Sample size determinations for Welch's test in oneway heteroscedastic ANOVA

22 Pages·2014·0.18 MB·English

72
BritishJournalofMathematicalandStatisticalPsychology(2014),67,72–93
©2013TheBritishPsychologicalSociety
www.wileyonlinelibrary.com
Sample size determinations for Welch’s test in
one-way heteroscedastic ANOVA
Show-Li Jan1 and Gwowen Shieh2*
1ChungYuanChristianUniversity,Taiwan,RepublicofChina
2NationalChiaoTungUniversity,Taiwan,RepublicofChina
Forone-wayfixedeffectsANOVA,itiswellknownthattheconventionalFtestofthe
equalityofmeansisnotrobusttounequalvariances,andnumerousmethodshavebeen
proposed for dealing with heteroscedasticity. On the basis of extensive empirical
evidence of Type I error control and power performance, Welch’s procedure is
frequentlyrecommendedasthemajoralternativetotheANOVAFtestundervariance
heterogeneity. To enhance its practical usefulness, this paper considers an important
aspectofWelch’smethodindeterminingthesamplesizenecessarytoachieveagiven
power.Simulationstudiesareconductedtocomparetwoapproximatepowerfunctions
ofWelch’stestfortheiraccuracyinsamplesizecalculationsoverawidevarietyofmodel
configurationswithheteroscedasticstructures.Thenumericalinvestigationsshowthat
Levy’s(1978a)approachisclearlymoreaccuratethantheformulaofLuhandGuo(2011)
fortherangeofmodelspecificationsconsideredhere.Accordingly,computerprograms
areprovidedtoimplementthetechniquerecommendedbyLevyforpowercalculation
and sample size determination within the context of the one-way heteroscedastic
ANOVAmodel.
1. Introduction
Theone-wayanalysisofvariance(ANOVA)Ftestisaprocedurewidelyusedfortestingthe
equality of means of independent normal distributions with homogeneous variances.
Thecorrespondingimplications,fromthebasicdiagnosticsofunderlyingassumptionsto
therequiredpowercalculationsandsamplesizedeterminations,havebeenextensively
addressed in the literature; see, for example, Howell (2010), Kirk (1995), Kutner,
Nachtsheim, Neter and Li (2005) and Scheff(cid:1)e (1959). However, the violation of the
independence,normality,andhomogeneityofvarianceassumptionseitherseparatelyor
inconjunctionwithoneanotherhasbeenthetargetofcriticisminapplicationsofANOVA
(Coombs,Algina&Oltman,1996;Glass,Peckham&Sanders,1972;Harwell,Rubinstein,
Hayes & Olds, 1992; Keselman et al., 1998). Specifically, the F test is not robust to all
*CorrespondenceshouldbeaddressedtoGwowenShieh,DepartmentofManagementScience,NationalChiao
TungUniversity,Hsinchu,Taiwan30010,RepublicofChina(e-mail:[email protected]).
DOI:10.1111/bmsp.12006
SamplesizedeterminationsforWelch’stestinone-wayheteroscedasticANOVA 73
degrees of unequal variances (Brown & Forsythe, 1974; Clinch & Keselman, 1982; De
Beuckelaer, 1996; Kohr & Games, 1974; Levy, 1978b; Wilcox, Charlin & Thompson,
1986),and theactual significancelevelandpowercanbedistorted even whensample
sizes are equal (Krutchkoff, 1986; Rogan & Keselman, 1977). Accordingly, various
parametricandnon-parametricalternativestothetraditionalFtesthavebeenproposedto
countertheeffectsofheteroscedasticity(Lix,Keselman&Keselman,1996).
Given the extensive Monte Carlo simulation studies conducted in this area, three
important aspects of these numerical evidences should be pointed out. First, the non-
parametricproceduresarealsosubstantiallyaffectedbyheterogeneousvariancesandare
generally inferior to the parametric approaches (Keselman, Rogan & Feir-Walsh, 1977;
Tomarken&Serlin,1986;Zimmerman,2000).Second,theparametrictestsofAlexander
and Govern (1994), Brown and Forsythe (1974), James (1951) and Welch (1951) have
been shown to provide accurate control of Type I error rate and competitive power
performance (Schneider & Penfield, 1997). However, there appears to be a lack of
consensusintheliteratureonwhichmethodismostappropriate.Essentially,thereisno
oneuniformlybestalternativetotheFtestunderheterogeneityofvariance(Dijkstra&
Werter,1981;Grissom,2000).Third,despitethefactthatnoapproachisideal,itisstillof
practicalimportancetohaveareliableandsimpletestprocedurethatissufficientlyrobust
toheteroscedasticitywhendistributionsarenormal.Onthebasisofthecomprehensive
appraisalsbyBrownandForsythe(1974),DeBeuckelaer(1996),Grissom(2000),Harwell
et al. (1992), Levy (1978b), Tomarken and Serlin (1986) and Wilcox et al. (1986), the
approximationofWelch(1951)isthemostwidelyrecommendedtechniquetocorrectfor
variance heterogeneity. In short, it has distinct advantages over other competing
approaches in its overall performance, computational ease, and general availability in
statisticalcomputerpackages.
Yetanotherproblemwiththecommonmethodsforanalysingthedatafromone-way
independent groups designs occurs when the distribution of each population is non-
normalinform.SeeCribbie,Fiksenbaum,KeselmanandWilcox(2012),LixandKeselman
(1998),Wilcox(2003)andWilcoxandKeselman(2003)formodernrobustmethodsand
updatedstrategieswhenthestandardassumptionsofnormalityandhomoscedasticityare
violated. In particular, the Welch test with robust estimators of trimmed means and
WinsorizedvarianceshasbeenshowntoprovideexcellentTypeIerrorcontrolandpower
performancewhendataarenon-normalandheterogeneous.However,wewillrestrictour
attentiontotheappropriateprocedurefortestingtheequalityofmeansofindependent
normaldistributionswithpossiblyunequalerrorvarianceshere.
ItisconceivablethatatestprocedurewithrobustTypeIerrorcontrolandexcellent
power performance is not sufficient for the purposes of research design and statistical
inference.Thecorrespondingpoweranalysisandsamplesizecomputationmustalsobe
consideredbeforeitcanbeadoptedasageneralmethodologyinpractice.Theoretically,
thenon-nulldistributionofatestprocedureisrequiredinordertoevaluatetheintrinsic
issuesofpoweranalysisandsamplesizeassessment.Buttoourbestknowledge,nopower
functionornon-nulldistributionhasbeenproposedfortheprescribedtestsofAlexander
and Govern (1994), Brown and Forsythe (1974) and James (1951). On the other hand,
several approximations have been described for the non-null distribution of Welch’s
(1951)testinLevy(1978a),LuhandGuo(2011)andKulinskaya,StaudteandGao(2003).
Althoughtheseresultspermitpowerandsamplesizeconsiderationsforthewell-known
Welch(1951)method,noresearchtodatehascomparedtheirdistinctcharacteristicsin
termsoftheoreticalprinciples,computationalrequirementsandempiricalperformance.
Butinfacttheirformulationsaremarkedlydifferentanddemandvaryingcomputational
74 Show-LiJanandGwowenShieh
efforts.Thusitisprudenttoexaminetheiruniquefeatureandfundamentaldiscrepancyin
ordertobetterunderstandtheselectionofanappropriateapproachtopoweranalysisand
samplesizedeterminationinone-factorANOVAstudies.
Insteadofanon-centralFdistribution,Kulinskayaet al.(2003)presentedachi-square-
based power approximation to the non-null distribution of Welch’s test. They showed
thattheshiftedandrescaledchi-squareapproximationismoreaccuratethanthestandard
chi-square transformation. However, there are two obvious disadvantages of their
approximatepowerfunction.First,thechi-square-basedformulationdoesnotconformto
the entrenched F test of homoscedastic ANOVA or Welch’s test of heteroscedastic
ANOVA. Second, the complexity of their proposed expression is overwhelming. It is
worthwhile to consider a more transparent procedure with fewer computational and
theoreticalhurdles.ThustheapproachofKulinskayaet al.(2003)willnotbeconsidered
furtherinthispaper.
Recently,LuhandGuo(2011)suggestedanon-centralFdistributiontoapproximate
the non-null distribution of Welch’s test. The non-centrality of their non-central F
distributionisadirectmodificationofthenon-centralityoftheusualFtest’sexactnon-null
Fdistributionunderbalanceddesignandhomogeneityofvariance.Inparticular,thenon-
centralityderivedinvolvesasimpleaverageofthevarianceandsamplesizeratiosofeach
group. The adapted formula at first sight provides a convenient approximation and is
computationallysimple.Notably,LuhandGuo (2011)concludedthattheirtechniqueis
suitableforobtainingtheadequatesamplesizesinheterogeneousANOVA.However,a
closer inspection of their numerical results reveals that the discrepancy between the
nominalpowerandsimulatedpower(orestimateoftruepower)issizeableforseveral
casesconsideredintheirsimulationstudy.Hence,theaccuracyoftheirproposedpower
functioninsamplesizeestimationisquestionable.Furtherexaminationsarerequiredto
demonstratetheunderlyingdrawbacksassociatedwiththeirapproximateprocedure.
According to the explication of power and sample size considerations for Welch’s
procedure presented above, the approximate technique proposed in Levy (1978a) has
beengiveninsufficientconsideration,thoughanotableexceptionisTomarkenandSerlin
(1986).DuetothecomplexityoftheoreticaljustificationforWelch’stestprocedure,no
explicit analytic form of the corresponding non-null distribution isavailable. However,
theapproximatenon-nulldistributionofLevy(1978a)canbeobtainedbyreplacingthe
sample means and variances in Welch’s test statistic with corresponding population
parameters. It was shown in the numerical comparisons of the estimated power and
simulatedpowerofLevy(1978a)thatthesuggestednon-centralFdistributionyieldsan
adequateapproximationtothenon-nulldistributionofWelch’statistic.Later,Tomarken
and Serlin (1986) also strongly recommended the non-central F approximation for
conductingpoweranalysesoftheWelchprocedure.ThustheformulaofLevy(1978a)is
ofgreat potential useandshould beproperly recognized.But theexplicationofLevy’s
non-centralFdistributionhasbeenconfinedtopowerexamination,andnosinglestudy
hasextendedtheinvestigationtosamplesizecalculation.Inviewofthelimitationsofthe
existingfindings,itisessentialtogeneralizeandassesstheeffectivenessofLevy’s(1978a)
approximateformulainsamplesizedeterminationwithmoderncomputingfacilitiesand
accessiblestatisticalsoftware.
ItisimportanttonotethattheapproximatepowerfunctionsofLevy’s(1978a)andLuh
andGuo(2011)bothrelyonanon-centralFdistribution,withidenticalnumeratorand
denominatordegreesoffreedom.Theonlydifferenceisintheirrespectivespecifications
of the non-centrality parameter. Because of the complex nature of the non-central
distributionandnon-centralityparameter,acompletetheoreticaltreatmentandanalytical
SamplesizedeterminationsforWelch’stestinone-wayheteroscedasticANOVA 75
evaluationisnotfeasible.However,therestillremainsnosimultaneouscomparisonofthe
empirical performance of the two approaches. In order to offer well-supported
recommendations on desirable sample sizes for heteroscedastic ANOVA models, this
paper appraises and compares the two approaches of Levy (1978a) and Luh and Guo
(2011)forpowercalculationsandsamplesizedeterminationsofWelch’stestprocedure.
Since optimal sample size determinations for Welch’s (1938) two-group test were
presentedinJanandShieh(2011),thispaperfocusesonthesituationswiththreeormore
treatment groups. Comprehensive empirical investigations were conducted to demon-
strate the potential advantages and disadvantages between the two methods under a
varietyofmeanstructures,variancepatterns,aswellasequalandunequalsamplesizes.
Our study reveals unique information that not only demonstrates the fundamental
deficiencyofexistinginvestigations,butalsoenhancestheusefulnessoftheWelchtestin
thecontextofANOVAundervarianceheterogeneity.Moreover,correspondingSASandR
computercodesarepresentedtofacilitatetherecommendedprocedureforcomputing
theachievedpowerlevelandrequiredsamplesizeinactualapplications.
2. TheWelchtest
Considertheone-wayheteroscedasticANOVAmodelinwhichtheobservationsX are
ij
assumed to be independent and normally distributed with expected values l and
i
variancesr2:
i
X (cid:1)Nðl;r2Þ; ð1Þ
ij i i
wherel andr2areunknownparameters,i = 1,…,g((cid:3) 2)andj = 1,…,N Totestthe
i i i
hypothesisthatalltreatmentmeansareequal,theclassicFtestisthemostwidelyused
statisticalprocedureassuminghomogeneityofvariance(r2 ¼r2 ¼...¼r2).However,
1 2 g
it has been shown in extensive studies that the conventional F test is sensitive to the
heteroscedasticityformulationdefinedin(1).OfthenumerousalternativestotheANOVA
Ftest,wefocusontheviableapproachproposedinWelch(1951)intheformof
P
g WðX(cid:3) (cid:4)X~Þ2=ðg(cid:4)1Þ
W ¼ i¼1 i i ; ð2Þ
1þ2ðg(cid:4)2ÞQ=ðg2(cid:4)1Þ
P P P
wherPe Wi¼Ni=Si2;Si2¼ PNj¼i1ðXij(cid:4)X(cid:3)iÞ2=ðNi(cid:4)1Þ;X(cid:3)i¼ Nj¼i1Xij=Ni;X~¼ gi¼1WiX(cid:3)i=U;
U ¼ g W, and Q¼ g ð1(cid:4)W=UÞ2=ðN (cid:4)1Þ. Under the null hypothesis H :
i¼1 i i¼1 i i 0
l = l = … = l ,Welch(1951)suggeststheapproximateFdistributionforW:
1 2 g
W (cid:1)(cid:5) Fðg(cid:4)1;^mÞ;
where Fðg(cid:4)1;^mÞ is the F distribution with g (cid:4) 1 and ^m¼ðg2(cid:4)1Þ=ð3QÞ degrees of
freedom.Hence,H0isrejectedatthesignificancelevelaifW >Fðg(cid:4)1Þ;^m;a,whereFðg(cid:4)1Þ;^m;a
is the upper 100 ath percentile of the F distribution Fðg(cid:4)1;^mÞ. Although numerical
evidenceconfirmstheaccurateTypeIerrorcontrolandsuperiorpowerperformanceof
Welch’s test, theoretical justification for the non-null distribution of W has rarely been
discussed.Especially,twonon-centralFapproximationsareconsideredinLevy(1978a)
andLuhandGuo(2011).LuhandGuo(2011)suggested
76 Show-LiJanandGwowenShieh
W (cid:1)(cid:5) Fðg(cid:4)1;m;K Þ;
LG
whereF(g (cid:4) 1,t,ΛLG)isthenon-centralFPdistributionwith(g (cid:4) 1)andt = (g2 (cid:4) 1)/
(3τ)Pdegrees of freedom, where s¼ gi¼1ð1(cid:4)xi=tÞ2=ðNi(cid:4)1Þ;xi ¼Ni=r2i, and
t¼ g x,andwithnon-centralityparameter
i¼1 i
P
g ðl (cid:4)l(cid:3)Þ2
K ¼g Pi¼1 i ;
LG g ð1=xÞ
i¼1 i
P
wherel(cid:3) ¼ g l=g.ThenthecorrespondingpowerfunctionofWelch’stestisofthe
i¼1 i
form
pðKLGÞ¼PfFðg(cid:4)1;m;KLGÞ>Fðg(cid:4)1Þ;m;ag: ð3Þ
Ontheotherhand,Levy(1978a)proposedtheapproximatenon-nulldistributionforW
givenby
W (cid:1)(cid:5) Fðg(cid:4)1;m;K Þ;
L
where
Xg
K ¼ xðl (cid:4)l~Þ2;
L i i
i¼1
P
andl~ ¼ g xl=t.Inthiscase,theassociatedpowerfunctionp(Λ )isexpressedas
i¼1 i i L
pðKLÞ¼PfFðg(cid:4)1;m;KLÞ>Fðg(cid:4)1Þ;m;ag: ð4Þ
Notethatthenon-centralityparameterΛ canbeexpressedas
LG
P
g ðl (cid:4)l~Þ2
K ¼Pi¼1 i :
LG g ð1=xÞ=g
i¼1 i
HPence, the heterPoscedastic variance property is only accommodated in the quantity
g ð1=xÞ=g¼ g ðr2=NÞ=g as a simple average of variances of group means.
i¼1 i i¼1 i i
Contrast this with the form of the non-centrality parameter of Levy’s (1978a) F
approximation. The variance heterogeneity directly employed to reflect the weight of
eachofthegroupmeansinΛ makesagreatdifferenceinpowerperformance.
L
It was demonstrated in Levy (1978a) that the actual power of Welch’s test
PfW >Fðg(cid:4)1Þ^m;agcanbewellapproximatedbyp(ΛL).AsnotedinTomarkenandSerlin
(1986), this procedure may prove useful in conducting power analysis for one-way
heteroscedasticANOVA.Moreover,itisofgreatinteresttoextendtheapproachtosample
sizedetermination,justasinthecaseofLuhandGuo(2011)withtheapproximatepower
functionp(Λ ).InspiteofthecomplexityinthedenominatordegreesoffreedomoftheF
LG
distribution, the power approximations in equations (3) and (4) closely resemble the
powerfunctionoftheANOVAFtest.Butthetwonon-centralityparametersΛ andΛ
L LG
differ considerably in their expressions, and thus the resulting behaviours of the two
powerfunctionsarepresumablydivergent.Wenextperformnumericalinvestigationsto
SamplesizedeterminationsforWelch’stestinone-wayheteroscedasticANOVA 77
evaluateandcomparetheaccuracyofthetwoformulasforcomputingsamplesizeunder
variousmodelconfigurationslikelytooccurinpractice.
3. Simulationstudies
In order to enhance the applicability of sample size methodology and the fundamental
usefulnessofWelch’sprocedure,twoMonteCarlosimulationstudieswereconductedto
investigatetheperformanceofthesamplesizecalculationwithrespecttothetwopower
functions described in Levy (1978a) and Luh and Guo (2011). With the approximate
powerformulasgiveninequations (3)and(4),thesamplesizes(N ,…,N )neededto
1 g
attainthespecifiedpower1 (cid:4) bcanbefoundbyasimpleiterativesearchforthechosen
significance level a and parameter values ðl;r2Þ, i = 1, …, g. Accordingly, the non-
i i
centralityparametersΛ andΛ definedin(3)and(4)canberewrittenas
LG L
K ¼N (cid:5)k and K ¼N (cid:5)k ; ð5Þ
LG T LG L T L
n o
P P P
rPespectively, where NT ¼P gi¼1Ni;kLG ¼Pg(cid:5) ig¼1ðli (cid:4)l(cid:3)Þ2= gi¼1ðr2j=qjÞ ;kL ¼
g qfðl (cid:4)l~Þ=rg2, l~ ¼ g ðql=r2Þ= g ðq=r2Þ and q = N/N for i = 1, …,
i¼1 i i i i¼1 i i i i¼1 j j i i T
g.Notethatk andk dependnotonthegroupsizesbutratherontheallocationratio
LG L
amongthegroups,andserveastheeffectsizemeasuresfortheapproximationsinLuhand
Guo(2011)andLevy(1978a),respectively.Astheremaybeseveralpossiblechoicesof
samplesizethatsatisfythechosenpowerlevelintheprocessofsamplesizecalculations,it
is constructive to consider an appropriate design with a priori designated sample size
ratios that leads toauniqueand optimal result. For ease ofillustration,thesample size
ratios (r1P, …, rg) are specified in advance with ri = Ni/N1i = 1, …, g. Note that
q ¼ r= g r,wherer = N/N fori = 1,…,g.Thusthetaskisconfinedtodeciding
i i j¼1 j i i 1
theminimumsamplesizeN (withN = N r,i = 2,…,g)requiredtoachievethedesired
1 i 1 i
powerlevel.
Each of the vital factors of mean pattern, variance characteristic, and sample size
structurehasbeenshowntoaffectthemagnitudeofnon-centralityandpower.Toprovide
asystematicdemonstration,fourpatternsofvariabilityinthemeanswereusedtoassess
powerandcomputesamplesize:(a)minimumvariability(onemeanateachextremeof
therange,andallothermeansatthemidpoint);(b)intermediatevariability(suchasmeans
equally spaced through the range); (c) maximum variability (half of the means at each
extremeoftherange);and(d)extremevariability(onemeanatoneextremeoftherange,
andallothermeansequalandattheotherextreme).Similarmeanconfigurations were
considered in Alexander and Govern (1994), Cohen (1988), De Beuckelaer (1996) and
TomarkenandSerlin(1986).Theempiricalexaminationconsistsoftwostudies,ofwhich
thefirstre-examinestheminimumvariabilitymeanpatternsinLuhandGuo(2011),and
thesecondevaluatestheothercasesofintermediate,maximumandextremevariability
thatwerenotconsideredinLuhandGuo(2011).
3.1. StudyI
3.1.1. Design
Forpurposesofcomparison,wereconsiderthemodelsettingswithg = 4and6inTable 1
ofLuhandGuo(2011)inwhichthemeanvaluesareofminimumvariabilitywithl = {1,0,
0, (cid:4)1} and {1, 0, 0, 0, 0, (cid:4)1}, respectively. The corresponding two variance settings,
78 Show-LiJanandGwowenShieh
representinghomogeneousandheterogeneousstructures,arer2 = {1,1,1,1}and{1,4,
9,16},and{1,1,1,1,1,1}and{1,1,4,4,9,9},respectively.Moreover,thesamplesize
ratio is fixed as the variance ratio r = N/N = r/r for i = 1, …, g. With these
i i 1 i 1
specifications,therequiredsamplesizeswerecomputedforthetwoapproacheswiththe
chosenpowervalueandsignificancelevel.Throughoutthisempiricalinvestigation,the
significancelevelissetata = .05.NotethatthesamplesizesofLuhandGuo’smethodare
calculated with the algorithm presented in Luh and Guo (2011), which involves some
further modification when applying the power function p(Λ ) in equation (3). In
LG
contrast,thesamplesizesforLevy’sprocedurearedeterminedwiththepowerfunctionp
(Λ )inequation (4).Inaddition,theactualorapproximatepowersarecalculatedwiththe
L
resulting sample sizes. The SAS/IML (SAS Institute, 2011) and R (R Development Core
Team,2006)programs employed toperformthesamplesizedeterminationandpower
calculation for Levy’s (1978a) procedure are presented in Appendices A–D. The
computed sample sizes and approximate powers are listed in Tables 1–3 for power
levels.7,.8and.9,respectively.Becauseoftheunderlyingmetricofintegersamplesizes,
thevaluesachievedaremarginallylargerthanthenominallevelforbothprocedures.The
onlytwoexceptionsoccurwiththevariancehomogeneitycasesofcomparativelysmall
samplesizesinTable 1.Thenforbothprocedures,estimatesofthetruepowerassociated
with given sample size and parameter configuration are computed via Monte Carlo
simulation of 10,000 independent data sets. For each replicate, (N , …, N ) normal
1 g
outcomes are generated with the one-way homoscedastic or heteroscedastic ANOVA
model.Next,theteststatisticWiscomputedandthesimulatedpoweristheproportionof
the 10,000 replicates whose test statistics W exceed the corresponding critical value
Fg(cid:4)1;^m;:05. For the procedure examined, the adequacy for power and sample size
calculation is determined by the difference between the simulated power and approx-
imatepowercomputedearlier.Thesimulatedpoweranddifferencearealsosummarized
inTables 1–3forthethreedesignatedpowerlevels.
3.1.2. Results
An inspection of the reported sample sizes in Tables 1–3 reveals that in general the
necessarysamplesizesforLuhandGuo’s(2011)methodarelargerthanthoseforLevy’s
(1978a)approach.ThereisonlyonecaseinTable 3wherethetwosetsofsamplesizesare
identical.Butevenwiththesamesamplesizes,thetwopowerfunctionsp(Λ )andp(Λ )
L LG
still give different approximate power values because of the distinct non-centrality
parameterformulations.Moreimportantly,thediscrepanciesbetweensimulatedpowers
and approximate powers indicate that the performance of Luh and Guo’s method is
noticeably unstable and in several cases disturbing. Specifically, the resulting errors in
Tables 1–3 range from .0131 to .1060. On the other hand, the errors associated with
Levy’s approach in Tables 1–3 clearly show that the approximate power formula of
equation (4)performsextremelywellbecauseallabsoluteerrorsarelessthan.01forthe
12casesexaminedhere.
3.2. StudyII
3.2.1. Design
To show a profound implication of the sample size procedures, further numerical
assessments were performed with different variability patterns in mean structure. By
SamplesizedeterminationsforWelch’stestinone-wayheteroscedasticANOVA 79
ominal erence 0036 0035 0027 0070
nn Diff .(cid:4) .(cid:4) .(cid:4) .(cid:4)
e
h
w
y(1978a) Simulatedpower .7760 .7094 .7725 .7082
v
e
L
d ay e
2011)an Lev proximatpower .7796 .7129 .7752 .7152
( p
o A
u
G
chesofLuhand Samplesizes (7,7,7,7) (10,20,30,40) (8,8,8,8,8,8) (10,10,20,20,30,30) 0,respectively.
a 1
o 0
ppr ce d.1
a n 0 2 7 4 n
e e 6 1 6 7 a
orth Differ .10 .10 .07 .09 333,
f 3
r .
we 0,
simulatedpo Guo Simulatedpower .8432 .8080 .8276 .8106 are.500,.098
ower,and Luhand proximatepower .7372 .7068 .7509 .7132 figurations
p p n
mate A elco
pproxi sizesures 8) 36,48) 9,9,9) 24,24,6) urmod
plesize,a Samplestruct (8,8,8, (12,24, (9,9,9, (12,12,36,3 forthefo
m kL
Computedsa70 variance }0,1(cid:4)}1,1,1}0,1(cid:4)}4,9,16}0,0,0,1(cid:4)}1,1,1,1,1}0,0,0,1(cid:4)}1,4,4,9,9 eeffectsizes
Table1. poweris. Meanand l={1,0,r={21,l={1,0,r={21,l={1,0,={r21,l={1,0,r={21, aTh.Note
80 Show-LiJanandGwowenShieh
ominal erence 0094 0049 0066 0050
nn Diff .(cid:4) .(cid:4) .(cid:4) .(cid:4)
e
h
w
d
8a) atewer 35 86 60 77
7 ulo 4 9 3 0
9 mp 8 7 8 8
1 . . . .
( Si
y
v
11)andLe aLevy proximatepower .8529 .8035 .8426 .8127
0 p
(2 A
o
u
chesofLuhandG Samplesizes (8,8,8,8) (12,24,36,48) (9,9,9,9,9,9) (12,12,24,24,36,36) 0,respectively.
a e 1
theappro Differenc .0583 .0606 .0405 .0814 3,and.10
r 3
o 3
powerf mulatedpower .8997 .8675 .8811 .8957 0980,.3
ed Si 0,.
wer,andsimulat LuhandGuo Approximatepower .8414 .8069 .8406 .8143 urationsare.50
g
o fi
matep ctures 0,10) 3,43) elcon
e,approxi esizesstru 9,9) 8,42,55) 0,10,10,1 5,29,29,4 efourmod
plesiz Sampl (9,9, (14,2 (10,1 (15,1 forth
m kL
Computedsa80 variance }0,1(cid:4)}1,1,1}0,1(cid:4)}4,9,16}0,0,0,1(cid:4)}1,1,1,1,1}0,0,0,1(cid:4)}1,4,4,9,9 eeffectsizes
Table2. poweris. Meanand l={1,0,r={21,l={1,0,={r21,l={1,0,r={21,l={1,0,={r21, a.ThNote
SamplesizedeterminationsforWelch’stestinone-wayheteroscedasticANOVA 81
ominal erence 0071 0010 0026 0063
nn Diff .(cid:4) .(cid:4) .(cid:4) .(cid:4)
e
h
w
978a) ulatedower 975 143 256 006
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