PROCEEDINGS OF SYMPOSIA IN APPLIED MATHEMATICS

VOLUME 1 NON-LINEAR PROBLEMS IN MECHANICS OF CONTINUA Edited by E. Reissner (Brown University, August 1947)

VOLUME 2 ELECTROMAGNETIC THEORY Edited by A. H. Taub (Massachusetts Institute of Technology, July 1948)

VOLUME 3 ELASTICITY Edited by R. V. Churchill (University of Michigan, June 1949)

VOLUME 4 FLUID DYNAMICS Edited by M. H. Martin (University of Maryland, June 1951)

VOLUME 5 WAVE MOTION AND VIBRATION THEORY Edited by A. E. Heins (Carnegie Institute of Technology, June 1952)

VOLUME 6 NUMERICAL ANALYSIS Edited by J. H. Curtiss (Santa Monica City College, August 195 3)

VOLUME 7 APPLIED PROBABILITY Edited by L. A. MacColl (Polytechnic Institute of Brooklyn, April 1955)

VOLUME 8 CALCULUS OF VARIATIONS AND ITS APPLICATIONS Edited by L. M. Graves (University of Chicago, April 1956)

VOLUME 9 ORBIT THEORY Edited by G. Birkhoff and R. E. Langer (New York University, April 1957)

VOLUME 10 COMBINATORIAL ANALYSIS Edited by R. Bellman and M. Hall, Jr. (Columbia University, April 1958)

VOLUME 11 NUCLEAR REACTOR THEORY Edited by G. Birkhoff and E. P. Wigner (New York City, April 1959)

VOLUME 12 STRUCTURE OF LANGUAGE AND ITS MATHEMATICAL ASPECTS Edited by R. Jakobson (New York City, April 1960)

VOLUME 13 HYDRODYNAMIC INSTABILITY Edited by R. Bellman, G. Birkhoff, C. C. Lin (New York City, April 1960)

VOLUME 14 MATHEMATICAL PROBLEMS IN THE BIOLOGICAL SCIENCES Edited by R. Bellman (New York City, April 1961)

VOLUME 15 EXPERIMENTAL ARITHMETIC, HIGH SPEED COMPUTING, AND MATHEMATICS Edited by N. C. Metropolis, A. H. Taub, J. Todd, C. B. Tompkins (Atlantic City and Chicago, April 1962)

VOLUME 16 STOCHASTIC PROCESSES IN MATHEMATICAL PHYSICS AND ENGI­NEERING Edited by R. Bellman (New York City, April 1963)

http://dx.doi.org/10.1090/psapm/028

VOLUME 17 APPLICATIONS OF NONLINEAR PARTIAL DIFFERENTIAL EQUA­TIONS IN MATHEMATICAL PHYSICS Edited by R. Finn (New York City, April 1964)

VOLUME 18 MAGNETO-FLUID AND PLASMA DYNAMICS Edited by H. Grad (New York City, April 1965)

VOLUME 19 MATHEMATICAL ASPECTS OF COMPUTER SCIENCE Edited by J. T. Schwartz (New York City, April 1966)

VOLUME 20 THE INFLUENCE OF COMPUTING ON MATHEMATICAL RESEARCH AND EDUCATION Edited by J. P. LaSalle (University of Montana, August 1973)

VOLUME 21 MATHEMATICAL ASPECTS OF PRODUCTION AND DISTRIBUTION OF ENERGY Edited by P. D. Lax (San Antonio, Texas, January 1976)

VOLUME 22 NUMERICAL ANALYSIS Edited by G. H. Golub and J. Oliger (Atlanta, Georgia, January 1978)

VOLUME 23 MODERN STATISTICS: METHODS AND APPLICATIONS Edited by R. V. Hogg (San Antonio, Texas, January 1980)

VOLUME 24 GAME THEORY AND ITS APPLICATIONS Edited by W. F. Lucas (Biloxi, Mississippi, January 1979)

VOLUME 25 OPERATIONS RESEARCH: MATHEMATICS AND MODELS Edited by S. I. Gass (Duluth, Minnesota, August 1979)

VOLUME 26 THE MATHEMATICS OF NETWORKS Edited by S. A. Burr (Pittsburgh, Pennsylvania, August 1981)

VOLUME 27 COMPUTED TOMOGRAPHY Edited by L. A. Shepp (Cincinnati, Ohio, January 1982)

VOLUME 28 STATISTICAL DATA ANALYSIS Edited by R. Gnanadesikan (Toronto, Ontario, August 1982)

AMS SHORT COURSE LECTURE NOTES published as a subseries of Proceedings of Symposia in Applied Mathematics

PROCEEDINGS OF SYMPOSIA IN APPLIED MATHEMATICS

Volume 28

STATISTICAL DATA ANALYSIS

AMERICAN MATHEMATICAL SOCIETY PROVIDENCE, RHODE ISLAND

LECTURE NOTES PREPARED FOR THE AMERICAN MATHEMATICAL SOCIETY SHORT COURSE

STATISTICAL DATA ANALYSIS

HELD IN TORONTO, ONTARIO, CANADA AUGUST 21 -22 , 1982

EDITED BY RAM GNANADESIKAN

The AMS Short Course Series is sponsored by the Society's Committee on Employment and Education Policy (CEEP). The series is under the direction of the Short Course Advisory Subcommittee of CEEP.

This volume was printed directly from copy prepared by the authors. It was typeset on a photocomposer driven by a computer running under the UNIX operating system. (UNIX is a trademark of Bell Laboratories.) The AMS expresses its gratitude for the substantial financial contribution made by Bell Laboratories in support of the Toronto Short Course on Statistical Data Analysis.

Library of Congress Cataloging in Publication Data

Main entry under title: Statistical data analysis.

(Proceedings of symposia in applied mathematics, ISSN 0160-7634; v. 28. AMS short course lecture notes)

Includes bibliographies. 1. Mathematical statistics—Addresses, essays, lectures. I. Gnanadesikan, Ram,

1932— . II. Series: Proceedings of symposia in applied mathematics; v. 28. III. Series: Proceedings of symposia in applied mathematics. AMS short course lecture notes. QA276.16.S82 1983 519.5 82-24308 ISBN 0-8218-0040-X

1980 Mathematics Subject Classification. Primary 62-07.

Copyright © 1983 by the American Mathematical Society.

Printed in the United States of America.

All rights reserved except those granted to the United States Government.

This book may not be reproduced in any form without the permission of the publishers.

TABLE OF CONTENTS

CHAPTER 1 Introduction (R. Gnanadesikan) 1 Bibliography 7

CHAPTER 2 Graphical Methods (P. A. Tukey) 8 2.1 Introduction 8 2.2 Looking at a single collection of numbers 9 2.3 Comparing two or more sets of numbers 11 2.4 Exploring distributional models for data 18 2.5 Looking at relationships among variables 23 2.6 Plots for higher-dimensional data 34 2.7 Concluding remarks 43

Bibliography 47

CHAPTER 3 Robust Methods (C. L. Mallows) 49 3.1 Introduction 49 3.2 The regression problem 51 3.3 Estimation of location: I 54 3.4 Outliers 57 3.5 Estimation of location: II 59 3.6 Location and scale 62 3.7 Robust regression 64 3.8 Robust smoothing 68 3.9 Other areas 71

Bibliography 71

CHAPTER 4 Multilinear Methods (J. B. Kruskal) 75 4.1 Introduction 75 4.2 A basic bilinear model 78 4.3 The rotation problem 81 4.4 Restrictions used to aid comparison 82

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4.5 Restrictions to find the true underlying factors 83 4.6 Singular value decomposition 84 4.7 Some more bilinear models 86 4.8 Trilinear models 89 4.9 An application 92

4.10 Appendix 1 — Relationship among some models 97 4.11 Appendix 2 — PARAFAC preprocessing 98

Bibliography 103

CHAPTER 5 A Case Study in Data Analysis (J. R. Kettenring) 105 5.1 Introduction 105 5.2 The data 106 5.3 Factor analysis and analysis of variance models 108 5.4 Factor analysis results 115

5.4.1 Two-factor model 115 5.4.2 Residuals 115 5.4.3 A robust fit 120 5.4.4 Parallel factors 120 5.4.5 Size effects 121 5.4.6 Recapitulation 124

5.5 Analysis of variance results 124 5.5.1 Estimates of main effects and two-way interactions 125 5.5.2 Decompositions of the two-way interactions 127 5.5.3 Assessing the significance of the decompositions 130 5.5.4 Decomposition of the three-way interaction 133 5.5.5 Recapitulation 134

5.6 Summary, perspective, and critique 136 5.7 Acknowledgement 139

Bibliography 139

CHAPTER 6 Summary and Conclusions (R. Gnanadesikan) 140 Bibliography 141

Preface

This book is an outcome of the 1982 American Mathematical Society Short Course given at Toronto. Statistical data analysis has been receiving a great deal of attention recently as evidenced by the fact that subsets of the authors of the present volume have given workshops or short courses on this topic at various meetings in the last two years, including those of the Mathematical Association of America and ICME-IV. The interest may be due to many things — practical importance of the topic, challenging research problems in a relatively young field, need for ideas and material for teaching courses on the subject.

Clearly neither the short course nor this book can provide enough details on all of the above facets of interest. However, the different chapters do try to address these aspects, although with varying degrees of emphasis. One hope of all the authors in publishing this book is that others will use this material as a starting point and, with the help of some of the references, be able to develop workshops, short courses and other educational forums on their own.

The authors are all employed by Bell Telephone Laboratories which provided support for the efforts of all of them. As the organizer of the short course and the editor of this book, it was my pleasure to organize and coordinate a well-meshed effort of colleagues instead of several individual contributions that would have had to stand on their own. I wish to thank all of the authors for their cooperation in this process.

All of us would also like to thank Susan A. Tarczynski for her outstanding help in the word processing services involved in producing this book.

November, 1982 R. Gnanadesikan, Editor

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