Power and Sample Size Calculation.docx
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Power and Sample Size Calculation.docx
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PowerandSampleSizeCalculation
PowerandSampleSizeCalculation
ByGaylaOlbrichtandYongWang
DefinitionandApplication
Statisticalpowerisdefinedastheprobabilityofrejectingthenullhypothesiswhilethealternativehypothesisistrue.Factorsthataffectstatisticalpowerincludethesamplesize,thespecificationoftheparameter(s)inthenullandalternativehypothesis,i.e.howfartheyarefromeachother,theprecisionoruncertaintytheresearcherallowsforthestudy(generallytheconfidenceorsignificancelevel)andthedistributionoftheparametertobeestimated.Forexample,ifaresearcherknowsthatthestatisticsinthestudyfollowaZorstandardnormaldistribution,therearetwoparametersthathe/sheneedstoestimate,thepopulationmean(μ)andthepopulationvariance(σ2).Mostofthetime,theresearcherknowoneoftheparametersandneedtoestimatetheother.Ifthatisnotthecase,someotherdistributionmaybeused,forexample,iftheresearcherdoesnotknowthepopulationvariance,he/shecanestimateitusingthesamplevarianceandthatendsupwithusingaTdistribution.
Inresearch,statisticalpowerisgenerallycalculatedfortwopurposes.
1.Itcanbecalculatedbeforedatacollectionbasedoninformationfrompreviousresearchtodecidethesamplesizeneededforthestudy.
2.Itcanalsobecalculatedafterdataanalysis.Itusuallyhappenswhentheresultturnsouttobenon-significant.Inthiscase,statisticalpoweriscalculatedtoverifywhetherthenon-significantresultisduetoreallynorelationinthesampleorduetoalackofstatisticalpower.
Statisticalpowerispositivelycorrelatedwiththesamplesize,whichmeansthatgiventheleveloftheotherfactors,alargersamplesizegivesgreaterpower.However,researchersarealsofacedwiththedecisiontomakeadifferencebetweenstatisticaldifferenceandscientificdifference.Althoughalargersamplesizeenablesresearcherstofindsmallerdifferencestatisticallysignificant,thatdifferencemaynotbelargeenoughbescientificallymeaningful.Therefore,asconsultants,wewouldliketorecommendthatourclientshaveanideaofwhattheywouldexpecttobeascientificallymeaningfuldifferencebeforedoingapoweranalysistodeterminetheactualsamplesizeneeded.
CalculationofStatisticalPower
Thepowerisaprobabilityanditisdefinedtobetheprobabilityofrejectingthenullhypothesiswhenthealternativehypothesisistrue.Afterpluggingintherequiredinformation,aresearchercangetafunctionthatdescribestherelationshipbetweenstatisticalpowerandsamplesizeandtheresearchercandecidewhichpowerleveltheypreferwiththeassociatedsamplesize.Thechoiceofsamplesizemayalsobeconstrainedbyfactorssuchasthefinancialbudgettheresearcherisfacedwith.Butgenerallyconsultantswouldliketorecommendthattheminimumpowerlevelissettobe0.80.
Insomeoccasions,calculationofpowerissimpleandcanbedonebyhand.StatisticalsoftwarepackagessuchasSASalsooffersawayofcalculatingpowerandsamplesize.
Theresearchersmusthavesomeinformationbeforetheycandothepowerandsamplesizecalculation.Theinformationincludespreviousknowledgeabouttheparameters(theirmeansandvariances)andwhatconfidenceorsignificancelevelisneededinthestudy.
HandCalculation.
Wewilluseanexampletoillustratehowaresearchercancalculatethesamplesizeneededforastudy.Giventhataresearcherhasthenullhypothesisthatμ=μ0andalternativehypothesisthatμ=μ1≠μ0,andthatthepopulationvarianceisknownasσ2.Also,heknowsthathewantstorejectthenullhypothesisatasignificancelevelofαwhichgivesacorrespondingZscore,calleditZα/2.Therefore,thepowerfunctionwillbe
P{Z>Zα/2orZ<-Zα/2|μ1}=1-Φ[Zα/2-(μ1-μ0)/(σ/n)]+Φ[-Zα/2-(μ1-μ0)/(σ/n)].
Thatisafunctionofthepowerandsamplesizegivenotherinformationknownandtheresearchercangetthecorrespondingsamplesizeforeachpowerlevel.
Forexample,iftheresearcherlearnsfromliteraturethatthepopulationfollowsanormaldistributionwithmeanof100andvarianceof100underthenullhypothesisandhe/sheexpectsthemeantobegreaterthan105orlessthan95underthenullhypothesisandhe/shewantsthetesttobesignificantat95%level,theresultingpowerfunctionwouldbe:
Power=1-Φ[-(105-100)/(10/n)]+Φ[--(95-100)/(10/n)],whichis,
Power=1-Φ[n/2]+Φ[-+n/2].
Thatfunctionshowsarelationshipbetweenpowerandsamplesize.Foreachlevelofsamplesize,thereisacorrespondingsamplesize.Forexample,ifn=20,thecorrespondingpowerlevelwouldbeabout0.97,5,thecorrespondingsamplesizewouldbe16.
UsingStatisticalPackage(SAS)
StatisticalpackageslikeSASenablesaresearchertodothepowercalculationeasily.Theprocedureinwhichpowerandsamplesizearecalculatedisspecifiedinthefollowingtext.
InSAS,statisticalpowerandsamplesizecalculationcanbedoneeitherthroughprogrameditororbyclickingthemenuthemenu.Inthelatter,asetofcodeisautomaticallygeneratedeverytimeacalculationisdone.
PROCPOWERandGLMPOWER
PROCPOWERandGLMPOWERarenewadditionstoSASasofversion9.0.Asofthiswriting,isnotcurrentlyinstalledonITaPmachines,butitcanbeinstalledonyourhomecomputerusingdisksavailableinStewardB14.MakesuretobringyourPurdueID.
Thetableonthefollowingpage(takenfromtheSAShelpfile)showsthetypesofanalysesofferedbyPROCPOWER.Atleastonestatementisrequired.Thesyntaxwithineachstatementvaries,however,thereissomesyntaxcommontoall.Thesecommonfeatureswillbeexpressedbyanexampleusingapairedt-test.MoreinformationoneachprocedurecanbefoundintheSAShelpfile.
Intheexample,assumethatapilotstudyhasbeendone,andthatthestandarddeviationofthedifferencebetweenthetwogroupshasbeenfoundtobe5,withameandifferenceof2.We’dliketocalculatetherequiredsamplesizeforanexperimentwith80%power.
procpower;
pairedmeanstest=diff
meandiff=2
stddev=5
npairs=.
power=.80;
run;
Analysis
Statement
Options
Multiplelinearregression:
TypeIIIFtest
MULTREG
Correlation:
Fisher'sztest
ONECORR
DIST=FISHERZ
Correlation:
ttest
ONECORR
DIST=T
Binomialproportion:
Exacttest
ONESAMPLEFREQ
TEST=EXACT
Binomialproportion:
ztest
ONESAMPLEFREQ
TEST=Z
Binomialproportion:
ztestwithcontinuityadjustment
ONESAMPLEFREQ
TEST=ADJZ
One-samplettest
ONESAMPLEMEANS
TEST=T
One-samplettestwithlognormaldata
ONESAMPLEMEANS
TEST=TDIST=LOGNORMAL
One-sampleequivalencetestformeanofnormaldata
ONESAMPLEMEANS
TEST=EQUIV
One-sampleequivalencetestformeanoflognormaldata
ONESAMPLEMEANS
TEST=EQUIVDIST=LOGNORMAL
Confidenceintervalforamean
ONESAMPLEMEANS
CI=T
One-wayANOVA:
One-degree-of-freedomcontrast
ONEWAYANOVA
TEST=CONTRAST
One-wayANOVA:
OverallFtest
ONEWAYANOVA
TEST=OVERALL
McNemarexactconditionaltest
PAIREDFREQ
McNemarnormalapproximationtest
PAIREDFREQ
DIST=NORMAL
Pairedttest
PAIREDMEANS
TEST=DIFF
Pairedttestofmeanratiowithlognormaldata
PAIREDMEANS
TEST=RATIO
Pairedadditiveequivalenceofmeandifferencewithnormaldata
PAIREDMEANS
TEST=EQUIV_DIFF
Pairedmultiplicativeequivalenceofmeanratiowithlognormaldata
PAIREDMEANS
TEST=EQUIV_RATIO
Confidenceintervalformeanofpaireddifferences
PAIREDMEANS
CI=DIFF
Pearsonchi-squaretestfortwoindependentproportions
TWOSAMPLEFREQ
TEST=PCHI
Fisher'sexacttestfortwoindependentproportions
TWOSAMPLEFREQ
TEST=FISHER
Likelihoodratiochi-squaretestfortwoindependentproportions
TWOSAMPLEFREQ
TEST=LRCHI
Two-samplettestassumingequalvariances
TWOSAMPLEMEANS
TEST=DIFF
Two-sampleSatterthwaitettestassumingunequalvariances
TWOSAMPLEMEANS
TEST=DIFF_SATT
Two-samplepooledttestofmeanratiowithlognormaldata
TWOSAMPLEMEANS
TEST=RATIO
Two-sampleadditiveequivalenceofmeandifferencewithnormaldata
TWOSAMPLEMEANS
TEST=EQUIV_DIFF
Two-samplemultiplicativeequivalenceofmeanratiowithlognormaldata
TWOSAMPLEMEANS
TEST=EQUIV_RATIO
Two-sampleconfidenceintervalformeandifference
TWOSAMPLEMEANS
CI=DIFF
Log-ranktestforcomparingtwosurvivalcurves
TWOSAMPLESURVIVAL
TEST=LOGRANK
Gehanranktestforcomparingtwosurvivalcurves
TWOSAMPLESURVIVAL
TEST=GEHAN
Tarone-Wareranktestforcomparingtwosurvivalcurves
TWOSAMPLESURVIVAL
TEST=TARONEWARE
PowerandSampleSizeCalculationUsingSASMenu
PowerandsamplesizecanalsobecalculatedusingthemenuinSAS.Whenusingthemenu,theusershouldspecifythechosendesignfortheunderlyingproject,andthenfillintherequiredparametersneededtodothecalculationforeachdesign.
Thegeneralprocedureofusingthemenuisasfollows:
1).OpenSAS
2).Gototheenhancededitorwindow.
3).Clickthe'solutions'buttononthemenu.
4).Inthesubmenu,click'analysis'.
5).Inthenextsubmenu,click'analyst',thenanewwindowwillpop-up.
6).Inthenewwindow,click'statistics'buttononthemenu.
7).Select'Samplesize',thenselectthedesignyouwanttouse.(thedesignsavailableinthatmenuinclude:
one-samplet-test,pairedt-test,twosamplet-testandone-wayANOVA).
8).Afteryouselectthedesignanotherwindowpops-upandasksyou
toinputtheneededoptionsandparameters.Ifyouneedtoknowtheneeded
samplesizeforyourresearch,youcanselect'NperGroup',theninputnumberoftreatments,correctedsumofsquare,thestandarddeviationandthealpha
level.Iftheresearcherwantstocalculatethesamplesizecorrespondingtoeachpowerlevel,he/shemaywanttospecifytherangeandintervalofpowerlevelinthe‘Powe
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