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Library of Congress Cataloging-in-Publication Data:

Myers, Raymond H.

Response surface methodology : process and product optimization using designed experiments.

- - 3rd ed. / Raymond H. Myers, Douglas C. Montgomery, Christine M. Anderson-Cook. p. cm. - - (Wiley series in probability and statistics)

Includes bibliographical references and index.

ISBN 978-0-470-17446-3 (cloth)

1. Experimental design. 2. Response surfaces (Statistics). I. Montgomery, Douglas C. II. Anderson-Cook, Christine M. III. Title.

QA279.M94 2008

519.507- -dc22

2008019012

PREFACE

This book deals with the exploration and optimization of response surfaces. This is a problem faced by experimenters in many technical fields, where, in general, the response variable of interest is y and there is a set of predictor variables x₁, x₂, …, xk. For example, y might be the viscosity of a polymer and x₁, x₂, and x₃ might be the reaction time, the reactor temperature, and the catalyst feed rate in the process. In some systems the nature of the relationship between y and the x’s might be known “exactly,” based on the underlying engineering, chemical, or physical principles. Then we could write a model of the form y = g(x₁, x₂, …, xk) + ε, where ε represents the “error” in the system. This type of relationship is often called a mechanistic model. We consider the more common situation where the underlying mechanism is not fully understood, and the experimenter must approximate the unknown function g with an appropriate empirical model y = f(x₁, x₂, …, xk) + ε. Usually the function f is a first-order or second-order polynomial. This empirical model is called a response surface model.

Identifying and fitting an appropriate response surface model from experimental data requires some knowledge of statistical experimental design fundamentals, regression modeling techniques, and elementary optimization methods. This book integrates all three of these topics into what has been popularly called response surface methodology (RSM).

We assume that the reader has some previous exposure to statistical methods and matrix algebra. Formal coursework in basic principles of experimental design and regression analysis would be helpful, but are not essential, because the important elements of these topics are presented early in the text. We have used this book in a graduate-level course on RSM for statisticians, engineers, and chemical/physical scientists. We have also used it in industrial short courses and seminars for individuals with a wide variety of technical backgrounds.

This third edition is a substantial revision of the book. We have rewritten many sections to incorporate new material, ideas, and examples, and to more fully explain some topics that were only briefly mentioned in previous editions. We have also woven the computer more tightly into the presentation, relying on JMP 7 and Design-Expert Version 7 for much of the computing, but also continuing to employ SAS for a few applications.

Chapters 1 through 4 contain the preliminary material essential to studying RSM. Chapter 1 is an introduction to the general field of RSM, describing typical applications such as (a) finding the levels of process variables that optimize a response of interest or (b) discovering what levels of these process variables will result in a product satisfying certain requirements or specifications on responses such as yield, molecular weight, purity, or viscosity. Chapter 2 is a summary of regression methods useful in response surface work, focusing on the basic ideas of least squares model fitting, diagnostic checking, and inference for the linear regression model. Chapters 3 and 4 describe two-level factorial and fractional factorial designs. These designs are essential for factor screening or identifying the correct set of process variables to use in the RSM study. They are also basic building blocks for many of the response surface designs discussed later in the text. Chapter 5 presents the method of steepest ascent, a simple but powerful optimization procedure used at the early stages of RSM to move the process from a region of relatively poor performance to one of greater potential. Chapter 6 introduces the analysis and optimization of a second-order response surface model. Both graphical and numerical techniques are presented. This chapter also includes techniques for the simultaneous optimization of several responses, a common problem in the application of RSM. Chapters 7 and 8 present detailed information on the choice of experimental designs for fitting response surface models. Chapter 7 is devoted to standard designs, including the central composite and Box–Behnken designs, and the important topic of blocking a response surface design. Chapter 8 covers small response surface designs, design optimality criteria, the use of computer-generated designs in RSM, and methods for evaluation of the prediction properties of response surface models constructed from various designs. We focus on variance dispersion graphs and fraction of design space plots, which are very important ways to summarize prediction properties. Chapter 9 contains more advanced RSM topics, including the use of mean square error as a design criterion, the effect of errors in controllable variables, RSM experiments for computer models, neural networks and RSM, split-plot type designs in a response surface setting, and the use of generalized linear models in the analysis of response surface experiments. Chapter 10 describes how the problem of robust parameter design originally proposed by Taguchi can be efficiently solved in the RSM framework. We show how RSM not only makes the original problem posed by Taguchi easier to solve, but also provides much more information to the analyst about process or system performance. This chapter also contains much information on robust parameter design and process robustness studies. Chapters 11 and 12 present techniques for designing and analyzing experiments that involve mixtures. A mixture experiment is a special type of response surface experiment in which the design factors are the components or ingredients of a mixture, and the response depends on the proportions of the ingredients that are present. Extensive sets of end-of-chapter problems are provided, along with a reference section.

The previous two editions of the text were written to emphasize methods that are useful in industry and that we have found useful in our own consulting experience. We have continued that applied focus in this new edition, though much new material has been added. We develop enough of the underlying theory to allow the reader to gain an understanding of the assumptions and conditions necessary to successfully apply RSM.

We are grateful to many individuals that have contributed meaningfully to this book. In particular, Dr. Bradley Jones, Mr. Pat Whitcomb, Dr. Geoff Vining, Dr. Soren Bisgaard, Dr. Connie Borror, Dr. Scott Kowalski, Dr. Dennis Lin, Dr. George Runger, and Dr. Enrique Del Castillo made many useful suggestions. Dr. Matt Carlyle and Dr. Enrique Del Castillo also provided some figures that were most helpful. We also thank the many classes of graduate students that have studied from the book and the instructors that have used the book. They have made many helpful comments and suggestions to improve the clarity of the presentation. We have tried to incorporate many of their suggestions. We also thank John Wiley & Sons for permission to use and adapt copyrighted material.

Blacksburg, Virginia | RAYMOND H. MYERS
Tempe, Arizona | DOUGLAS C. MONTGOMERY
Los Alamos, New Mexico | CHRISTINE M. ANDERSON-COOK
March, 2008 |

1

INTRODUCTION

1.1 RESPONSE SURFACE METHODOLOGY

Response surface methodology (RSM) is a collection of statistical and mathematical techniques useful for developing, improving, and optimizing processes. It also has important applications in the design, development, and formulation of new products, as well as in the improvement of existing product designs.

The most extensive applications of RSM are in the industrial world, particularly in situations where several input variables potentially influence some performance measure or quality characteristic of the product or process. This performance measure or quality characteristic is called the response. It is typically measured on a continuous scale, although attribute responses, ranks, and sensory responses are not unusual. Most real-world applications of RSM will involve more than one response. The input variables are sometimes called independent variables, and they are subject to the control of the engineer or scientist, at least for purposes of a test or an experiment.

Figure 1.1 shows graphically the relationship between the response variable yield (y) in a chemical process and the two process variables (or independent variables) reaction time (ξ₁) and reaction temperature (ξ₂). Note that for each value of ξ₁ and ξ₂ there is a corresponding value of yield y, and that we may view these values of the response yield as a surface lying above the time–temperature plane, as in Fig. 1.1a. It is this graphical perspective of the problem environment that has led to the term response surface methodology. It is also convenient to view the response surface in the two-dimensional time–temperature plane, as in Fig. 1.1b. In this presentation we are looking down at the time–temperature plane and connecting all points that have the same yield to produce contour lines of constant response. This type of display is called a contour plot.

Figure 1.1 (a) A theoretical response surface showing the relationship between yield of a chemical process and the process variables reaction time (ξ₁) and reaction temperature (ξ₂). (b) A contour plot of the theoretical response surface.

Clearly, if we could easily construct the graphical displays in Fig. 1.1, optimization of this process would be very straightforward. By inspection of the plot, we note that yield is maximized in the vicinity of time ξ₁ = 4 hr and temperature ξ₂ = 525ºC. Unfortunately, in most practical situations, the true response function in Fig. 1.1 is unknown. The field of response surface methodology consists of the experimental strategy for exploring the space of the process or independent variables (here the variables ξ₁ and ξ₂), empirical statistical modeling to develop an appropriate approximating relationship between the yield and the process variables, and optimization methods for finding the levels or values of the process variables ξ₁ and ξ₂ that produce desirable values of the responses (in this case that maximize yield).

1.1.1 Approximating Response Functions

In general, suppose that the scientist or engineer (whom we will refer to as the experimenter) is concerned with a product, process, or system involving a response y that depends on the controllable input variables ξ₁, ξ₂,…, ξk. The relationship is

(1.1)

where the form of the true response function f is unknown and perhaps very complicated, and ε is a term that represents other sources of variability not accounted for in f. Thus ε includes effects such as measurement error on the response, other sources of variation that are inherent in the process or system (background noise, or common/special cause variation in the language of statistical process control), the effect of other (possibly unknown) variables, and so on. We will treat ε as a statistical error, often assuming it to have a normal distribution with mean zero and variance σ². If the mean of ε is zero, then

(1.2)

The variables ξ1, ξ₂,…, ξk in Equation 1.2 are usually called the natural variables, because they are expressed in the natural units of measurement, such as degrees Celsius (ºC), pounds per square inch (psi), or grams per liter for concentration. In much RSM work it is convenient to transform the natural variables to coded variables x₁, x₂,…, xk, which are usually defined to be dimensionless with mean zero and the same spread or standard deviation. In terms of the coded variables, the true response function (1.2) is now written as

(1.3)

Because the form of the true response function f is unknown, we must approximate it. In fact, successful use of RSM is critically dependent upon the experimenter’s ability to develop a suitable approximation for f. Usually, a low-order polynomial in some relatively small region of the independent variable space is appropriate. In many cases, either a first-order or a second-order model is used. For the case of two independent variables, the first-order model in terms of the coded variables is

(1.4)

Figure 1.2 shows the three-dimensional response surface and the two-dimensional contour plot for a particular case of the first-order model, namely,

Figure 1.2 (a) Response surface for the first-order model = 50 + 8x₁ + 3x₂. (b) Contour plot for the first-order model.

In three dimensions, the response surface is a plane lying above the x₁, x₂ space. The contour plot shows that the first-order model can be represented as parallel straight lines of constant response in the x₁, x₂ plane.

The first-order model is likely to be appropriate when the experimenter is interested in approximating the true response surface over a relatively small region of the independent variable space in a location where there is little curvature in f. For example, consider a small region around the point A in Fig. 1.1b; the first-order model would likely be appropriate here.

The form of the first-order model in Equation 1.4 is sometimes called a main effects model, because it includes only the main effects of the two variables x₁ and x₂. If there is an interaction between these variables, it can be added to the model easily as follows:

(1.5)

This is the first-order model with interaction. Figure 1.3 shows the three-dimensional response surface and the contour plot for the special case

Figure 1.3 (a) Response surface for the first-order model with interaction = 50 + 8x₁ + 3x₂ – 4x₁x₂. (b) Contour plot for the first-order model with interaction.

Notice that adding the interaction term – 4x₁x₂ introduces curvature into the response function.

Often the curvature in the true response surface is strong enough that the first-order model (even with the interaction term included) is inadequate. A second-order model will likely be required in these situations. For the case of two variables, the second-order model is

(1.6)

This model would likely be useful as an approximation to the true response surface in a relatively small region around the point B in Fig. 1.1b, where there is substantial curvature in the true response function f.

Figure 1.4 presents the response surface and contour plot for the special case of the second-order model

Figure 1.4 (a) Response surface for the second-order model (b) Contour plot for the second-order model.

Notice the mound-shaped response surface and elliptical contours generated by this model. Such a response surface could arise in approximating a response such as yield, where we would expect to be operating near a maximum point on the surface.

The second-order model is widely used in response surface methodology for several reasons. Among these are the following:

1. The second-order model is very flexible. It can take on a wide variety of functional forms, so it will often work well as an approximation to the true response surface. Figure 1.5 shows several different response surfaces and contour plots that can be generated by a second-order model.

2. It is easy to estimate the parameters (the β’s) in the second-order model. The method of least squares, which is presented in Chapter 2, can be used for this purpose.

3. There is considerable practical experience indicating that second-order models work well in solving real response surface problems.

Figure 1.5 Some examples of types of surfaces defined by the second-order model in two variables x₁ and x₂. (Adapted with permission from Empirical Model Building and Response Surfaces, G. E. P. Box and N. R. Draper, John Wiley & Sons, New York, 1987.)

In general, the first-order model is

(1.7)

and the second-order model is

(1.8)

In some situations, approximating polynomials of order higher than two are used. The general motivation for a polynomial approximation for the true response function f is based on the Taylor series expansion around the point x₁₀, x₂₀,…, xk₀. For example, the first-order model is developed from the first-order Taylor series expansion

(1.9)

where x refers to the vector of independent variables and x₀ is the vector of independent variables at the specific point x₁₀, x₂₀,…, xk₀. In Equation 1.9 we have only included the first-order terms in the expansion, so if we let β₀ = f (x₁₀, x₂₀,…, xk₀), β₁ = (f / x₁),…, βk = (f =xk), we have the first-order approximating model in Equation 1.7. If we were to include second-order terms in Equation 1.9, this would lead to the second-order approximating model in Equation 1.8.

Finally, note that there is a close connection between RSM and linear regression analysis. For example, consider the model

The β’s are a set of unknown parameters. To estimate the values of these parameters, we must collect data on the system we are studying. Regression analysis is a branch of statistical model building that uses these data to estimate the β’s. Because, in general, polynomial models are linear functions of the unknown β’s, we refer to the technique as linear regression analysis. We will also see that it is very important to plan the data collection phase of a response surface study carefully. In fact, special types of experimental designs, called response surface designs, are valuable in this regard. A substantial part of this book is devoted to response surface designs.

1.1.2 The Sequential Nature of RSM

Most applications of RSM are sequential in nature. That is, at first some ideas are generated concerning which factors or variables are likely to be important in the response surface study. This usually leads to an experiment designed to investigate these factors with a view toward eliminating the unimportant ones. This type of experiment is usually called a screening experiment. Often at the outset of a response surface study there is a rather long list of variables that could be important in explaining the response. The objective of factor screening is to reduce this list of candidate variables to a relative few so that subsequent experiments will be more efficient and require fewer runs or tests. We refer to a screening experiment as phase zero of a response surface study. You should never undertake a response surface analysis until a screening experiment has been performed to identify the important factors.

Once the important independent variables are identified, phase one of the response surface study begins. In this phase, the experimenter’s objective is to determine if the current levels or settings of the independent variables result in a value of the response that is near the optimum (such as the point B in Fig. 1.1b), or if the process is operating in some other region that is (possibly) remote from the optimum (such as the point A in Fig. 1.1b). If the current settings or levels of the independent variables are not consistent with optimum performance, then the experimenter must determine a set of adjustments to the process variables that will move the process toward the optimum. This phase of response surface methodology makes considerable use of the first-order model and an optimization technique called the method of steepest ascent. These techniques will be discussed and illustrated in Chapter 5.

Phase two of a response surface study begins when the process is near the optimum. At this point the experimenter usually wants a model that will accurately approximate the true response function within a relatively small region around the optimum. Because the true response surface usually exhibits curvature near the optimum (refer to Fig. 1.1), a second-order model (or very occasionally some higher-order polynomial) will be used. Once an appropriate approximating model has been obtained, this model may be analyzed to determine the optimum conditions for the process. Chapter 6 will present techniques for the analysis of the second-order model and the determination of optimum conditions.

This sequential experimental process is usually performed within some region of the independent variable space called the operability region. For the chemical process illustrated in Fig. 1.1, the operability region is 0 hr < ξ₁ ≤ 7 hr and 100ºC ≤ ξ₂ ≤ 800ºC. Suppose we are currently operating at the levels ξ₁ = 2.5 hr and ξ₂ = 500ºC, shown as point A in Fig. 1.6. Now it is unlikely that we would want to explore the entire region of operability with a single experiment. Instead, we usually define a smaller region of interest or region of experimentation around the point A within the larger region of operability. Typically, this region of experimentation is either a cuboidal region, as shown around the point A in Fig. 1.6, or a spherical region, as shown around point B.

Figure 1.6 The region of operability and the region of experimentation.

The sequential nature of response surface methodology allows the experimenter to learn about the process or system under study as the investigation proceeds. This ensures that over the course of the RSM application the experimenter will learn the answers to questions such as (1) the location of the region of the optimum, (2) the type of approximating function required, (3) the proper choice of experimental designs, (4) how much replication is necessary, and (5) whether or not transformations on the responses or any of the process variables are required. A substantial portion of this book—Chapters 3, 4, 7, and 8—is devoted to designed experiments useful in RSM.

1.1.3 Objectives and Typical Applications of RSM

Response surface methodology is useful in the solution of many types of industrial problems. Generally, these problems fall into three categories:

1. Mapping a Response Surface over a Particular Region of Interest. Consider the chemical process in Fig. 1.1b. Normally, this process would operate at a particular setting of reaction time and reaction temperature. However, some changes to these normal operating levels might occasionally be necessary, perhaps to produce a product that meets other specific customer requirement. If the true unknown response function has been approximated over a region around the current operating conditions with a suitable fitted response surface (say a second-order surface), then the process engineer can predict in advance the changes in yield that will result from any readjustments to time and temperature.

2. Optimization of the Response. In the industrial world, a very important problem is determining the conditions that optimize the process. In the chemical process of Fig. 1.1b, this implies determining the levels of time and temperature that result in maximum yield. An RSM study that began near point A in Fig. 1.1b would eventually lead the experimenter to the region near point B. A second-order model could then be used to approximate the yield response in a narrow region around point B, and from examination of this approximating response surface the optimum levels or condition for time and temperature could be chosen.

3. Selection of Operating Conditions to Achieve Specifications or Customer Requirements. In most response surface problems there are several responses that must in some sense be simultaneously considered. For example, in the chemical process of Fig. 1.1, suppose that in addition to yield, there are two other responses: cost and concentration. We would like to maintain yield above 70%, while simultaneously keeping the cost below $34/pound; however, the customer has imposed specifications for concentration such that this important physical property must be 65 ± 3 g/liter.

One way that we could solve this problem is to obtain response surfaces for all three responses—yield, cost, and concentration—and then superimpose the contours for these responses in the time–temperature plane, as illustrated in Fig. 1.7. In this figure we have shown the contours for yield = 70%, cost = $34/pound, concentration = 62 g/liter, and concentration = 68 g/liter. The unshaded region in this figure represents the region containing operating conditions that simultaneously satisfy all requirements on the process.

Figure 1.7 The unshaded region showing the conditions for which yield ≥70%, cost ≤$34/pound, and 62 g/liter ≤ concentration ≤ 68 g/liter.

In practice, complex process optimization problems such as this can often be solved by superimposing appropriate response surface contours. However, it is not unusual to encounter problems with more than two process variables and more complex response requirements to satisfy. In such problems, other optimization methods that are more effective than overlaying contour plots will be necessary. We will discuss methodology for solving these types of problems in Chapter 6.

1.1.4 RSM and the Philosophy of Quality Improvement

During the last few decades, industrial organizations in the United States and Europe have become keenly interested in quality and process improvement. Statistical methods, including statistical process control (SPC) and design of experiments, play a key role in this activity. Quality improvement is most effective when it occurs early in the product and process development cycle. It is very difficult and often expensive to manufacture a poorly designed product. Industries such as semiconductors and electronics, aerospace, automotive, biotechnology and pharmaceuticals, medical devices, chemical, and process industries are all examples where experimental design methodology has resulted in shorter design and development time for new products, as well as products that are easier to manufacture, have higher reliability, have enhanced field performance, and meet or exceed customer requirements.

RSM is an important branch of experimental design in this regard. RSM is a critical technology in developing new processes, optimizing their performance, and improving the design and/or formulation of new products. It is often an important concurrent engineering tool, in that product design, process development, quality, manufacturing engineering, and operations personnel often work together in a team environment to apply RSM. The objectives of quality improvement, including reduction of variability and improved product and process performance, can often be accomplished directly using RSM.

1.2 PRODUCT DESIGN AND FORMULATION (MIXTURE PROBLEMS)

Many product design and development activities involve formulation problems, in which two or more ingredients are mixed together. For example, suppose we are developing a new household cleaning product. This product is formulated by mixing several chemical surfactants together. The product engineer or scientist would like to find an appropriate blend of the ingredients so that the grease-cutting capability of the cleaner is good, and so that it generates an appropriate level of foam when in use. In this situation the response variables—namely, grease-cutting ability and amount of foam—depend on the percentages or proportions of the individual chemical surfactants (the ingredients) that are present in the product formulation.

There are many industrial problems where the response variables of interest in the product are a function of the proportions of the different ingredients used in its formulation. This is a special type of response surface problem called a mixture problem.

While we traditionally think of mixture problems in the product design or formulation environment, they occur in many other settings. Consider plasma etching of silicon wafers, a common manufacturing process in the semiconductor industry. Etching is usually accomplished by introducing a blend of gases inside a chamber containing the wafers. The measured responses include the etch rate, the uniformity of the etch, and the selectivity (a measure of the relative etch rates of the different materials on the wafer). All of these responses are a function of the proportions of the different ingredients blended together in the etching chamber.

There are special experimental design techniques and model-building methods for mixture problems. These techniques are discussed in Chapters 11 and 12.

1.3 ROBUST DESIGN AND PROCESS ROBUSTNESS STUDIES

It is well known that variation in key performance characteristics can result in poor product and process quality. During the 1980s, considerable attention was given to this problem, and methodology was developed for using experimental design, specifically for the following:

1. For designing products or processes so that they are robust to environment conditions.

2. For designing or developing products so that they are robust to component variation.

3. For minimizing variability in the output response of a product around a target value.

By robust, we mean that the product or process performs consistently on target and is relatively insensitive to factors that are difficult to control.

Professor Genichi Taguchi used the term robust parameter design (or RPD) to describe his approach to this important class of industrial problems. Essentially, robust parameter design methodology strives to reduce product or process variation by choosing levels of controllable factors (or parameters) that make the system insensitive (or robust) to changes in a set of uncontrollable factors that represent most of the sources of variability. Taguchi referred to these uncontrollable factors as noise factors. These are the environmental factors such as humidity levels, changes in raw material properties, product aging, and component variability referred to in 1 and 2 above. We usually assume that these noise factors are uncontrollable in the field, but can be controlled during product or process development for purposes of a designed experiment.

Considerable attention has been focused on the methodology advocated by Taguchi, and a number of flaws in his approach have been discovered. However, there are many useful concepts in his philosophy, and it is relatively easy to incorporate these within the framework of response surface methodology. In Chapter 10 we will present the response surface approach to robust design and process robustness studies.

1.4 USEFUL REFERENCES ON RSM

The origin of RSM is the seminal paper by Box and Wilson (1951). They also describe the application of RSM to chemical processes. This paper had a profound impact on industrial applications of experimental design, and was the motivation of much of the research in the field. Many of the key research and applications papers are cited in this book.

There have also been four review papers published on RSM: Hill and Hunter (1966), Mead and Pike (1975), Myers et al. (1989) and Myers et al. (2004). The paper by Myers (1999) on future directions in RSM offers a view of research needs in the field. There are also two other full-length books on the subject: Box and Draper (1987) and Khuri and Cornell (1996). A second edition of the Box and Draper book was published in 2007 with a slightly different title [Box and Draper (2007)]. The monograph by Myers (1976) was the first book devoted exclusively to RSM.

2

BUILDING EMPIRICAL MODELS

2.1 LINEAR REGRESSION MODELS

The practical application of response surface methodology (RSM) requires developing an approximating model for the true response surface. The underlying true response surface is typically driven by some unknown physical mechanism. The approximating model is based on observed data from the process or system and is an empirical model. Multiple regression is a collection of statistical techniques useful for building the types of empirical models required in RSM.

As an example, suppose that we wish to develop an empirical model relating the effective life of a cutting tool to the cutting speed and the tool angle. A first-order response surface model that might describe this relationship is

(2.1)

where y represents the tool life, x₁ represents the cutting speed, and x₂ represents the tool angle. This is a multiple linear regression model with two independent variables. We often call the independent variables predictor variables or regressors. The term “linear” is used because Equation 2.1 is a linear function of the unknown parameters β₀, β₁, and β₂. The model describes a plane in the two-dimensional x₁, x₂ space. The parameter β₀ fixes the intercept of the plane. We sometimes call β₁ and β₂ partial regression coefficients, because β₁ measures the expected change in y per unit change in x₁ when x₂ is held constant, and β₂ measures the expected change in y per unit change in x₂ when x₁ is held constant.

In general, the response variable y may be related to k regressor variables. The model

(2.2)

is called a multiple linear regression model with k regressor variables. The parameters βj, j = 0, 1, … , k, are called the regression coefficients. This model describes a hyperplane in the k-dimensional space of the regressor variables {xj}. The parameter βj represents the expected change in response y per unit change in xj when all the remaining independent variables xi (i ≠ j) are held constant.

Models that are more complex in appearance than Equation 2.2 may often still be analyzed by multiple linear regression techniques. For example, considering adding an interaction term to the first-order model in two variables, say

(2.3)

If we let x₃ = x₁x₂ and β₃ = β₁₂, then Equation 2.3 can be written as

(2.4)

which is a standard multiple linear regression model with three regressors. As another example, consider the second-order response surface model in two variables:

(2.5)

If we let and β₅ = β₁₂, then this becomes

(2.6)

which is a linear regression model. In general, any regression model that is linear in the parameters (the β-values) is a linear regression model, regardless of the shape of the response surface that it generates.

In this chapter we will present and illustrate methods for estimating the parameters in multiple linear regression models. This is often called model fitting. We will also discuss methods for testing hypotheses and constructing confidence intervals for these models, as well as for checking the adequacy of the model fit. Our focus is primarily on those aspects of regression analysis useful in RSM. For more complete presentations of regression, refer to Montgomery, Peck, and Vining (2006) and Myers (1990).

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