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Use R! Series Editors: Robert Gentleman Kurt Hornik Giovanni Parmigiani

Use R! Albert: Bayesian Computation with R Bivand/Pebesma/Gómez-Rubio: Applied Spatial Data Analysis with R Cook/Swayne: Interactive and Dynamic Graphics for Data Analysis: With R and GGobi Hahne/Huber/Gentleman/Falcon: Bioconductor Case Studies Paradis: Analysis of Phylogenetics and Evolution with R Pfaff: Analysis of Integrated and Cointegrated Time Series with R Ritz/Streibig: Nonlinear Regression with R Sarkar: Lattice: Multivariate Data Visualization with R Spector: Data Manipulation with R

Christian Ritz Jens Carl Streibig Nonlinear Regression with R 123

Jens Carl Streibig Department of Agriculture and Ecology (Crop Science) Faculty of Life Sciences University of Copenhagen Hoejbakkegaard Allé 13 DK-2630 Taastrup Denmark jcs@life.ku.dk Kurt Hornik Department für Statistik und Mathematik Wirtschaftsuniversität Wien Augasse 2-6 A-1090 Wien Austria Christian Ritz Department of Basic Sciences and Environment (Statistics) Faculty of Life Sciences University of Copenhagen Thorvaldsensvej 40 DK-1871 Frederiksberg C Denmark ritz@life.ku.dk Series Editors: Robert Gentleman Program in Computational Biology Division of Public Health Sciences Fred Hutchinson Cancer Research Center 1100 Fairview Ave. N, M2-B876 Seattle, Washington 98109-1024 USA Giovanni Parmigiani The Sidney Kimmel Comprehensive Cancer Center at Johns Hopkins University 550 North Broadway Baltimore, MD 21205-2011 USA ISBN: 978-0-387-09615-5 DOI: 10.1007/978-0-387-09616-2 e-ISBN: 978-0-387-09616-2 2008 Springer Science+Business Media, LLC Library of Congress Control Number: 2008938643 c All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identiﬁed as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper springer.com

To Ydun Marie Ritz and in memory of Erik Ritz

Preface This book is about nonlinear regression analysis with R, in particular, how to use the function nls() and related functions and methods. Range of the book Nonlinear regression may be a conﬁned and narrow topic within statistics. However, the use of nonlinear regression is seen in many applied sciences, ranging from biology to engineering to medicine and pharmacology. Therefore, this book covers a wide range of areas in the examples used. Appendix A lists the disciplines from which data are used in this book. What not to expect This book is not a textbook on nonlinear regression. Basic concepts will be brieﬂy introduced, but the reader in need of more explanations will have to consult comprehensive books on nonlinear regression such as Bates and Watts (1988) or Seber and Wild (1989). Instead, this book may be particularly well- suited as an accompanying text, explaining in detail how to carry out nonlinear regression with R. However, we also believe that the book is useful as a stand- alone, self-study text for the experimenter or researcher who already has some experience with R and at the same time is familiar with linear regression and related basic statistical concepts. Prerequisites Experience with R at a level corresponding to the ﬁrst few chapters in Dal- gaard (2002) should be suﬃcient: The user should be acquainted with the basic objects in R such as vectors, data frames, and lists, as well as basic plotting and statistics functions for making scatter plots, calculating descrip- tive statistics, and doing linear regression.

VIII Preface How to read the book Chapter 2 is essential for getting started on using nls(). Section 3.2 and Chapter 4 are at times somewhat technical regarding the use of R, whereas Section 6.3 is technical on a statistical level. These parts of the book could be skipped on a ﬁrst reading. The R extension package nlrwr is support software for this book, and it is available at CRAN: http://cran.r-project.org/web/packages/nlrwr/index.html. All datasets and functions used are available upon loading nlrwr. All code snippets used in the book are also found in the scripts folder that comes with the package. This means that all R code snippets shown in this book can be run once the support package nlrwr has been installed and loaded. Appendix A provides a list of all datasets used, with a reference to the package where they are found, and Appendix C lists the main functions used in this book together with a package reference. Acknowledgments We would like to thank Claire della Vedova and Christian Pipper for proof- reading parts of the book. The ﬁrst author also wishes to thank the partici- pants of the short course Non-linear regression with R, held at the Faculty of Life Sciences, University of Copenhagen, in September 2007, for useful com- ments and suggestions. We are also grateful to Spencer Graves for his valuable comments on an almost ﬁnal version of the book. All remaining breaches or errors rest solely on the authors. This volume has beneﬁtted vastly from the many comments and sugges- tions from the anonymous reviewers of earlier versions. We are also thankful to John Kimmel for his encouragement and guidance throughout this book project. Finally, we would like to thank the R Core Development Team for making all this happen by developing a great open source project. The book has been written using LATEX 2εand Sweave, yet another powerful invention in the wake of the R project. Copenhagen July 2008 Christian Ritz Jens C. Streibig

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 A stock-recruitment model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Competition between plant biotypes . . . . . . . . . . . . . . . . . . . . . . . 1.3 Grouped dose-response data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 3 4 2 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Getting started with nls() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2.1 Introducing the data example . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.2 Model ﬁtting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.3 Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2.4 Making plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2.5 Illustrating the estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3 Generalised linear models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3 Starting Values and Self-starters . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.1 Finding starting values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.1.1 Graphical exploration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.1.2 Searching a grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 Using self-starter functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2.1 Built-in self-starter functions for nls() . . . . . . . . . . . . . . . 30 3.2.2 Deﬁning a self-starter function for nls() . . . . . . . . . . . . . 31 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4 More on nls() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.1 Arguments and methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.2 Supplying gradient information . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.2.1 Manual supply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

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