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Introduction to Experimental Design Read chapter Read first chapter

2018 | OriginalPaper | Chapter

16. Introduction to Response-Surface Methodology

Authors : Paul D. Berger, Robert E. Maurer, Giovana B. Celli

Published in: Experimental Design

Publisher: Springer International Publishing

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Abstract

Until now, we have considered how a dependent variable, yield, or response depends on specific levels of independent variables or factors. The factors could be categorical or numerical; however, we did note that they often differ in how the sum of squares for the factor is more usefully partitioned into orthogonal components. For example, a numerical factor might be broken down into orthogonal polynomials (introduced in Chap. 12). For categorical factors, methods introduced in Chap. 5 are typically employed. In the past two chapters, we have considered linear relationships and fitting optimal straight lines to the data, usually for situations in which the data values are not derived from designed experiments. Now, we consider experimental design techniques that find the optimal combination of factor levels for situations in which the feasible levels of each factor are continuous. (Throughout the text, the dependent variable, Y, has been assumed to be continuous.) The techniques are called response-surface methods or response-surface methodology (RSM).

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previous chapter Multiple Linear Regression
next chapter Introduction to Mixture Designs
Appendix
Available only for authorised users
Footnotes
1
G. E. P. Box and K. B. Wilson (1951), “On the Experimental Attainment of Optimum Conditions.” Journal of the Royal Statistical Society, Series B, vol. 13, pp. 1–45.
 
2
By “flat,” we mean from a curvature-of-the-earth perspective; we are not referring to the issue of hills and valleys.
 
3
In general, the only time that stage two requires more than one experiment is the unfortunate, but not unlikely, situation of a “saddle point,” or some other undesirable happenstance. This issue is discussed later in the chapter.
 
4
This example, and a broad outline of some of the features used for the illustration of the RSM process, were suggested by the discussion in C. R. Hicks, Fundamental Concepts in the Design of Experiments, 3rd edition, New York, Holt, Rinehart and Winston, 1982.
 
5
In the real world, issues involving the packaging and other aspects of banana preservation are quite complex. There are many considerations and a large number of factors involved. However, as we noted, our example is greatly simplified, but captures the features we wish to illustrate. The example is loosely based on one author’s experience designing a complex experiment addressing some of these issues for a well-known harvester and shipper of bananas. Incidentally, in general, brown spots on bananas are not unhealthy or inedible – only unsightly!
 
6
We obtain the transformation for U as follows: let U = a + bX 1. When X 1 = 1/4, U =  −1; this gives −1 = a + b(1/4) = a + b/4; when X 1 = 1/2 , U =  +1; this gives 1 = a + b(1/2) = a + b/2. We have two equations with two unknowns, a and b. We solve these and find a = −3 and b = 8. The second equation is found in a similar fashion.
 
7
Latitude refers to the angle that measures north-south position from the equatorial plane, whereas longitude is the angle that measures west-east position from the Greenwich Meridian.
 
8
We repeat that this will be the last experiment, unless the situation is an “unlucky” one, in which there is either a saddle point, or local maxima in addition to the global maximum.
 
9
G. E. P. Box and J. S. Hunter (1957), “Multi-factor Experimental Designs for Exploring Response Surfaces.” Annals of Mathematical Statistics, vol. 28, n. 1, pp. 195–241.
 
10
In theory, the point could be a minimum point, but, unless the surface is very irregular, this is very unlikely after being directionally guided by a series of steepest-ascent analyses. A saddle point represents a stationary situation where there is neither a maximum nor a minimum – this term is used due to the resemblance of the surface to a saddle. That is, if you are on a horse, sitting on a saddle, facing north-south, then from a north-south direction, you are sitting at a minimum point, while from an east-west direction, you are sitting at a maximum point.
 
11
Note that the fractional-factorial design described is of resolution five.
 
12
The authors don’t know where this world is; if we did, we’d buy property there!
 
13
Remember: We are describing what the NASA team did, not necessarily what we would have done; then again, we weren’t there, and they made the judgment call. This thought applies to many of the subsequent steps the NASA researchers chose to follow.
 
14
Actually, it was the taxpayers’ money.
 
15
Of course, search techniques were available back in the late 1960s on mainframe computers. However, they were not readily accessible nor user-friendly. Today, one might question whether the calculus step shouldn’t always be replaced by Solver or its equivalent.
 
16
G. Derringer and R. Suich (1980), “Simultaneous Optimization of Several Response Variables.” Journal of Quality Technology, vol. 12, n. 4, pp. 214–219.
 
Metadata
Title
Introduction to Response-Surface Methodology
Authors
Paul D. Berger
Robert E. Maurer
Giovana B. Celli
Copyright Year
2018
Book
Experimental Design

Print ISBN: 978-3-319-64582-7

Electronic ISBN: 978-3-319-64583-4

Copyright Year: 2018

DOI
https://doi.org/10.1007/978-3-319-64583-4_16

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