Multiple Comparisons Theory and Methods

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Multiple Comparisons Theory and Methods
Preface xi
1 Introduction to simultaneous statistical inference 1
1.1 The one-way model 1
1.1.1 Example: treatment x age combination 2
1.2 Modeling 4
1.3 Simultaneous confidence intervals 7
1.3.1 The Studentized maximum modulus method 7
1.3.2 Example: treatment x age combination (continued) 9
1.3.3 A product inequality method 10
1.3.4 Example: treatment x age combination (continued) 12
1.3.5 The Bonferroni inequality method 13
1.3.6 Example: treatment x age combination (continued) 13
1.3.7 Scheffe's method 14
1.3.8 Example: treatment x age combination (continued) 16
1.4 Simultaneous testing 16
1.4.1 The Shaffer-Holm stepwise testing method 18
1.5 Unequal variances 20
1.6 Nonparametric methods 21
1.7 Deduced inference versus direct inference 21
1.8 Exercises 22
2 Classification of multiple comparison methods 24
2.1 Types of multiple comparisons inference 24
2.1.1 Direct inference versus deduced inference 26
2.2 Strength of multiple comparisons inference 27
2.2.1 Confidence intervals methods 28
2.2.2 Confident directions methods 28
2.2.3 Confident inequalities methods 29
2.2.4 Tests of homogeneity 29
2.2.5 Individual comparison methods 30
2.3 Inferential tasks of multiple comparison methods 33
2.3.1 Example: effect of smoking on pulmonary health 33
2.3.2 Example: Bovine growth hormone toxicity studies
2.4 Choosing a multiple comparison method
2.5 Exercises
Contents
36
39
3 Multiple comparisons with a control 43
3.1 One-sided multiple comparisons with a control 43
3.1.1 Balanced one-way model 43
3.1.1.1 Dunnett's method 44
3.1.1.2 The stepdown method of Naik, Marcus,
Peritz and Gabriel 45
3.1.1.3 Dunnett and Tamhane's stepup method 50
3.1.2 Example: smoking and pulmonary health (continued) 52
3.1.3 Unbalanced one-way model 54
3.1.3.1 Dunnett's method 55
3.1.3.2 The Miller-Winer method 56
3.1.3.3 Methods based on probabilistic inequalities 57
3.1.4 Example: smoking and pulmonary health (continued) 59
3.2 Two-sided multiple comparisons with a control 60
3.2.1 Balanced one-way model 60
3.2.1.1 Dunnett's method 60
3.2.2 Example: bovine growth hormone safety (continued) 61
3.2.3 Unbalanced one-way model 63
3.2.3.1 Dunnett's method 63
3.2.3.2 The Miller-Winer method 64
3.2.3.3 Methods based on probabilistic inequalities 65
3.2.4 Example: human food safety of recombinant insulinlike
growth factor-! (riGF-1) 66
3.3 Nonparametric methods 67
3.3.1 Pairwise ranking methods 69
3.3.2 Joint ranking methods 73
3.4 Other approaches to stepwise testing 75
3.5 Appendix: detailed proofs 76
3.5.1 Proof of the validity of the stepdown method 76
3.5.2 Proof of the validity of the stepup method 78
3.6 Exercises 79
4 Multiple comparisons with the best 81
4.1 Constrained multiple comparison with the best 82
4.1.1 Balanced one-way model 83
4.1.2 Example: insect traps 86
4.1.3 Comparison with the sample best 87
4.1.4 Unbalanced one-way model 89
4.1.5 Example: SAT scores 94
4.1.6 Connection with bioequivalence 95
4.1.7 Connection with ranking and selection
4.2 Unconstrained multiple comparisons with the best 103
4.2.1 Balanced One-way Model 103
4.2.1.1 Deduction from all-pairwise comparisons 103
4.2.1.2 The Edwards-Hsu method 105
4.2.2 Example: insect traps (continued) 107
4.2.3 Comparison with the sample best 108
4.2.4 Unbalanced one-way model 109
4.2.4.1 Deduction from all-pairwise comparisons 110
4.2.4.2 The Edwards-Hsu method 111
4.2.5 Example: SAT scores (continued) 113
4.3 Nonparametric methods 115
4.4 Exercises 116
5 All-pairwise comparisons 119
5.1 Balanced one-way model 119
5.1.1 Tukey's method 119
5.1.2 Bofinger's confident directions method 121
5.1.3 Hayter's one-sided comparisons 123
5.1.4 Multiple range tests 125
5.1.4.1 Newman-Keuls multiple range test 127
5.1.4.2 The Methods of Ryan, Einot and Gabriel,
and Welsch 128
5.1.4.3 Duncan's multiple range test 129
5.1.5 Scheffe's method 130
5.1.6 A general stepdown testing scheme, with application
to multiple F-tests 132
5.1.7 The closed testing method, with applications to
closed multiple range tests 137
5.1.8 Fisher's least significant difference methods 139
5.1.9 Example: insect traps 140
5.2 Unbalanced one-way model 144
5.2.1 The Tukey-Kramer method 145
5.2.1.1 Graphical representation of the Tukey-
Kramer method 146
5.2.2 Hayter's method 153
5.2.3 Scheffe's method 154
5.2.4 Stepdown tests 157
5.2.5 The Miller-Winer method 158
5.2.6 Fisher's least significant difference methods 158
5.2.7 Methods based on probabilistic inequalities 159
5.2.8 Example: SAT scores 160
5.3 Nonparametric methods 165
5.3.1 Pairwise ranking methods 165
5.3.2 Joint ranking methods
6 Abuses and misconceptions in multiple comparisons 175
6.1 Not adjusting for multiplicity 175
6.2 Inflation of strength of inference 175
6.3 Conditional inference 177
6.4 Post hoc comparisons 179
6.5 Recommendations 180
7 Multiple comparisons in the general linear model 181
7.1 Models with one-way structure 185
7.1.1 Variance-balanced designs 187
7.1.2 Orthogonal designs 188
7.1.2.1 Two-way model with proportional cell frequencies
188
7.1.2.2 One-way ANCOVA model with different
slopes 191
7.1.2.3 Example: Pooling batches in drug stability
studies 192
7.2 Multiple comparisons with a control 195
7.2.1 Deterministic approximation methods 197
7.2.1.1 Methods based on probabilistic inequalities 197
7.2.1.2 The factor analytic method 200
7.2.1.3 The linear programming method 204
7.2.2 Stochastic approximation methods 209
7.2.2.1 The method of Edwards and Berry 209
7.2.2.2 A variance reduction method 210
7.2.2.3 Example: blood pressure in dogs 210
7.3 Multiple comparisons with the best 213
7.4 All-pairwise comparisons 215
7.4.1 Methods based on probabilistic inequalities 216
7.4.2 Stochastic approximation methods 218
7.5 Scheffe's method for all contrasts 218
7.6 Nonparametric methods 220
7.6.1 Pairwise ranking methods based on signed ranks 220
7.6.2 Joint ranking methods based on Friedman rank sums 222
7.7 Two-way mixed models 223
7.8 One-way repeated measurement models 225
7.9 Exercises 226
A Some useful probabilistic inequalities 227
A.1 An inequality for conditionally independent random variabies
227
A.2 The Bonferroni inequality 228
A.3 Slepian's inequality 228
A.4 Sidak's inequality 229
A.5 The Hunter-Worsley Inequality 229
B Some useful geometric lemmas 231
B.1 Projecting spheres 231
B.2 Projecting rectangles 233
B.3 Deriving confidence sets by pivoting tests 233
C Sample size computations 236
C.1 Sample size computation for Tukey's MCA method 237
C.2 Sample size computation for MCB 238
C.3 Sample size computation for Dunnett's MCC method 240
C.4 An Example 240
D Accessing computer codes 243
D .1 Online access to codes 243
D.2 Critical value computations 243
D.3 Sample size computations 244
E Tables of critical values 245
Bibliography 258
Author index 271
Subject index 27

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