Power and Sample Size Calculation.docx

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