Table Of ContentSpringerBriefs in Statistics
Tejas A. Desai
Important Applications
of the Behrens-Fisher
Statistic and the False
Discovery Rate
SpringerBriefs in Statistics
SpringerBriefs present concise summaries of cutting-edge research and practical
applications across a wide spectrum of fields. Featuring compact volumes of 50
to 125 pages, the series covers a range of content from professional to academic.
Typicaltopicsmightinclude:
• Atimelyreportofstate-of-theartanalyticaltechniques
• A bridge between new research results, as published in journal articles, and a
contextualliteraturereview
• Asnapshotofahotoremergingtopic
• Anin-depthcasestudyorclinicalexample
• Apresentationofcoreconceptsthatstudentsmustunderstandinordertomake
independentcontributions
SpringerBriefs in Statistics showcase emerging theory, empirical research, and
practicalapplicationinStatisticsfromaglobalauthorcommunity.
SpringerBriefs are characterized by fast, global electronic dissemination, stan-
dard publishing contracts, standardized manuscript preparation and formatting
guidelines,andexpeditedproductionschedules.
Moreinformationaboutthisseriesathttps://link.springer.com/bookseries/8921
Tejas A. Desai
Important Applications of the
Behrens-Fisher Statistic and
the False Discovery Rate
TejasA.Desai
Thaltej,Gujarat,India
ISSN2191-544X ISSN2191-5458 (electronic)
SpringerBriefsinStatistics
ISBN978-3-030-99887-5 ISBN978-3-030-99888-2 (eBook)
https://doi.org/10.1007/978-3-030-99888-2
©TheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerlandAG2022
Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether
thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuse
ofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,and
transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar
ordissimilarmethodologynowknownorhereafterdeveloped.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook
arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor
theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany
errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional
claimsinpublishedmapsandinstitutionalaffiliations.
ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
Contents
1 TheBehrens-FisherStatistic................................................. 1
References...................................................................... 2
2 One-Way ANOVA with Possible Heteroscedasticity of
VariancesandNon-NormalityofErrors ................................... 3
2.1 TheOne-WayANOVAProblemandtheBehrens-FisherStatistic .... 3
2.2 OurMethodologyforOne-WayANOVAWhenk ≥3 ................ 6
2.3 TheClassicalANOVAFTest ........................................... 6
2.4 TheKruskal-WallisTest................................................. 7
2.5 ComparativeAnalysesforOne-WayANOVA.......................... 7
References...................................................................... 13
3 One-WayMANOVAwithPossibleHeteroscedasticityand
Non-NormalityofErrors..................................................... 15
3.1 TheOne-WayMANOVAProblemandOurMethodology
ForIt...................................................................... 15
3.2 TheClassicalWilks’Lambda ........................................... 16
3.3 ARobustStatisticforOne-WayMANOVA ............................ 16
3.4 ComparativeAnalysesforOne-WayMANOVA ....................... 18
3.5 DataAnalysisExample:TheOsloTransectData ...................... 37
References...................................................................... 49
4 StatisticalAnalysisofs×rTablesWhentheColumnVariable
IsOrdinal...................................................................... 51
4.1 MethodologyforaSingle2×r Table.................................. 51
4.2 SimulationsforaSingle2×r Table ................................... 52
4.3 ExampleofaBehrens-FisherAnalysisofaSingle2×r Table....... 56
4.4 Setsof2×r Tables ..................................................... 58
4.4.1 Methodology..................................................... 58
4.4.2 SimulationsofSetsof2×r Tables............................ 60
4.4.3 ExampleofanAnalysisofaSetofTwo2×r Tables......... 65
v
vi Contents
4.5 Behrens-Fisher Analysis of s × r Tables, s ≥ 3 and
ColumnVariableOrdinal................................................ 68
4.5.1 ASingles×r TableWhens≥3andColumn
VariableOrdinal ................................................. 68
4.5.2 SimulationsofaSingleSetofs×r Table,s>2
andColumnVariableOrdinal................................... 69
4.5.3 SimulationsofMoreThanOnes×r Table,s>2,
andColumnVariableOrdinal................................... 73
4.5.4 ExampleofanAnalysisofa3×3 TableWhenthe
ColumnVariableIsOrdinal..................................... 79
References...................................................................... 82
5 ConcludingRemarks ......................................................... 83
Chapter 1
The Behrens-Fisher Statistic
Abstract Inthisbriefchapter,weintroducetheBehrens-Fisherstatisticfortesting
equalityofmeans,anddemonstratethatitisasymptoticallystandardnormal.
Inmanyappliedsettings,oneisinterestedincomparingmeansoftwodistributions.
TheclassicalANOVAFtestforcomparingtwomeansassumesnormalerrorsand
homoscedasticity. The nullhypothesis of nonparametric testssuch astheKruskal-
Wallis test assumes that the two distributions are the same, whereas in reality two
distritbutionscanhavethesamemeanbutdifferentscaleparameters.Let(cid:2)μ and(cid:2)μ
1 2
betheunbiasedorasymptoticallyunbiasedsamplemeansofthetwodistributions,
respectively. Let(cid:2)s2 and(cid:2)s2 be the respective unbiased or asymptotically unbiased
1 2
estimatesofvariancesofthetwodistributions.WethenproposetheBehrens-Fisher
statistic(Behrens,1929;Fisher,1935)forthissituation:
(cid:2)μ −(cid:2)μ
BF = (cid:3)1 2 (1.1)
(cid:2)s2 (cid:2)s2
1 + 2
n n
1 2
NotethatasaresultoftheCentralLimitTheorem,
(cid:2)μ −μ (cid:2)μ −μ
1(cid:3) 1 −D→N(0,1)asn −→∞and 2(cid:3) 2 −D→N(0,1)asn −→∞
1 2
(cid:2)s2 (cid:2)s2
1 2
n n
1 2
©TheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerlandAG2022 1
T.A.Desai,ImportantApplicationsoftheBehrens-FisherStatisticandtheFalse
DiscoveryRate,SpringerBriefsinStatistics,
https://doi.org/10.1007/978-3-030-99888-2_1
2 1 TheBehrens-FisherStatistic
Therefore,
(cid:4) (cid:5) (cid:4) (cid:5)
(cid:2)s2 D (cid:2)s2 D
((cid:2)μ1−μ1)−N 0,n1 −→0asn1−→∞and((cid:2)μ2−μ2)−N 0,n2 −→0asn2−→∞
1 2
Therefore,
(cid:4) (cid:5)
(cid:2)s2 (cid:2)s2
D
((cid:2)μ −μ )−((cid:2)μ −μ )−N 0, 1 + 2 −→0asn −→∞andn −→∞
1 1 2 2 n n 1 2
1 2
Therefore,
((cid:2)μ −(cid:2)μ )−(μ −μ )
1 (cid:3)2 1 2 −D→N(0,1)asn −→∞andn −→∞
1 2
(cid:2)s2 (cid:2)s2
1 + 2
n n
1 2
Underthenullhypothesisthatμ =μ ,
1 2
(cid:2)μ −(cid:2)μ
(cid:3)1 2 −D→N(0,1)asn −→∞andn −→∞
1 2
(cid:2)s2 (cid:2)s2
1 + 2
n n
1 2
Wehaveshownabovethatunderthenullhypothesisofnodifferenceinmeans,
theBehrens-Fisherstatisticisasymptoticallystandardnormal.Notethatnowherein
theaboveargumentdidweassumethattheunderlyingresponsevariableisnormalor
continuous.Infact,oneorbothoftheresponsevariablescanbeordinallydiscrete.
Neither did we assume homoscedasticity. Thus, the Behrens-Fisher statistic is a
truly nonparametric statistic for detecting differences between two means. Also,
the mathematical form of the Behrens-Fisher statistic is very intuitive and easy to
understand.
References
Behrens, W.U.: Ein Beitrag zur Fehlerberechnung bei wenigen Beobachtungen. Land-
wirtschaftlicheJahrbücher68,807–837(1929).Berlin:WiegandtandHempel
Fisher,R.A.:Thefiducialargumentinstatisticalinference.Ann.Eugenics8(4),391–398(1935)
Chapter 2
One-Way ANOVA with Possible
Heteroscedasticity of Variances and
Non-Normality of Errors
Abstract Inthischapter,westartoutbypositingtheBehrens-Fisherstatisticasthe
statisticofchoicefortestingequalityoftwomeans.After,numericallyshowingthat
theCLTappliestothisstatistic,weconsidertheproblemoftestingequalityofmore
thantwomeans.TheFDR(thefalsediscoveryrate)isusedhereforthatpurposefor
adjusting the p-values yielded by all possible pairwise comparisons. Type I errors
andpowerfordifferentnullhypothesesandalternativesusingtheFDR,theclassical
ANOVAFstatisticandalsotheKruskal-Wallis statisticareshown.
2.1 The One-Way ANOVAProblem andthe Behrens-Fisher
Statistic
Theone-wayANOVAproblemistheproblemoftestingtheequalityofkmeansof
kunivariatedistributionsSothenullhypothesisis
H :μ =μ =...=μ
o 1 2 k
versusthealternative
Ha :μ1,...,μk arenotallequal
First,weconsiderthecasek =2.Let
T(cid:2) =(cid:2)μ −(cid:2)μ
n 1,n 2,n
where (cid:2)μ1,n and (cid:2)μ2,n are the two sample means calculated from two independent
univariatesamplesofsizen.Then
(cid:2)σ2 (cid:2)σ2
V(cid:2)(T(cid:2))= 1,n + 2,n
n
n n
©TheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerlandAG2022 3
T.A.Desai,ImportantApplicationsoftheBehrens-FisherStatisticandtheFalse
DiscoveryRate,SpringerBriefsinStatistics,
https://doi.org/10.1007/978-3-030-99888-2_2
