1st Edition

Simultaneous Inference in Regression

By Wei Liu Copyright 2010
    292 Pages 89 B/W Illustrations
    by CRC Press

    292 Pages 89 B/W Illustrations
    by CRC Press

    Simultaneous confidence bands enable more intuitive and detailed inference of regression analysis than the standard inferential methods of parameter estimation and hypothesis testing. Simultaneous Inference in Regression provides a thorough overview of the construction methods and applications of simultaneous confidence bands for various inferential purposes. It supplies examples and MATLAB® programs that make it easy to apply the methods to your own data analysis. The MATLAB programs, along with color figures, are available for download on www.personal.soton.ac.uk/wl/mybook.html

    Most of the book focuses on normal-error linear regression models. The author presents simultaneous confidence bands for a simple regression line, a multiple linear regression model, and polynomial regression models. He also uses simultaneous confidence bands to assess part of a multiple linear regression model with the zero function, to compare two regression models, and to evaluate more than two regression models. The final chapter demonstrates the use of simultaneous confidence bands in generalized linear regression models, such as logistic regression models.

    This book shows how to employ simultaneous confidence bands to make useful inferences in regression analysis. The topics discussed can be extended to functions other than parametric regression functions, offering novel opportunities for research beyond linear regression models.

    Introduction to Linear Regression Analysis
    Linear regression models
    Parameter estimation
    Testing hypotheses
    Confidence and prediction intervals

    Confidence Bands for One Simple Regression Model
    Preliminaries
    Hyperbolic bands
    Three-segment bands
    Two-segment bands
    Other confidence bands
    Extensions and restrictions of a confidence band
    Comparison of confidence bands
    Confidence bands for percentiles and tolerance bands
    Bayesian simultaneous credible bands

    Confidence Bands for One Multiple Regression Model
    Hyperbolic bands over the whole space
    Hyperbolic bands over a rectangular region
    Constant width bands over a rectangular region
    Hyperbolic bands over an ellipsoidal region
    Constant-width bands over an ellipsoidal region
    Other confidence bands

    Assessing Part of a Regression Model
    Partial F test approach
    Hyperbolic confidence bands
    Assessing equivalence to the zero function

    Comparison of Two Regression Models
    Partial F test approach
    Hyperbolic bands over the whole space
    Confidence bands over a rectangular region
    Confidence bands over an ellipsoidal region
    Assessing the equivalence of two models

    Comparison of More Than Two Regression Models
    Partial F test approach
    Hyperbolic confidence bands for all contrasts
    Bands for finite contrasts over rectangular region
    Bands for finite contrasts over ellipsoidal region
    Equivalence of more than two models

    Confidence Bands for Polynomial Regression
    Confidence bands for one model
    Confidence bands for part of a polynomial model
    Comparison of two polynomial models
    Comparison of more than two polynomial models

    Confidence Bands for Logistic Regression
    Introduction to logistic regression
    Bands for one model
    Bands for comparing two models
    Bands for comparing more than two models

    Appendix A: Approximation of the Percentile of a Random Variable
    Appendix B: Computation of Projection π
    (t,P,Xr)
    Appendix C: Computation of Projection
    π*(t,W,X2)
    Appendix D: Principle of Intersection-Union Test
    Appendix E: Computation of the K-Functions in Chapter 7

    Bibliography

    Index

    Biography

    Wei Liu is a professor of statistics at the University of Southampton, UK. Dr. Liu has published more than 80 papers in peer-reviewed journals, including Annals of Statistics, Journal of the American Statistical Association, Journal of the Royal Statistical Society, Biometrika, and Biometrics. His research encompasses multiple comparison, simultaneous inference, and sequential methods.

    I like in particular this book since almost all results are given with proofs and that these proofs are easy to understand having some knowledge in linear models. It is especially helpful that the geometric ideas behind the proofs are worked out and presented with many illustrations. Together with the comprehensive review of the published literature and the presentation of some unsolved problems, this book is very valuable for researchers. It is also very recommendable for practitioners since the ideas of the concepts are worked out clearly and many examples and figures are presented where the confidence bands are applied to real data. MATLAB programs for all methods can be downloaded from the author’s website.
    —Christine Müller, Biometrical Journal 53 (2011)

    This book provides a comprehensive discussion of methods for determining simultaneous confidence bands in regression. … The book provides a valuable up-to-date review of work in this area.
    —David J. Hand, International Statistical Review (2011), 79

    … definitely fills a significant niche, providing practitioners with powerful inference tools for parametric regression and stimulating further research in this important area.
    — Tatyana Krivobokova, Georg-August- Univer,citat Gottingen, Royal Statistics Society, February 2012