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Summary
Summary
The authoritative reference on nonparametric methods for evaluating longitudinal data in factorial designs
Broadening the range of techniques that can be used to evaluate longitudinal data, Nonparametric Analysis of Longitudinal Data in Factorial Experiments presents nonparametric methods of evaluation that supplement the generalized linear models approach. Emphasizing the practical application of these methods in statistical procedures, this book provides a unified approach for the analysis of factorial designs involving longitudinal data that is appropriate for metric data, count data, ordered categorical data, and dichotomous data.
Topics covered include nonparametric models, effects and hypotheses in experimental design, estimators for relative effects, experiments for one and several groups of subjects, multifactorial experiments, dependent replications, and experiments with numerous time points. The basic mathematical principles for the methods introduced here are described in theory, consistent with the book's minimal math requirements. Simple approximations for small data sets are provided, as well as ample chapter exercises to test skills, an appendix that includes original data for the examples used throughout the book, and downloadable SAS-IML macros for implementing the more extensive calculations. All applications are designed to be useful in many fields.
Generously supplemented with more than 110 graphs and tables, Nonparametric Analysis of Longitudinal Data in Factorial Experiments is an essential reference for statisticians and biometricians, researchers in clinical trials, psychological studies, and in the fields of forestry, agriculture, sociology, ecology, and biology, as well as graduate students in statistics and biostatistics.
Author Notes
EDGAR BRUNNER is Head of the Department of Medical Statistics at the University of Gottingen, Germany.
SEBASTIAN DOMHOF is Assistant Professor in the Department of Medical Statistics at the University of Gottingen, Germany.
FRANK LANGER is a Statistician at Lilly Deutschland GmbH, Bad Homburg, Germany.
Table of Contents
Preface | p. xi |
Acknowledgments | p. xv |
1 Introduction | p. 1 |
1.1 Motivation | p. 1 |
1.2 Overview of This Book | p. 2 |
1.3 Examples | p. 4 |
1.3.1 Panic Disorder Study I | p. 4 |
1.3.2 Panic Disorder Study II | p. 5 |
1.3.3 Solling Spruce Forest Roof Project | p. 6 |
1.3.4 Water-Maze Test | p. 8 |
1.3.5 [alpha]-Amylase Study | p. 9 |
1.3.6 Cortisol Concentration in Plasma | p. 10 |
1.3.7 Plasma-Renin Activity | p. 12 |
1.3.8 [gamma]-GT Study | p. 13 |
1.3.9 Stem-Cell Concentrate Study | p. 14 |
1.3.10 Shoulder Tip Pain Study | p. 15 |
1.3.11 Body-Weight Evolution of Wistar Rats | p. 17 |
1.3.12 PCT-Concentration Study | p. 17 |
1.3.13 Postoperative Edema | p. 18 |
1.3.14 Removal of Cardioplegical Solution (HTK Study) | p. 19 |
2 Models | p. 21 |
2.1 Nonparametric Models | p. 21 |
2.1.1 Historical Development | p. 21 |
2.1.2 Definition of the Model | p. 22 |
2.1.3 Notation | p. 24 |
2.1.4 Examples and Specific Designs | p. 25 |
2.2 Covariance Structures | p. 32 |
3 Effects and Hypotheses | p. 35 |
3.1 Nonparametric Effects | p. 35 |
3.1.1 Relative Summary Effects | p. 35 |
3.1.2 Relative Marginal Effects | p. 38 |
3.2 Nonparametric Hypotheses | p. 40 |
3.2.1 Hypotheses on the Distribution Functions | p. 40 |
3.2.2 Hypotheses on Relative Effects | p. 42 |
4 Estimators for Relative Effects | p. 45 |
4.1 The (Normalized) Empirical Distribution Function | p. 45 |
4.2 Ranks | p. 47 |
4.3 Estimation of Relative Treatment Effects | p. 52 |
4.4 Asymptotic Distributions of the Estimators | p. 55 |
4.4.1 Relative Marginal Effects | p. 55 |
4.4.2 Relative Summary Effects | p. 58 |
4.5 Confidence Intervals | p. 60 |
4.6 Graphical Representation of the Results | p. 62 |
4.6.1 Metric Data | p. 63 |
4.6.2 Ordered Categorical Data | p. 64 |
Problems | p. 65 |
5 Test Statistics | p. 67 |
5.1 Statistics for Normal Distributions | p. 67 |
5.2 Wald-Type Statistic (WTS) | p. 69 |
5.3 Hotelling's T[superscript 2] Statistic | p. 70 |
5.4 ANOVA-Type Statistic (ATS) | p. 71 |
5.5 Comparison of the Statistics Q[subscript n], Z[subscript R] and F[subscript n] | p. 73 |
5.6 Statistics for Patterned Alternatives | p. 75 |
5.7 Consistency of Statistics | p. 75 |
5.8 Comparison with the Rank Transform (RT) Method | p. 80 |
6 Software | p. 83 |
6.1 Special SAS Macros | p. 83 |
6.2 SAS Standard Procedures | p. 84 |
7 Experiments for One Group of Subjects | p. 87 |
7.1 Two Time Points | p. 87 |
7.1.1 The Hypothesis H[superscript F subscript 0] : F[subscript 1] = F[subscript 2] | p. 88 |
7.1.2 The Wilcoxon Signed Rank Test | p. 90 |
7.1.3 The Nonparametric Behrens-Fisher Situation | p. 92 |
7.1.4 Confidence Intervals | p. 93 |
7.1.5 Application to the Stem-Cell Concentrate Study Data | p. 94 |
7.2 More than Two Time Points | p. 95 |
7.2.1 Models and Hypotheses | p. 95 |
7.2.2 Global Alternatives | p. 98 |
7.2.3 Confidence Intervals | p. 98 |
7.2.4 Examples | p. 99 |
7.2.5 Patterned Alternatives | p. 102 |
7.2.6 Application to the Data from the Panic Disorder Study I | p. 103 |
7.2.7 Application to the Data from the [alpha]-Amylase Study | p. 104 |
7.2.8 Missing Values | p. 105 |
7.2.9 Singular Covariance Matrices | p. 107 |
7.3 Software | p. 110 |
7.3.1 Instructions on the Use of the Macro LD_F1 | p. 110 |
7.3.2 Instructions on the Use of the Macro LD_CI | p. 113 |
7.3.3 SAS Standard Procedures | p. 114 |
Problems | p. 117 |
8 Experiments for Several Groups of Subjects | p. 119 |
8.1 Two Groups and Two Time Points | p. 120 |
8.1.1 Nonparametric Effects and Hypotheses | p. 120 |
8.1.2 Statistics | p. 121 |
8.1.3 Application to the Stem-Cell Concentrate Data | p. 124 |
8.2 The Two-Period Cross-Over Design | p. 128 |
8.3 Several Groups and Several Time Points | p. 130 |
8.3.1 Technical Formulation of Hypotheses | p. 131 |
8.3.2 Interpretation of the Nonparametric Hypotheses | p. 133 |
8.3.3 Group Effects | p. 134 |
8.3.4 Time Effects | p. 137 |
8.3.5 Interactions | p. 140 |
8.3.6 Missing Values and Singular Covariance Matrices | p. 143 |
8.3.7 Examples | p. 144 |
8.4 Methods for Summary Effects | p. 152 |
8.4.1 Effects, Hypotheses and Statistics | p. 152 |
8.4.2 Application to the PRA Study | p. 153 |
8.5 Software | p. 157 |
8.5.1 Instructions on the Use of the Macro F1_LD_F1 | p. 157 |
8.5.2 Instructions on the Use of the Macro LD_CI | p. 165 |
8.5.3 Instruction on the Use of the Macro OWL | p. 166 |
8.5.4 SAS Standard Procedures | p. 168 |
Problems | p. 171 |
9 Dependent Replications | p. 175 |
9.1 Models | p. 175 |
9.2 Examples | p. 178 |
9.3 Software | p. 183 |
9.3.1 SAS Macros | p. 183 |
9.3.2 SAS Standard Procedures | p. 184 |
Problems | p. 184 |
10 Multifactorial Experiments | p. 187 |
10.1 Introduction | p. 187 |
10.2 Examples | p. 188 |
10.2.1 F2-LD-F1 | p. 188 |
10.2.2 LD-F2 | p. 188 |
10.2.3 F1-LD-F2 | p. 189 |
10.3 General Techniques | p. 189 |
10.4 Analysis of Some Examples | p. 190 |
10.5 Macros for Multifactorial Experiments | p. 195 |
10.5.1 Instructions on the Use of the Macro F2_LD_F1 | p. 195 |
10.5.2 Instructions on the Use of the Macro LD_F2 | p. 199 |
10.5.3 Instructions on the Use of the Macro F1_LD_F2 | p. 202 |
10.5.4 Instructions on the Use of the Macro LD_CI | p. 206 |
10.6 SAS Standard Procedures | p. 206 |
10.6.1 F2-LD-F1 | p. 207 |
10.6.2 LD-F2 | p. 208 |
Problems | p. 210 |
11 Numerous Time Points | p. 211 |
11.1 General Discussion | p. 211 |
11.2 Reduction of the Design | p. 212 |
11.3 Examples | p. 213 |
11.3.1 Body-Weight Evolution of Wistar Rats | p. 213 |
11.3.2 Cortisol Concentration in Plasma | p. 215 |
Problems | p. 217 |
References | p. 219 |
Appendix A Original Data | p. 225 |
A.1 Panic Disorder Study I | p. 225 |
A.2 Panic Disorder Study II | p. 226 |
A.3 SO[subscript 4]-Concentration | p. 227 |
A.4 Vitality of Treetops | p. 230 |
A.5 Water-Maze Test | p. 231 |
A.6 [alpha]-Amylase Study | p. 232 |
A.7 Cortisol Concentration in Plasma | p. 233 |
A.8 Plasma-Renin Activity in Serum | p. 236 |
A.9 [gamma]-GT Study | p. 238 |
A.10 Stem-Cell Concentrate Study | p. 239 |
A.11 Shoulder Tip Pain Study | p. 241 |
A.12 Body-Weight Evolution of Wistar Rats | p. 242 |
A.13 Postoperative Edema | p. 244 |
A.14 PCT-Concentration Study | p. 246 |
A.15 HTK Study | p. 248 |
Appendix B Results from the SAS Macros | p. 249 |
B.1 Confidence-Intervals for the [alpha]-Amylase Study | p. 249 |
B.2 Confidence-Intervals for the Shoulder Tip Pain Study | p. 250 |
B.3 Confidence-Intervals for the Plasma-Renin Study | p. 251 |
Glossary | p. 253 |
Index | p. 257 |