Empirical model-building and response surfaces

696 Pages·1987·27.624 MB·English

by  George E. P Box

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Y (ene Varig Mey ee, tN,
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Apt9 °
mpirical Model-Building
nd Response Surfaces
Digitized by the Internet Archive
In 2021 with funding from
Kahle/Austin Foundation
https://archive.org/details/empiricalmodelou0000boxg
Empirical Model-Building
and Response Surfaces
GEORGE E. P. BOX
NORMAN R. DRAPER
John Wiley & Sons
New York . Chichester . Brisbane . Toronto . Singapore
Copyright © 1987 by John Wiley & Sons, Inc.
All rights reserved. Published simultaneously in Canada.
Reproduction or translation of any part of this work
beyond that permitted by Section 107 or 108 of the
1976 United States Copyright Act without the permission
of the copyright owner is unlawful. Requests for
permission or further information should be addressed to
the Permissions Department, John Wiley & Sons, Inc.
Library of Congress Cataloging-in-Publication Data:
Box, George E. P.
Empirical model-building and response surfaces.
(Wiley series in probability and mathematical
statistics. Applied probability and statistics)
Bibliography: p.
Includes indexes.
1. Experimental design. 2. Response surfaces
(Statistics) I. Draper, Norman Richard. II. Title.
III. Series.
QA279.B67 1986 519.5 86-4064
ISBN 0-471-81033-9
Printed in the United States of America
109876543 i
Preface
The experimenter frequently faces the task of exploring the relationship
between some response y and a number of predictor variables x =
(x, X2,..., X,)’. Thus, for example, the response y might be the reaction
time of a subject and three variables x,, x,, x, might be hours of sleep
deprivation, amount of exercise, and dose of a certain drug. Or, in a
chemical experiment, the response y might be the yield of product and
X1, Xz, xX, might be the temperature, time, and pressure at which the
reaction was conducted.
Various degrees of knowledge or ignorance may exist about the nature of
such relationships. At one extreme it may be that the relationship is
“exactly known” from physical considerations and consequently a mathe-
matical function can be written down which, apart from experimental error
&, is believed to exactly represent the relationship between the response y
and the variables x. We can then write down a mechanistic model of the
form y = f(x) + «. When such knowledge of the physical mechanism
underlying a relationship is known or even suspected, it should be used. A
number of interesting problems concerning checking of model fit, model
discrimination, and choice of experimental design occur when mechanistic
models are being considered but these are not a major topic of discussion
here.
Many examples exist, however, where knowledge of the mechanistic
model is lacking and all that is known is that, apart from experimental
error, the relationship between y and the various x’s is likely to be smooth.
Such relationships can be explored experimentally and useful conclusions
drawn. In particular, if we have only one variable x, observations of y at
various levels of x may be plotted and a smooth curve drawn through the
points by eye. Alternatively, some smooth function such as a straight line or
a quadratic curve may be fitted, for example, by least squares. If we denote
such an approximating (or graduating) function by g(x) we may refer to
the relationship y = g(x) + € as an empirical model.
i PREFACE
It is well known that the usefulness of a simple function such as a
straight line or a quadratic curve is greatly enhanced if we allow for the
possibility that, before making the plot, an appropriate transformation in y
or in x or in both is made. For example, a relationship might be simplified
by plotting log y against x, y against log x, or log y against log x; in other
examples, a square root or reciprocal transformation, rather than a log,
might prove valuable.
It is possible to use statistical estimation procedures both to choose the
adjustable constants in g(x) (determining, for example, the slope and
constant of the fitted line) as well as the appropriate transformation for y
or for x. Such fitting processes can be thought of as a means of matching a
(possibly transformed) low degree polynomial g(x) to the data in much the
same way that a French curve is adjusted to the data to obtain a smooth fit.
In this way a “mathematical French curve” g(x) is made to substitute
locally for the true but unknown function f(x).
_ When there is more than one x variable, simple graphical plotting is not
available. However, we can extend the idea of the fitting of a “mathemati-
cal French curve” to more than one dimension. A class of such graduating
functions explored in this book consists of the polynomials of degree one
and two in x,,X>,...,X,, again allowing for the possibility that both
response y and each of the variables x,, x,,..., x, might first be subjected
to transformation. The response surfaces so generated may be put to a
number of somewhat different uses:
1. To show how a particular response y is affected by a set of variables x
over some specific region of interest.
2. To discover what settings of the x’s will give a product simultaneously
satisfying specifications for a number of responses y,, y>,...,; Ym (€-.-,
yield, impurity, color, texture, and so on).
3. To explore the space of the x variables to define the maximum
response and to determine the nature of this maximum.
In problems 2 and 3 it is very likely that movement away from the initial
experimental region may be necessary before the objective is obtained. The
method of steepest ascent is used to move from a region of low response to
one of higher response using a local first-order approximation to the
response function (a first degree polynomial in the x’s, possibly trans-
formed). Possibly after one or more applications of steepest ascent, a region
of improved response may be found where second-order effects pre-
dominate and further exploration will then be possible in terms of a second
degree polynomial in the x’s, again possibly transformed. When this region
contains a maximum it is important not only to estimate its location but