Testing Statistical Hypotheses of Equivalence
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Summary
Equivalence testing has grown significantly in importance over the last two decades, especially as its relevance to a variety of applications has become understood. Yet published work on the general methodology remains scattered in specialists' journals, and for the most part, it focuses on the relatively narrow topic of bioequivalence assessment.
With a far broader perspective, Testing Statistical Hypotheses of Equivalence provides the first comprehensive treatment of statistical equivalence testing. The author addresses a spectrum of specific, two-sided equivalence testing problems, from the one-sample problem with normally distributed observations of fixed known variance to problems involving several samples and multivariate data. The treatment includes a concise review of basic mathematical results on optimal tests for equivalence, and the author makes available on the Internet a collection of computer programs that allow easy implementation of the methods presented.
In a field as complex and rich in potential applications as equivalence testing, Testing Statistical Hypotheses of Equivalence stands alone as a coherent reference that furnishes both the theoretical and practical tools needed for dealing with equivalence trials of any complexity and in any phase.
Table of Contents
INTRODUCTION
Demonstration of Equivalence as a Basic Problem of Applied Statistics
Formalization of the Statistical Notion of Equivalence
Major Fields of Application of Equivalence Tests
Choosing the Main Distributional Parameter
Numerical Specification of the Limits of Equivalence
METHODS FOR ONE-SIDED EQUIVALENCE PROBLEMS
Objective of a One-Sided Equivalence Trial and Formulation of Hypotheses
Standard Solution for Location Parameter Families
Exact Two-Sample Tests for Binomial Distributions
GENERAL APPROACHES TO THE CONSTRUCTION OF TESTS FOR EQUIVALENCE IN THE STRICT SENSE
The Principle of Confidence Interval Inclusion
Bayesian Tests for Equivalence
The Classical Approach to Deriving Optimal Parametric Tests for Equivalence Hypotheses
Construction of Asymptotic Tests for Equivalence
EQUIVALENCE TESTS FOR SELECTED ONE-PARAMETER PROBLEMS
The One-Sample Problem with Normally Distributed Observations of Known Variance
Test for Equivalence of a Hazard Rate to Some Given Reference Value with Exponentially Distributed Survival Times
Testing for Equivalence of a Single Binomial Proportion to a Fixed Reference Success Probability
Confidence-Interval Inclusion Rules as Asymptotically UMP Tests for Equivalence
EQUIVALENCE TESTS FOR DESIGNS WITH PAIRED OBSERVATIONS
Sign Test for Equivalence
Equivalence Tests for the McNemar Setting
Paired t-Test for Equivalence
Signed Rank Test for Equivalence
A Generalization of the Signed Rank Test for Equivalence for Noncontinuous Data
EQUIVALENCE TESTS FOR TWO UNRELATED SAMPLES
Two-Sample t-Test for Equivalence
Mann-Whitney Test for Equivalence
A distribution-Free Two-Sample Equivalence Test Allowing for Arbitrary Patterns of Ties
Testing for Dispersion Equivalence of Two Gaussian Distributions
Equivalence Tests for Two Binomial Samples
Log-Rank Test for Equivalence of Two Survivor Functions
MULTI-SAMPLE TESTS FOR EQUIVALENCE
The Intersection-Union Principle as a General Solution to Multi-Sample Equivalence Problems
F-Test for Equivalence of k Normal Distributions
Modified Studentized Range Test for Equivalence
Testing for Dispersion Equivalence of More than Two Gaussian Distributions
A Nonparametric k-Sample Test for Equivalence
TESTS FOR ESTABLISHING GOODNESS OF FIT
Testing for Equivalence of a Single Multinomial Distribution with a
Fully Specified Reference Distribution
Testing for Approximate Collapsibility of Multiway Contingency Tables
Establishing Goodness of Fit of Linear Models for Normally Distributed Data
THE ASSESSMENT OF BIO-EQUIVALENCE
Introduction
Methods of Testing for Average Bioequivalence
Individual Bioequivalence: Criteria and Testing Procedures
Approaches to Defining and Establishing Population Bioequivalence
Bioequivalence Assessment as a Problem of Comparing Bivariate Distributions
APPENDIX
REFERENCES
INDEX
Editorial Reviews
"The main value of the book is in its rather comprehensive and explicit treatment of various tests for equivalence problems. The book is very carefully written and mathematically correct … applied (bio-)statisticians may find this book a helpful manual for various testing procedures in the field of equivalence testing. These procedures are very carefully described and are accompanied by SAS of Fortran code. More mathematically oriented readers may regard this book as a good source of examples, and the thorough discussion of relevant practical questions in the field might stimulate them to doing further research in this area."
—Journal of the American Statistical Association, March 2004
"The book is well organized for the applied statistician with an interest in the theoretical background for statistical procedures. What is especially useful about this book for the applied statistician is that the author weaves into the presentation of the equivalence tests, discussions of power, and sample size as well as simulation results that evaluate the small sample properties of the tests. In addition, a location on the World Wide Web where one can find SAS programs to facilitate the implementation of these tests is provided … Wellek has provided a text on equivalence testing that the applied statistician will find useful both as a theory underlying equivalence testing as well as a source of equivalence tests and tools for their implementation for a wide variety of situations."
—Journal of Biopharmaceutical Statistics
"… a thorough and wide-ranging exposition of statistical methods for the testing of hypotheses of equivalence … The content of the book and the presentation of the material are praiseworthy for several reasons. Throughout the book, the focus is on optimal tests … . Where exact methods are not available, asymptotically equivalent tests are presented. The small sample behaviour, both size and power, of asymptotic tests is thoroughly studied through simulation. This must have been an enormous amount of work, but the effort certainly was not in vain; one is convinced that one can apply the methods presented with confidence. Even more deserving of praise is the fact that computer programs (mostly in SAS language) implementing various tests and power calculations are provided … In summary, this is an excellent book and strongly recommended. It is required reading for all biostatisticians and developers of statistical software who work in this arena. It will be useful when teaching statistical test theory and in practice. …"
—Robert Schall, Clinical Trials, 2004
"The publication of a book specifically dealing with the statistical issues and methods surrounding the assessment of equivalence is timely … the primary objective of this book is stated to be: ‘a systematic and comprehensive account of testing procedures for the alternative hypothesis’. It achieves this well."
—Current Medical Research and Opinion, 2004
"… this book provides the reader with the opportunity to do equivalence testing for various statistical problems. The author has helpfully provided a collection of computer programs (in SAS and Fortran) that allow for easy implementation of the methods presented … I was left feeling that this book provides a good grounding in statistical techniques to do hypothesis testing in equivalence trials."
—Pharmaceutical Statistics, 2003
"Wellek has done a commendable service to the academic community by providing the first comprehensive treatment of testing hypotheses of equivalence, since most of the work in this area so far is confined to research journals. … Overall, this excellent book has great potential for applications and should interest students and researchers alike. Summing Up: Highly recommended."
—D.V. Chopra, CHOICE
"… this book has explained well with examples and case studies the topics … Researchers will find this book thought-provoking."
—Journal of Statistical Computation & Simulation
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