Field Effects Analysis of Variance

183 Pages·1978·9.243 MB·English

by  Lloyd Fisher, J. McDonald

Probability and Mathematical Statistics
A Series of Monographs and Textbooks
Editors Z. W. Birnbaum E. Lukacs
University of Washington Bowling Green State University
Seattle, Washington Bowling Green, Ohio
S. Vajda. Probabilistic Programming. 1972
Sheldon M. Ross. Introduction to Probability Models. 1972
Robert B. Ash. Real Analysis and Probability. 1972
V. V. Fedorov. Theory of Optimal Experiments. 1972
K. V. Mardia. Statistics of Directional Data. 1972
H. Dym and H. P. McKean. Fourier Series and Integrals. 1972
Tatsuo Kawata. Fourier Analysis in Probability Theory. 1972
Fritz Oberhettinger. Fourier Transforms of Distributions and Their Inverses: A
Collection of Tables. 1973
Paul Erdös and Joel Spencer. Probabilistic Methods in Combinatorics. 1973
K. Sarkadi and I. Vincze. Mathematical Methods of Statistical Quality Control. 1973
Michael R. Anderberg. Cluster Analysis for Applications. 1973
W. Hengartner and R. Theodorescu. Concentration Functions. 1973
Kai Lai Chung. A Course in Probability Theory, Second Edition. 1974
L. H. Koopmans. The Spectral Analysis of Time Series. 1974
L. E. Maistrov. Probability Theory: A Historical Sketch. 1974
William F. Stout. Almost Sure Convergence. 1974
E. J. McShane. Stochastic Calculus and Stochastic Models. 1974
Z. Govindarajulu. Sequential Statistical Procedures. 1975
Robert B. Ash and Melvin F. Gardner. Topics in Stochastic Processes. 1975
Avner Friedman, Stochastic Differential Equations and Applications, Volume 1,
1975; Volume 2.1975
Roger Cuppens. Decomposition of Multivariate Probabilities. 1975
Eugene Lukacs. Stochastic Convergence, Second Edition. 1975
H. Dym and H. P. McKean. Gaussian Processes, Function Theory, and the Inverse
Spectral Problem. 1976
N. C. Giri. Multivariate Statistical Inference. 1977
Lloyd Fisher and John McDonald. Fixed Effects Analysis of Variance. 1978
In Preparation
Sidney C. Port and Charles J. Stone. Brownian Motion and Classical Potential Theory.
Fixed Effects
Analysis of Variance
LLOYD FISHER
JOHN MCDONALD
Department of Biostatistics
University of Washington
Seattle, Washington
ACADEMIC PRESS New York San Francisco London 1978
A Subsidiary of Har court Brace Jovanovich, Publishers
COPYRIGHT © 1978, BY ACADEMIC PRESS, INC.
ALL RIGHTS RESERVED.
NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR
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OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY
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ACADEMIC PRESS, INC.
Ill Fifth Avenue, New York, New York 10003
United Kingdom Edition published by
ACADEMIC PRESS, INC. (LONDON) LTD.
24/28 Oval Road, London NW1 7DX
Library of Congress Cataloging in Publication Data
Fisher, Lloyd, Date
Fixed effects analysis of variance.
(Probability and mathematical statistics
; )
1. Analysis of variance. I. McDonald, John,
Date joint author. II. Title.
QA279.F57 519.5'35 77-92238
ISBN 0-12-257350-1
PRINTED IN THE UNITED STATES OF AMERICA
TO
Ginny and the Christian fellowship of
Gerald, Ian, Carl, Busso, John, Ed, Art, and Ross
LF
TO
my family and friends for putting up with the perpetual student
JMcD
PREFACE
This book is designed as a reference and also as a textbook for a one-
quarter course for statistics students at the upper-division undergraduate
or the beginning graduate level. It was written because in the authors'
opinion there is no book suitable for a one-quarter course on a moderately
advanced theoretical level. Most theoretical textbooks are too long and
are designed for a one-year course. WJhen the books are used for a one-
quarter course, the course is almost exclusively theory without examples,
which often has a detrimental effect on the motivation of the students.
Textbooks designed for a one-quarter length course tend to be on a theo-
retical level that is too low. Thus, this book has been designed to bridge
the one-quarter gap between books such as those by Günther and Scheffé,
for example. Care has been taken to use examples from the literature in
order to give students the emotional as well as mental realization that the
techniques are indeed used for day-to-day data analysis.
Prerequisites are a background in probability and statistics at roughly
the level of Hogg and Craig. It is useful if the probability and statistics
course has covered the concepts of hypothesis testing, p values, confidence
intervals, and moment generating functions. The reader should also have
a background in linear algebra. At the University of Washington this
background, although required, has usually been rusted somewhat by the
time interval between the linear algebra course and the analysis of variance
course. Also, the linear algebra courses often do not cover certain useful
topics, such as projection operators and quadratic forms, that are crucial
for the analysis of variance. For this reason, an appendix reviewing results
of linear algebra is included. There is particular emphasis on projection
operators, a topic slighted in some linear algebra books, and acquaintance
with the results of Appendix 1 will save the reader many hours which might
xi
xii Preface
be spent thumbing through the diverse linear algebra literature. The appen-
dix, which gives this background, is at a higher level than is needed for
most of the text. It is felt that when this book is used as a textbook the
instructor may need to adjust the lectures to the level and material of the
linear algebra course taught at the particular school.
The finalc hapter in the book is a short introduction to multiple regression.
This was included since the general linear model is available, waiting to be
used at the end of the text. It is felt that this chapter will enable the reader
to make the connection between analysis of variance and multiple regression
techniques as they are studied.
Results of Appendix 1 are prefaced by A; for example, Theorem 1 of the
appendix is referred to in the text as Theorem Al.
This book is not intended to be a terminal point in the education of the
reader. It is hoped that this book will enable one to tackle the more advanced
texts by Scheffé, The Analysis of Variance, Searle, Linear Models, and
Kempthorne, Design and Analysis of Experiments, as well as books of a
more applied bent, such as Winer, Statistical Principles in Experimental
Design.
The scopfe of this book has been quite modest as befits a book written
for a one-quarter course. Some extensions of the theory pointing the way
to generalization and to other techniques are given in the problems at the
ends of the chapters.
A CKNO WLEDGMENTS
The authors wish to acknowledge the kind cooperation and permission
of the many authors and publishers of examples cited in the text. Thanks
are due to The Society for Child Development, Inc. for giving permission
to use their copyrighted material that appeared in Child Development 46,
258-262; 47, 237-241, 532-534 as used in Example 2.2 and Problem 6.14.
Thanks are also due to the American Journal of Nursing Company for
permission to use their copyrighted material from Nursing Research that
appears in Example 2.1. We also wish to thank Marcel Dekker, Inc., and
Professors Odeh, Owen, and Birnbaum for allowing the use of the tables
given in Appendix 2.
Students who worked through versions of the text for two consecutive
years are thanked for their kind comments and sufferance. Thanks are also
due to Ms. Janice Alcorn and Ms. Cindy Bush. Finally, one author (LF)
would like to thank his family, who were patient during many hours of work
on the text at home.
xiii
INTRODUCTION
1
The subject matter we will consider is extremely beautiful. The mathe-
matical theory of the fixed effects analysis of variance is elegant and can
give considerable insight into much of statistical theory after the material
has been mastered. Also, when applied to appropriate data, it gives a very
concise and elegant method of analyzing quite complex situations. Unfor-
tunately, between the elegant theory and elegant application there is a fair
amount of detail and unaesthetic algebraic manipulation. If one keeps sight
of the beginning and the end results of the process, however, the algebraic
manipulations should not prove too much of a nuisance.1
The term "analysis of variance," and indeed much of the fundamental
work in the subject, is the work of Sir Ronald A. Fisher. In the words of
Fisher:
... the analysis of variance, that is, by the separation of variance as-
cribable to one group of causes from the variance ascribable to other
groups.2
This, then, is the thrust of the subject we are going to study: to see which
part of variation in data might reasonably be attributed to various causes
and which part of the variability seems to be due to uncontrollable variation,
whether biological, physical measurement error, or otherwise.
1 Beauty and elegance are in the eye of the beholder, and this view, of course, represents that
of the authors of this material.
2 R. A. Fisher, Statistical Methods for Research Workers, 11th ed., revised, p. 211. Hafner,
New York, 1950
1
2 Chapter 1
It is interesting to consider Fisher's background, which enabled him to
develop the analysis of variance. Mahalanobis says of him:
Love of mathematics dominated his educational career; and he was for-
tunate in coming under the tuition of a brilliant mathematical teacher,
W. N. Roe of Stanmore Park, also well known in England as a Somerset-
shire cricketer. At Harrow he worked under C. H. P. Mayo and W. N.
Roseveare. The peculiar circumstances of the teaching by the latter
will be of interest to statisticians familiar with Fisher's geometrical
methods. On account of his eyes he was forbidden to work under
artificial light, and when he would go to work with Mr. Roseveare in
the evenings the instruction was given purely by ear without the use
of paper and pencil or any other visual aid. To this early training may
be attributed Fisher's tendency to use hypergeometrical representations,
and his great power in penetrating problems requiring geometrical
intuition. It is, perhaps, a consequence of this that throughout life, his
solutions have been singularly independent of symbolism. He does not
usually attempt to write down the analysis until the problem is solved
in his mind, and sometimes, he confesses, after the key to the solution
has been forgotten. I have already mentioned that Fisher was not
acquainted with Students' work when he wrote his 1914 paper on the
exact distribution of the correlation coefficient. Here he introduced
for the first time the brilliant technique of representing a sample of
size n by a point in a space of n dimensions. Such representation has
proved extremely useful in subsequent work not only in the theory of
distribution, but also in other fields of statistical theory such as the
work of J. Neyman and E. S. Pearson.3
In line with these comments we find that the mathematics needed to
understand the analysis of variance involves working in n-dimensional
Euclidean space. While the reader of this text is expected to have had some
background in linear algebra, some review may prove necessary (see Appen-
dix 1).
The usefulness of Fisher's mathematical background resulted from the
fact that he was basically concerned with experimentation and applications.
It is a common misapprehension among mathematicians that new
statistical methods can be developed solely by the process of pure
reasoning without contact with sorted numerical data. It was part of
Fisher's strength that although an early mathematician, he held no
brief for this belief. He was himself a skilled and accurate computer,
3 P. C. Mahalanobis, Professor Ronald Aimer Fisher, Biometrics 20, 238-251 (June 1964).
Introduction 3
and did not hesitate to embark on the analysis of any body of data that
interested him. The rapid advancement of experimental design and
analysis owe much to this.4
Fisher's contribution to the analysis of variance is summarized as follows:
Perhaps no technique is more characteristic of Fisher, more closely
associated with his name, or more central to his pattern of thought
than the analysis of variance. Arising almost incidentally during his
efforts to make sense of agricultural experiments whose designs were
inadequate for sound inference, this rapidly became the standard way
of analyzing results from the next generation of experiments5.
In this text we shall present the theoretical ideas and some applications of
the analysis of variance. The emphasis will be on understanding the theoreti-
cal background behind the analysis of variance. We shall, however, attempt to
interject enough examples so the reader will see how the theoretical material
is related to applications and will be in a position to pursue the subject
further. Near the end of this text, we shall see that the models we are con-
sidering are appropriate not only for the analysis of variance but for under-
standing many other multivariate statistical topics. Thus, we shall touch
upon topics known as multiple regression analysis, and certain other aspects
of multivariate statistics such as multiple and partial correlation coefficients.
4 F. Yates, Appreciations of R. A. Fisher, Biometrics 20, No. 2, 312-313 (June 1964).
5 D. J. Finney, Sir Ronald Fisher's contributions to biométrie statistics, Biometrics 20,
322-329 (June 1964).