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(continuedafterindex)
A.S. Hedayat N.J.A. Sloane
John Stufken
Orthogonal Arrays
Theory and Applications
, Springer
A.S. Hedayat N.J.A. SIoane John Stufken
Department of Mathematics, AT& T Labs Research Department of Statistics
Statistics, and Computer 180 Park A venue Iowa State University
Science Florham Park, NJ 07932- Ames, IA 500 11
University of Illinois 0971 USA
Chicago, IL 60607-7045 USA [email protected]
USA [email protected]
[email protected]
Library of Congress Cataloging-in-Publieation Data
Hedayat, A.
Ortbogonal arrays : theory and applieations I A.S. Hedayat,
N.J.A. Sioane, John Stufken.
p. em. - (Springer series in statisties)
Includes bibliographieal referenees and indexes.
ISBN 978-1-4612-7158-1 ISBN 978-1-4612-1478-6 (eBook)
DOI 10.1007/978-1-4612-1478-6
1. Ortbogonal arrays. 1. Sioane, N. J. A. (Neil James
Alexander), 1939- . II. Stufken, John. III. Title. IV. Series.
QAI6S.H43 1999
SI1'.6-de21 99-30377
Printed on acid-free paper.
© 1999 Springer Science+Business Media New York
Originally published by Springer-Verlag New York Inc. in 1999
Softcover reprint ofthe hardcover Ist edition 1999
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9 8 7 6 S 4 3 2 1
ISBN 978-1-4612-7158-1
Dedicated to
Batool, Leyla and Yashar Hedayat,
Susanna Cuyler Sloane,
Lili, Sharon and Byron Stufken
Preface
Who should read this book'? Anyone who is running experiments,
whetherin achemistry lab oramanufacturingplant (trying to make
those alloys stronger), or in agricultural or medical research. Any
one interestedin one ofthe mostfascinating areas ofdiscrete math
ematics, connected to statistics and coding theory, with applications
to computer science and cryptography. This is the first book on the
subject since its introduction more than fifty years ago, and can be
used as a graduate text or as a reference work. It features all ofthe
key results, many very useful tables, and a large number ofexercises
and research problems. Most ofthe arrays that can be obtained by
the methods in this book are available electronically.
Orthogonal arrays are beautiful and useful. They are essential in statistics
and they are used in computer science and cryptography. In statistics they
are primarily used in designing experiments, which simply means that they
are immensely important in all areas of human investigation: for example in
medicine, agriculture and manufacturing.
Your automobile lasts longer today because oforthogonal arrays.l
The mathematical theory is extremely beautiful: orthogonal arrays are re
lated to combinatorics, finite fields, geometry and error-correcting codes. The
definition of an orthogonal array is simple and natural, and we know many
elegant constructions - yet there are at least as many unsolved problems.
l'VT"L ....__. ~K'Tr-n" D__L__ ~I__ ...... ...nne ........... l1A 110 Ie.....,.. .....1........ C'.......4-:....... l' Q \
viii Preface
Here is an example ofan orthogonal array ofstrength 2:
0 0 0 0 0 0 0 0 0 0 0
1 1 1 0 1 1 0 1 0 0 0
0 1 1 1 0 1 1 0 1 0 0
0 0 1 1 1 0 1 1 0 1 0
0 0 0 1 1 1 0 1 1 0 1
1 0 0 0 1 1 1 0 1 1 0
0 1 0 0 0 1 1 1 0 1 1
1 0 1 0 0 0 1 1 1 0 1
1 1 0 1 0 0 0 1 1 1 0
0 1 1 0 1 0 0 0 1 1 1
1 0 1 1 0 1 0 0 0 1 1
1 1 0 1 1 0 1 0 0 0 1
Pick any two columns, say the first and the last:
0 0
1 0
0 0
0 0
0 1
1 0
0 1
1 1
1 0
0 1
1 1
1 1
Each ofthe four possible rows we might see there,
o o
0, 1, 1 0, 1 1,
does appear, and they all appear the same number of times (three times, in
fact). That's the property that makes it an orthogonal array.
Only O's and 1's appear in that array, but for use in statistics
o
or 1
in the first column might be replaced by
"butter" or "margarine",
and in the second column by
"sugar" or "no sugar" ,
Preface ix
and so on. Or
"slow cooling" or "fast cooling" ,
"catalyst" or "no catalyst" ,
etc., depending on the application.
Since only O's and l's appear, this is called a 2-1evel array. There are 11
columns, which means we can vary the levels of up to 11 different variables,
and 12 rows, which means we are going to bake 12 different cakes, or produce
12 different samplesofthe alloy. In short, we call this array an
OA(12,11,2,2) .
The first "2" indicates the number of levels, and the second "2" the strength,
which is the number of columns where we are guaranteed to see all the possi
bilities an equal number of times. In an orthogonal array of strength 3 (with
two levels), in any three columns we would see each ofthe eight possibilities
000,001,010,011,100,101,110,111
equally often. (The formal definition is given in the first chapter.)
As already mentioned, the main applications of orthogonal arrays are in
planningexperiments. The rows ofthearray represent theexperimentsor tests
tobeperformed-cakestobebaked, samplesofalloytobeproduced, integrated
circuits to be etched, test plots ofcrops to be grown, and so on.
The columns of the orthogonal array correspond to the different variables
whose effects are being analyzed. The entries in the array specify the levels at
which the variables are to be applied. Ifa row ofthe orthogonal array reads
110100...
this could mean that in that test the first, second, fourth variables (where the
l'soccur) areto beset at their "high" levels, and thethird, fifth, sixthvariables
(where the O's occur) at their "low" levels.
By basing the experiment on an orthogonal array of strength t we ensure
that all possible combinations ofup tot ofthe variablesoccur togetherequally
often.
The aim here is to investigate not onlytheeffectsofthe individual variables
(orfactors) ontheoutcome, butalsohowthevariablesinteract. Obviously,even
with a moderate numberoffactors and a small numberoflevelsfor each factor,
the number of possible level combinations for the factors increases rapidly. It
may therefore not be feasible to make even one observation at each ofthe level
combinations. In such cases observations are made at only some of the level
x Preface
combinations, and the purpose ofthe orthogonal array is to specifywhich level
combinations are to be used. Such experiments are called "fractional factorial"
experiments. While there are nowadaysother applicationsoforthogonal arrays
in statistics (for example in computer experiments2 and survey sampling3),
the principal application is in the selection oflevel combinations for fractional
factorial experiments.
Since the rows of an orthogonal array represent runs (or tests or samples)
- which require money, time, and other resources- there are always practical
constraints on the number ofrows that can be used in an experiment. Finding
the smallest possible number ofrows is a problem ofeminent importance. On
the other hand, for a given number of runs we may want to know the largest
number ofcolumns that can be used in an orthogonal array, since this will tell
us how many variables can be studied. We also want the strength to be large,
though in many real-life applications this is set at 2, 3 or 4.
Then the main questions we ask are:
• for whichvaluesofthenumbersofrows, columns, strength and levelsdoes
an orthogonal array exist?
• how can we construct the array, ifit exists?
We have already mentioned the use oforthogonal arrays in automobile de
sign. The following are some typical applications in medicine:
• Pharmaceutical companies use orthogonal arrays to investigate the sta
bility and shelflifeofdrugs, which usually involve many different factors.
The use oforthogonal arrays can lead to more economical tests and pro
vide better statistical information.
• Multiple drug therapy is widely practiced in order to treat either a single
medical disorder or to treat several concurrently existing ailments in the
same patient. But many drugs interact with each other. Bioavailability
studiesareusedtoevaluatethecombinationsofvariousdoses. Orthogonal
arrays can help to identifythe medicallysafeand effective regions sothat
physicianscanadjustdosagelevelsto avoidor minimizeinteractionswhen
using multiple drugs.
• Orthogonalandrelatedarrays havebeenusedinclinicaltrialswhichstudy
howdrugsareabsorbed, distributed, metabolizedandeliminatedfrom the
body. Orthogonal arrayscan be useful for studyingthe effectsofmultiple
factors on these drug characteristics. Orthogonal arraysor closely related
20wen (1992).
3McCarthy (1969), Wu (1991).
