Table Of ContentThe
Anal~s
of Variance
f\
HENR Y SCHEFFE
Professor of Statistics
University of California, Berkeley
UNDP/SF/FAO
FlSHERIES P"OJECT NO. 70
LIBRARY
New York· John Wiley &: Sons, Inc.
London . Sydney
The Analysis
of Variance
A WILEY PUBLICATION IN MATHEMATICAL STATISTICS
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Copyright © 1959 by John Wiley & Sons, Inc.
AU Rights Reserved
This book or any part thereof must not
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Copyright, Canada, 1959, International Copyright, 1959
John Wiley & Sons, Inc., Proprietor
All Foreign Rights Reserved
Reproduction in whole or in part forbidden.
FIFTH PnIN'rlNO, JANUARY, 1967
Library of Congress Catalog Card Number: 59-14994
Printed in the United States of America
'fo
Maud Susan Sherwood
(1 f~ - ,'6.
WILEY SERIES IN PROBABILITY
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Preface
In this book I have tried to elucidate in a unified way what appears
to me at present to be the basic theory of the analysis of variance.
This necessitates considering several different mathematical models for
the subject. The theory of Part I, namely that for fixed-effects models
with independent ohservations of equal variance, I judge to be jelled
into a fairly permanent form, but th theory of Part II, namely that
under other models, I expect will undergo considerable extension and
revIsIon. Perhaps this presentation will help stimulate the needed
growth. What I feel most apologetic ahout is the little I have to offer
the read('r on the unbalanced cases of the random-effects models and
mixed models. These cannot be generally avoided in planning bio
logical experiments, especially in genetics, the situation being unlike
that in physical science. This gap in the theory I have not been able
to fill.
The mathematical background necessary for the reader to under
stand this book is a course in calculus at some time in the past, and at
least occasional usc of some mathematical notation in the present.
Very little of the calculus is actually employed, but the reader who
never had it would be unlikely to have developed sufficient ease in the
necessary language of mathematics. Most of the derivations in the
book are of an algebraic nature. To facilitate the derivations in Chs.
1, 2, and 6, vector and matrix methods arc extensively employed. The
exposition of the needed vector and matrix algebra in Apps. I and II
should make the book self-contained for the reader with the minimal
mathematical background indicated above. The reader not at home
with matrix notation should write out in longhand without this nota
tion some of the first equations he encounters in this notation. Then
soon he will reach the stage where matrix formulations are not only
easi r to look at and to write, but also to think in.
My decision to use matrix notation may be further justified in the
following way. It is well known that one unifying and insightful way
of regarding the analysis of variance is from the geometrical viewpoint:
it may be viewed as a method of resolving the vector of observations
vii
VIII PREFACE
into vectors lying in certain specifi d spaces corresponding to different
sources of variation in the observations, and to I;!ach of which a mean
ingful interpretation can be given. For understanding the geometry
of such resolutions and the geometrical interpretation of the statistics
used to test whether the magnitudes of some of the component vectors
associated with different sources are significant, the concept of orthogo
nality of vectors and spaces is indispensable. The easiest way of
defining, applying, and manipulating this geometric concept is, I
believe, through the use of matrix notation.
The statistical background necessary for the reader is knowledge
equivalent to that aimed at in a sound year course in statistics stressing
the concepts of elementary probablility, confidence intervals, and the
power (or operating characteristics) of tests, and including use of the
t-, x2, and F-distributions.
This book contains 117 problc:ms at the ends of the chapters and
appendices, of which 38 require numerical computations with "real"
data. The variety of applications in these 38 problems should give
some idea of the·broad applicability of the analysis of variance, even
though the problems were chosen only because they furnish suitable
examples of the methods described in the text, and with no conscious
attempt at inclusion of many substantive fields. The importance of
carrying through a considerable amount of numerical work is greater
here than it is in learning most branches of statistics. Indeed, some
practitioners of the analysis of variance would regard the computa
tional techniques as the most important part of the subject, and con
sider as perverted my emphasis on the choice of mathematical models.
I realize that many practitioners have developed reliable intuitive and
verbal paths to the correct analysis in given situations without defining
the model, but I find it easier to follow the path to which I am con
strained by the choice of model; the approach of choosing the model
and then making the analysis dictated by it seems to me also to be
simpler to teach, as well as more appropriate for a book on the theory
of the subject.
The book is intended as a text for a one-semester or two-quarter
course at the senior or graduate level, and for self-study. At Berkeley
in a semester graduate course meeting for three lecture hours and two
laboratory sessions per week the material in the book is covered except
for ehs. 5,8, and 9, which are included with other topics in a course on
the design of experiments for which this course is prerequisite. In
future we will·expect the student before starting this course to have
acquired a knowledge of matrix algebra at least equal to that obtain
able from an elementary course, or to have worked his way through
PREFACE IX
Apps. I and II of this book. For a shorter course eh. 6 and parts of
Ch. 7 might also be omitted.
The following topics are omitted, since one purpose of the book is
to serve as a text for a Course in the analysis of variance, and these
omissions are usually cov red in other courses in statistics depart
ments: the multivariate generalization of the exclusively univariate
tl.eory developed here, sequential methods in the analysis of variance,
and non parametric t hpory, except for the permutation tests based on
the F-statistics for the Latin-square and incomplete-blocks designs.
For the same reason the design of experiments is touched on only
incidentally, and the theories of confounding, fractional replication,
response-surface exploration, and the more complicated experimental
designs receive no mention. However, the omission of the decision
throry approach to the subject is mainly for another reason: Except
possibly for one problem (experiments designed for choosing the best
of a set of treatments, see sec. 3.7), this approach seems to me to have
yielded as yet no important new useful methods in this, perhaps the
most widely used, branch of statistics, where typically many possible
decisions from a set of data are considen:d. *
I earnestly hope this book will be suitable for self-study, the route
which many users of statistical methods have had to follow because
the subject is still not available, or not encouraged, in many college
and university programs of training for scientific and engineering pro
fessions. For the reader wishing to master the subject in this way the
above remark about the importance of some numerical computation is
especially pertinent, and I urge him to work most of the 38 problems
involving data. If he cannot find access to a desk calculating machine
he will generally have to calculate directly from the definitions of the
various sums of squares and not, because of the consequent loss of
significant figures, from the computing formulas given in the book and
intended for use on machines, as explained at the end of sec. 3.1. As
for the remaining problems, the reader without benefit of a teacher
should not feel discouraged if he cannot solve them all, for they vary in
mathematical difficulty from being easy for all to being easy only for
professional mathematicians. The lone reader may also find it helpful.
when the argument is geometrical or permits a geometrical interpreta
tion, to sketch figures, like Fig. 2.9.1. which suggest the n-dimensional·
geometrical relationships in a two-dimensional representation.
I am also hopeful that the reader will not experience the abundance
• However, recent work by Kiefer and Wolfowitz (1958) opens the possibility that
a game-theoretic approach to the problems of optimum experimental design may yield
new solutions which are useful and computable.
