advanced review response surface methodology

Advanced Review

Response surface methodologyAndre I. Khuri1∗ and Siuli Mukhopadhyay2

The purpose of this article is to provide a survey of the various stages in thedevelopment of response surface methodology (RSM). The coverage of thesestages is organized in three parts that describe the evolution of RSM since itsintroduction in the early 1950s. Part I covers the period, 1951–1975, during which theso-called classical RSM was developed. This includes a review of basic experimentaldesigns for fitting linear response surface models, in addition to a description ofmethods for the determination of optimum operating conditions. Part II, whichcovers the period, 1976–1999, discusses more recent modeling techniques in RSM,in addition to a coverage of Taguchi’s robust parameter design and its responsesurface alternative approach. Part III provides a coverage of further extensionsand research directions in modern RSM. This includes discussions concerningresponse surface models with random effects, generalized linear models, andgraphical techniques for comparing response surface designs. 2010 John Wiley &Sons, Inc. WIREs Comp Stat 2010 2 128–149

PART I. THE FOUNDATIONAL YEARS:1951–1975

An Introduction and Some Preliminaries

Response surface methodology (RSM) consistsof a group of mathematical and statistical

techniques used in the development of an adequatefunctional relationship between a response of interest,y, and a number of associated control (or input)variables denoted by x1, x2, . . . , xk. In general, such arelationship is unknown but can be approximated bya low-degree polynomial model of the form

y = f ′(x)β + ε (1)

where x = (x1, x2, . . . , xk)′, f (x) is a vector function ofp elements that consists of powers and cross- productsof powers of x1, x2, . . . , xk up to a certain degreedenoted by d (≥ 1), β is a vector of p unknownconstant coefficients referred to as parameters, andε is a random experimental error assumed to havea zero mean. This is conditioned on the belief thatmodel (1) provides an adequate representation of theresponse. In this case, the quantity f ′(x)β represents

∗Correspondence to: [email protected] of Statistics, University of Florida, Gainesville, FL32611, USA2Department of Mathematics, Indian Institute of TechnologyBombay, Powai, Mumbai 400076, India

DOI: 10.1002/wics.73

the mean response, that is, the expected value of y,and is denoted by µ(x).

Two important models are commonly used inRSM. These are special cases of model (1) and includethe first-degree model (d = 1),

y = β0 +k∑

i=1

β ixi + ε (2)

and the second-degree model (d = 2)

y = β0 +k∑

i=1

β ixi +∑ ∑

i<j

β ijxixj +k∑

i=1

β iix2i + ε.

(3)

The purpose of considering a model such as (1) isthreefold:

1. To establish a relationship, albeit approximate,between y and x1, x2, . . . , xk that can be used topredict response values for given settings of thecontrol variables.

2. To determine, through hypothesis testing, signif-icance of the factors whose levels are representedby x1, x2, . . . , xk.

3. To determine the optimum settings ofx1, x2, . . . , xk that result in the maximum (orminimum) response over a certain region ofinterest.

128 2010 John Wiley & Sons, Inc. Volume 2, March/Apr i l 2010

WIREs Computational Statistics Response surface methodology

In order to achieve the above three objectives, a seriesof n experiments should first be carried out, in eachof which the response y is measured (or observed) forspecified settings of the control variables. The totalityof these settings constitutes the so-called responsesurface design, or just design, which can be representedby a matrix, denoted by D, of order n × k called thedesign matrix,

D =

x11 x12 . . . x1kx22 x22 . . . x2k. . . . . .

. . . . . .

. . . . . .

xn1 xn2 . . . xnk

(4)

where xui denotes the uth design setting of xi

(i = 1, 2, . . . , k; u = 1, 2, . . . , n). Each row of Drepresents a point, referred to as a design point,in a k-dimensional Euclidean space. Let yu denotethe response value obtained as a result of applyingthe uth setting of x, namely xu = (xu1, xu2, . . . , xuk)′(u = 1, 2, . . . , n). From Eq. (1), we then have

yu = f ′(xu)β + εu, u = 1, 2, . . . , n (5)

where εu denotes the error term at the uthexperimental run. Model (5) can be expressed inmatrix form as

y = Xβ + ε (6)

where y = (y1, y2, . . . , yn)′, X is a matrix of ordern × p whose uth row is f ′(xu), and ε = (ε1, ε2,. . . , εn)′. Note that the first column of X is the columnof ones 1n.

Assuming that ε has a zero mean and avariance–covariance matrix given by σ 2In, the so-called ordinary least-squares estimator of β is (seee.g., Ref 1, Section 2.3)

β = (X ′X)−1X ′y. (7)

The variance–covariance matrix of β is then of theform

Var(β) = (X ′X)−1X ′(σ 2In)X(X ′X)−1

= σ 2 (X ′X)−1. (8)

Using β, an estimate, µ(xu), of the mean response atxu is obtained by replacing β by β, that is,

µ(xu) = f ′(xu)β, u = 1, 2, . . . , n. (9)

The quantity f ′(xu)β also gives the so-called predictedresponse, y(xu), at the uth design point (u =1, 2, . . . , n). In general, at any point, x, in anexperimental region, denoted by R, the predictedresponse y(x) is

y(x) = f ′(x)β, x ∈ R. (10)

Since β is an unbiased estimator of β, y(x) is anunbiased estimator of f ′(x)β, which is the meanresponse at x ∈ R. Using Eq. (8), the variance of y(x)is of the form

Var[y(x)] = σ 2 f ′(x)(X ′X)−1f (x). (11)

The proper choice of design is very important in anyresponse surface investigation. This is true because thequality of prediction, as measured by the size of theprediction variance, depends on the design matrix Das can be seen from formula (11). Furthermore, thedetermination of the optimum response amounts tofinding the optimal value of y(x) over the region R.It is therefore imperative that the prediction variancein Eq. (11) be as small as possible provided that thepostulated model in Eq. (1) does not suffer from lackof fit (for a study of lack of fit of a fitted responsemodel, see e.g., Ref 1, Section 2.6).

Some Common Design PropertiesThe choice of design depends on the properties itis required, or desired, to have. Some of the designproperties considered in the early development ofRSM include the following:

OrthogonalityA design D is said to be orthogonal if the matrix X ′Xis diagonal, where X is the model matrix in Eq. (6).The advantage of this approach is that the elementsof β will be uncorrelated because the off-diagonalelements of Var(β) in Eq. (8) will be zero. If theerror vector ε in Eq. (6) is assumed to be normallydistributed as N(0, σ 2In), then these elements will bealso stochastically independent. This makes it easierto test the significance of the unknown parameters inthe model.

RotatabilityA design D is said to be rotatable if the predictionvariance in Eq. (11) is constant at all points that areequidistant from the design center, which, by a propercoding of the control variables, can be chosen to be thepoint at the origin of the k-dimensional coordinates

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system. It follows that Var[y(x)] is constant at allpoints that fall on the surface of a hyperspherecentered at the origin, if the design is rotatable.The advantage of this property is that the predictionvariance remains unchanged under any rotation ofthe coordinate axes. In addition, if optimization ofy(x) is desired on concentric hyperspheres, as in theapplication of ridge analysis, which will be discussedlater, then it would be desirable for the design to berotatable. This makes it easier to compare the valuesof y(x) on a given hypersphere as all such values havethe same variance.

The necessary and sufficient condition for adesign to be rotatable was given by Box and Hunter.2

More recently, Khuri3 introduced a measure ofrotatability as a function of the so-called moments ofthe design under consideration (see e.g., Appendix 2Bin Ref 1). The function is expressible as a percentagetaking large values for a high degree of rotatability.The value 100 is attained when the design is rotatable.The advantages of this measure are:

1. The ability to compare designs on the basis ofrotatability.

2. The assessment of the extent of departure fromrotatability of a design whose rotatability maybe ‘sacrificed’ to satisfy another desirable designproperty.

3. The ability to improve rotatability by a properaugmentation of a nonrotatable design.

Uniform PrecisionA rotatable design is said to have the additionaluniform precision property if Var[y(x)] at the originis equal to its value at a distance of one from theorigin. This property, which was also introducedby Box and Hunter,2 provides for an approximateuniform distribution of the prediction variance insidea hypersphere of radius one. This helps in producingsome stability in the prediction variance in the vicinityof the design center.

Design RobustnessBox and Draper4 listed several additional designproperties that pertain to detection of lack of fit,generation of satisfactory distribution of informationthroughout the experimental region, estimation of theerror variance, insensitivity to outliers and to errorsmade in the actual implementation of the settingsof the control variables. These properties provideguidelines for the choice of a design (i.e., a ‘wishlist’). It is not, however, expected that a single designwill satisfy all of these properties. A design is said to

be robust if its properties are not severely impactedby failures to satisfy the assumptions made about themodel and the error distribution.

Design OptimalityOptimal designs are those that are constructed on thebasis of a certain optimality criterion that pertainsto the ‘closeness’ of the predicted response, y(x), tothe mean response, µ(x), over a certain region ofinterest denoted by R. The design criteria that addressthe minimization of the variance associated with theestimation of model (1)’s unknown parameters arecalled variance-related criteria. The most prominentof such criteria is the D-optimality criterion thatmaximizes the determinant of the matrix X ′X. Thisamounts to the minimization of the size of theconfidence region on the vector β in model (6).Actually, this criterion results in the so-called discreteD-optimal design as compared with the continuous D-optimal design, which was introduced by Jack Kiefer.The latter design is based on the notion that a designrepresents a probability measure defined on the regionR. A discrete design is then treated as a special caseconsisting of a collection of n points in R that are notnecessarily distinct.

Another variance-related criterion that is closelyrelated to D-optimality is the G-optimality criterionwhich requires the minimization of the maximumover R of the prediction variance in Eq. (11).Kiefer developed a continuous counterpart of thiscriterion and showed that it is equivalent tothe continuous D-optimality criterion. This resultis based on the equivalence theorem proved byKiefer and Wolfowitz.5 Other less-known variance-related criteria include A-optimality and E-optimality.See Ref 6, Chapter 4, for a description of thesecriteria.

These variance-related criteria are often referredto as alphabetic optimality. They are meaningfulwhen the fitted model in Eq. (1) represents the truerelationship connecting y to its control variables.

Designs for First- and Second-DegreeModelsAs was pointed out earlier in the Section onIntroduction and Some Preliminaries, the first-degree model in Eq. (2) and second-degree model inEq. (3) are the most-frequently used approximatingpolynomial models in classical RSM. Designs forfitting first-degree models are called first-order designsand those for fitting second-degree models are referredto as second-order designs.



First-Order DesignsThe most common first-order designs are 2k factorial(k is the number of control variables), Plackett–Bur-man, and simplex designs.

The 2k Factorial DesignIn a 2k factorial design, each control variable ismeasured at two levels, which can be coded to takethe values, −1, 1, that correspond to the so-called lowand high levels, respectively, of each variable. Thisdesign consists of all possible combinations of suchlevels of the k factors. Thus, each row of the designmatrix D in Eq. (4) consists of all 1s, all −1s, or acombination of 1s and −1s and represents a particulartreatment combination. In this case, the number, n,of experimental runs is equal to 2k provided thatno single design point is replicated more than once.For example, in a chemical experiment, the controlvariables are x1 = temperature of a reaction measuredat 250, 300 (◦C), x2 = pressure set at 10, 16 (psi), andx3 = time of the reaction taken at 4, 8 (minutes). Thecoded settings, ±1, for x1, x2, x3 are attained throughthe linear transformation,

x1 = temperature − 27525

(12)

x2 = pressure − 133

(13)

x3 = time − 62

. (14)

The corresponding 23 design matrix is of order 8 × 3of the form

D =

−1 −1 −11 −1 −1

−1 1 −1−1 −1 1

1 1 −11 −1 1

−1 1 11 1 1

. (15)

If k is large (k ≥ 5), the 2k design requires alarge number of design points. Since the number ofunknown parameters in Eq. (2) is only k + 1, fractionsof 2k can be considered to fit such a model. Forexample, we can consider a one-half fraction designthat consists of one-half the number of points of a 2k

design, or a one-fourth fraction design that consistsof one- fourth the number of points of a 2k design.In general, a 2−mth fraction of a 2k design consistsof 2k−m points from a full 2k design. Here, m is apositive integer such that 2k−m ≥ k + 1 so that all the

k + 1 parameters in model (2) can be estimated. Theconstruction of fractions of a 2k design is carried outin a particular manner, a description of which can befound in several experimental design textbooks, suchas Refs 7–9. See also Chapter 3 in Ref 1.

The Plackett–Burman DesignThe Plackett–Burman design allows two levels for eachof the k control variables, just like a 2k design, butrequires a much smaller number of experimental runs,especially if k is large. It is therefore more economicalthan the 2k design. Its number, n, of design points isequal to k + 1, which is the same as the number ofparameters in model (2). In this respect, the designis said to be saturated because its number of designpoints is equal to the number of parameters to beestimated in the model. Furthermore, this design isavailable only when n is a multiple of 4. Therefore, itcan be used when the number, k, of control variablesis equal to 3, 7, 11, 15,....

To construct a Plackett–Burman design in kvariables, a row is first selected whose elements areequal to −1 or 1 such that the number of 1s is k+1

2 andthe number of −1s is k−1

2 . The next k − 1 rows aregenerated from the first row by shifting it cyclicallyone place to the right k − 1 times. Then, a row ofnegative ones is added at the bottom of the design.For example, for k = 7, the design matrix, D, haseight points whose coordinates are x1, x2, . . . , x7 andis of the form

D =

1 1 1 −1 1 −1 −1−1 1 1 1 −1 1 −1−1 −1 1 1 1 −1 1

1 −1 −1 1 1 1 −1−1 1 −1 −1 1 1 1

1 −1 1 −1 −1 1 11 1 −1 1 −1 −1 1

−1 −1 −1 −1 −1 −1 −1

. (16)

Design arrangements for k = 3, 7, 11, . . . , 99 factorscan be found in Ref 10.

The Simplex DesignThe simplex design is also a saturated design withn = k + 1 points. Its design points are located at thevertices of a k-dimensional regular-sided figure, or asimplex, characterized by the fact that that the angle,θ , which any two points make with the design center(located at the origin of the coordinates system) is suchthat cos θ = −1

k . For example, for k = 2, the simplexdesign consists of the vertices of an equilateral trianglewhose center is (0, 0), and for k = 3, the design pointsare the vertices of a tetrahedron centered at (0, 0, 0).



Box11 presented a procedure for constructinga simplex design using a particular pattern of aone-factor-at-a-time design. This procedure is alsoexplained in Ref 1, Section 3.3.5. The simplexdesign is a less frequently used design than 2k orPlackett–Burman designs. This is because the actualsettings in a simplex design are, in general, difficult toattain exactly in a real experimental situation.

All the above designs (2k or fractions of,Plackett–Burman, simplex) share the same propertyof being orthogonal. For a first-order design,orthogonality can be achieved if the design matrixD is such that D′D is diagonal. We may recall thatan orthogonal design causes the variance–covariancematrix of β, the least-squares estimator of the vector β

of unknown parameters in the model, to be diagonal (ifthe error vector ε is assumed to have a zero mean anda variance–covariance matrix σ 2In). This means thatthe elements of β are uncorrelated, hence independentunder the normality assumption on ε. Furthermore,it can be shown that under an orthogonal design,the variances of the elements of β have minimumvalues (see Section 3.3 in Ref 1). This means thatan orthogonal first-order design provides maximumprecision for estimating the unknown parameters inmodel (2).

Second-Order DesignsThe number of parameters in the second-degreemodel in Eq. (3) is p = 1 + 2k + 1

2k(k − 1). Hence,the number of distinct design points of a second-order design must be at least equal to p. The designsettings are usually coded so that 1

n

∑nu=1 xui = 0 and

1n

∑nu=1 x2

ui = 1, i = 1, 2, . . . , k, where n is the numberof experimental runs and xui is the uth setting of theith control variable (u = 1, 2, . . . , n).

The most frequently used second-order designsare the 3k factorial, central composite, and theBox–Behnken designs.

The 3k Factorial DesignThe 3k factorial design consists of all the combinationsof the levels of the k control variables which havethree levels each. If the levels are equally spaced,then they can be coded so that they correspond to−1, 0, 1. The number of experimental runs for thisdesign is 3k, which can be very large for a largek. Fractions of a 3k design can be considered toreduce the cost of running such an experiment. Ageneral procedure for constructing fractions of 3k

is described in Montgomery9 (Chapter 9). See alsoRef 12, Appendix 2.

The Central Composite Design (CCD)This is perhaps the most popular of all second-orderdesigns. It was first introduced in Ref 13. This designconsists of the following three portions:

1. A complete (or a fraction of) 2k factorial designwhose factors’ levels are coded as −1, 1. This iscalled the factorial portion.

2. An axial portion consisting of 2k pointsarranged so that two points are chosen on theaxis of each control variable at a distance of α

from the design center (chosen as the point atthe origin of the coordinates system).

3. n0 center points.

Thus, the total number of design points in a CCDis n = 2k + 2k + n0. For example, a CCD for k = 2,α = √

2, n0 = 2 has the form

D =

−1 −11 −1

−1 11 1

−√2 0√2 00 −√

20

√2

0 00 0

. (17)

We note that the CCD is obtained by augmentinga first-order design, namely, the 2k factorial withadditional experimental runs, namely, the 2k axialpoints and the n0 center-point replications. Thus, thisdesign is developed in a manner consistent with thesequential nature of a response surface investigationin starting with a first-order design, to fit a first-degreemodel, followed by the addition of design points tofit the larger second-degree model. The first-orderdesign serves in a preliminary phase to get initialinformation about the response system and to assessthe importance of the factors in a given experiment.The additional experimental runs are chosen for thepurpose of getting more information that can lead tothe determination of optimum operating conditions onthe control variables using the second-degree model.

The values of α (or the axial parameter) and n0,the number of center-point replications, are chosen sothat the CCD can acquire certain desirable properties.For example, choosing α = F1/4, where F denotesthe number of points in the factorial portion, causesthe CCD to be rotatable. The value of n0 can



then be chosen so that the CCD can achieve eitherthe orthogonality property or the uniform precisionproperty. Note that orthogonality of a second-orderdesign is attainable only after expressing model (3)in terms of orthogonal polynomials as explained inBox and Hunter2 (pp. 200–201). See also Khuri andCornell1 (Section 4.3). In particular, Table 4.3 inKhuri and Cornell’s book can be used to determinethe value of n0 for a rotatable CCD to have eitherthe additional orthogonality property or the uniformprecision property.

The Box–Behnken DesignThis design was developed by Box and Behnken.14

It provides three levels for each factor and consistsof a particular subset of the factorial combinationsfrom the 3k factorial design. The actual constructionof such a design is described in the three RSM booksBox and Draper,15 Section 15.4, Khuri and Cornell,1

Section 4.5.2, and Myers and Montgomery,16 Sec-tion 7.4.7.

The use of the Box–Behnken design is popular inindustrial research because it is an economical designand requires only three levels for each factor wherethe settings are −1, 0, 1. Some Box–Behnken designsare rotatable, but, in general, this design is not alwaysrotatable. Box and Behnken14 list a number of designarrangements for k = 3, 4, 5, 6, 7, 9, 10, 11, 12, and16 factors.

Other second-order designs are available butare not as frequently used as the ones we havealready mentioned. Some of these designs includeHoke17 designs, Box–Draper saturated designs (seeRef 18), uniform shell designs by Doehlert,19 andhybrid designs by Roquemore.20

Determination of Optimum ConditionsOne of the main objectives of RSM is the deter-mination of the optimum settings of the controlvariables that result in a maximum (or a mini-mum) response over a certain region of interest,R. This requires having a ‘good’ fitting model thatprovides an adequate representation of the meanresponse because such a model is to be utilized todetermine the value of the optimum. Optimizationtechniques used in RSM depend on the nature of thefitted model. For first-degree models, the method ofsteepest ascent (or descent) is a viable technique forsequentially moving toward the optimum response.This method is explained in detail in Myers andMontgomery,16 Khuri and Cornell1 (Chapter 5), andBox and Draper15 (Chapter 6). Myers and Khuri21

developed certain improvements regarding the stop-ping rule used in the execution of this method.

Since the first-degree model is usually used at thepreliminary stage of a response surface investigation,we shall only mention here optimization techniquesthat are applicable to second-degree models. Suchmodels are used after a series of experiments havebeen sequentially carried out leading up to a regionthat is believed to contain the location of the optimumresponse.

Optimization of a Second-Degree ModelLet us consider the second-degree model in Eq. (3),which can be written as

y = β0 + x′β∗ + x′Bx + ε (18)

where x = (x1, x2, . . . , xk)′, β∗ = (β1, β2, . . . , βk)′, andB is a symmetric matrix of order k × k whose ithdiagonal element is β ii (i = 1, 2, . . . , k), and its (i, j)thoff-diagonal element is 1

2β ij (i, j = 1, 2, . . . , k; i �= j).If n observations are obtained on y using a designmatrix D as in Eq. (4), then Eq. (18) can be writtenin vector form as in Eq. (6), where the parametervector β consists of β0 and the elements of β∗ and B.Assuming that E(ε) = 0 and Var(ε) = σ 2In, the least-squares estimate of β is β as given in Eq. (7). Thepredicted response at a point x in the region R is thenof the form

y(x) = β0 + x′β∗ + x′Bx (19)

where β0 and the elements of β∗ and B are theleast-squares estimates of β0 and the correspondingelements of β∗ and B, respectively.

The Method of Ridge AnalysisThis is a useful procedure for optimizing the predictedresponse based on the fitted second-degree model inEq. (19). It was introduced by Hoerl22 and formalizedby Draper.23 This method optimizes y(x) in Eq. (19)subject to x being on the surface of a hypersphere ofradius r and centered at the origin, namely,

k∑i=1

x2i = r2. (20)

This constrained optimization is conducted usingseveral values of r corresponding to hyperspherescontained within the region R. The rationale for doingthis is to get information about the optimum at variousdistances from the origin within R.

Since this optimization is subject to the equalityconstraint given by Eq. (20), the method of Lagrange



multipliers can be used to search for the optimum. Letus therefore consider the function

H = β0 + x′β∗ + x′Bx − λ(x′x − r2) (21)

where λ is a Lagrange multiplier. Differentiating Hwith respect to x and equating the derivative to zero,we get

β∗ + 2(Bx − λx) = 0. (22)

Solving for x, we obtain

x = −12

(B − λIn)−1β∗. (23)

The solution in Eq. (23) represents just a stationarypoint of y(x). A maximum (minimum) is achieved atthis point if the Hessian matrix, that is, the matrix∂

∂x[

∂H∂x′

]of second-order partial derivatives of H with

respect to x is negative definite (positive definite).From Eq. (22), this matrix is given by

∂

∂x

[∂H∂x′

]= 2(B − λIn). (24)

Therefore, to achieve a maximum, Draper23 suggestedthat λ be chosen larger than the largest eigenvalue ofB. Such a choice causes B − λIn to be negative definite.Choosing λ smaller than the smallest eigenvalue of Bcauses B − λIn to be positive definite, which resultsin a minimum. Thus, by choosing several values of λ

in this fashion, we can, for each λ, find the locationof the optimum (maximum or minimum) by usingformula (23) and hence obtain the value of x′x = r2.The solution from Eq. (23) is feasible provided that rcorresponds to a hypersphere that falls entirely withinthe region R. The optimal value of y(x) is computedby substituting x from Eq. (23) into the right-handside of Eq. (19). This process generates plots of y andxi against r (i = 1, 2, . . . , k). These plots are useful indetermining, at any given distance r from the origin,the value of the optimum as well as its location. Moredetails concerning this method can be found in Myersand Montgomery,16 Khuri and Cornell1 (Section 5.7),and Box and Draper15 (Chapter 19).

Since the method of ridge analysis optimizesy(x) on concentric hyperspheres within the regionR, its application is meaningful provided that theprediction variance in formula (11) is constant onthe surface of any given hypersphere. This calls forthe use of a rotatable design to fit model (18).If, however, the design is not rotatable, then theprediction variance can vary appreciably on thesurface of a hypersphere, which may lead to poor

estimates of the optimum response. For this reason,Khuri and Myers24 proposed a modification of themethod of ridge analysis whereby the optimization ofy(x) is carried out under an added constraint on thesize of the prediction variance. This modification canproduce better optimization results when the designused to fit model (18) is not rotatable. More recently,Paul and Khuri25 extended the use of Khuri andMyers’ modification to linear models where the errorvariances are heterogeneous and also to generalizedlinear models.

PART II. FURTHER DEVELOPMENTSAND THE TAGUCHI ERA: 1976–1999

Multiresponse ExperimentsIn a multiresponse experiment, measurements onseveral responses are obtained for each setting of agroup of control variables. Examples of multiresponseexperiments are numerous, for example, a chemicalengineer may be interested in maximizing theyield while minimizing the cost of a certainchemical process. Refs 26–28 cited several papers inwhich multiresponse experiments were studied. Whileanalyzing the data from a multiresponse experiment,special attention should be given to the correlatednature of the data within experimental runs. Usually,it is assumed that the responses are correlated withinruns but independent otherwise.

Suppose that n is the number of experimentalruns and q is the number of responses. Then the ithresponse may be modeled as (see Ref 1, pp. 252–254)

yi = Xiβi + εi, i = 1, . . . , q (25)

where yi is an n × 1 vector of observations on the ithresponse, Xi is an n × pi known matrix of rank pi, βiis a vector of pi unknown regression parameters, andεi is a vector of random errors associated with the ithresponse. Using matrix notation, the above model canbe expressed as

Y = Xβ + ε (26)

where Y = [y′1, . . . , y′

q]′, X is the block-diagonalmatrix, diag(X1, . . . , Xq), β = [β′

1, . . . , β′q]′, and ε =

[ε′1, . . . , ε′

q]′. It is assumed that E(ε) = 0 and thevariance–covariance matrix Var(ε) = � ⊗ I, where Iis the n × n identity matrix. The best linear unbiasedestimate of β is given by

β = [X′(�−1 ⊗ I)X]−1X′(�−1 ⊗ I)Y. (27)



An estimate of � is used to find β when � is unknown.A commonly used estimate of � is the one proposedby Zellner.29

In selecting a design optimality criterion formultiresponse experiments, one needs to consider allthe responses simultaneously. Draper and Hunter30

proposed a criterion for the estimation of theunknown parameters in a multiresponse situation.Their criterion was used for selecting additionalexperimental runs after a certain number of runshave already been chosen. The approach used wasBayesian, with the unknown parameter vector havinga uniform prior distribution. The variance–covariancematrix of the responses was assumed to be known.Box and Draper31 later extended Draper andHunter’s30 criterion by considering response surfacedesigns with blocks. The most common criterion formultiresponse designs is the D-optimality criterion.One such criterion was developed by Fedorov32 forlinear multiresponse designs. Fedorov’s procedure wassequential in nature and was used for constructingD-optimal designs. However, it required knowledgeof the variance–covariance matrix associated with theseveral response variables. Wijesinha and Khuri33 latermodified Fedorov’s procedure by using an estimate of� at each step of the sequential process. Some recentworks on D-optimal designs for linear multiresponsemodels include those of Krafft and Schaefer,34

Bischoff,35 Chang,36,37 Imhof,38 and Atashgah andSeifi.39 Locally D-optimal designs for describing thebehavior of a biological system were constructedby Hatzis and Larntz40 for nonlinear multiresponsemodels. Other design criteria for linear multiresponsemodels include the power criterion of Wijesinha andKhuri41 and the robustness criterion of Wijesinha andKhuri42 and Yue.43

The model given in Eq. (26) is said to sufferfrom lack of fit if it does not represent the q true meanresponses adequately. Due to the correlated natureof the responses in a multiresponse situation, lackof fit of one response may affect the fit of the otherresponses. Khuri44 proposed a multivariate test of lackof fit that considers all the q responses simultaneously.His test was based on Roy’s union intersectionprinciple45 (Chapters 4 and 5) and required thatreplicated observations on all responses be taken atsome points of the experimental region. Levy andNeill46 considered additional multivariate lack of fittests and used simulations to compare the powerfunctions of these tests.

One important objective of multiresponse exper-imentation is to determine the optimum operatingconditions on the control variables that lead to thesimultaneous optimization of the predicted values of

the responses within a region of interest. In mul-tiresponse experiments, the meaning of ‘optimum’ issometimes unclear as there is no unique way to orderthe multiresponse data. Conditions that are optimalfor one response may be far from optimal or evenphysically impractical for the other responses from theexperimental point of view. For example, in a dose–re-sponse experiment, where both efficacy and toxicityresponses are measured at each dose, the experimentermay wish to find the dose level of the drug(s) whichsimultaneously maximize efficacy while minimizingtoxicity. Common knowledge is that as the dose of adrug increases, so do its efficacy and toxic side effects.This implies that the efficacy response is optimized ata higher dose level, whereas the toxicity response isminimized at a lower dose level. Thus, it is difficultto identify dose levels which are optimal for bothresponses. The problem of simultaneous optimiza-tion for linear multiresponse models was addressed inRefs 47–50.

Lind et al.47 developed a graphical approach inwhich contours of all the responses were superimposedon each other and the region where operatingconditions were ‘near’ optimal for all the responseswas identified. As the number of responses and controlfactors increases, finding the optimum graphicallybecomes infeasible. Myers and Carter51 proposedthe dual response system consisting of a primaryresponse and a secondary response. The procedureinvolved determining the operating conditions forwhich the primary response was optimized whilethe secondary response was constrained to be equalto some prespecified value. Refs 52–55 providedvarious extensions to the dual response approach.Harrington48 developed the desirability approachto multiresponse optimization. In his algorithm,exponential type transformations were used totransform each of the responses into desirabilityfunctions. Derringer and Suich49 later generalizedthe transformations and developed more flexibledesirability functions. The individual desirabilityfunctions were then incorporated into a singlefunction, which gave the desirability for the whole setof responses. Both Refs 48 and 49 used the geometricmean of the individual desirability functions toconstruct a single overall desirability function, whichwas maximized to determine the operating conditions.Del Castillo et al.56 modified the desirability approachof Ref 49 such that both the desirability function andits first derivative were continuous. However, theirapproach ignored variations and correlations existingamong the responses. Wu57 presented an approachbased on the modified double-exponential desirability



function taking into account correlations among theresponses.

Multiresponse optimization using the so-calledgeneralized distance approach was introduced byKhuri and Conlon.50 The main characteristic of thisapproach was that it took into account the heterogene-ity of the variances of the responses and also the corre-lated nature of the responses. If the individual optimawere not attained at the same settings of the con-trol factors, then compromise conditions on the inputvariables that are ‘favorable’ to all the mean responseswere determined. The deviation from the ideal opti-mum was measured by a distance function expressedin terms of the estimated mean responses along withtheir variance–covariance matrix. By minimizing sucha distance function, Khuri and Conlon arrived ata set of conditions for a ‘compromise optimum’.Vining58 proposed a mean squared error method todetermine the compromise optimum for a multire-sponse experiment. Pignatiello59 and Ames et al.60

also proposed approaches based on the squared errorloss. A comprehensive survey of the various meth-ods of multiresponse optimization was presentedin Ref 61.

Taguchi’s Robust Parameter DesignRobust parameter design is a well-established engi-neering technique to increase the quality of a productby making it robust/insensitive to the uncontrollablevariations present in the production process. Since theintroduction of parameter design in the United Statesby Genichi Taguchi during the 1980s, a multitude ofpapers by Kackar,62 Taguchi and Wu,63 Taguchi,64

Nair and Shoemaker65 and books authored by Khuriand Cornell (Chapter 11),1 Taguchi,66 Phadke,67 Wuand Hamada,68 and Myers and Montgomery,69 andseveral other authors, have been written on the topic.Review articles by Myers et al.70 and Robinson et al.71

cite several papers based on robust parameter designs.Taguchi proposed that the input variables in an exper-iment were of two types, (1) control factors: easyto control and (2) noise factors: difficult to control.These difficult-to-control noise factors are the causeof variations in a production process. The main aimof parameter design is to determine the settings ofthe control factors for which the process response isrobust to the variability in the system caused by thenoise factors. To achieve this goal, Taguchi advocatedthe use of crossed arrays by crossing an orthogonalarray of control variables (inner array) with an orthog-onal array of noise variables (outer array). Taguchiidentified that there were three specific goals in anexperiment:

1. The smaller, the better: minimizing the response.

2. The larger, the better: maximizing the response.

3. Target is best: achieving a given target value.

For each of the different goals, Taguchi definedperformance criteria known as signal-to-noise (S/N)ratios that took into account both the process meanand the variance. Each set of settings of the controlvariables contained n runs in the noise variables fromthe outer array. For each of the three different goals,he defined the S/N ratios as follows:

1. The smaller, the better: −10 log[1n

∑ni=1 y2

i ].

2. The larger, the better: −10 log[1/n∑n

i=1 1/y2i ].

3. Target is best: −10 log( s2

y2 ), where y is the sample

mean and s2 is the sample variance.

All the above S/N ratios are to be maximized.Although the Taguchi method was a significant steptoward quality improvement, it received a numberof criticisms. It was pointed out (see Ref 69) that inthe Taguchi methodology (1), interactions among thecontrol factors were not estimated, (2) large numbersof experimental runs were required, (3) S/N ratioswere unable to distinguish between inputs affectingprocess mean from those affecting the variance.Some of the authors who discussed the Taguchimethodology in detail and offered criticisms wereMyers and Montgomery,69 Box,72,73 Easterling,74

Pignatiello and Ramberg,75 Nair and Pregibon,76,Welch et al.77, and Nair.78

Response Surface Approach to RobustParameter DesignTwo response surface approaches to parameter designwere introduced during the 1990s. The approacheswere (1) the dual response approach and (2) thesingle model approach. In the dual response approach,separate models were fitted to the process meanand the process variance. While in the single modelapproach, as the name suggests, a single modelcontaining both the noise and the control variableswas fitted to the process response.

Vining and Myers79 were the first to proposethat Taguchi’s aim of keeping the mean on targetwhile minimizing the process variance could alsobe achieved in a response surface framework. Theyfitted separate second-degree models to the process



mean (µ) and the process variance (σ 2),

µ = b0 + x′b + x′Bx, (28)

σ 2 = c0 + x′c + x′Cx (29)

where b0, b, B, c0, c, C were the estimates of thecoefficients. Using the dual response optimization ofRef 51, σ 2 was minimized while keeping the processmean at target.

Del Castillo and Montgomery80 suggested theuse of nonlinear programming to solve a similaroptimization problem as proposed by Vining andMyers,79 but replacing the equality constraintsby inequalities. Criticizing the use of Lagrangianmultipliers and equality constraints, Lin and Tu81

proposed a procedure based on the mean squarederror (MSE) criterion. Using the same examples asdiscussed by Vining and Myers79 and Del Castilloand Montgomery,80 they showed that more reductionin the variance was possible by introducing a littlebias. One criticism of using the MSE was that norestriction was placed on the distance of the meanfrom the target. Addressing this problem, Copelandand Nelson82 minimized the variance while keepingthe distance between the mean and the target lessthan some specified quantity, �. For processes whereit was important to keep the mean near the target,� was chosen to be small. Various other extensionsto the dual response approach were suggested byFan,55 Kim and Lin,83 Del Castillo et al.,84 Koksoyand Doganaksoy,85 Kim and Cho,86 and Tang andXu.87

To overcome the shortcomings (requirement oftoo many runs and being unable to fit interactionterms) of Taguchi’s crossed array, Welch et al.77

proposed a combined array, which was a singleexperimental design for both the control and the noisevariables. The combined array was shown to be moreeconomical than the crossed arrays of Taguchi (seeRefs 77, 88, 89). Myers et al.90 used the combinedarray of Welch et al.77 to fit a single model containingboth the control and noise variables to the responsevariable,

y(x, z) = β0 + g′(x)β + z′δ + g′(x)�z + ε (30)

where x and z are the control and noise variables,respectively. In the above model, g′(x) is a row of thedesign matrix containing polynomial and interactionterms in the control variables, β and δ are the vectorsof regression coefficients for the control and noisevariables, respectively, and � contains the interactioncoefficients. They showed that although a single modelin the noise and control variables was fitted to the

response, there were still two response surfaces for theprocess mean and the variance,

E[y(x, z)] = β0 + g′(x)β (31)

and

Var[y(x, z)] = [δ′ + g′(x)�]Var(z)[δ′ + g′(x)�]′ + σ 2e .

(32)

To solve the parameter design problem, they chosethe estimated squared error loss as the performancecriterion,

E[y(x, z) − T]2 (33)

where T was the prespecified target value, andminimized it with respect to x. Myers et al.90

proposed a linear mixed effects approach, in whichthe elements of δ and � in Eq. (30) were treated asrandom. Aggarwal and Bansal91 and Aggarwal et al.92

considered robust parameter designs involving bothquantitative and qualitative factors. Brenneman andMyers93 considered the single model in the control andnoise variables to model the response. In their model,the noise variables were considered to be categoricalin nature.

PART III. EXTENSIONS AND NEWDIRECTIONS: 2000 ONWARDS

Response Surface Models with RandomEffectsThe response surface models we have consideredso far include only fixed polynomial effects. Thesemodels are suitable whenever the levels of the factorsconsidered in a given experiment are of particularinterest to the experimenter, for example, thetemperature and concentrations of various chemicalsin a certain chemical reaction. There are, however,other experimental situations where, in addition to themain control factors, the response may be subject tovariations due to the presence of some random effects.For example, the raw material used in a productionprocess may be obtained in batches selected atrandom from the warehouse supply. Because batchesmay differ in quality, the response model shouldinclude a random effect to account for the batch-to-batch variability. In this section, we considerresponse surface models which, in addition to thefixed polynomial effects, include random effects.

Let µ(x) denote the mean of a response variable,y, at a point x = (x1, x2, . . . , xk)′. It is assumed that



µ(x) is represented by a polynomial model of degreed (≥1) of the form

µ(x) = f ′(x)β. (34)

Suppose that the experimental runs used to estimatethe model parameters in Eq. (34) are heterogeneousdue to an extraneous source of variation referredto as a block effect. The experimental runs aretherefore divided into b blocks of sizes n1, n2, . . . , nb.Let n = ∑b

i=1 ni be the total number of observations.The block effect is considered random, and the actualresponse value at the uth run is represented by themodel

yu = f ′(xu)β + z′uγ + g′(xu)�zu + εu, u = 1, 2, . . . , n

(35)

where g′(x) is such that f ′(x) = [1, g′(xu)], xu is thevalue of x at the uth run, zu = (zu1, zu2, . . . , zub)′,where zui is an indicator variable taking the value 1if the uth trial is in the ith block and the value 0otherwise, γ = (γ 1, γ 2, . . . , γ b)′, where γ i denotes theeffect of the ith block, and εu is a random experimentalerror (i = 1, 2, . . . , b; u = 1, 2, . . . , n). The matrix �

contains interaction coefficients between the blocksand the fixed polynomial terms in the model. Becausethe polynomial portion in Eq. (35) is fixed and theelements of γ and � are random, model (35) isconsidered to be a mixed-effects model. It can beexpressed in vector form as

y = Xβ + Zγ +p∑

j=2

U jδj + ε (36)

where the matrix X is of order n × p and is the sameas in model (6), Z is a block-diagonal matrix of ordern × b of the form

Z = diag(1n1 , 1n2 , . . . , 1nb). (37)

U j is a matrix of order n × b whose ith column isobtained by multiplying the elements of the jth columnof X with the corresponding elements of the ith columnof Z (i = 1, 2, . . . , b; j = 2, 3, . . . , p), δj is a vector ofinteraction coefficients between the blocks and the jthpolynomial term (j = 2, 3, . . . , p) in model (35). Notethat δj is the same as the transpose of the jth row of �

in Eq. (35).We assume that γ , δ2, δ3, . . . , δp are normally

and independently distributed with zero means andvariance–covariance matrices σ 2

γ Ib, σ 22Ib, . . . , σ 2

pIb,respectively. The random error vector, ε, is also

assumed to be independent of all the other randomeffects and is distributed as N(0, σ 2

ε In). Consequently,the mean of y and its variance–covariance matrix aregiven by

E(y) = Xβ, (38)

Var(y) = σ 2γ ZZ′ +

p∑j=2

σ 2j U jU j

′ + σ 2ε In. (39)

On the basis of Eqs. (38) and (39), the best linearunbiased estimator of β is the generalized least-squaresestimator (GLSE), β,

β = (X ′�−1X)−1X ′�−1y (40)

where

� = 1σ 2

ε

Var(y)

= σ 2γ

σ 2ε

ZZ′ +p∑

j=2

σ 2j

σ 2ε

U jU j′ + In. (41)

The variance–covariance matrix of β is

Var(β) = (X ′�−1X)−1σ 2ε . (42)

The GLSE of β requires knowledge of

the ratios of variance components,σ2

γ

σ2ε,

σ22

σ2ε, . . . ,

σ2p

σ2ε.

Because the variance components, σ 2γ , σ 2

2, . . . , σ 2p, σ 2

ε

are unknown, they must first be estimated. Letσ 2

γ , σ 22, . . . , σ 2

ε denote the corresponding estimates ofthe variance components. Substituting these estimatesin Eq. (41), we get

� = σ 2γ

σ 2ε

ZZ′ +p∑

j=2

σ 2j

σ 2ε

U jU j′ + In. (43)

Using � in place of � in Eq. (40), we get the so-calledestimated generalized least-squares estimator (EGLSE)of β, denoted by β

∗,

β∗ = (X ′�

−1X)−1X ′�

−1y. (44)

The corresponding estimated variance–covariancematrix of β

∗is approximately given by

Var(β∗) ≈ (X ′�

−1X)−1σ 2

ε . (45)



Furthermore, the predicted response at a point x in aregion R is

y∗(x) = f ′(x)β∗. (46)

This also gives an estimate of the mean response µ(x)in Eq. (34). The corresponding prediction variance isapproximately given by

Var[y∗(x)] ≈ σ 2ε f ′(x)(X ′�

−1X)−1f (x). (47)

Estimates of the variance components can be obtainedby using either the method of maximum likelihood(ML) or the method of restricted maximum likelihood(REML). These estimates can be easily obtained byusing PROC MIXED in SAS.94

Tests concerning the fixed effects (i.e., theelements of β) in the mixed model (35) can becarried out by using the ESGLSE of β and itsestimated variance–covariance matrix in Eq. (45).More specifically, to test, for example, the hypothesis,

H0 : a′β = c (48)

where a and c are given constants, the correspondingtest statistic is

t = a′β∗ − c

[a′(X ′�−1

X)−1a σ 2ε ]1/2

(49)

which, under H0, has approximately the t-distributionwith ν degrees of freedom. Several methods areavailable in PROC MIXED in SAS for estimatingν. The preferred method is the one based on Kenwardand Roger’s95 procedure.

Of secondary importance is testing the signifi-cance of the random effects in the mixed model (35).The test statistic for testing the hypothesis,

H0 : σ 2γ = 0 (50)

is given by the F-ratio,

F = R(γ |β, δ2, δ3, . . . , δp)(b − 1)MSE

(51)

where MSE is the error (residual) mean square formodel (35), and R(γ |β, δ2, δ3, . . . , δp) is the type IIIsum of squares for the γ -effect (block effect). UnderH0, F has the F-distribution with b − 1 and m degreesof freedom, where m = n − b − (p − 1)b. Similarly, totest the hypothesis

H0j : σ 2j = 0, j = 2, 3, . . . , p (52)

we can use the F-ratio,

Fj = Type III S.S. for δj

(b − 1)MSE, j = 2, 3, . . . , p (53)

which, under H0j, has the F-distribution with b − 1and m degrees of freedom. Type III sum of squares forδj is the sum of squares obtained by adjusting δj forall the remaining effects in model (35). More detailsconcerning these tests can be found in Ref 96.

It should be noted that in the case of themixed model in Eq. (35), the process variance (i.e.,the variance–covariance matrix of y) in eq. (39) is nolonger of the form σ 2

ε In, as is the case with classicalresponse surface models with only fixed effects wherethe response variance is assumed constant throughoutthe region R. Furthermore, the process variancedepends on the settings of the control variables, andhence on the chosen design, as can be seen fromformula (39). This lack of homogeneity in the valuesof the response variance should be taken into accountwhen searching for the optimum response over R onthe basis of the predicted response expression givenin Eq. (46). In particular, the size of the predictionvariance in Eq. (47) plays an important role indetermining the operating conditions on the controlvariables that lead to optimum values of y∗(x) over R.This calls for an application of modified ridge analysisby Khuri and Myers,24 which was mentioned earlier.More details concerning such response optimizationcan be found in Ref 96 and also in Khuri and Cornell1

(Section 8.3.3).More recently, the analysis of the mixed model

in Eq. (35) under the added assumption that theexperimental error variance is different for thedifferent blocks was outlined in Ref 97. This providesan extension of the methodology presented in thissection to experimental situations where the errorvariance can change from one block to another due tosome extraneous sources of variation such as machinemalfunction in a production process over a period oftime.

Introduction to Generalized Linear ModelsGeneralized linear models (GLMs) were first intro-duced by Nelder and Wedderburn98 as an extension ofthe class of linear models. They are used to fit discreteas well as continuous data having a variety of parentdistributions. The traditional assumptions of normal-ity and homogeneous variances of the response data,usually made in an analysis of variance (or regression)situation, are no longer needed. A classic book onGLMs is the one by McCullagh and Nelder.99 In addi-tion, the more recent books by Lindsey,100 Dobson,101



McCulloch and Searle,102 and Myers et al.103 provideadded insight into the application and usefulness ofGLMs.

In a GLM situation, the response variable,y, is assumed to follow a distribution from theexponential family. This includes the normal as wellas the binomial, Poisson and gamma distributions.The mean response is modeled as a function of theform, µ(x) = h[f′(x)β], where x = (x1, . . . , xk)′, f(x) isa known vector function of order p × 1 and β is avector of p unknown parameters. The function f′(x)βis called the linear predictor and is usually denotedby η(x). It is assumed that h(·) is a strictly monotonefunction. Using the inverse of the function h(·), onecan express η(x) as g[µ(x)]. The function g(·) is calledthe link function.

Estimation of β is based on the method ofmaximum likelihood using an iterative weightedleast-squares procedure (see Ref 99, pp. 40–43). Theestimate of β is then used to estimate the meanresponse µ(x).

A good design is one which has a lowprediction variance or low mean-squared error ofprediction (MSEP) (expressions for the predictionvariance and MSEP are given in Ref 104). However,the prediction variance and the MSEP for GLMsdepend on the unknown parameters of the fittedmodel. Thus, to minimize either criterion, we requiresome prior knowledge about β. This leads to thedesign dependence problem of GLMs. Some commonapproaches to this problem are listed below. A detailedreview of design issues for GLMs can be found inRef 105.

• Locally optimal designs: designs for GLMsdepend on the unknown parameters of the fittedmodel. Due to this dependence, the constructionof a design requires some prior knowledge ofthe parameters. If initial values of the parametersare assumed, then a design obtained on the basisof an optimality criterion, such as D-optimalityor A-optimality, is called locally optimal. Theadequacy of such a design depends on how closethe initial values are to the true values of theparameters. A key reference in this area is theone by Mathew and Sinha106 concerning designsfor a logistic regression model. Other relatedwork include those of Abdelbasit and Plackett,107

Minkin,108 Khan and Yazdi,109 Wu,110 and Sitterand Wu.111

• Sequential designs: in this approach, experimen-tation is not stopped at the initial stage. Instead,using the information obtained, initial estimates

of the parameters are updated and used to findadditional design points in the subsequent stages.This process is carried out till convergence isachieved with respect to some optimality cri-terion, for example, D-optimality. Sequentialdesigns were proposed by Wu,112 Sitter andForbes,113 Sitter and Wu,114 among others.

• Bayesian designs: in the Bayesian approach, aprior distribution is assumed on the parametervector, β (β as in the linear predictor),which is then incorporated into an appropriatedesign criterion by integrating it over theprior distribution. For example, one criterionmaximizes the average over the prior distributionof the logarithm of the determinant of Fisher’sinformation matrix. This criterion is equivalentto D-optimality in linear models. Bayesianversions of other alphabetic optimality criteriacan also be used such as A-optimality. Oneof the early papers on Bayesian D-optimalitycriterion is the one by Zacks.115 Later, theBayesian approach was discussed by severalauthors including Chaloner,116 Chaloner andLarntz,117,118 and Chaloner and Verdinelli.119

Designs for a family of exponential modelswere presented by Dette and Sperlich120 andMukhopadhyay and Haines121 (see Ref 122).Atkinson et al.123 developed D- and Ds- (optimalfor a subset of parameters) optimal Bayesiandesigns for a compartmental model.

• Quantile dispersion graphs (QDGs) approach:this approach was recently introduced byRobinson and Khuri124 in a logistic regressionsituation. In this graphical technique, designswere compared on the basis of their quantiledispersion profiles. Since in small samples, theparameter estimates are often biased, Robinsonand Khuri124 considered the mean-squared errorof prediction (MSEP) as a criterion for comparingdesigns. Khuri and Mukhopadhyay104 laterapplied the QDG approach to compare designsfor log-linear models representing Poisson-distributed data.

• Robust design approach: in this approach, aminmax procedure is used to obtain designsrobust to poor initial parameter estimates.Sitter125 applied the minmax procedure to binarydata and used the D-optimality and the Fielleroptimality criteria to select designs. It was shownby Sitter125 that his D-optimal designs were morerobust to poor initial parameters than locally D-optimal designs for binary data. An extension ofSitter’s work was given by King and Wong.126



Some recent works on designs for GLMs include thoseby Dror and Steinberg,127 Woods et al.,128 and Russellet al.129 All of these papers are focused on GLMs withseveral independent variables. Dror and Steinberg127

and Russell et al.129 used clustering techniques forconstructing optimal designs.

All the above references discuss design issues forGLMs with a single response. Very little work hasbeen done on multiresponse or multivariate GLMs,particularly in the design area. Such models areconsidered whenever several response variables canbe measured for each setting of a group of controlvariables, and the response variables are adequatelyrepresented by GLMs. Books by Fahrmeir and Tutz130

and McCullagh and Nelder99 discuss the analysis ofmultivariate GLMs.

In multivariate generalized linear models(GLMs), the q-dimensional vector of responses, y, isassumed to follow a distribution from the exponentialfamily. The mean response µ(x) = [µ1(x), . . . , µq(x)]′at a given point x in the region of interest, R, is relatedto the linear predictor η(x) = [η1(x), . . . , ηq(x)]′ by thelink function g : Rq → Rq,

η(x) = Z′(x)β = g[µ(x)] (54)

where x = (x1, . . . , xk)′, Z(x) = ⊕qi=1fi(x), fi(x) is a

known vector function of x, β is a vector of unknownparameters. If the inverse of g, denoted by h, exists,where h : Rq → Rq, then

µ(x) = h[η(x)] = h[Z′(x)β]. (55)

Estimation of β is based on the method ofmaximum likelihood using an iterative weightedleast-squares procedure (see Ref 130, p. 106). Thevariance–covariance matrix, Var(β), is dependent onthe unknown parameter vector β. This causes thedesign dependence problem in multivariate GLMs.Some of the key references for optimal designs inmultivariate GLMs are Refs 131 and 132. Heiseand Myers131 studied optimal designs for bivariatelogistic regression, whereas Zocchi and Atkinson’s132

work was based on optimal designs for multinomiallogistic models. A recent work by Mukhopadhyayand Khuri133 compares designs for multivariate GLMsusing the technique of quantile dispersion graphs.

The problem of optimization in a GLMenvironment is not as well developed as in the caseof linear models. In single-response GLMs, Paul andKhuri25 used modified ridge analysis to carry outoptimization of the response. Instead of optimizing themean response directly, Paul and Khuri25 optimizedthe linear predictor. Mukhopadhyay and Khuri134

used the generalized distance approach, initiallydeveloped for the simultaneous optimization of severallinear response surface models, for optimization in amultivariate GLM situation.

Application of GLMs to the robust parameterdesign problem has been discussed by several authorsincluding Nelder and Lee,135 Engel and Huele,136

Brinkley et al.,137 Hamada and Nelder,138 Nelderand Lee,139 Lee and Nelder,140 and Myers et al.141

Nelder and Lee135 modeled the mean and the varianceseparately using GLMs. Both mean and variance werefunctions of the control factors. Nelder and Lee139

modeled the mean as a function of both the controland the noise variables, whereas the variance wasa function of the control variables only. Engel andHuele136 adopted the single response model of Myerset al.90 and assumed nonconstant error variances. Intheir paper, the process variance depended on thenoise variables as well as the residual variance. Theymodeled the residual variance using an exponentialmodel which guaranteed positive variance estimates.This exponential model was previously used by Boxand Meyer,142 Grego,143 and Chan and Mak.144 Arecent paper by Robinson et al.145 proposed the useof generalized mixed models in a situation where theresponse was nonnormal and the noise variable was arandom effect.

Graphical Procedures for Assessingthe Prediction Capability of a ResponseSurface DesignStandard design optimality criteria usually base theirevaluations on a single number, like D-efficiency, butdo not consider the quality of prediction throughoutthe experimental region. However, the predictioncapability of any response surface design does notremain constant throughout the experimental region.Thus, rather than relying on a single-number designcriterion, a study of the prediction capability of thedesign throughout the design region should give moreinformation about the design’s performance. Threegraphical methods have been proposed to study theperformance of a design throughout the experimentalregion.

1. Variance dispersion graphs: Giovannitti-Jensenand Myers146 and Myers et al.147 proposedthe graphical technique of variance dispersiongraphs (VDGs). VDGs are two-dimensionalplots displaying the maximum, minimum, andaverage of the prediction variance on concentricspheres, chosen within the experimental region,



against their radii. The prediction variance is thevariance of the predicted response,

y(x) = f′(x)β (56)

as given in formula (10), where f(x) is a knownvector function of x = (x1, . . . , xk)′ and β is theleast-squares estimate of β in Eq. (7). As inEq. (11), the prediction variance is

Var[y(x)] = f′(x)(X′X)−1f(x)σ 2. (57)

A scaled version of the prediction variance (SPV)is given by

N Var[y(x)]σ 2 = Nf′(x)(X′X)−1f(x) (58)

where N is the number of experimental runs.The average prediction variance, Vr, over thesurface, Ur, of a sphere of radius r is

Vr = Nψ

σ 2

∫Ur

Var[y(x)] dx (59)

and the maximum and minimum predictionvariances, respectively, are given by

maxx∈Ur

N Var[y(x)]σ 2 , min

x∈Ur

N Var[y(x)]σ 2 (60)

where Ur = {x :∑k

i=1 x2i = r2} and ψ−1 =∫

Urdx is the surface area of Ur. A design is con-

sidered to be good if it has low and stable valuesof Vr throughout the experimental region. Themaximum and minimum prediction variancereflect the extent of variability in the predictionvariance values over Ur. A big gap between themaximum and minimum values implies thatthe variance function is not stable over theregion. The software for constructing VDGswas discussed by Vining.148 Borkowski149

determined the maximum, minimum, andaverage prediction variances for central com-posite and Box–Behnken designs analyticallyand showed that they were functions of theradius and the design parameters only. Trincaand Gilmour150 further extended the VDGapproach and applied it to blocked responsesurface designs. Borror et al.151 used the VDGapproach to compare robust parameter designs.Similar to the VDG approach, Vining andMyers152 proposed a graphical approach toevaluate and compare response surface designson the basis of the mean-squared error of

prediction. Vining et al.153 made use of VDGsin mixture experiments.

2. Fraction of design space plots: Zahran et al.154

proposed fraction of design space (FDS) plotswhere the prediction variance is plotted againstthe fraction of the design space that hasprediction variance at or below the given value.They argued that VDGs provide some but notall information on SPV of a design, becausetwo designs may have the same VDG patternbut different SPV distributions. This happensbecause VDGs fail to weight the information ineach sphere by the proportion of design spaceit represents. Zahran et al.154 defined the FDScriterion as

FDS = 1φ

∫A

dx (61)

where A = {x : V(x) < Q}, V(x) = N Var[y(x)]σ2 , φ

is the total volume of the experimental region,Q is some specified quantity.

3. Quantile plots: Khuri et al.155 proposed thequantile plots (QPs) approach for evaluatingand comparing several response surface designsbased on linear models. In this graphicaltechnique, the distribution of SPV on a givensphere was studied in terms of its quantiles.Khuri et al.155 showed through examples thattwo designs can have the same VDG pattern butmay display very different distributions of SPV.They stated that this occurred because VDGsprovide information only on the extreme valuesand average value of the SPV but not on thedistribution of SPV on a given sphere. To obtainthe QPs on the surface, Ur, of a sphere of radiusr, several points were first generated randomlyon Ur. Spherical coordinates were used for therandom generation of points. The values of theSPV function, V(x), at points x on U(r) werecomputed. All these values formed a sampleT(r). The quantiles of T(r) were obtained andthe pth quantile of T(r) was denoted by Qr(p).Plots of Qr(p) against p gave the QPs on U(r).Khuri et al.156 used the QPs to compare designsfor mixture models on constrained regions.

Graphical Methods for ComparingResponse Surface DesignsQuantile dispersion graphs (QDGs) were proposedby Khuri157 for comparing designs for estimatingvariance components in an analysis of variance(ANOVA) situation. The exact distribution of the



variance component estimator was determined interms of its quantiles. These quantiles were dependenton the unknown variance components of the ANOVAmodel. Plots of the maxima and the minima ofthe quantiles over a subset of the parameter spaceproduced the QDGs. These graphs assessed thequality of the ANOVA estimators and allowedcomparison of designs with respect to their estimationcapabilities. Lee and Khuri158,159 extended the use ofQDGs to unbalanced random one-way and two-waymodels, respectively. They used QDGs to comparedesigns based on ANOVA and maximum likelihoodestimation procedures. QDGs were used by Khuriand Lee160 to evaluate and compare the predictioncapabilities of nonlinear designs with one controlvariable throughout the region of interest R. Morerecently, Saha and Khuri161 used QDGs to comparedesigns for response surface models with randomblock effects.

Robinson and Khuri124 generalized the workof Khuri and Lee160 by addressing nonnormalityand nonconstant error variance. They considered theproblem of discriminating among designs for logisticregression models using QDGs based on the MSEP.Khuri and Mukhopadhyay104 later applied the QDGapproach to compare the prediction capabilities ofdesigns for Poisson regression models, also using theMSEP criterion. The MSEP incorporates both theprediction variance and the prediction bias, whichresults from using maximum likelihood estimates(MLEs) of the parameters of the fitted model. Asin any design criterion for GLMs, the MSEP and itsquantiles depend on the unknown parameters of themodel. For a given design, quantiles of the MSEPwere obtained within a region of interest. To comparedesigns using QDGs in a region R, several pointswere generated on concentric surfaces, denoted by Rν ,which were obtained by shrinking the boundary of Rby a factor ν. The value of the MSEP was computedfor each x on Rν and β in a parameter space, C. An‘initial’ data set on the response was used to constructthe parameter space C. Quantiles of MSEP for a givendesign D were computed, and the pth quantile wasdenoted by QD(p, β, ν) for 0 ≤ p ≤ 1. These quantilesprovided a description of the distribution of MSEP forvalues of x on Rν . The dependence of the quantileson β was investigated by computing QD(p, β, ν) forseveral values of β that formed a grid, C, inside C.Subsequently, the minimum and maximum values ofQD(p, β, ν) over the values of β in C were obtained

QminD (p, ν) = min

β∈C{QD(p, β, ν)} (62)

QmaxD (p, ν) = max

β∈C{QD(p, β, ν)}. (63)

Plotting these values against p resulted in the QDGsof the MSEP over Rν . Using several values of ν, theentire region R was covered. Given several designsto compare, a design that displays close and smallvalues of Qmax

D and QminD over the range of p is

considered desirable because it has good predictioncapability and is robust to the changes in β. Khuriand Mukhopadhyay104 also studied the effect of thechoice of the link function and/or the nature of theresponse distribution on the shape of the QDGs for agiven design.

Mukhopadhyay and Khuri133 extended theapplication of GLMs to compare response surfacedesigns for multivariate GLMs. Since the MSEP is amatrix in the multivariate situation, they considered ascalar-valued function of the MSEP, namely the largesteigenvalue of the MSEP matrix (EMSEP), as theircomparison criterion. Similar to the MSEP, EMSEPalso depends on x and β. As in the univariate case,quantiles of EMSEP were computed on concentricregions Rν , for a given design D. Mukhopadhyayand Khuri133 chose the parameter space C to be the(1 − α)100% confidence region of β. Subsequently,the minimum and maximum quantiles were computedover the values of β in a grid of points from C andplotted against p to obtain the QDGs. The authorsillustrated their proposed methodology using a dataset from a combination drug therapy study on malemice taken from Gennings et al.,162 pp. 429–451 (seeSection 6 in Mukhopadhyay and Khuri.133)

A numerical example. In a drug therapyexperiment on male mice, the pain relieving effects oftwo drugs, morphine sulfate (x1), and 9-tetrahydro-cannabinol (x2), on two binary responses, pain relief(y1), and side effect (y2), were studied. The responsey1 takes the value 1 if relief occurs, otherwise it takesthe value zero. The response y2 is equal to one if aharmful side effect develops, otherwise it is equal tozero. A 5 × 7 factorial design with six mice in eachrun was considered. The experimental region R wasrectangular in shape with R : {0 ≤ x1 ≤ 8, 0 ≤ x2 ≤15}. The following first-degree models were used to fitthe data

η1(x) = β1 + β2x1 + β3x2,

η2(x) = β4 + β5x1 + β6x2,

η3(x) = β7 + β8x1 + β9x2. (64)

Here, η(x) = [η1(x), η2(x), η3(x)]′ is the three-dimensional linear predictor as explained in Eq. (54).



FIGURE 1 | Comparisonof the QDGs for designs D1

(5 × 7 factorial) and D2

(32 factorial), forp = 0(0.05)1 andν = 1, 0.9, 0.8, 0.6.

0.0 0.4 0.6 0.8 1.0

0.00

0.01

0.02

0.03

0.00

0.01

0.02

0.03

Probabilities (p)

Qua

ntile

s of

EM

SE

P

nu = 1

D1D2

Probabilities (p)

Qua

ntile

s of

EM

SE

P

nu = 0.9

D1D2

0.00

0.01

0.02

0.03

Probabilities (p)

Qua

ntile

s of

EM

SE

P

nu = 0.8

D1D2

0.00

0.01

0.02

0.03

Probabilities (p)

Qua

ntile

s of

EM

SE

P

nu = 0.6

D1D2

0.2

0.0 0.4 0.6 0.8 1.00.2 0.0 0.4 0.6 0.8 1.00.2

0.0 0.4 0.6 0.8 1.00.2

The corresponding link function used was

µi(x) = exp[ηi(x)]

1 + ∑3l=1 exp[ηl(x)]

, i = 1, 2, 3 (65)

where ηi(x) is the estimate of ηi(x) for i = 1, 2, 3.Note that µi(x) is the ith element of µ(x) =[µ1(x), µ2(x), µ3(x)]′, where for a given x, µ1(x) isthe probability that y1 = 1 and y2 = 1, µ2(x) is theprobability that y1 = 1 and y2 = 0, and µ3(x) is theprobability that y1 = 0 and y2 = 1. The original 5 × 7factorial design (D1) was compared with D2, a 32 fac-torial design with the center point (4, 7.5) replicated

three times and all other points replicated four times.The EMSEP values were computed for x on concentricrectangles Rν and β ∈ C. The 95% confidence regionof β was chosen to be C and C was a set of 500 ran-domly chosen points from C. The QDGs comparingdesigns D1 and D2 on the basis of the quantiles ofEMSEP values are shown in Figure 1. From Figure 1,it can be noted that both designs are robust to thechanges in the parameter values, and overall, D2 hasbetter prediction capability than D1 for almost all val-ues of ν and p. More details concerning this examplecan be found in Sections 5 and 6 in Ref 133.

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