Pairwise Multiple Comparisons: Theory and Computation

107 Pages·2019·1.915 MB·English

by  Taka-aki Shiraishi, Hiroshi Sugiura, Shin-ichi Matsuda

SPRINGER BRIEFS IN STATISTICS
JSS RESEARCH SERIES IN STATISTICS
Taka-aki Shiraishi
Hiroshi Sugiura
Shin-ichi Matsuda
Pairwise Multiple
Comparisons
Theory and
Computation
SpringerBriefs in Statistics
JSS Research Series in Statistics
Editors-in-Chief
Naoto Kunitomo, Graduate School of Economics, Meiji University, Bunkyo-ku,
Tokyo, Japan
Akimichi Takemura, The Center for Data Science Education and Research,
Shiga University, Bunkyo-ku, Tokyo, Japan
Series Editors
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Nakano-ku, Tokyo, Japan
Tomoyuki Higuchi, The Institute of Statistical Mathematics, Tachikawa, Tokyo,
Japan
Toshimitsu Hamasaki, Office of Biostatistics and Data Management, National
Cerebral and Cardiovascular Center, Suita, Osaka, Japan
Shigeyuki Matsui, Graduate School of Medicine, Nagoya University, Nagoya,
Aichi, Japan
Manabu Iwasaki, School of Data Science, Yokohama City University, Yokohama,
Tokyo, Japan
Yasuhiro Omori, Graduate School of Economics, The University of Tokyo,
Bunkyo-ku, Tokyo, Japan
Masafumi Akahira, Institute of Mathematics, University of Tsukuba, Tsukuba,
Ibaraki, Japan
Takahiro Hoshino, Department of Economics, Keio University, Tokyo, Japan
Masanobu Taniguchi, Department of Mathematical Sciences/School,
Waseda University/Science & Engineering, Shinjuku-ku, Japan
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Taka-aki Shiraishi Hiroshi Sugiura
(cid:129) (cid:129)
Shin-ichi Matsuda
Pairwise Multiple
Comparisons
Theory and Computation
123
Taka-aki Shiraishi Hiroshi Sugiura
Faculty of Science andEngineering Faculty of Science andEngineering
Nanzan University Nanzan University
Nagoya,Aichi, Japan Nagoya,Aichi, Japan
Shin-ichi Matsuda
Faculty of Science andEngineering
Nanzan University
Nagoya,Aichi, Japan
ISSN 2191-544X ISSN 2191-5458 (electronic)
SpringerBriefs inStatistics
ISSN 2364-0057 ISSN 2364-0065 (electronic)
JSSResearch Series in Statistics
ISBN978-981-15-0065-7 ISBN978-981-15-0066-4 (eBook)
https://doi.org/10.1007/978-981-15-0066-4
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Preface
Analysis of variance methods are commonly used statistical procedures in
multi-sample models. However,in the analysis of variance, since the homogeneity
of means is tested and the confidence regions of the mean vector are given by an
interiorofellipses,specificcomparisonsofmeansarenotdrawn.Multipletestsand
simultaneous confidence intervals specify differences in means. Therefore, fields
including medicine, pharmacy, biology, and psychology use multiple comparison
proceduresfordataanalyses.Tukey(1953),Miller(1981),HochbergandTamhane
(1987), Hsu (1996), and Bretz et al. (2010) are some technical books on multiple
comparisons. The present book discusses progressive multiple comparisons.
The detailed discussion of this monograph focuses on all-pairwise multiple
comparisonsofmeansinmulti-samplemodels.Closedtestingproceduresbasedon
maximum absolute values of some two-sample t-test statistics and based on F-test
statisticsareintroducedinhomoscedasticmulti-samplemodels.Theresultssuggest
that(i)multistepprocedures aremoreeffectivethan single-step procedures andthe
Ryan–Einot–Gabriel–Welsch(REGW)testsand(ii)confidenceregionsinducedby
multistepproceduresareequivalenttosimultaneous confidenceintervals.Next,we
introduce the multistep test procedure superior to the single-step Games–Howell
procedure in heteroscedastic multi-sample models. Under simple ordered restric-
tions of means, we also discuss closed testing procedures based on maximum
values of two-sample one-sided t-test statistics and based on Bartholomew’s
statistics. Furthermore, we introduce distribution-free procedures. Simulation
studies are performed under the null hypothesis and some alternative hypotheses.
Although single-step multiple comparison procedures are utilized in general, the
closed testing procedures stated in the present book are fairly more powerful than
thesingle-stepprocedures.Inordertoexecutethemultiplecomparisonprocedures,
the upper 100a percentiles of the complicated distributions are required. Classical
integralformulassuchastheSimpson’sruleandtheGaussianrulehavebeenused
for the calculation of the integral transform that appears in statistical calculations.
However, these formulas are not effective for the complicated distribution. As a
numericalcalculation,theauthorsintroducetheSincmethodwhichistheoptimum
in terms of accuracy and computational cost.
v
vi Preface
Shiraishi writes Chaps. 1–4 about multiple comparison procedures. These
chapters are translated into English from Japanese of Shiraishi and Sugiura (2018,
Kyoritsu Shuppan Co., Ltd.). We obtain the permission of Kyoritsu Shuppan Co.,
Ltd. and publish these chapters. Matsuda writes computer simulations for com-
paring the simulated power of multiple comparison tests and statistical analysis of
rawdatainChaps.5and6,respectively.SugiurawritesChap.7aboutcomputation
ofdistributionfunctionsforstatisticsundersimpleorderrestrictions.Thischapteris
translated into English from Japanese of Shiraishi and Sugiura (2015, J. Japan
Statistical Society; Japanese Issue). We obtain the permission of Japan Statistical
Society and publish Chap. 7. Shiraishi writes Chap. 8 as related topics.
Nagoya, Japan Taka-aki Shiraishi
June 2019 Hiroshi Sugiura
Shin-ichi Matsuda
Acknowledgements Theauthorsaregratefultotworefereesforvaluablecomments.Wewould
like to thank Prof. Takemitsu Hasegawa of the University of Fukui for careful reading and
criticisms of the manuscript of Chap. 7. He also made helpful suggestions on improving the
presentationofChap.7.ThisresearchwassupportedinpartbyaGrant-in-AidforCo-operative
Research(C)18K11204and19K11870fromtheJapaneseMinistryofEducation.Wewouldlike
tothankEditage(www.editage.com)forEnglishlanguageediting.
References
BretzF,HothornT,WestfallP(2010)MultiplecomparisonsusingR.Chapman&Hall,London
HochbergY,TamhaneAC(1987)Multiplecomparisonprocedures.Wiley,NewYork
HsuJC(1996)Multiplecomparisons-theoryandmethods.Chapman&Hall,London
MillerRG(1981)Simultaneousstatisticalinference,2ndedn.Springer,Berlin
ShiraishiT,SugiuraH(2015)Theupper100aHthpercentilesofthedistributionsusedinmultiple
comparisonproceduresunderasimpleorderrestriction.JJapanStatSoc.JapaneseIssue44,
271–314(inJapanese)
Shiraishi T, Sugiura H (2018) Theory of multiple comparison procedures and its computation.
Kyoritsu-ShuppanCo.,Ltd.(inJapanese)
TukeyJW(1953)Theproblemofmultiplecomparisons.ThecollectedworksofJohnW.Tukey
(1994),volumeVIII:multiplecomparisons.Chapman&Hall,London
Contents
1 All-Pairwise Comparisons in Homoscedastic
Multi-sample Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The Tukey–Kramer Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Closed Testing Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 Multiple Comparisons in Heteroscedastic Multi-sample Models . . . . 13
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 The Games–Howell Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Closed Testing Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4 Asymptotic Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Multiple Comparison Procedures Under Simple Order
Restrictions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Multiple Comparisons Under Equal Sample Sizes . . . . . . . . . . . . 24
3.3 Closed Testing Procedures Under Unequal Sample Sizes . . . . . . . 32
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4 Nonparametric Procedures Based on Rank Statistics . . . . . . . . . . . . 35
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2 The Single-Step Procedures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.3 Closed Testing Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.4 Multiple Comparisons Under Simple Order Restrictions . . . . . . . . 40
4.4.1 Multiple Comparisons Under Equal Sample Sizes . . . . . . . 40
4.4.2 Multiple Comparisons Under Unequal Sample Sizes . . . . . 42
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
vii
viii Contents
5 Comparison of Simulated Power Among Multiple Comparison
Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.2 Simulation Settings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.4 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
6 Application of Multiple Comparison Tests to Real Data. . . . . . . . . . 49
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.3 Critical Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.4 Application to Real Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
6.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
7 Computation of Distribution Functions for Statistics Under
Simple Order Restrictions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
7.1 Function Family G and Sinc Approximation . . . . . . . . . . . . . . . . 57
7.1.1 Function Family G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
7.1.2 Sinc Approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
7.1.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
7.1.4 Error Analysis for Finite Sinc Approximation . . . . . . . . . . 69
7.1.5 DE Formula (Double Exponential Formula) . . . . . . . . . . . 71
7.2 Computation of Statistic Values of Hayter Type. . . . . . . . . . . . . . 73
7.2.1 Distribution Functions of Hayter Statistic
and Their Nature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
7.2.2 Computation of the Distribution Functions
and the Upper 100aH% Point. . . . . . . . . . . . . . . . . . . . . . 74
7.2.3 Numerical Computation of the Density Function
gðsjmÞ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
7.3 Computation of Level Probability . . . . . . . . . . . . . . . . . . . . . . . . 82
7.3.1 Fundamental Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 83
7.3.2 Computation of the Table Q. . . . . . . . . . . . . . . . . . . . . . . 85
7.3.3 Computation of the Table P. . . . . . . . . . . . . . . . . . . . . . . 88
7.3.4 Computation of Integrals . . . . . . . . . . . . . . . . . . . . . . . . . 89
7.3.5 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 92
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
8 Related Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
8.1 Multiple Comparisons Under Simple Order Restrictions . . . . . . . . 95
8.2 Two-Way Layouts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
8.3 Bernoulli Responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 101
Chapter 1
All-Pairwise Comparisons in
Homoscedastic Multi-sample Models
Abstract We consider multiple comparison procedures among mean effects in
homoscedastick-samplemodels.Weproposeclosedtestingproceduresbasedonthe
maximumvaluesofsometwo-samplet-teststatisticsandbasedon F-teststatistics.
Theresultsrevealthattheproposedproceduresaremorepowerfulthansingle-step
proceduresandtheREGW(Ryan-Einot-Gabriel-Welsch)typetests.
1.1 Introduction
Weconsiderhomoscedastick-samplemodelsundernormality.(X ,...,X )isa
i1 ini
randomsampleofsizen fromtheithnormalpopulationwithmeanμ andvariance
i i
σ2(i =1,...,k),thatis, P(X ≤ x)=(cid:3)((x −μ )/σ),where(cid:3)(x)isastandard
ij i
normal distribution function. Further, X ’s are assumed to be independent. Then
ij (cid:2)
unbiased estimators for μ , overall mean ν = k n μ /n, and σ2, respectively,
aregivenbyμˆi = X¯i·,νˆ =i X¯··,and i=1 i i
VE = 1 (cid:3)k (cid:3)ni (Xij −X¯i·)2, (1.1)
m
i=1 j=1
(cid:2) (cid:2) (cid:2)
where X¯i· :=(1/ni) nji=1Xij, X¯·· :=(1/n) ik=1 nji=1Xij,
(cid:3)k
m :=n−k, andn := n . (1.2)
i
i=1
(cid:2)
Theratio Ft := ik=1ni(X¯i·−X¯··)2/{(k−1)VE}isusedtotestthenullhypothesis
ofnotreatmenteffects,
H : μ =···=μ , (1.3)
0 1 k
asfollows.WerejectH atlevelαifF > Fk−1(α),whereFk−1(α)denotestheupper
0 t m m
100α%pointofF-distributionwithdegreesoffreedom(k−1,m).Forspecifiedi,i(cid:3)
suchthat1≤i <i(cid:3) ≤k,ifweareinterestedintesting
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T.-a.Shiraishietal.,PairwiseMultipleComparisons,
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