MODELS AND DESIGNS FOR GENERALIZATIONS OF MIXTURE
EXPERIMENTS WHERE THE RESPONSE DEPENDS ON THE TOTAL AMOUNT
BY
GREGORY FRANK PIEPEL
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1985
To Polly
and
Erin
ACKNOWLEDGMENTS
I would like to express my appreciation to Dr. John
Cornell for serving as my dissertation advisor, and for
providing stimulating discussions and comments on my
research and the field of mixture experiments in general. I
would like to thank Dr. Andr4 Khuri, Dr. Frank Martin, and
Dr. Esam Ahmed for serving on my committee. I would also
like to thank Dr. Randy Carter, who served on my Part C and
oral defense examining committees. Words of thanks also go
to Cynthia Zimmerman for her expert job of typing this
manuscript and to Joe Branch for his work in preparing the
artwork for the figures.
Finally, I would like to thank my wife, Polly, and
daughter, Erin, for their support and encouragement over the
course of my studies and research.
iii
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS......................... .. ............... iii
ABSTRACT .................... ............ .......... .vii
CHAPTER
ONE INTRODUCTION........ .................. ... ..... 1
1.1 The Response Surface Problem...............1
1.2 Mixture ExperimentsAn Introduction........6
1.3 The Subject of This Research
Generalizations and Extensions of
Mixture Experiments........................9
TWO LITERATURE REVIEWMIXTURE AND MIXTUREPROCESS
VARIABLE EXPERIMENTS............................. 10
2.1 Models for Mixture Experiments.............10
2.2 Mixture Experiment Designs.................18
2.3 MixtureProcess Variable Experiments.......26
THREE MODELS FOR MIXTUREAMOUNT EXPERIMENTS............ 2
3.1 An Introduction to MixtureAmount
Experiments............................ 52
3.2 Including the Total Amount in Mixture
Models................................... 34
3.3 MixtureAmount Models Based on
Scheffe Canonical Polynomials..............36
3.4 MixtureAmount Models Based on
Other Mixture Model Forms.................. 45
3.5 MixtureAmount ModelsA Summary...........51
FOUR DESIGNS FOR MIXTUREAMOUNT EXPERIMENTS............54
4.1 Developing Designs for MixtureAmount
Experiments .............................. 54
4.2 Fractionating Designs for Mixture
Amount Experiments.........................61
FIVE MODELS AND DESIGNS BASED ON THE COMPONENT
AMOUNTS ...................................... .88
5.1 Standard Designs and Polynomial Models
Based on the Component Amounts.............89
5.2 Models and Designs for Experiments
Where the Component Amounts Have a
MixtureLike Restriction....................91
SIX COMPARISON OF MIXTUREAMOUNT, COMPONENT AMOUNT,
AND COMPONENTWISE MIXTURE EXPERIMENTS.......... 101
6.1 Comparison of Constraint Regions.......... 101
6.2 Comparison of Models......................114
6.3 Comparison of Designs..................... 124
6.4 Comparing the Predictive Ability of
MixtureAmount and Component Amount
Models ...................................137
SEVEN EXAMPLES OF MIXTUREAMOUNT, COMPONENT AMOUNT,
AND COMPONENTWISE MIXTURE EXPERIMENTS...........146
7.1 A MixtureAmount Experiment Example.......146
7.2 A Component Amount Experiment Example.....155
7.3 A ComponentWise Mixture Experiment
Example..................................162
EIGHT SUMMARY AND CONCLUSIONS........................167
8.1 Summary ............. ... .......... ....... 168
8.2 Recommendations........................ 174
APPENDICES
A SCHEFFE CANONICAL POLYNOMIAL MIXTUREAMOUNT
MODELS........................... .... ......... 178
A.1 Models in Which the Components Blend
Linearly ....... ................... .....178
A.2 Models in Which the Components Blend
Nonlinearly.. ..... ....................... 179
B MIXTUREAMOUNT MODELS WHEN THE CANONICAL
POLYNOMIAL FORM IS NOT THE SAME AT ALL LEVELS
OF TOTAL AMOUNT..................................183
B.1 Linear and Quadratic Blending at Two
Amounts .............. .................. . 183
B.2 Linear, Quadratic, and Quadratic
Blending at Three Amounts..................185
B.3 Linear, Linear, and Quadratic
Blending at Three Amounts..................189
B.4 SpecialCubic Blending....................190
C THREE COMPONENT DNOPTIMAL DESIGNS FOR
VARIOUS CANONICAL POLYNOMIAL MIXTUREAMOUNT
MODELS ...... ............. ...................... 193
C.1 Two Levels of Amount.......................194
C.2 Three Levels of Amount....................195
D CONSIDERATIONS IN CHOOSING AMONG TWO OR MORE
DNOPTIMAL DESIGNS.................................. 218
D.1 Other Optimality Criteria and Parameter
Variances ........ ............................ 218
D.2 Parameter Estimates as Functions of the
Observations............................... 221
E DERIVATION OF EQUATION (6.14)....................229
F DERIVATION OF EQUATION (6.17)................... 231
G INCLUDING PROCESS VARIABLES IN MIXTUREAMOUNT
EXPERIMENTS ........................ ............ 233
REFERENCES ............ ........... ..... ........... ........236
BIOGRAPHICAL SKETCH... .................................. 241
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
MODELS AND DESIGNS FOR GENERALIZATIONS OF MIXTURE
EXPERIMENTS WHERE THE RESPONSE DEPENDS ON THE TOTAL AMOUNT
BY
GREGORY FRANK PIEPEL
May, 1985
Chairman: John A. Cornell
Major Department: Statistics
The definition of a mixture experiment requires that
the response depend only on the proportions of the
components present in the mixture and not on the total
amount of the mixture. This definition is extended to
encompass experiments where the response may also depend on
the total amount of the mixture. Experiments of this type
are referred to as general mixture experiments.
Three types of general mixture experiments (mixture
amount, component amount, and componentwise mixture) are
discussed. Designs and models for these experiments are
presented and compared.
A mixtureamount experiment consists of a series of
usual mixture experiments conducted at each of two or more
vii
levels of total amount. Mixtureamount models are developed
by writing the parameters of mixture models as functions of
the total amount. This class of models is quite broad in
that it includes models that are appropriate when the
components blend differently at the different levels of
total amount as well as models that are appropriate when the
effect of the total amount is not the same with all
component blending properties. Designs for both
unconstrained and constrained mixtureamount experiments are
discussed, as are techniques for fractionating mixture
amount designs.
Component amount experiments utilize standard response
surface designs and polynomial models in the component
amounts. Componentwise mixture experiments are similar to
usual mixture experiments, except that the level of total
amount is not fixed and therefore may have an effect on the
response. Componentwise mixture models and designs can be
specified in terms of the component amounts or in terms of
componentwise proportions.
Several real and hypothetical examples are utilized to
illustrate and compare the mixtureamount, component amount,
and componentwise mixture designs and models. Recommenda
tions are given as to when each of the three experimental
approaches should be used.
viii
CHAPTER ONE
INTRODUCTION
1.1 The Response Surface Problem
In a general response surface problem, interest centers
around an observable response y which is a function of q
predictor variables xl, x2, xq. The predictor
variables are quantitative and continuous and their values
are assumed to be controlled by the experimenter. The
response y is quantitative and continuous. The functional
relationship between the predictor variables and the
response may be expressed as
Yk = f(xkl'xk2',.. Xkq) + k k=1,..., N, (1.1)
where yk is the kth of N observations of the response in an
experiment, xki is the value of the ith predictor variable
for the kth observation, and sk is the experimental error
contained in the kth observation.
The form of the function f in (1.1) is usually not
known and may be quite complex. In practice, an
approximating function is identified with as simple a form
as possible; often first or seconddegree polynomials in the
predictor variables xl, x2, . are adequate. The
selection of an appropriate approximating function (often
2
referred to as model selection) is the first step in solving
a response surface problem. Usually a model linear in the
parameters is chosen.
A linear response surface model may be written in
matrix notation as
y = X8 + (1.2)
where y is an Nxl vector of observed response values, X is
an Nxp matrix of known constants (N 2 p 2 q), 8 is a pxl
vector of unknown parameters, and E is an Nxl vector of
random errors. It is usually assumed that E(e) = Q and
Var(E) = a2V, where V is a diagonal matrix. Most often in
practice V = IN (the NxN identity matrix containing ones on
the main diagonal and zeros elsewhere). Since E(E) = Q, the
model (1.2) can alternately be expressed as
n = E(y) = XB (1.3)
Another step in solving the response surface problem is
to estimate the parameters 8 and refine the model form if
necessary. If we assume Var(E) = c2IN, then the ordinary
least squares estimator of 8 is given by
8 = (X'X)'X'y (1.4)
and has variance
12
Var(8) = (X X) a (1.5)
The portion of response surface analysis involving
model selection, parameter estimation, and model refinement
is known as regression analysis. Hence, the model
parameters are called regression coefficients and the
response surface model is called the regression model.
Once a fitted regression model y = X8 is obtained, the
next step is to test it for adequacy of fit. If it is found
to be adequate, it can then be used to make predictions of
expected response values for any set of predictor variable
values x, x2, . xq within the experimental region.
If we let XO represent this set of values expanded to
resemble the terms in the model, the predicted value and its
variance for the expected mean response at x0 are
y(x ) = X 0 (1.6)
and
Var[y(x )] = Var(x 8)
0, 1
= x(X X)1 2 (1.7)
Under the assumption Var(s) = a2V (where V is a known
diagonal matrix, not necessarily the identity matrix), s is
estimated by weighted least squares, yielding
8 = (XV' X) X'V (1.8)
Var(B) = (X V1 X) 2 (1.9)
and
Varly(x )] = x (X V' X) 0 (1.10)
~ 00
For more details on these formulas and regression analysis
in general, see Draper and Smith (1981) or Montgomery and
Peck (1982).
Note that the parameter estimators, parameter estimator
variances, and prediction variances in (1.4) through (1.10)
all depend on the Nxp matrix X, which is referred to as the
(expanded) design matrix. Clearly the experimental design
chosen is of great importance in determining the fitted
model and its properties. Box and Draper (1975) gave 14
criteria to consider in choosing a response surface
design. Myers (1971) presented several classes of response
surface designs which support the fitting of first and
seconddegree polynomial models in the predictor
variables. Among the designs discussed by Myers are the 2q
and 3q factorials, the 2qk fractional factorials, and the
central composite designs.
In recent years, computeraided design of response
surface experiments has received much attention. A design
criterion of interest is chosen and points are selected for
the design from a candidate list so as to optimize the
design criterion selected. Several design criteria of
interest are:
1. Doptimality seeks to maximize det(X'X) or
equivalently minimize det[(X X) '.
2. Goptimality seeks to minimize the maximum
prediction variance over a specified set of design
points.
3. Voptimality seeks to minimize the average
prediction variance over a specified set of design
points.
4. Aoptimality seeks to minimize trace [(X'X)1].
Designs consisting of N points obtained by using these
optimality criteria are referred to as DN, GN, VN, and AN
optimal designs. These design criteria and computer
programs for implementing them are discussed by St. John and
Draper (1975), Mitchell (1974), and Welch (1984).
In summary, the major parts of a response surface
analysis are:
1. Selection of an appropriate model to approximate
the response surface over the region of interest.
2. Development of a design which supports the fitting
of the selected model form and provides for testing
the adequacy of fit of the model.
3. Fitting the chosen model, testing it for adequacy
of fit, and revising the model if necessary.
4. Determination of the levels or ranges of the
predictor variables that yield the optimum response
value.
In the following chapters, we will be concerned mainly with
the first two items listed above, that is, model selection
and design development.
1.2 Mixture ExperimentsAn Introduction
A mixture experiment involves mixing two or more
components (ingredients) together to form some end product,
and then measuring or observing one or more properties of
the resulting mixture or end product. In the usual
definition of a mixture experiment (Cornell 1981, Scheffe
1958), the properties of the mixture are assumed to depend
on the proportions of the components present and not on the
total amount of the mixture. Some examples of mixture
experiments are:
1. Sandwich fish patties made using mullet,
sheepshead, and croaker (Cornell and Deng 1982).
The texture of the fish patties was one of several
responses of interest.
2. Coatings (paints) made from blending a prime
pigment, vehicle, and two extender pigments (Hesler
and Lofstrom 1981). Hiding power and scrubbability
were the properties of interest.
3. Waste glasses obtained by mixing SiO2, B203, Al20,
CaO, MgO, Na20, ZnO, Ti02, Cr203, FepO and NiO
(Chick, Piepel, Mellinger et al. 198T). Leach
rates, viscosity, conductivity, and crystallinity
were several of the glass properties investigated.
A mixture experiment problem is clearly a response surface
problem, with the proportions of the components in a mixture
being the predictor variables.
In a mixture experiment (as defined above), the
response to a mixture of q components is a function of the
proportions xj, x2, . xq of components in the
mixture. Since xi represents the proportion of the ith
component in the mixture, the following constraints hold:
q
0 < x. < 1 (i=1,2,. ..,q); x. = 1 (1.11)
i=1 1
Mixture experiments having only these constraints are
referred to as unconstrained mixture experiments. Physical,
theoretical, or economic considerations often impose
additional constraints in the form of lower and upper bounds
on the levels of components
0 < Li < x < U 1 (i=1,2,...,q) (1.12)
Experiments where these additional constraints are imposed
on the xi are referred to as constrained mixture experi
ments.
The region of mixture component combinations defined by
constraints (1.11) and (1.12) is referred to as the
constraint region. Geometrically, restriction (1.11)
defines the constraint region as a regular (q1)dimensional
simplex. In general, restrictions (1.12) reduce the
constraint region given by (1.11) to an irregular
(q1)dimensional hyperpolyhedron. For further discussion
of the geometry of mixture experiments, see Crosier (1984)
and Piepel (1983).
In constrained mixture experiments, it is often
desirable (see Kurotori 1966, Gorman 1970, St. John 1984,
Crosier 1984) to transform the components to new variables
referred to as pseudocomponents. If at least one component
has a nonzero lower bound, then the pseudocomponent values
xi may be obtained from the original component values xi by
x. L
xi = i = 1,2,...,q (1.13)
1 E L
j=1
q
where E L < 1. Crosier (1984) referred to this as the
j=1
Lpseudocomponent transformation. If at least one variable
has a nonunity upper bound, then pseudocomponent values may
be obtained by the Upseudocomponent transformation,
U. x.
xi = q i = 1,2,...,q (1.14)
Z U 1
j=1
q
where E U. > 1. Crosier (1984) presented additional
j=1 3
discussion on the use of these two pseudocomponent trans
formations and gave guidelines for choosing between them.
Models and designs for mixture experiments are reviewed
in Chapter 2. Before proceeding to that material, however,
the purpose and subject of this research is presented.
1.3 The Subject of This ResearchGeneralizations
and Extensions of Mixture Experiments
The purpose of this research is to consider extensions
and generalizations of the usual mixture experiment
described in Section 1.2. As a first step, the following
general definition is presented.
Definition: A general mixture experiment is an
experiment in which two or more components
(ingredients) are mixed together and a property
(response) of the resulting mixture is measured.
The response is assumed to be a function of the
proportions of the components present in the
mixture and possibly the total amount of the
mixture.
The usual mixture experiment, as defined in Section 1.2, is
obviously a special case of the general mixture experiment,
where the total amount of the mixture does not affect the
response. It will be seen in the following chapters that
several quite different types of experiments also satisfy
the definition of a general mixture experiment. Models and
designs for these situations will be presented, discussed,
and compared.
Models and designs for usual mixture experiments and
mixture experiments with process variables form the basis
for much of the work to follow. These topics are reviewed
in Chapter 2.
CHAPTER TWO
LITERATURE REVIEWMIXTURE AND MIXTUREPROCESS
VARIABLE EXPERIMENTS
This chapter reviews models and designs for (usual)
mixture experiments and for mixture experiments with process
variables.
2.1 Models for Mixture Experiments
Scheffe (1958) developed canonical forms of polynomial
models for mixture experiments by substituting the mixture
q
constraint E xi = 1 into certain terms in the standard
i=1
polynomial models and then simplifying. For example, with
q = 2 mixture components, the standard seconddegree
polynomial model is
n = 0 x + a2x2 + ,12x1x2
2 2
+ a11x1 222 .(2.1)
Multiplying the constant term by unity and applying the
2
mixture restriction x + x2 = 1 to the x. terms yields
1 2 1
2 2
10 = 0(x1 + x2), = x1(1 x2) and x2 = x2(1 x1).
Hence, (2.1) can be reduced to the form
n = S 1x1 + 622 + 12x1x2 (2.2)
where a1 = o0 + a1 + al1' 2 = a0 + 2 + a22, and
312 = a12 all a22
The general forms of the first, second, and third
degree canonical polynomial models in q mixture components
are
q
n = E Bixi (2.5)
i=1
q q
n = E Six + Z i 8 xx. (2.4)
i=1 i
and
q q q
n = Sixi + EE s jx + r ij i.x .j(xi xj)
i=1 i
q
+ E Bij kx x xk (2.5)
i
The specialcubic canonical polynomial model is a reduced
form of the full cubic model (2.5) obtained by deleting the
5ijxixj(xi xj) terms.
In each of the above model forms, the first q terms,
81x1 + . + 8qxq, represent the linear blending of the
components while the remaining terms represent nonlinear
blending of the components. We shall refer to these
phenomena throughout, as the linear and nonlinear blending
properties of the components.
Scheffe's canonical polynomial models are widely used
and have been shown to adequately approximate many types of
mixture response surfaces. However, there are certain types
of mixture surfaces for which the canonical polynomial
models are not adequate. For example, when one or more of
the mixture components have an additive effect, Becker
(1968) recommended that homogeneous models of degree one be
used. [A function f(x,y, . ,z) is homogeneous of degree
n if f(tx,ty, . ,tz) = tnf(x,y, . ,z) for every
t > 0.] Three such models, which Becker referred to as H1,
H2, and H3, are given by
q
H1: n = E .x +
i= 1
q
Z 8ij min(xi,x ) +
i
(2.6)
q
H2: n = B ix +
i=1 1
+ 812...q
q
H3: n = B ixi +
i=i1
+ 812...q
If the denominator
defined to be zero.
H1, H2, and H3 (the
are often adequate.
q 21
Z 8 ij xix./(x+x) + . .
i
x1x2... Xq/(x1+X2+...+x )q1
S1/2
E Z 8 (x.x.)
ij (1 x
i
(x x2...x )/q .
(2.7)
+ . .
(2.8)
of a term in H2 is zero, that term is
In practice, the secondorder forms of
first two sets of terms in each model)
Snee (1973) discussed the types of
+ 812...q min(x x2, ...,x q) ,
curvature generated by these secondorder Becker models over
the region 0 < xi < 1.
Becker further noted that the forms of H1, H2, and H3
implicitly assume the response surface attains its maximum
(or minimum) at the centroid of the simplex. He suggested
alternate forms of the models for situations where this is
not the case (see Becker 1968 or Cornell 1981).
Draper and St. John (1977a) proposed several mixture
models which consist of Scheffe canonical polynomial models
plus inverse terms of the form xi1. For example, the first
and seconddegree models with inverse terms are
q q 1
1
n = Bixi + Z 8_ix (2.9)
i=1 i=1
q q q
1
n = i + E 8ij.xix + 8 ix (2.10)
i=1 i
Inverse terms may be added to any Scheffe polynomial model
in a similar manner.
An 1
An inverse term xi proves helpful in situations where
an extreme change in the response (f(x) ") occurs as the
proportion of a component tends to its lower bound of
zero. For similar situations where a component has a non
zero lower bound Li, inverse terms of the form (xi Li)1
may be used. If f(x) +  as xi + Ui (where Ui represents
the component's upper bound such that 0 < L.< U. < 1), then
1 1
inverse terms of the form (Ui xi1 are appropriate.
inverse terms of the form (U. x.) are appropriate.
It is assumed when using any of the above models with
inverse terms that the experimental region itself does not
include the boundary of any component i that causes f(x) +
. as xi Li or Ui. See Draper and St. John (1977a) for
further discussion of this assumption.
In some mixture experiments, interest centers around
ratios of component proportions and how the response depends
on these ratios (see e.g., Hackler, Kriegel, and Hader 1956
or Kenworthy 1963). Snee (1973) noted that models based on
ratios are useful alternatives to the Scheffe and Becker
models for particular types of surfaces because the ratio
models describe a different type of curvature than do the
other models.
A ratio model is developed by replacing the set of
component proportions xi, i=1,2, . ,q, with an
equivalent set of ratio variables rj, j=1,2, . ,q1. In
general, there are many possibilities. For example, with
three components the following equivalent transformations
(among others) are possible:
Transformation rC r2
R1 x1/x2 x2/x3
R2 x1/x2 x3/x2
R3 x1 x2/x3
R4 x1/(x2 + x ) x2/x3
It is seen that ratios may not be defined if certain
component proportions take on zero values in the
denominator. In such cases, Snee (1973) suggested adding a
small positive quantity c to each xi so that the denominator
is always greater than zero.
The above example illustrates that only q1 ratio
variables are needed to replace the q component proportions
q
(owing to the mixture restriction Z x. = 1). Because of
i=1 2
this reduction in the number of variables, the ratio
variables are mathematically independent. Hence, standard
polynomial models in the ratio variables, such as
q1
n = a0 + Z ajrj (2.11)
j=1
and
q1 q1 q1 2
n = 0 + rr + E ajk rk j la r (2.12)
j=1 j
may be used. Further discussion of ratio models may be
found in Snee (1973) and Cornell (1981).
Becker (1978) presented additional models for mixture
experiments with additive or inactive components. For
q = 3, Becker suggested the following model form when at
least one component is inactive:
n = O + 1x1/(x1+x2) + '2x2/(x2+x3) + B3x3/(x3+x1)
q
+ E z ijhij(xi,x ) + 8123hl23(x1,x2,x3) (2.15)
i
Here hij and h123 are specified functions which are
homogeneous of degree zero. Inactivity of a component xi is
suspected when Bi = 0, Sij = 0 for jPi, and 8123= 0. The
hij and h123 functions suggested by Becker are
xs / x s
hi(xi,xj) i (2.14)
ij i j/ \i j/
1, 1 13
h123(x1,x2,x ) = x1 x2 txx2 ( (2.15)
(77 1 2 x1+2 ) 2+x5) x x
He also noted that when si or ti is negative, hij or h123
takes on an extremely large value near the boundary
xi = 0. Models of the form (2.13) are then alternatives to
the inverse term models of Draper and St. John (1977a)
discussed earlier.
Becker (1978) also made a general model suggestion for
mixture experiments with additive components, extending his
earlier work (Becker 1968). This suggestion is to consider
the model
q q
n 8.x. + Z E 8.. (x +x )hi (x ,x ) + . .
i=1' i
+ 812...q(x1+...+xq)h12...q(x1 ...,x q) (2.16)
where the functions hij(xi,xj), hijk(xixj,xk), etc. are
homogeneous of degree zero. This model can be simplified by
deleting higher order terms. Becker (1978) gave some
suggestions for the h functions, and noted that the H2 and
H3 models of (2.7) and (2.8) are of the form (2.16).
Aitchison and BaconShone (1984) presented the
polynomial models
q1
n = 8 + Bizi (2.17)
i=1
q1 q1
S= + E 8.z + Z i z (2.18)
i=1 i
where zi = log(xi/xq), i=1,2, . ,q1. Rewriting (2.17)
and (2.18) in terms of the component proportions xi gives
the symmetric model forms
q q
n = +r iilog xi ( 0i = 0) (2.19)
i=1 i=1
q q 2
S= + og x + E i6 (log xilog xj) (2.20)
i=1 i
where the 8ij are functions of the yij. Note that the above
models are not directly applicable when the component pro
portions take on zero values, since log xi +  as xi + 0.
This behavior suggests models (2.19) and (2.20) as alterna
tives to the inverse term models of Draper and St. John
(1977a) when the component proportions approach but do not
equal zerovalued boundaries. Substituting xi Li or
Ui xi for xi in (2.19) or (2.20) yields models useful for
constrained mixture experiments where f(x) + as xi Li
or U.i
Aitchison and BaconShone pointed out that
s. = 0 (i=1,...,c); Bi = 0 (1 < i < j < c) (2.21)
indicates the inactivity of components 1, 2, . c.
They also noted that
.. = 0 (i=1,2,...,c; j=c+1,...,q) (2.22)
indicates that components 1, 2, . c are additive with
respect to components c+1, c+2, . q. This is a more
general concept of additivity than that considered by Becker
(1968, 1978), where he implied
8i = 0 (i=1,2,...,c; j=i+1,...,q) (2.23)
indicates that components 1, 2, . c are additive.
2.2 Mixture Experiment Designs
As with any response surface problem, choosing an
experimental design is an important part of a mixture
experiment. Designs for both constrained and unconstrained
mixture experiments are reviewed.
Scheffe (1958) proposed the {q,m} simplexlattice
designs for exploring the full qcomponent simplex region in
an unconstrained mixture experiment. The fq,m}
simplexlattice design (m=1,2, . .) consists of the
(q+m1) points in the simplex (1.11) that represent all
possible mixtures obtainable when the proportion of each
component can take on the values O, 1/m, 2/m, . 1.
Examples of some {q,m} simplexlattices are given in Figure
2.1.
The simplexlattice gives an equally spaced distribu
tion of points over the simplex (1.11) and enables a Scheffe
canonical polynomial of degree m in the xi to be fitted
exactly. For example, a {q,2} simplexlattice supports the
fitting of the Scheffe canonical polynomial model (2.4),
while a {q,3} simplexlattice supports the fitting of model
(2.5).
Scheffe (1965) presented the simplexcentroid designs
and associated "special" canonical polynomial models for
unconstrained mixture experiments. The simplexcentroid
design consists of 2q 1 points: the q pure components,
q
the (2) twocomponent blends with equal proportions of 1/2
q
for each of the proportions present, the (3) threecomponent
blends with equal proportions of 1/3 for each of the compo
nents present, . and the qcomponent blend with equal
proportions of 1/q for all components. The simplexcentroid
design contains blends involving every subset of the q
components where the components present in any blend occur
in equal proportions. Examples are given in Figure 2.2.
The simplexcentroid design has an associated "special"
canonical polynomial model
(2/3 o, 1/3)
(0,1,0) /
X2
(1/2 o, 02)
(0/3 0,2/3)
(0,0,1)
{3,2}
{4,2}
(3,3}
{4,3}
Figure 2.1 Some {q,m} SimplexLattice Designs for Three and
Four Components
X3 X2
Figure 2.2 SimplexCentroid Designs for (a) Three
Components and (b) Four Components
q q q
n = E S.xX + E s ij.x x + xxijkXix
i=1 i
+. . + 812...qX1X2...Xq (2.24)
This model contains 2q 1 terms and hence provides an exact
fit to data collected at the points of the corresponding
simplexcentroid design. The specialcubic model
3 3
S= E ixi + 8 ijxixj + 123x1 x2x3 (2.25)
i=1 i
is an example of (2.24) for the case q = 3.
Becker (1978) proposed radial designs for mixture
experiments for the purpose of detecting inactive components
or components with additive effects. He defined a radial
design as one inwhich all points lie on straight lines
(rays) extending from one or more focal points. For
unconstrained mixture experiments, focal points of interest
are the vertices of the simplex (xi = 1, xj = 0, jWi)
corresponding to those components thought to have additive
effects or thought to be inactive components. For
constrained mixture problems, focal points might be the
simplex vertices, the vertices of the pseudocomponent
simplex, or other points depending on how the concept of a
component effect is to be defined (see Piepel 1982).
The radial designs of Becker (1978) are an extension of
the axial designs proposed by Cornell (1975). The axis of
component i is the imaginary line extending from the vertex
xi = 1, xj = 0, j i, to the point xi = 0, x, = 1/(q1), j i,
on the opposite boundary. The points of an axial design lie
only on the component axes. Cornell and Gorman (1978)
illustrated the use of an axial design for detecting an
additive blending component. Other uses for axial designs
have been discussed by Cornell (1975, 1977).
Draper and St. John (1977b) presented DNoptimal
designs corresponding to their mixture models with inverse
terms (2.9) and (2.10), for three and four components. The
points of support upon which the DNoptimal designs for
models (2.9) and (2.10) with three components are based, are
shown in Figure 2.3. For the designs displayed in Figure
2.3, it was assumed that xi 1 0.05, i=1,2,3 to avoid the
problems that occur in xj1 when xi = 0.
Standard response surface designs such as factorials
and central composite designs are appropriate for fitting
polynomial models in ratio variables. Kenworthy (1963)
discussed factorial designs for situations where each ratio
variable is of the form rk = xi/xj. Hackler, Kriegel, and
Hader (1956), Donelson and Wilson (1960), Kissell and
Marshall (1962), and Kissell (1967) all used central
composite designs in situations where their ratio variables
were defined using ratio functions of the form ri =
x./(xi+x j + .).
(.90,.05,.05)
(.78,.17,.05)
(.2, .6,.2)
(.17,.78,.05)
(.05,.90,.05).
x2
(.2,.2,.6)
(.17,.05,.78)
.(.05,.05,.90)
(.05,.78,.17)
(.82,.05,.13)
(.72,.14,.14)
(.475,.05,.475)
(.14,.14,.72)
(.13,.05,.82)
(.05,.05,.90)
(.05,.82,.13)
Figure 2.3.
(.05, .13,.82)
Points of Support of DNOptimal Designs for
Models 2.9 (a) and 2.10 (b)
(.14,
(.13,
(.05,
Aitchison and BaconShone (1984) did not discuss
designs for their logratio models. However, it is clear
from the polynomial forms (2.17) and (2.18) that standard
response surface designs (such as factorials and central
composite designs) in the zi = log(xi/xj) are applicable.
Most of the designs discussed above were originally
proposed for unconstrained mixture experiments. For certain
constrained mixture experiments where the constraint region
is again a regular simplex, the designs discussed so far can
be adapted by using a pseudocomponent transformation.
However, most constrained mixture experiments have an
irregular hyperpolyhedron as a constraint region. We now
discuss designs for this type of constrained mixture
experiment.
McLean and Anderson (1966) presented an algorithm for
generating the vertices of the constraint region given the
lower (Li) and upper (Ui) bounds for the component
proportions. For fitting the Scheffe seconddegree
canonical polynomial model (2.4), they suggested using the
vertices and face centroids of the constraint region, and
referred to such designs as extreme vertices designs.
Snee and Marquardt (1974) presented an algorithm called
XVERT, for generating vertices of a constraint region.
Additionally, the XVERT algorithm helps one choose an
efficient experimental design for fitting the firstdegree
canonical polynomial model (2.3), by selecting only a subset
of all extreme vertices. The approach of Snee and Marquardt
includes the use of the DN, GN, and ANoptimality criteria
in choosing efficient designs.
Nigam, Gupta, and Gupta (1983) presented the XVERT1
algorithm for generating designs for fitting the first
degree canonical polynomial model (2.3). The XVERT1
algorithm is considerably faster than XVERT since it does
not depend on any of the optimality criteria (DN, GN, VN,
and ANoptimality) in building a design. Resulting designs
compare well with those developed by XVERT with respect to
measures such as Gefficiency, det(X'X), and tr[(X'X)1].
Snee (1975) discussed the development of mixture
designs for fitting the seconddegree canonical polynomial
model (2.4) in constrained regions. For 3 q 5, Snee
suggested the design should consist of the vertices, all
constraint plane centroids, the overall centroid, and the
centroids of long edges. For q 5, Snee suggested using a
computeraided design approach to select design points from
a list of candidates comprised of the vertices, edge cen
troids, face centroids, and overall centroid.
Goel (1980) discussed the UNIEXP algorithm which
assigns points uniformly over the constraint region. Goel
claimed that designs generated with the UNIEXP algorithm
compare favorably with those developed by the computeraided
design approach of Snee (1975).
Saxena and Nigam (1977) presented a transformational
approach for adapting the symmetric simplex designs of Murty
and Das (1968) to constrained mixture experiments. Murthy
and Murty (1983) discussed a transformational approach for
adapting factorial (fractional or complete) designs for
constrained mixture experiments. Both approaches differ
from the approach of Snee (1975) in that some points are
placed inside the region, whereas Snee's design points are
placed primarily on the constraint region boundaries.
2.3 MixtureProcess Variable ExDeriments
In mixture experimentation, it may be of interest to
observe changes in the response values caused by varying the
levels of n process variables in addition to the q mixture
component proportions. Scheffe (1963) gave an example where
the response is "road octane number" of a blend of
gasolines, and the make and speed of the car might be varied
as well as the proportions of the gasolines.
Since the main purpose of Scheffe's article was to
introduce the simplexcentroid design, he naturally
suggested a simplexcentroid x (k1 x k2 x . x kn)
factorial arrangement for mixtureprocess variable
experiments (where k represents the number of levels of the
jth process variable). Such an arrangement can be thought
of as a simplexcentroid design at each of the k1 x k2 x
. x k factorial points or alternately as an (kI x k2 x
. . x kn) factorial at each of the 2q 1 simplexcentroid
points.
Scheffe also discussed an associated model for the
simplexcentroid x (k1 x k2 x .. x kn) design. The
notation gets rather messy if the process variables are
considered as classification variables, and there are more
than two levels of each factor. For process variables zl,
z2, . zn measured on a continuous scale, the model can
be represented somewhat easier. As an illustration, the
model for q = 3 components and n = 2 process variables,
where each process variable is set at two levels, is given
by
n ax x + a 0x3 + 8 2xxx2 + B xx + B8 x2x
1X1 1 2 42 3 3 812X12 13 1 3 23 2 3
2 .
+ 8 3xqx2x + [8x + 8 + 2 832
+8123 x1x2x z 2 3 j= x1 x2 2 23 2x
12
+ 8123xx2x3]zlz2 (2.26)
In general, when there are n process variables each at two
levels, the complete canonical polynomial contains 2q+n 2n
terms and is of degree q+n in the x's and z's.
Note in (2.26) that there are no terms involving only
the process variables (main effects of the process variables
or interactions containing only process variables). This is
due to the identity x + x2 + + xq =1.
The number of points in the simplexcentroid x fac
torial design increases rapidly with the number of mixture
and process variables q and n. Scheffe (1963) discussed two
fractionation methods for reducing the number of points in a
simplexcentroid x 2n design. The first method is somewhat
complicated and will not be discussed here. The second
fractionation method sets up a 1:1 correspondence between
the (2q 1)2n points of the simplexcentroid x 2n design
and the points of a 2q+n design (after removing the 2n
points corresponding to those combinations where all of the
q mixture components are absent). A fraction of the sim
plexcentroid x 2n design is obtained by taking the points
corresponding to those in a fraction of the 2q+n design.
Points in the resulting design that have one mixture com
ponent present correspond to pure mixtures, points with two
mixture components present correspond to binary mixtures
with each component proportion equaling 1/2, . and so
on. The process variable combinations of high and low
levels of the zi at these mixture points are interpreted as
usual. As an example, for q = 5 and n = 2 let A, B, C, D,
and E represent the mixture components and let F and G
represent the two process variables. Then the point "acdg"
in the 27 design is the mixture composition
(1/3,0,1/3,1/3,0) run at the low level of process variable F
and at the high level of process variable G.
Cornell and Gorman (1984) presented various fractional
design plans for mixtureprocess variable experiments with
q = 2 or 3 components and with n = 3 process variables each
at two levels. They utilized Scheffe's second fractionation
method and considered designs for fitting mixtureprocess
variable models containing fewer than the 2q+n 2n terms in
the complete model.
It was noted earlier that there are no terms in
Scheffe's mixtureprocess variable models [e.g. (2.26)]
involving only the process variables. Gorman and Cornell
(1982) discussed reparametrized model forms that do contain
such terms. They introduced a simple example to illustrate
their work. For a twocomponent mixture experiment with one
process variable (at two levels), they considered the
canonical polynomial model
0 0 0
= 81x + 1 2x2 + 812x1x2
+ (B11 x 2 12x2)z1 (2.27)
The effects of the process variable z1 are contained in the
coefficients 61, B2, and 812. If zI has the same (constant)
11 21
effect on all compositions, then S1 = a2 and 812 = O,
yielding the reduced model
0X + X + x x + 1(2.28)
n = 81 + 8x2 1212 O+ 1 (2.28)
1 1 1
where = 81 = Note that the terms in (2.28) are not a
0 1 2
subset of the terms in (2.27), specifically S8z is not con
tained in (2.27). Gorman and Cornell also noted that one
can get a distorted view of the effects and significance of
zi by considering (2.27).
To arrive at a reduced form of the combined model in
the xi's and zi's, Gorman and Cornell suggested repara
metrizing the general form of the mixtureprocess variable
q
canonical polynomial model by first substituting 1 Z x.
j=2 J
for xl in all crossproduct terms involving x, (alone) with
the process variables and then rewriting the terms in the
model. For the above example, substituting 1 x2 for x1 in
the term 8xizi1 in (2.27) yields
o 0 o 1
n = 8 X1 + 82x2 + 2X1X2 + 81(1 x2)z1
1 1
= sBx1 + S x + 2X1X2 + 62x2zx
+ 812x1x2z1 + z1 (2.29)
1 1 1 1 1 1
where 8 8 and = B Hence, 6 represents the
difference between the effect of z1 on the linear blending
of x and x while 81 represents the effect of z on the
2 1 0 1
linear blending of xj. Note that when the terms 6x2z1 and
12xlx2z1 are omitted from (2.29), we obtain (2.28). Hence,
the reparametrized model is suitable for obtaining reduced
model forms through subset selection procedures. Also, note
that the reparametrized form (2.29) now has a term with z,
alone. But, as noted above, the coefficient 81 of zi
represents the effect of z, on the linear blending of xj,
not an overall main effect of z1 (unless both 61 and 812 are
zero, in which case 80 is a measure of the overall main
effect of zl).
In closing this section, it should be noted that
Scheffe's mixtureprocess variable models are still
applicable for constrained mixtureprocess variable
experiments. The concept of a mixture x factorial design
(and a fraction thereof) is valid and can be used in
situations where the mixture design is defined for studying
the response surface over a constrained region.
CHAPTER THREE
MODELS FOR MIXTUREAMOUNT EXPERIMENTS
In the usual definition of a mixture experiment
(Cornell 1981, Scheffe 1958), the response is said to depend
only on the proportions of the components present in the
mixture and not on the total amount of the mixture. This
definition has often prompted the question, "If the total
amount of the mixture also affects the response, do we still
have a mixture experiment?" Based on the above definition,
the answer is no. However, a mixture experiment in which
the amount of the mixture varies and affects the response is
a general mixture experiment (as defined in Section 1.3).
3.1 An Introduction to MixtureAmount Experiments
A general mixture experiment in which a (usual) mixture
experiment is conducted at each of several total amounts
will be referred to as a mixtureamount experiment. An
example of a mixtureamount experiment is the application of
fertilizer, where the amount (level) of fertilizer applied
is allowed to vary and the different levels can affect the
yield as much as the fertilizer formulation. Another
example is the treatment of a disease with drugs, where both
32
the amount and composition of the drug affect the speed and
quality of recovery that occurs.
This generalization of the definition raises many
questions about the design, modeling, and analysis of
mixtureamount experiments. For example:
1. Are the blending properties of the mixture compo
nents affected by varying the total amount of the
mixture? If so, how?
2. If the blending properties of the component are not
affected by the total amount, what effect if any
does varying the total amount have on the response?
3. What model forms are appropriate for measuring the
component blending properties and the total amount
effects mentioned in questions 1 and 2 above?
4. What type of designs should be used to develop
models to answer the above questions?
5. Finally, if there are process variables in the mix
ture experiment, how are their effects affected, if
at all, by varying the total amount of the mixture?
In this chapter, models and designs that relate to questions
1, 2, 3, and 4 are discussed. Mixtureamount experiments
with process variables are discussed briefly in Appendix G.
Before proceeding with model development, several
simple hypothetical situations are presented to illustrate
what is meant in the first two questions above by the total
amount affecting the component blending properties. Con
sider a mixtureamount experiment with q = 2 components and
a total amount variable A at two levels (say A1 < A2).
Suppose the two components blend linearly at both amounts.
Several possible situations are shown in Figure 3.1, where
the pure component proportions are denoted by (xl,x2) =
(1,0) and (0,1). Figure 3.1(a) illustrates the case where
changing the level of the total amount has no effect on the
response (the lines are coincident), while in Figure 3.1(b)
increasing the amount from A1 to A2 increases the response
at all mixtures by a constant amount. Note that Figure
3.1(b) illustrates a situation where the total amount does
not affect the component blending properties but does affect
the response. Figures 3.1(c) and (d) illustrate cases where
the total amount does affect the component blending proper
ties. In Figure 3.1(c), we see that an increase in the
value of the response results from raising the level of the
amount of the mixture, and the effect of raising the amount
becomes larger as the proportion of component 2 in the
mixture increases. Figure 3.1(d) represents a situation
where changing the amount has a considerable effect on the
blending properties of the two components; at A1, increasing
the proportion x2 produces an increase in the response
value, while at A2 it results in a decrease in the response
value.
3.2 Including the Total Amount in Mixture Models
Since the response in a typical mixture experiment does
not depend on the total amount, the usual mixture model
forms must be modified to incorporate amount effects for
AA2 A
Al
c(d)
on .
I AI
(1,0) (0,1) (1,0) (0,1)
(a) (b)
A2
A1
A2
Al
(1,0) (0,1) (1,0) (0,1)
(c) (d)
Figure 3.1. Plots of Several TwoComponent Blending Systems
at Two Total Amounts A1 and A2
fitting data from a mixtureamount experiment. A technique
for doing so is suggested by recognizing the similarity of a
mixtureamount experiment to a mixture experiment with one
process variable. Likewise, a mixtureamount experiment
with n process variables is similar to a mixture experiment
with n+1 process variables.
Scheffe (1963) developed models for mixture experiments
with process variables by considering the parameters of his
canonical polynomial mixture models as being dependent on
the process variable effects (these models were presented in
Section 2.3). This same technique can be adapted for
mixtureamount experiments with or without process
variables. Mixtureamount experiments without process
variables are discussed in this chapter. The extension to
mixtureamount experiments with process variables is
discussed in Appendix G.
3.3 MixtureAmount Models Based on Scheffe
Canonical Polynomials
Scheff6's canonical polynomials (see Section 2.1) have
been shown to be a versatile class of equations for modeling
mixture response surfaces. Since a mixtureamount experi
ment is just a series of mixture experiments run at each of
several amounts A1, A2, . Ar, r > 2, it is natural to
envision fitting the entire experimental data set as a
series of smaller experiments which are performed at each
amount Ai. We consider the fitting of a Scheffe canonical
polynomial model such as (2.3) or (2.4) at each amount.
To begin the development, suppose a particular Scheffe
canonical polynomial model form, denoted by nC, adequately
describes the component blending at each of the r levels of
A. If the total amount of the mixture affects the response,
the parameters of nC vary as A varies, i.e., the parameters
of nC depend on A. This dependence can be modeled (for each
parameter 8m in nC) using the standard polynomial form
r1
Sm(A) = + z k(A, (3.1)
k=1
where Sm(A) denotes that the parameter is considered to be a
function of A, and A denotes a coded version of A.
Although (3.1) implies an (r1)th degree polynomial can be
used if desired, a seconddegree polynomial will often
suffice in practice. Nonpolynomial functions of A that
might also be appropriate for certain applications are
r1
B (A) = 0 + Z Ak (3.2)
Sm ,k=1m
or
r1
S(A) = O + k (log A)k (3.3)
k=1
In practice, A1 and log A in the above equations would
usually be coded, as was A in (3.1).
By writing the parameters in a Scheffe canonical
polynomial model as functions of A, a new model is obtained
that enables us to measure the effects of total amount on
the blending properties of the components. This model
derivation technique may be applied to any of the mixture
models reviewed in Section 2.1 (see Section 3.4). Any model
obtained in this manner will be referred to as a mixture
amount model.
As an example, let us derive the form of a quadratic by
quadratic mixtureamount model where q = 2 and r = 3. Then,
nC is of the form (2.4) and the 8m's are of the form (3.1),
which yields
n = B1(A)x1 + B2(A)x2 + B12(A)x1x2
12 12 12 10 2
= [% + 0 A + 1(1 ) x [82 + 82A + 82(A ) ]x2
= Bxi + 2x2 + B2x1lx2
2 k k k k
+ E [x + 2 + B12x1x2](A ) (5.4)
k=1
Note that the subscript of a a parameter in (3.4) refers to
the components that are present in the associated term while
the superscript refers to the power of the A variable for
that term.
When the levels A,, A2, A3 are coded to have zero mean
(usually 1, 0, +1 if the levels are equally spaced), the
terms in the combined model (3.4) have the following
interpretation:
i) soX1 + BOX + 8 2X1x2 represents the linear and
nonlinear blending properties of the mixture
components at the average level of total amount,
ii) [BLx1 + 8 x2 1 l2X12]A' represents the linear
effect of total amount on the linear and nonlinear
blending properties of the mixture components,
iii) [Szx1 + 2x2 2xlx2](A')2 represents the
quadratic effect of total amount on the linear and
nonlinear blending properties of the mixture
components.
Thus the coefficients 8 and 8j of the terms xi(A )k and
xixj(A')k, k=1,2, in (3.4) are measures of the effects of
changing the amount of the mixture on the linear and
nonlinear blending properties of the mixture components (at
the average level of total amount).
When the levels of A' and (A')2 are coded to be the
coefficients of orthogonal polynomials [i.e., when A and
(A')2 in (3.4) are replaced by the first and seconddegree
orthogonal polynomials P1(A) and P2(A)], the interpretations
of the coefficients change somewhat. Under this coding, the
coefficients 8$, B2, and B12 measure the linear and
nonlinear blending properties of the components averaged
over the levels of total amount. The coefficients 8 and
8j, k=1,2, are measures of the effects of changing the
amount of the mixture on the linear and nonlinear blending
properties of the mixture components (averaged over the
levels of total amount). See Section B.2 of Appendix B for
an example that illustrates the above interpretations.
For general q and r, a model of the form (3.4) is
written as
q q 0
n = E 8.x + r Z 8 ijxixj
i=1 i
r1 q k
+ E [ E x + E 8 .x x ](A ) (3.5)
k=1 i= i
Depending on the way in which changing the total amount
affects the component blending for a particular application,
all of the terms in (3.5) may or may not be needed. Several
reduced forms of (3.5) that may be appropriate for various
applications are listed and discussed in Appendix A.
Suppose now the amount of the mixture does not affect
the blending properties of the components but does have an
effect on the value of the response. For the model of
(3.5), this implies that A has a constant linear effect for
1 1 1 1
all compositions (which forces 81 = = .. = 1 and 12
1 1
= 81 = 1 = 0); that A has a constant quadratic
13 =q1,q
effect for all compositions (which forces 82 = 2 =
82 and 82 = _2 1 = 0); . ; and that A
S 12 q1,q
has a constant (r1)th degree effect for all compositions
(which forces 8r1 = B1 = 1 and 8 =1 = r1
1 2 q 12 13
= Sr q = O). In this case, the model of (3.5) takes
the reduced form
q 0q 0 r1 k k
n = SOx. + E B .x.x. + S (A ) (3.6)
i=1 1 1 i
where the 8k (= = ), k=1,2, . ,r
0 1 q
represent the linear, quadratic, . (r1)th degree
effects of total amount on the response. Several reduced
forms of (3.6) that are of the most practical interest are
also presented and discussed in Appendix A.
Note that the terms of (3.6) are not a subset of the
terms of (3.5); specifically the terms with A alone
(80(A')k, k=1,2, . ,r1) arenot contained in (3.5).
This means that a subset regression procedure cannot be used
on (3.5) to arrive at the form (3.6). This problem may be
alleviated by reparametrizing (3.5) as suggested by
Gorman and Cornell (1982). The reparametrization involves
q1 k k
replacing xq with 1 xi in the terms 8 x (A), k=1,2,
4 i=1 'qq
. ,r1, of (3.5) and simplifying. For the q = 2, r = 3
example considered earlier (3.4) is reparametrized as
0 0 0
n = 1 x1 + 2 + 812X1X2
2 k k kk
+ z [11X1 + 2(1 x1) + 12XX2(A
k=1
2
2 k k k k
k k k k k 1
x2. Although (3.7) now contains the terms +A' and 8((A')2
where 6A and ()2 and e, note 2, k=1,2. Hence, ( repre
measure the linear and quadraticnear effects of total amount,
amounrespectively, on the linear blending properties of x, and do not
measurements the overall difference between the quadratic effects of total
of total amount unlesson the linear blending properties of x and
x(. Although (7)7) now contains the terms OA. and 8O(A
The Scheffe canonical polynomial mixtureamount models
considered Ad thus far were all developed under the assumption
measure the linesame canonical polynomial form id quadratic effects of appropriate forunt,
despctively, onbing the component blending at each total amount.
measure the overall linear and quadratic effects of total
amount [unless we find that 61 =
The Scheffe canonical polynomial mixtureamount models
considered thus far were all developed under the assumption
that the same canonical polynomial form is appropriate for
describing the component blending at each total amount.
Situations where the appropriate forms of the canonical
polynomials at each level of total amount are different are
also of interest. Mixtureamount model forms for these
situations will now be discussed.
Let us suppose one of the Scheffe canonical polynomial
forms (linear, quadratic, cubic, etc.) is appropriate for
describing component blending at each level of A, and that
the appropriate forms are not the same for all levels of
A. Further, consider the most complicated form (i.e.,
highest degree) that is needed at one of the levels of A.
Then, if the form of the mixtureamount model is derived
using this "most complicated" canonical polynomial, it will
be an adequate (but overparametrized) form for fitting data
from the mixtureamount experiment. The appropriate mix
tureamount model form is a reduced form of the "adequate"
mixtureamount model. The nature of these model reductions
are determined for several situations in Appendix B.
Several canonical polynomial mixtureamount models of
practical interest have been discussed in this section and
are also discussed in Appendices A and B. To determine if
one model is better than another, or if one model is most
appropriate for a particular application, one can perform a
series of full vs. reduced model tests,
(SSEduced SSEful)/(er e f)
F* = SSEu/e (5.8)
where SSE reduced, SSEfull, er and ef are the sum of squares
for error and the error degrees of freedom for the reduced
and full models, respectively. As an example, suppose in a
mixtureamount experiment that the component blending is
nonlinear (quadratic), and that the total amount has at most
a linear effect on the component blending properties. For
this situation, we might consider the models
q 0q 0 1'
Model 1: n = S S.x. + z 8 .x.x. + $ A
i=l1 I i
q 0 q 0 q '
Model 2: n = E B.x. + r Z 8B .x.x. + EZ .x.A
i=1 i
q 0 q0 q 1
Model 3: n = Sixi + E E ijxix + S xi A
i=1 i
q 1
+ Z S ij.x.x A ,
i
which are models (A6), (A7), and (A8) in Appendix A. We
begin by fitting models 3 and 2, treating them as the "full"
and "reduced" models respectively, and performing the test
(3.8). The test is a measure of the significance of the
q(q1)/2 terms 80xixjA', 1 i < j < q, in model 3 over and
above the contribution of the terms contained in model 2.
The error degrees of freedom are ef = Nq(q+1) and er =
Nq(q+3)/2, respectively. If the test is significant, model
3 is selected. If the test is not significant, then models
2 and I are compared, treating them as "full" and "reduced,"
respectively. The full vs. reduced model test (3.8) can
also be used to compare the models discussed in Appendix B
to the corresponding models without parameter restrictions.
Another model selection approach is to fit the repara
metrized form of a "full" mixtureamount model (such as
model 3 in the above example) and use variable selection
techniques such as allpossiblesubsets regression or step
wise regression to determine the most appropriate model.
However, the reduced models of Appendix B are not obtainable
using this approach.
3.4 MixtureAmount Models Based on Other
Mixture Model Forms
In the previous section, mixtureamount models were
developed by writing the parameters of Scheffe canonical
polynomial models as functions of the total amount A. This
technique may also be used with any of the other mixture
models taken from the literature, many of which were
presented in Section 2.1. Expressions such as (3.1), (3.2),
(3.3) or any other appropriate function of A may be used for
the parameters of the mixture model chosen. Any such model
obtained by this technique is referred to as a mixture
amount model.
As an example, assume the inverse term model (2.9) is
appropriate at each of two amounts, Al and A2. Using am's
of the form (3.1) yields the mixtureamount model
q9 1
n = S S.(A)x. + 8 (A)x
q Aq 0 x1
i=.1 1 i=1 
q0 q 0 1
6 8.x. + a .ix
i=1 i=1
q1 q 1 1
+ E S.ixA + x A (3.9)
i=1 i=1 i
Recall that A denotes a coded form of the total amount
variable A. When the levels Al and A2 are coded to have
mean zero, the terms in (3.9) have the following interpreta
tions:
q 0 q 0 1
(i) 2 80xi and Z B_0 respectively represent the
i=1 i=1
linear and nonlinear blending properties of the
mixture components at the average level of total
amount,
q q 1 I
(ii) E 1 xiA and a 8xi A respectively represent
i=1 i=1
the linear effects of total amount on the linear
and nonlinear blending properties of the
components.
The phrase "nonlinear blending" in the above interpretations
refers to an extreme increase or decrease in the response
value as the value of xi approaches zero.
As another example, assume that the secondorder form
of Becker's H3 model (2.8) is appropriate at each of three
levels of A and that total amount has a logarithmic effect
on component blending properties. The appropriate mixture
amount model is given by
q q
n = Z S.(A)xi + E 8 .(A)(xi x )1/2
i=1 i
q 0 1 q
= + (log A)]xi + E E [8 +81 (log A) ](x.xj)
i=1 1 i i
0 q 0 1/2
Z Sixi + E Sij (xix )
i=1 i
q 1 q 1 1/2'
+ 8x.(log A) + E 1 (x.x.)1/2(log A) (3.10)
i=1 1 i
The notation (log A) above denotes a coded form of log A.
When the three levels log A1, log AZ, and log A3 are coded
to have mean zero, the terms in (3.10) have the following
interpretations:
q 0 q 0 1/2
(i) x and Z 8 i(xxj)1/ respectively repre
i=1 i
sent the linear and nonlinear blending properties
of the mixture components at the average level of
log (total amount),
q q 1 1/2
(ii) E Bixi(log A) and S B ij(xix ) (log A)
i=1 i
respectively represent the logarithmic effects of
total amount on the linear and nonlinear blending
properties of the mixture components.
All of the techniques discussed in Section 3.3.,
Appendix A, and Appendix B for deriving or reducing the
Scheffe canonical polynomial mixtureamount models are
applicable for any of the other types of mixtureamount
models. The example models (3.9) and (3.10) will be used to
illustrate this point.
Models such as (3.9) and (3.10) are appropriate for
situations where the total amount affects the linear and
nonlinear component blending properties similarly [e.g., in
(3.9) it is assumed that the total amount has a linear
effect on both the linear and nonlinear blending
properties]. For situations where this is not the case,
reduced models (similar to those presented in Appendix A for
Scheffe canonical mixtureamount models) may be needed. For
example, the reduced form of (3.9),
q 0 q 0 1 q
xn = E + Z X + Zi ixiA (3.11)
i=1 i= i=1
is appropriate if the total amount has a linear effect on
the linear component blending properties but does not affect
the nonlinear blending properties.
If the total amount does not affect the blending prop
erties but does affect the response, the appropriate models
for the two examples are
q 0 q 0 1 1
n = S x. + E 8 ,x. + 88 A(.12)
i=1 i=I 
and
q 0 q 0 1/2 1 A)
n = x. + E Z S (.x.x)12 + s(log A)' (3.15)
i=1 i
These models are reduced forms of (3.9) and (3.10),
respectively. However, the terms in (3.12) and (3.13) are
not subsets of the terms in (3.9) and (3.10). The Gorman
and Cornell (1982) reparametrization technique [reexpres
q1 1
sing x as 1 z x. in the terms 8 x A of (3.9) and
q i1 i=1
sqx (log A)' of (3.10)] discussed in Section 3.3 is
applicable here.
For each of (3.9) and (3.10), it is implicitly assumed
that the same mixture model is valid at each level of A.
For situations where this is not the case, the appropriate
parameter restrictions can be obtained as was done for the
Scheffe canonical polynomial mixtureamount models in
Appendix B. As an illustration, consider the situation
specified by the mixture models
"A =l = ax1 + bx2
1
nA' =' cx1 + dx2 + ex (3.14)
where the two levels of A are coded as 1 and +1 (A denotes
the coded version of the total amount variable A). The
appropriate mixtureamount model for this situation is of
the form
0 0 0 1 1 xA' 1
n = 81x + + 1x1 1+ A + 82x2A
1 1 '
+ 8 x1 A (3.15)
with as yet unknown parameter restrictions. Substituting
the data
X1 x2 A
1 0 1 a
.05 .95 1 .05a + .95b
.50 .50 1 .50a + .50b
1 0 1 c+e
.05 .95 1 .05c + .95d + 20e
.50 .50 1 .50c + .50d +2e
into (3.15) and solving the resulting system of equations
yields the parameter estimates
0 a+c 0 b+d O e
81 a 2 2 = 2 1 2
1 ca 1 db 1 e (1
81 = 2 82 = 2 81 = (3.16)
The appropriate parameter restriction for this situation is
thus seen to be all = sI. If the inverse nonlinear
blending occurred at the low level of A instead of the high
level, the parameter restriction would be al8 = O$1
3.5 MixtureAmount ModelsA Summary
A mixtureamount model is developed by writing the
parameters of any (usual) mixture model as functions of the
total amount of the mixture. This modeling technique is
very flexible in that any mixture model (e.g., a Scheffe
canonical polynomial, one of Becker's models, a model with
inverse terms, a ratio model, a logratio model, etc.) can
be used, and the parameters may be written as any function
of A. The simplest application of this mixtureamount
modeling technique is to choose a mixture model which is
assumed to be adequate at all levels of amount to be
considered and assume that the parameters of this model are
all expressible as a common function of A. However, the
technique does not require that the mixture model
appropriate at each level of amount be the same nor does it
require that each parameter be expressible as the same
function of A. Reduced forms of mixtureamount models
obtained by this technique provide for many of these
situations (see Appendices A and B).
The considerable flexibility of the mixtureamount
modeling technique and the resultant vast number of models
to be considered raises questions about the practical
aspects of selecting an appropriate mixtureamount model. A
natural model selection approach is suggested when data from
a complete (not fractional) mixtureamount experiment is
available. Since a mixtureamount experiment is defined as
being a series of usual mixture experiments run at each of
several amounts, it is natural to first select (using the
data) an appropriate mixture model separately for each
amount. Often these individual models will all belong to a
particular family (canonical polynomials, inverseterm
models, etc.), in which case a "largest" member of the
family adequate for all levels of amount could be fitted.
Then graphical or weighted least squares (WLS) regression
techniques can be used to investigate the form of functional
dependence on A for each parameter. The information gained
by selecting (fitting) an appropriate model at each level of
A can then be used (as in Appendices A and/or B) to select
the appropriate mixtureamount model. If only two levels of
A are considered in the mixtureamount experiment, the
graphical or WLS regression techniques will not be helpful
in choosing the functional form of parameter dependence on
A. Prior knowledge about the system may suggest a form such
as (3.2) or (3.3) rather than the linear form (3.1).
If the available data are from a fractional mixture
amount experiment or are not from a mixtureamount
experiment at all, the above "natural" approach to model
selection may not be appropriate. In such situations, the
sequential "full vs. reduced" model procedure discussed at
the end of Section 3.3 is appropriate. The Gorman and
Cornell "reparametrization followed by variable selection"
technique discussed in Section 3.3 may also be of help in
such situations. The practical aspects of selecting a
mixtureamount model will be considered further in Chapter 7
where several examples will be presented.
Finally, note that mixtureamount models in general are
tools for answering the first two questions posed in Section
3.1. That is, if the blending properties of mixture com
ponents are affected by varying the total amount, then a
mixtureamount model is appropriate for modeling the
response. By coding the levels of A (or A1, log A, etc.)
to have mean zero, mixtureamount models provide a descrip
tion of the blending properties of the components at the
average level of A (or A1, log A, etc.) and explain how the
total amount affects these component blending properties.
By using orthogonal polynomial functions of A (or A1, log
A, etc.) in mixtureamount models, descriptions of the
component blending properties averaged over the levels of A
(or A1, log A, etc.) and how the total amount affects these
properties are obtained. If the blending properties of the
components are not affected by the total amount, then a
reduced model is appropriate and explains how varying the
amount affects the response (if at all).
CHAPTER FOUR
DESIGNS FOR MIXTUREAMOUNT EXPERIMENTS
Designs for both unconstrained and constrained mixture
amount experiments are presented in this chapter. An
unconstrained mixtureamount experiment is one in which the
component proportions xi vary between 0 and 1. A
constrained mixtureamount experiment is one in which at
least one component proportion is restricted by a nonzero
lower bound or a nonunity upper bound, or by both.
In Section 4.1, a general approach to developing
designs for mixtureamount experiments is presented and
guidelines for selecting the levels of total amount to be
investigated are given. Techniques for fractionating
mixtureamount designs are discussed in Section 4.2.
4.1 Developing Designs for MixtureAmount Experiments
Since a mixtureamount experiment is defined as a
series of mixture experiments at several levels of total
amount, it is natural to propose as mixtureamount designs
those designs obtained by constructing a usual mixture
design at each level of total amount. Usual mixture designs
for both unconstrained and constrained mixture experiments
were discussed in Section 2.2.
Defining a mixtureamount design as a series of
separate mixture designs allows us some degree of
flexibility in specifying an overall design, since the
mixture designs set up at each level of total amount may or
may not be the same. The family of mixture designs needed
will depend on the family of mixture models selected as well
as whether or not the component proportions are
constrained. In practice, unless a great deal is known
about the component blending properties and the effect total
amount has on these properties, the same mixture model is
usually considered at each level of total amount. Then, the
same mixture design (corresponding to the mixture model
under consideration) is constructed at each level of total
amount. However, in situations where it is known beforehand
that components blend differently at different amounts, or
where additional investigation into the component blending
at one amount is desired, one can choose to run different
mixture designs at different total amounts. Also,
fractionated designs (which we discuss in Section 4.2) can
be viewed as different mixture designs at each level of
total amount.
As an illustration of the design development process,
consider a q = 3, r = 3 unconstrained mixtureamount
experiment where the experimenter does not anticipate having
additive or inactive components, nor does he expect extreme
response behavior as component proportions approach zero.
Based on this knowledge, the experimenter selects the
specialcubic canonical polynomial as being an appropriate
model for describing component blending at each of the three
levels of total amount. An appropriate mixtureamount
design is then a threecomponent simplexcentroid mixture
design set up at each of the three levels of total amount
(see Figure 4.1). However, the experimenter may be curious
as to whether or not the special cubic is adequate (i.e., Is
it an underestimate of a full cubic surface?), but cannot
afford to run a larger mixture design at each level of total
amount. As an alternative, he may choose to run a {3,3}
simplexlattice design (for measuring the full cubic shape
of the surface) at one of the levels of total amount, say at
the middle level, while keeping the simplexcentroid designs
at the high and low levels of total amount (see Figure 4.2).
As a second illustration, consider a constrained
mixtureamount experiment with three components and two
levels of total amount, where the component proportions are
constrained by .1 x,1 .4, .1 < x2 .3, and .35 x3 <
.75. An appropriate design for the specialcubic by linear
mixtureamount model
1 3 h h h ( h
n = z [ Z Bix. + zE 8 .ixix. + 8 x x ](A ) (4.1)
h=O i=1 i
consists of the vertices, centroids of the longest edges,
and the overall centroid of the constraint region at each of
Figure 4.1. MixtureAmount Design Consisting of a Three
Component SimplexCentroid Design at Each of
Three Amounts
Figure 4.2. MixtureAmount Design Consisting of a {3,3}
SimplexLattice Design at the Middle Level and
a SimplexCentroid Design at the Low and High
Levels of Total Amount
the two levels of amount (based on the recommendation of
Snee 1975see Section 2.2). The points of this design are
listed in Table 4.1 and are pictured in Figure 4.3.
Another important aspect of developing designs for
mixtureamount experiments is the choice of spacing for the
levels of total amount. If only two levels of A are to be
investigated, they should be chosen far enough apart to
allow the total amount effect to be detected. However, if
it is suspected that the effect of A could be quadratic but
only a linear effect is desired, the two levels should be
close enough so that the assumption of a linear effect of A
is valid.
When a higherthanlinear effect of A is to be
investigated and more than two levels of A are to be used,
the choice of spacing for the levels of A will depend on
what is known (or guessed) about the effect of total amount
on the response. If it is believed that a polynomial
function of A will adequately explain the effect of total
amount, the levels of A should be equally spaced. If it is
believed that a functional form such as (3.2) or (3.3) will
adequately explain the effect of total amount, the levels of
A should be equally spaced on a log A or A1 scale,
respectively. Regardless of the scale chosen, the equally
spaced levels should be spread far enough apart to yield
detectable differences in response as the level of A
changes.
Table 4.1.
Pt(a)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
Design Points for Fitting a SpecialCubic Model
in a ThreeComponent Constrained MixtureAmount
Experiment at Two Levels of Amount
x1
.10
.10
.15
.40
.40
.35
.225
.10
.275
.40
.25
.10
.10
.15
.40
.40
.35
.225
.10
.275
.40
.25
.30
.15
.10
.10
.25
.30
.30
.225
.10
.175
.20
.30
.15
.10
.10
.25
.30
.30
.225
.10
.175
.20
_ 3
XL
.60
.75
.75
.50
.35
.35
.475
.675
.625
.425
.55
.60
.75
.75
.50
.35
.35
.475
.675
.625
.425
.55
A
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
(a) These point numbers are used in Figure
4.3.
C
,4
cl
0
C
*a
e4
o
a,
0
c
o0
4
1
C)
cn
cli
0 C
CL
4
0)
o4
O4
1
00C
*ca)
03 &0
C C
< <
i i
(U (
'4
4.2 Fractionating Designs.for MixtureAmount
Experiments
As q (the number of mixture components) and r (the
number of levels of total amount) increase, the total number
of design points in mixtureamount experiments can become
excessive. The total number of design points can be reduced
by running only a subset (fraction) of the points in a
complete mixtureamount design. Since a reduction in the
total number of design points can result in a considerable
savings in terms of cost and time of experimentation,
methods for fractionating mixtureamount designs are now
discussed.
Fractionating mixtureamount designs is fairly
straightforward for those particular situations where the
overall design is a factorial design. Factorial mixture
designs are appropriate for mixture models in ratio
variables [e.g. (2.11) or (2.12)] or in logratio variables
[e.g. (2.17) or (2.18)]. Running such a factorial mixture
design at each of several levels of total amount yields a
factorial mixtureamount design. If the q1 mathematically
independent ratio or logratio variables are each
investigated at two levels and A is also investigated at two
levels, an appropriate mixtureamount design is a 2q1 x 2 =
2q factorial design. Similarly, a 3q factorial mixture
amount design is appropriate if the q1 ratio or logratio
variables and the total amount variable A are each
investigated at 3 levels. Fractionation methods for 2q and
3q designs are well known and many such fractional designs
have been tabled (e.g., see Cochran and Cox 1957).
Fractionation methods for the 2k3m series of factorial
designs are discussed briefly in Appendix G with respect to
mixtureamountprocess variable experiments, but the
techniques are applicable here also.
The second fractionation method of Scheffe (1963),
discussed in Section 2.3, can be used to reduce the number
of points in simplexcentroid x 2 mixtureamount designs
(designs in which a q component simplexcentroid design is
set up at each of two levels of total amount). In general,
a simplexcentroid x 2 mixtureamount design supports
fitting a mixtureamount model of the general form
1 q h q h q h
S= C ix + Z xx + xx I+ 8ijkXiXjXk
h=O i=1 i
h h
++ x 2...qX1 ...x q](A )h( (4.2)
However, fractions of a simplexcentroid x 2 mixtureamount
design will not support fitting this full model. The value
of q and degree of fractionation will determine the reduced
forms of (4.2) that can be fitted.
As an example, a onehalf fraction of the three
component simplexcentroid x 2 mixtureamount design is
listed in Table 4.2 and is pictured in Figure 4.4. This
sevenpoint design supports fitting either the specialcubic
Table 4.2.
x1
1/2
1/2
0
1
0
0
1/3
OneHalf Fraction(a) of a SimplexCentroid x 2
MixtureAmount Design for Three Components
x2
1/2
0
1/2
0
1/3
x3
0
1/2
1/2
0
0
1
1/3
A
1
1
1
1
1
1
1
(a) Fraction obtained using I = +ABCD as the defining
contrast. Switching the levels of A' yields the
I = ABCD fraction.
A' = L
A' = +1
Figure 4.4. Graphical Display of Design in Table 4.2
mixture model, or, the seventerm mixtureamount model
3 3 0 )
n = E Sx. + E Z B .x.x + 80A (4.3)
i=1 i
Fitting the specialcubic mixture model is only appropriate
if the total amount does not affect the response, while
fitting (4.3) is only appropriate if the nonlinear blending
is quadratic and the total amount has a linear effect on the
response (but does not affect the component blending
properties). Hence, by taking a onehalf fraction of the
complete design for q = 3 and r = 2, we forfeit the ability
to detect whether or not the total amount affects the
component blending.
As a second example, consider the onehalf fraction of
the fourcomponent simplexcentroid x 2 mixtureamount
design which is listed in Table 4.3 and is pictured in
Figure 4.5. This 15point design supports fitting the 15
term mixture model
4 O 4 0 4 0
n E i .x. + E x x. + E E B xijk.x ix
i=1 i
+ 0 234x2xx4 (4.4)
1234xi 2 '
or the 15term mixtureamount models
4 0 4 0
n = Si x + z Z 8 x.x.
i=1 i
4 0
+ EL 8ijkxixjxk + A (4.5)
i
or
4 4
n= Bx. + EE i. x x
i=1 i
4
+ 1 1 '
+ x 8Sx.A + {one 8 .x.x.A term} (4.6)
i=1 J
These three models are appropriate under different
assumptions about component blending and how the total
amount affects the response, if at all. Model (4.6) is the
only one of the three that allows for the component blending
properties being affected by total amount, and does so at
the cost of assuming there is no special cubic or quartic
blending among the four components. Since the face
centroids (1/3,1/3,1/3,0), . (0,1/3,1/3,1/3) and the
overall centroid (1/4,1/4,1/4,1/4) are included in the
simplexcentroid design for the purpose of estimating the
special cubic and quartic blending properties, it seems
apparent that the onehalf fraction (in Table 4.3), is not
optimal for fitting model (4.6). That this indeed is the
case is noted by observing that one could do better by
replacing the face and overall centroids with the remaining
Table 4.3.
x1
1/2
1/2
1/2
0
0
0
1/4
1
0
0
0
1/3
1/3
1/3
0
OneHalf Fraction(a) of a SimplexCentroid x 2
MixtureAmount Design for Four Components
x2
1/2
0
0
1/2
1/2
0
1/4
0
1
0
0
1/3
1/3
0
1/3
0
1/2
0
1/2
0
1/2
1/4
0
0
1
0
1/3
0
1/3
1/3
x
A
0
0
1/2
0
1/2
1/2
1/4
0
0
0
1
0
1/3
1/3
1/3
(a) Fraction obtained using I = +ABCDE
contrast. Switching the levels of
I = ABCDE fraction.
A' = 1
as the defining
A' yields the
A' = +1
Figure 4.5. Graphical Display of Design in Table 4.3
vertex points and another edge centroid (see later in this
section for a discussion of such designs).
The above two examples illustrate that the second
fractionation method of Scheffe can be used to fractionate
the simplexcentroid x 2 mixtureamount designs. However,
depending on the type of component blending to be
investigated, these fractions provide at best a portion of
the information about the effects of total amount on
component blending and at worst no information about the
effect of A on the response. If all higher order component
blending terms (such as cubic, quartic, . .) are to be
included in the model, these fractions provide no
information about how the total amount affects the response
(if at all). If some of the higher order component blending
properties may be assumed to be negligible, then these
fractions do provide some information about how the total
amount affects the response (or the component blending).
However, for situations in which higher order component
blending properties are assumed to be negligible, fractional
designs with better characteristics than those provided by
the method of Scheffe can be obtained using a computeraided
design approach. One such approach based on DNoptimality
is discussed below.
The fractionation methods discussed so far are appli
cable only for certain types of mixtureamount designs.
However, the computeraided design approach, introduced in
68
Section 1.1, provides a method for fractionating any
mixtureamount design for both unconstrained and constrained
mixtureamount experiments. Recall that the computeraided
design approach involves choosing a criterion of interest
(e.g., DN, GN, VN, or ANoptimality) and then selecting
points for the design from a candidate list so as to opti
mize the design criterion chosen. For design fractionation
purposes, the candidate points are the points of any
mixtureamount design to be fractionated. The DNoptimality
criterion (which seeks to maximize det(X'X), where X is the
Npoint expanded design matrix associated with the mixture
amount model to be fitted) is chosen for this work because
of its popularity and the availability of Mitchell's (1974)
DETMAX computer program to implement it. Although the
DETMAX algorithm does not guarantee generation of a DJ
optimal design, it often does so; when it does not, the
resulting design is near DNoptimal.
We discuss the development of DNoptimal designs for
canonical polynomial mixtureamount models. The development
for other families of mixtureamount models proceeds in much
the same way.
The candidate points for a given design/model are
usually the points of the associated complete mixtureamount
design. Several examples are given below.
O The candidate points for the models (Al) (A5) in
Appendix A are (assuming an unconstrained mixture
amount experiment) the simplex vertices at each
level of total amount. Since there are no
nonlinear blending terms in these models, the DN
optimal design will not contain binary, ternary,
. etc. mixtures even if included in the
candidate list. For a constrained mixtureamount
experiment, the candidate points would consist of
the constraint region vertices at each level of
total amount.
0 The candidate points for models (A6) (A14) in
Appendix A are (for an unconstrained mixtureamount
experiment) the simplex vertices (1,0, . ,0),
. (0,0, ,1) and the edge centroids
(..5 .5,0, . ,0), . (0, . ,0,.5,.5).
The face centroids (1/3,1/3,1/3,0, . ,0),
. (0,0, . ,1/3,1/3,1/3) would be included
if the mixtureamount model under consideration
contains special cubic terms. For a constrained
mixtureamount experiment, the candidate points
would consist of the constraint region vertices and
edge centroids at each level of total amount. The
twodimensional face centroids would be included if
the mixtureamount model contains special cubic
terms.
0 The candidate points for a full cubic canonical
polynomial mixtureamount model in an unconstrained
mixtureamount experiment are the points of a {q,3}
simplexlattice at each level of total amount.
The DNoptimal (or near DNoptimal) designs for several
of the canonical polynomial mixtureamount models of
Appendix A were obtained using the DETMAX program for three
component unconstrained mixtureamount experiments with two
and three levels of total amount. Some of the many possible
DNoptimal designs for the models considered are given in
Appendix C. The results for q = 3 suggest procedures for
developing DNoptimal designs (without the need of a
computer program such as DETMAX) for unconstrained
mixtureamount experiments for all values of q 3. The
procedures for two levels of amount are given in Tables 4.4
 4.9 and for three levels of amount in Tables 4.10 
4.12. The following terms are used in these tables:
0 positionsThe possible geometric locations of the
design points regardless of the level of total
amount.
0 pointA specific candidate point chosen for the
design.
O full setAll candidate points included exactly
once in the design.
The procedures in Tables 4.4 4.12 are written in a
way that facilitates the generation of a sequence of DN
optimal designs as N increases, with each design being
obtainable by adding one or more points to the preceding
design. The designs from Figures C.1 C.9 in Appendix C
serve as examples of the procedures in Tables 4.4 4.12 for
the case q = 3.
The procedures in Tables 4.4 4.12 describe how to
generate DNoptimal designs for p < N < C+p, where p is the
number of parameters in the particular model and C is the
number of candidate points for the design. For each of the
nine models considered, an N = C+p design consists of the C
candidate points plus an N = p design. Hence, the
procedures cycle and are applicable for developing DN
optimal designs for any value of N 2 p.
Table 4.4. Sequential DNOptimal Design Development
Procedure for Model (A6) in Appendix A
Candidate Points
Simplex vertices and edge centroids at the two levels of A
(assumed coded as 1 and +1). There are C = q(q+1)
candidate points.
Model
n = SOx. + E .x.x. + A
i=1 i
N
aq(q+1)
2
p+1 to C
C+1 to C+p
Procedure*
The smallest possible DNoptimal design
for this model contains points that cover
all positions once with one position
covered twice (once at each of the two
levels of A). The positions covered once
may be at either of the two levels of A.
Add points to cover the remaining posi
tions at each level of A (without repli
cating points) until a full set of
candidate points is obtained.
Add additional points to cover each posi
tion once. Note that an N = C+p design is
a full set plus an N = p design. Hence,
the procedure cycles, continuing as above.
* See Figure C.1 in Appendix C for examples of designs
generated by this procedure for the case q = 3.
Table 4.5. Sequential DNOptimal Design Development
Procedure for Model (A7) in Appendix A
Candidate Points
Simplex vertices and edge centroids at the two levels of A
(assumed coded as 1 and +1). There are C = q(q+1)
candidate points.
Model
q 0 q
n = Z .x. + E
i=1 i
N
2 + q
p+1 to C
C+1 to C+2p
C+2q+1 to C+p
0 q 1
Sijoxix + x 8ixiA
Procedure*
The smallest possible D optimal design
for this model contains the vertices at
both levels of A and points which cover
the edge centroid positions once. The
edge centroids may be chosen at either
level of amount so long as each position
is covered.
Add the remaining edge centroid points
until a full set of candidate points is
obtained.
Add additional points to cover each vertex
position once, then twice (without repli
cating among the additional points).
These points serve as second replicates of
the vertex positions at each level of A.
Add points to cover each edge centroid
position once. Note that an N = C+p
design is just a full set of candidate
points plus an N = p design. Hence, the
procedure cycles, continuing as above.
* See Figure C.2 in Appendix C for examples of designs
generated by this procedure for the case q = 3.
Table 4.6. Sequential DNOptimal Design Development
Procedure for Model (A8) in Appendix A
Candidate Points
Simplex vertices and edge centroids at the two levels of A
(assumed coded as 1 and +1). There are C = q(q+1)
candidate points.
Model
q q q0 q 1 q 1
1n = BZ x. + Z B ..x.x. + x.A + Z Z B .x.x.A
i=1 i
N
p = C
C+1 to 2C
Procedure*
The smallest possible DNoptimal design
for this model consists of a full set of
candidate points.
Add additional points until a second full
set is obtained. In choosing additional
points, it is not necessary to cover each
position once before covering a position
twice (once at each of the two levels of
A). However, points should not be
replicated within the additional points.
Note that the procedure cycles, continuing
as above.
* See Figure C.3 in Appendix C for examples of designs
generated by this procedure for the case q = 3.
74
Table 4.7. Sequential DNOptimal Design Development
Procedure for the SpecialCubic by Constant
MixtureAmount Model Below
Candidate Points
Simplex vertices, edge centroids, and two dimensional face
centroids at the two levels of A (assumed coded as 1 and
+1). There are C = (q +5q)/3 candidate points.
Model
q0 q0 q0 1
S= E 8 x + E ..x.x + EZ i kXiXX + 50A
i=1 i< i
N
p = C/2 + 1
p+1 to C
C+1 to C+p
Procedure*
The smallest possible DNoptimal design
for this model contains points that cover
all positions once with one position
covered twice.
Add points to cover the remaining
positions twice (without replicating
points) until a full set of candidate
points is obtained.
Add additional points to cover each
position once. Note that an N = C+p
design is just a full set plus an N = p
design. Hence, the procedure cycles,
continuing as above.
* See Figure C.4 in Appendix C for examples of designs
generated by this procedure for the case q = 3.
Table 4.8. Sequential DNOptimal Design Development
Procedure for the SpecialCubic by Linear
MixtureAmount Model Below
Candidate Points
Simplex vertices, edge centroids, and two dimensional face
centroids at the two levels of A (assumed coded as 1 and
+1). There are C = (q +5q)/3 candidate points.
Model
q 0q 0 q0 1q 1
n = x. + E x Z B .x.x. + E 8 x + E .x.A
i=1 1 1 i
N
p = C/2 + q
p+1 to p+(2)
p+( )+1 to C
C+1 to C+2q
C+2q+1 to
C+2q+(q)
C+2q+(q)+1 to
C+p
Procedure*
The smallest possible DNoptimal design
for this model contains points chosen to
cover all positions once and the vertices
twice (once at each of the two levels of
A).
Add the remaining edge centroids.
Add the remaining face centroids until a
full set of candidate points is obtained.
Add additional points to cover the vertex
positions once, then twice.
Add additional points to cover the edge
centroid positions once.
Add additional points to cover the face
centroid positions once. Note that an
N = C+p design is just a full set of
candidate points plus an N = p design.
Hence, the procedure cycles, continuing
as above.
* See Figure C.5 in Appendix C for examples of designs
generated by this procedure for the case q = 3.
Table 4.9.
Sequential DNOptimal Design Development
Procedure for the SpecialCubic by Linear
MixtureAmount Model Below
Candidate Points
Simplex vertices, edge centroids, and two dimensional face
centroids at the two levels of A (assumed coded as 1 and
+1). There are C = (q+5q)/3 candidate points.
Model
1 q h q h q h h
n = 1 [ hx + E jxx. + E 8jk iXj x ]( )h
h=O i=1 i
N
p = C
C+1 to 2C
Procedure*
The smallest possible DNoptimal design
for this model consists of a full set of
candidate points.
Add additional points until a second full
set is obtained. In choosing additional
points, it is not necessary to cover each
position once before covering a position
twice (once at each of the two levels of
A). However, points should not be repli
cated within the additional points. Note
that the procedure cycles, continuing as
above.
* See Figure C.6 in Appendix C for examples of designs
generated by this procedure for the case q = 3.
Table 4.10.
Sequential (Near)D1Optimal Design Development
Procedure for Model (A12) in Appendix A
Candidate Points
Simplex vertices and edge centroids at the three levels of A
(assumed coded as 1, 0, and +1). There are C = 3q(q+1)/2
candidate points.
Model
q 0
n =r Z 8x
i=1
q 1 q 2 2
Si.x.A + x x(A )
i=1 i=1
N
S= q(q+) 2q
p+1 to p+(q)
p+(q)+1 to C
C+1 to C+3q
Procedure*
The smallest possible DNoptimal design
for this model contains the vertices at
all three levels of A and covers the edge
centroid positions once.
Add points to cover the edge centroid
positions twice. Two edge centroids at
each of the three levels of A gives the
smallest variances for parameter
estimates.
Add the remaining edge centroids to
complete a full set of candidate points.
Add vertex points until all vertices are
included again (replicated twice).
Slightly larger determinants are obtained
if one first covers the q vertex positions
once, then twice, and finally three times
(1ll vertices). Variances of the sO and
8 are smaller if the vertex points are
concentrated at A' = 0, while the
variances of the 8' are smaller if the
vertex points are concentrated at A' = 1
and A' = +1.
q B
+ ZE .x.x. +
i
78
Table 4.10.continued.
N
C+3q+1 to C+p
Procedure*
Add points to cover the edge centroid
positions once. Note that an N = C+p
design is just a full set of candidate
points plus an N = p design. Hence, the
procedure cycles, continuing as above.
* See Figure C.7 in Appendix C for examples of designs
generated by this procedure for the case q = 3.
79
Table 4.11.
Sequential (Near)DOptimal Design Development
Procedure for Model (A13) in Appendix A
Candidate Points
Simplex vertices and edge centroids at the three levels of A
(assumed coded as 1, 0, and +1). There are C = 3q(q+1)/2
candidate points.
Model
q 0
n = s.x. + s s
i=1 i i
q i
+ s ixi .xj.A
i
N
p = q2 + 2q
p+1 to C
C+1 to C+2q
(3q2+7q+2)/2 to
6q+4(q)1
2
0
8 .x.x. +
lijx3i
q 2
+ E 8
i=1 1
q 1
E 8 .A
i=1 1 1
xi(A' )2
Procedure*
The smallest possible DNoptimal design
for this model contains the vertices at
all three levels of A plus the edge
centroids on the A' = 1 and A' = +1
simplexes.
Add edge centroids on the A' = 0 simplex
to obtain a full set of candidate points.
Add vertices on the A' = 1 and A' = +1
simplexes. Slightly larger determinants
are obtained if one first covers the q
vertex positions once, then twice.
Designs for N in this range are not easy
to describe. They are not sequentially
obtainable from the above designs as they
do not contain a full set of candidate
points. See Figure C.8 in Appendix C for
some examples when q = 3.
80
Table 4.11.continued.
N
Procedure*
6q + 4(q)
6q+4(q)+1 to
C+p
This unique design consists of the
vertices at all three levels of A twice,
and the edge centroids on the A' = 1 and
A' = +1 simplexes twice. Note that it
does not contain a full set of candidate
points.
Add edge centroids on the A' = 0 simplex.
Note that an N = C+p design is just a full
set of candidate points plus an N = p
design. Hence, the procedure cycles,
continuing as above.
* See Figure C.8 in Appendix C for examples of designs
generated by this procedure for the case q = 3.
Table 4.12. Sequential DNOptimal Design Development
Procedure for Model (A14) in Appendix A
Candidate Points
Simplex vertices and edge centroids at the three levels of A
(assumed coded as 1, 0, and +1). There are C = 3q(q+1)/2
candidate points.
Model
q 0 q0 q 1 q 1
I = E X. + ijx.x. + Z i.xA + Z Z a xix.A
i=1 i
q 2 q 2 2
+ Six.(A ) + E B ijix (A )
i=1 i
N
p =C
C+1 to C+p
Procedure*
The smallest possible DNoptimal design
for this model consists of one full set of
candidate points.
Add additional points until a second full
set is obtained. It is not necessary to
cover all positions once (or twice) before
covering some positions twice (or three
times). Concentrating the new points on
the A' = 1 and A' = t1 simplexes lowers
the variances of the B while
concentrating points on the AO = 0 smplex
lowers the variances of the a and B.
For larger N, the procedure cycles,
continuing as above.
* See Figure C.9 in Appendix C for examples of designs
generated by this procedure for the case q = 3.
A potential criticism for the use of DNoptimal designs
is that they are specific to the model under considera
tion. However, this is not totally true here. To see this,
consider the procedures in Tables 4.4 4.6 and note that
the procedures in Tables 4.4 and 4.5 are more complicated
(restrictive) than the procedure of Table 4.6. The simpli
city of the procedure in Table 4.6 is a result of the
associated mixtureamount model having xiA and xixjA terms
corresponding to the xi and xixj terms. The models in
Tables 4.4 and 4.5 are reduced forms of the model in Table
4.6 and do not have this "symmetry of terms" property. The
"nonsymmetry of terms" for the models in Tables 4.4 and 4.5
is why the corresponding procedures for developing DN
optimal designs are not as straightforward as the procedure
of Table 4.6. The point of this discussion is that designs
developed by the more restrictive procedures in Tables 4.4
and 4.5 also satisfy the procedure of Table 4.6; that is,
the procedures of Tables 4.4 and 4.5 generate designs that
are not only DNoptimal for their corresponding models, but
are also DNoptimal for the model of Table 4.6. This is
true only for designs containing N > C points, since a
minimum of C points is needed to support fitting the model
of Table 4.6.
On the other hand, designs generated by the procedure
of Table 4.4 are in general not DNoptimal for the model of
Table 4.5 (and vice versa), although it may be possible to
construct such designs for certain values of N. As an
example, when q = 3 the designs for 9 N < 14 displayed in
the first design columns of Figures C.1 and C.2 are DN
optimal for the models of both tables.
The above discussion uses the procedures and models of
Tables 4.4, 4.5, and 4.6 to illustrate that DNoptimal
designs obtained by using these procedures may be optimal
for more than one model. Similar results hold for the
procedures and models of Tables 4.7 4.12; specifically,
O the procedures of Tables 4.7 and 4.8 yield designs
that are also DNoptimal for the model of Table 4.9
0 the procedures of Tables 4.10 and 4.11 yield
designs that are also DNoptimal for the model of
Table 4.12.
It is clear from the procedures in Tables 4.4 4.12
(and the examples for the case q = 3 in Appendix C) that
there is often more than one DNoptimal design for a given
model and value of N. To choose among several such designs,
we might consider other properties or characteristics of the
designs. One characteristic already considered is whether
the design is DNoptimal for more than one model. Another
design characteristic that might be of interest is how the
parameter estimators depend on the observations at the
design points. For many designs, some of the parameter
estimators will depend on the form of the model while others
may not. It might also be of interest to consider the
parameter estimator variances. Properties of interest might
be criterion based measures such as tr[(X'X)1],
max[x'(X'X)1x], or avg[x'(X'X)lx], where the maximum or
average is computed over the candidate points. Examples of
how these characteristics and properties might be used to
choose among several DNoptimal designs are given in
Appendix D for some of the threecomponent designs from
Appendix C.
The DNoptimal computeraided design approach can also
be used to fractionate designs for constrained mixture
amount experiments. However, because of the unlimited ways
in which the mixture component proportions can be con
strained, it is not possible to develop general procedures
as we did for unconstrained mixtureamount experiments. One
must have and use a computer program (such as DETMAX) for
each particular application. As an example, consider the
three component constrained mixtureamount design given
earlier in Table 4.1 (and pictured in Figure 4.3).
Fractions of this design for several values of N are
presented graphically in Figure 4.6. The designs were
obtained using DETMAX for the specialcubic by linear
mixtureamount model
O 3 O 0 13
i = i i x + E E 8 x.x. + 8 x12x3 + E .x.A (4.7)
i=1 i
To summarize, in this section we have discussed several
techniques for fractionating mixtureamount designs for
4
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unconstrained mixtureamount experiments, including a
computeraided design approach. This approach was also used
to fractionate a constrained mixtureamount design. It is
clear that the computeraided design approach is quite
powerful and can be used to fractionate any mixtureamount
design (including the ratio or logratio variable
designs). The DNoptimality criterion was chosen for use
here because of its popularity and the availability of the
DETMAX program (Mitchell 1974) to implement it.
CHAPTER FIVE
MODELS AND DESIGNS BASED ON THE COMPONENT AMOUNTS
Mixtureamount experiments, introduced in the previous
chapter, were seen to be a type of general mixture
experiment in which the experimenter wishes to understand
not only how the components blend with one another, but also
if and how the amount of the mixture affects the component
blending. Mixtureamount models and designs were formulated
in terms of the component proportions and the levels of
total amount of the mixture in such a way as to provide this
information to the experimenter. However, oftentimes
experimenters formulate their questions concerning the
effects of the components on the response by expressing
their models and designs in terms of the amounts of
individual components. For example, in a fertilizer study
the experimenter may only want to know how much of each
component is to be present in the fertilizer in order to
maximize the crop yield.
In this chapter, we shall discuss two types of general
mixture experiments where the models and designs may be
expressed in terms of the component amounts. For each type
of experiment, the respective designs and models are
mentioned.
88
1
5.1 Standard Designs and Polynomial Models Based
on the Component Amounts
Let us consider an experimental approach in which the
controllable variables are the amounts of the individual
components, denoted by ai, i=1,2, . ,q. Typically, an
experimenter wishing to model the response as a function of
the individual component amounts, would select either a
first or seconddegree polynomial model of the form
q
n = a0 + Z aiai (5.1)
i=1l
q q 2 q
n = C0 + Z aa i+ E La ia + E aiaa (5.2)
i=1 i=1 i
Similar models are obtained by substituting log ai (or other
functions of the ai) for the ai in (5.1) and (5.2). These
models, or those in (5.1) and (5.2), would be fitted to data
collected at the points of any standard response surface
design (as discussed in Section 1.1). Studies of this type
were performed by Hader et al. (1957), Moore et al. (1957),
Suich and Derringer (1977), and Valencia (1983).
An experiment conducted using the above standard
design, component amount model approach will be referred to
as a component amount (CA) experiment. A component amount
experiment is a type of general mixture experiment (as
defined in Section 1.3). To see this, first note that the
amount of an individual component (ai) may be written as the
product of the proportion of the component in the mixture
(xi) and the total amount of the mixture (A): ai = xiA,
i=1,2, . ,q. Although the response in a component
amount experiment is nominally assumed to be a function of
the component amount variables, the relationship ai = xiA
allows us to view the response as a function of the
component proportions and the total amount of the mixture,
i.e.,
n = f(al,a2,...,a )
= f(x1A,x2A,...,xqA)
= g(x ,x2,...,x ,A) (5.3)
Recalling the definition (in Section 1.3) of a general
mixture experiment as one in which the response is assumed
to be a function of the component proportions and possibly
the total amount of the mixture, (5.3) shows that a compo
nent amount experiment is a type of general mixture experi
ment.
It is of interest to note that the component amount
variables ai in a component amount experiment are mathemati
cally independent. A type of experiment formulated in terms
of the component amounts where this is not the case is
discussed in the next section.
91
5.2 Models and Designs for Experiments Where the
Component Amounts Have a MixtureLike Restriction
A different experimental approach based on the
component amounts is discussed in this section. This
approach arises in situations where the possible
combinations of component amounts are restricted by a linear
constraint on the ai. The approach is introduced with the
following hypothetical twocomponent example.
A softdrink company would like to determine the blend
of two artificial sweeteners (S1 and S2, say) that yields
the best taste (minimum intensity of aftertaste) when used
in a diet drink. From previous experience, the company
knows that the optimum amounts of the individual sweeteners
S1 and S2 when used alone in the drink are 9 and 12
mg/fl.oz., respectively. An experiment is set up where
average aftertaste rating values are collected from the
combinations of the two sweeteners (al,a2) = (9,0),
(6.75,3), (4.5,6), (2.25,9), and (0,12). The data collected
are to be used for fitting the model
n = a1 + 1 a2 + aa2 12ala2 (5.4)
which will in turn be used to determine the best combination
of sweeteners S1 and S2.
To see what makes this experimental approach different
from the mixtureamount and component amount approaches,
first consider the plot of the sweetener combinations given
in Figure 5.1. The combinations all lie on the line
a2 = (4/3)al + 12 (5.5)
The company has chosen a desired sweetness level based on
the amounts al = 9 and a2 = 12, and the level is constant on
the line (5.5). They wish only to consider combinations of
the sweeteners along this line. The line (5.5) places a
restriction on the amounts a, and a2 of the two sweeteners
and thus they are not mathematically independent as is the
case with the component amount variables in the component
amount approach.
The restriction (5.5) on the component amount vari
ables, rewritten as
a1/9 + a2/12 = 1 (5.6)
q
is reminiscent of the restriction Z x. = 1 in a mixture or
i=1 '
mixtureamount experiment. Since the total amounts of the
five combinations chosen for the experiment are different,
it is natural to compare this approach to the mixtureamount
approach. There is a clear difference between the two
approaches; with a mixtureamount approach the component
blends are performed at each of two or more levels of total
amount, while with this approach each blend (combination) is
performed at exactly one amount.
