19 p value 0 5262alternativehypothesis truedifferenceinmeansisnotequalto095percentconfidenceinterval 1 01802790 5380279sampleestimates meanofthedifferences 0 24 extensions someadditionalexamplesofpairedsamplesare differencesbetweenleftandrighteyedifferencesbetweendominantandrecessivehandmatchedsampleswhenyouhavemorethantwosamples techniquesaccountforthecorrelationbetweenthesamplesmultivariate longitudinaldata unpairedsamples oftenitisimpracticaltodesignstudytousethesamepatientsforbothgroupex comparisonofcholesterolinmalesandfemalesex timeconstraintssincethesamplesarenotpaired wecannotusethedifferencebetweentheindividualsamplesmustadjustpreviousanalysis example anotheraspectofthetumorvolumestudywastryingtocomparethetumorvolumeamongpatientswithdifferentformsofcancer theaveragetumorsizeisimportanttoknowtheeffectoftreatmentcanbedetermined inthisstudy patientswithbrain breastandlivertumors butinitiallywewillonlycomparethebrainandbreastcancers allofthetumorsweremeasuredusingtherecistmethod nullhypothesis thenullhypothesisisthatthereisnodifferencebetweenthevolumeofthetumorinthetwoformsofcancerh0 mbrain mbreast ormbrain mbreast 0moregenerally wecantestifthedifferencebetweentwogroupsisaspecificvalue m1 m2 dthisoccurswhencomparingtwotreatmentgroupsandweareinterestedifthetwogroupsaredifferentbyaspecificamount eachpatientcontributesoneobservationcanestimatefromthesamplemeanandstandarddeviationinbraincancergroupwithmeanandstandarddeviationinbreastcancergroupwitharethetwogroupsthesame h0 m1 m2 orm1 m2 0todeterminethis wearegoingtolookatwealsoneedtoknow differenceinthesamplemeans wearegoingtousethedifferenceofthemeansasourteststatistic butweneedtoestimatethevarianceofthisdifferencetodetermineifthedifferenceissignificantbasicformofteststatistic standarddeviationsknownunknowntheestimateofthestandarddeviationchangeswhenthesampleshaveequalvarianceorthesampleshaveunequalvariance equalvariance sometimeswewillbewillingtoassumethatthevarianceinthetwogroupsisequal ifweknowthisvariance wecanusethez statisticoftenwehavetoestimates2withthesamplevariancefromeachofthesamples sincewehavetwoestimatesofonequantity wepoolthetwoestimates equalvariancecontinued theestimateofsisgivenby thet statisticbasedonthepooledvarianceisverysimilartothez statisticasalways thet statistichasat distributionwithdegreesoffreedom forthetumorvolumestudy therewere20braincancersubjectsand28breastcancersubjectsthesummarystatisticsandhistogramforthedataaregivenherewhatcanyousayaboutthedistributions doestheequalvarianceassumptionseemvalidinthiscase hypothesistest h0 meanbraintumorsize meanbreasttumorsizetwoindependentsampleswithequalvariance alpha 0 05p value 0 046rejectnullhypothesisconclusion thereisasignificantdifferenceinthesizeofbrainandbreastcancertumors rcode ifweonlyhadtheteststatisticsabove wecancalculatetheteststatisticandthencompareittothet distributionusingpt 2 054 df 46 todeterminetheareainthelowertailhowdoweconvertthisintotheappropriatep value withthefulldata wecanusedata read table cancer dat header t gr data 1 size data 2 t test size gr 0 size gr 1 var equal t routput twosamplet testdata size gr 0 andsize gr 1 t 2 054 df 46 p value 0 04568alternativehypothesis truedifferenceinmeansisnotequalto095percentconfidenceinterval 2 65174438 0 02682705sampleestimates meanofxmeanofy16 1500017 48929 unequalvariance often weareunwillingtoassumethatthevariancesareequalwenowwritetheteststatisticas thedistributionofthisstatisticisdifficulttoderiveandweapproximatethedistributionusingat distributionwithndegreesoffreedom thisiscalledthesatterthwaiteorwelchapproximationwhenyoucompleteatwo samplet testinrandthevariancesarenotassumedequal thisapproximationisused example forthecomparisonofthebraincancerstothelivercancers thevariancesaremuchmoredifferent let susetheunequalvariancetwosamplet testinthiscase example h0 meanbraintumorsize meanlivertumorsizetwoindependentsampleswithequalvariance alpha 0 05p value 0 0044rejectnullhypothesisconclusion thereisasignificantdifferenceinthesizeofthebrainandlivertumorsize routput t test size gr 0 size gr 2 welchtwosamplet testdata size gr 0 andsize gr 2 t 3 1666 df 22 48 p value 0 00439alternativehypothesis truedifferenceinmeansisnotequalto095percentconfidenceinterval 5 288291 1 105827sampleestimates meanofxmeanofy16 1500019 34706 practice getthetvdatasetfromthecoursefolderwewanttocomparetheamountoftvtheboysandgirlswatch performthemostappropriatetest boysarecodedas0andgirlsarecodedas1 canwetestifthevariancesareequal sincewecanneverbesureifthevariancesareequal couldwetestiftheyareequal ofcoursewecan but rememberthereiserrorineverystatisticaltestsometimesitisjustpreferredtousetheunequalvarianceunlessthereisagoodreason equalityofvariance h0 s12 s22totestthishypothesis weusethesamplevariances ifoneofthevariancesismuchlargerthantheother thisisevidenceagainstthenullaswediscussedacoupleclassesago testofequality onewaytotestifthetwovariancesareequalistocheckiftheratioisequalto1 h0 ratio 1 underthenull theratiosimplifiestotheratioof2chi squarerandomvariableshasanf distributionthef distributionisdefinedbythenumeratoranddenominatordegreesoffreedomherewehaveanf distributionwithn1 1andn2 1degreesoffreedomthisworksbetterwith f distribution hereisthef distributionwith5and500degreesoffreedomnotetheskewofthedistribution example var test size gr 1 size gr 0 ftesttocomparetwovariancesdata size gr 1 andsize gr 0 f 1 719 numdf 27 denomdf 19 p value 0 2247alternativehypothesis trueratioofvariancesisnotequalto195percentconfidenceinterval 0 7103353 904512sampleestimates ratioofvariances1 719033 var test size gr 2 size gr 0 ftesttocomparetwovariancesdata size gr 2 andsize gr 0 f 4