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Models and designs for generalizations of mixture experiments where the response depends on the total amount

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Models and designs for generalizations of mixture experiments where the response depends on the total amount
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Piepel, Gregory Frank, 1954-
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1985
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Full Text
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. Andr 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 Experiments An Introduction 6
1.3 The Subject of This Research
Generalizations and Extensions of
Mixture Experiments 9
TWO LITERATURE REVIEWMIXTURE AND MIXTURE-PROCESS
VARIABLE EXPERIMENTS 10
2.1 Models for Mixture Experiments 10
2.2 Mixture Experiment Designs 18
2.3 Mixture-Process Variable Experiments 26
THREE MODELS FOR MIXTURE-AMOUNT EXPERIMENTS 32
3.1 An Introduction to Mixture-Amount
Experiments 32
3.2 Including the Total Amount in Mixture
Models 34
3.3 Mixture-Amount Models Based on
Scheff Canonical Polynomials 36
3.4 Mixture-Amount Models Based on
Other Mixture Model Forms 45
3.5 Mixture-Amount ModelsA Summary 51
FOUR DESIGNS FOR MIXTURE-AMOUNT EXPERIMENTS 54
4.1 Developing Designs for Mixture-Amount
Experiments 54
4.2 Fractionating Designs for Mixture-
Amount Experiments 61
IV


FIVE MODELS AND DESIGNS BASED ON THE COMPONENT
AMOUNTS 38
5.1 Standard Designs and Polynomial Models
Based on the Component Amounts 39
5-2 Models and Designs for Experiments
Where the Component Amounts Have a
Mixture-Like Restriction 91
SIX COMPARISON OF MIXTURE-AMOUNT, COMPONENT AMOUNT,
AND COMPONENT-WISE 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
Mixture-Amount and Component Amount
Models 137
SEVEN EXAMPLES OF MIXTURE-AMOUNT, COMPONENT AMOUNT,
AND COMPONENT-WISE MIXTURE EXPERIMENTS 146
7.1 A Mixture-Amount Experiment Example ....... 146
7.2 A Component Amount Experiment Example 155
7.3 A Component-Wise Mixture Experiment
Example 162
EIGHT SUMMARY AND CONCLUSIONS 1 67
3.1 Summary 168
8.2 Recommendations 174
APPENDICES
A SCHEFFE CANONICAL POLYNOMIAL MIXTURE-AMOUNT
MODELS 178
A.1 Models in Which the Components Blend
Linearly 173
A.2 Models in Which the Components Blend
Nonlinearly 179
B MIXTURE-AMOUNT MODELS WHEN THE CANONICAL
POLYNOMIAL FORM IS NOT THE SAME AT ALL LEVELS
OF TOTAL AMOUNT 183
B.1 Linear and Quadratic 31ending at Two
Amounts 183
B.2 Linear, Quadratic, and Quadratic
Blending at Three Amounts 185
v


3.3 Linear, Linear, and Quadratic
31ending at Three Amounts 189
B.4 Special-Cubic Blending 190
C THREE COMPONENT DN-0PTIMAL DESIGNS FOR
VARIOUS CANONICAL POLYNOMIAL MIXTURE-AMOUNT
MODELS 193
C.1 Two Levels of Amount 194
C.2 Three Levels of Amount 195
D CONSIDERATIONS IN CHOOSING AMONG TWO OR MORE
DN-0PTIMAL 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 MIXTURE-AMOUNT
EXPERIMENTS 233
REFERENCES 236
BIOGRAPHICAL SKETCH 241
vi


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 component-wise mixture) are
discussed. Designs and models for these experiments are
presented and compared.
A mixture-amount experiment consists of a series of
usual mixture experiments conducted at each of two or more
vu


levels of total amount. Mixture-amount 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 mixture-amount 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. Component-wise 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. Component-wise mixture models and designs can be
specified in terms of the component amounts or in terms of
component-wise proportions.
Several real and hypothetical examples are utilized to
illustrate and compare the mixture-amount, component amount,
and component-wise 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 x^, X£, . 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
(1.1)
yk = f(xk1xk2
where y^ 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 e^ 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 second-degree polynomials in the
predictor variables x^, . x are adequate. The
selection of an appropriate approximating function (often
1


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 = XB + e (1.2)
where y is an Nx1 vector of observed response values, X is
an Nxp matrix of known constants (N > p > q), S is a px1
vector of unknown parameters, and £ is an Nx1 vector of
random errors. It is usually assumed that E(£) = Q and
Var(£) = 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) = 0, 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 3 and refine the model form if
necessary. If we assume Var(e) = a^I^, then the ordinary
least squares estimator of S is given by
8 = (X'x)1x'y (1.4)
and has variance


3
Var(3) = (X'x)"1a2
(1.5)
The portion of response surface analysis involving
modal 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.
A -A
Once a fitted regression model y = XB 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 x1, x2, xq within the experimental region.
If we let Xq 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 Xq are
y(xQ) = xQ B
(1 .6)
and
Var[y(xQ) ] = Varixg 8)
- 2
(1.7)
Under the assumption Var(e) = a^V (where V is a known
diagonal matrix, not necessarily the identity matrix), 3 is
estimated by weighted least squares, yielding


4
3 = (x'v"1XrVv"1y ,
(1 .8)
Var(S) = (x'V"1X)~1a2
(1 .9)
and
Var[y(xQ)] = Xg(x'v 1X) 1 xQ cr2.
(1.10)
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
second-degree polynomial models in the predictor
variables. Among the designs discussed by Myers are the 2^
and factorials, the 2q-lc fractional factorials, and the
central composite designs.
In recent years, computer-aided design of response
surface experiments has received much attention. A design
criterion of interest is chosen and points are selected for


5
the design from a candidate list so as to optimize the.
design criterion selected. Several design criteria of
interest are:
1. D-ootimality seeks to maximize det(x'x) or
equivalently minimize det[(X X)-1].
2. G-optimality seeks to minimize the maximum
prediction variance over a specified set of design
points.
3. V-optimality seeks to minimize the average
prediction variance over a specified set of design
points.
4. A-ootimality seeks to minimize trace [(X X)1].
Designs consisting of N points obtained by using these
optimality criteria are referred to as D^, GN, V^, and AiNj-
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.


6
In mhe 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 SiQp, BpO^, AI0O3,
CaO, MgO, NapO, ZnO, TiO? CrpQx, FepO-z, 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


7
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 x-j x?, . Xq of components in the
mixture. Since Xj_ represents the proportion of the ith
component in the mixture, the following constraints hold:
q
0 < X. < 1 (i1,2 q); Z x. = 1 (1.11)
1 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 < xi < < 1 (i=1,2,...,q) (1.12)
Experiments where these additional constraints are imposed
on the x^ 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 (q-1)-dimensional
simplex. In general, restrictions (1.12) reduce the
constraint region given by (1.11) to an irregular


8
(q-1)-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 oseudocomponents. If at least one component
has a nonzero lower bound, then the pseudocomponent values
x| may be obtained from the original component values x-¡_ by
i i
q
1 E L,
j = 1 3
i = 1,2,
>q ,
(1 .13)
q
where E L. < 1. Crosier (1984) referred to this as the
3=1 3
L-pseudocoraponent transformation. If at least one variable
has a nonunity upper bound, then pseudocomponent values may
be obtained by the U-pseudocomponent transformation,
x.
i
Ui *
q
3=1
i = 1,2
, q ,
(1 .14)
q
where £ U. > 1. Crosier (1984) presented additional
3 = 1 J
discussion on the use of these two pseudocomponent trans
formations and gave guidelines for choosing between them.


9
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 MIXTURE-PROCESS
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
A
Scheffe (1958) developed canonical forms of polynomial
models for mixture experiments by substituting the mixture
q
constraint Z x. = 1 into certain terms in the standard
i =1 1
polynomial models and then simplifying. For example, with
q = 2 mixture components, the standard second-degree
polynomial model is
n aQ + a1x1 + a2x2 + a12X1X2
+ a11x1 + a22X2 *
(2.1 )
Multiplying the constant term by unity and applying the
2
mixture restriction x1 + x2 = 1 to the xi terms yields
2 2
oIq = (x ^ + x2), = x ^ (1 x2) and x^ = x2(1 x^).
Hence, (2.1) can be reduced to the form
n = S1x1 + S2X2 + S12x1x2
(2.2)
10


11
where s1 = 32 = a0 + a2 + a22 and
s12 = a12 a11 a22
The general forms of the first, second, and third-
degree canonical polynomial models in q mixture components
are
n
. s 3ixi
i=1
(2.3)
n
q
e
i=1
3iXi
q
z z
i<2
8 x. x .
lj i J
(2.4)
and
n
q q q
E f^x. + E E Bi1XiX + E E 5 X.X (X
i=1 11 i<0 J i<0 J J
q
+ E E E 8
i . X X .X,
ljk i j k
(2.5)
The special-cubic canonical polynomial model is a reduced
form of the full cubic model (2.5) obtained by deleting the
6i-sx.-x^(x x.:) terms.
In each of the above model forms, the first q terms,
8^x^ + . + SqXq, 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


12
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 affect, 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 q
H1 : n = E 3.x. + E Z 8.. min(x.,x.)
i=1 1 1 i + fl12...q min(x1x2* *,Xq}
;2.6)
q q v2-i
H2: n = E 8,x. + E E B x.x,/(x.+x.) + .
i-1 i * a12...q
(2.7)
q q 1/2
H3: n = E 3 . x + E E 3 i (x i x .) +
i-1 i * S12...q <*iVV
1/q
(2.8)
If the denominator of a term in H2 is zero, that terra is
defined to be zero. In practice, the second-order forms of
H1, H2, and H3 (the first two sets of terms in each model)
are often adequate. Snee (1973) discussed the types of


13
curvature generated by these second-order Becker models over
the region 0 < x^ < 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 x^. For example, the first
and second-degree models with inverse terms are
q q-
n = I 0.x. + l 6 .x, (2.9)
i-1 1 1 i=1 1
q q q
n = ZB.x. +EI8..X.X.+ Z S .x. (2.10)
i-1 11 i *
Inverse terms may be added to any Scheffe polynomial model
in a similar manner.
An inverse term x^1 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-
A
zero lower bound Lj, inverse terms of the form (x^ L^)_
may be used. If f(x) as x, > (where represents
the component's upper bound such that 0 < L^< Ih < 1), then
-1
inverse terms of the form (U^ x^) are appropriate.


14
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)
oo as x^ > or 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 Scheff 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 x^, i = 1,2, . ,q, with an
equivalent set of ratio variables r ^, j=1,2, . ,q-1 In
general, there are many possibilities. For example, with
three components the following equivalent transformations
(among others) are possible:
Transformation
R1
R2
R3
R4
It is seen that ratios may
r1
r2
x1/x2
x2/x3
x1 /x2
x^/x2
X1
x2/x3
x1/(x2 + x^)
x2/x.
not be defined if certain


15
component proportions take on zero values in the
denominator. In such cases, Snee (1975) suggested adding a
small positive quantity c to each x-j_ so that the denominator
is always greater than zero.
The above example illustrates that only q-1 ratio
variables are needed to replace the q component proportions
q
(owing to the mixture restriction z x- =1)* Because o
i = 1
this reduction in the number of variables, the ratio
variables are mathematically independent. Hence, standard
polynomial models in the ratio variables, such as
q-1
n = a0 + Z a.c.
d-1 J 3
and
(2.11 )
n = a,
q-1
s a,r
d*i J J
q-1
+ Z Z a
j jkrjrk +
q-1 2
(2.12)
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 = S0 + s1x1/(x1+x2) + s2x2/(x2+x5) + S3x3/(x3+x1)
q
+ z z
i 3. .h. (x x .)
ij 10 l 0
6123h123 ^ X1,X2
(2.13)


16
Here h.. and hare specified functions which are
i 25
homogeneous of degree zero. Inactivity of a component x^ is
suspected when 3^ = 0, 3^^ = 0 for j^i, and 8^23 0* The
and ¡1^23 functions suggested by Becker are
hid(xi
(2.14)
^123 ^ X1x2x3^
i X2 \'2
\X2 + X3 J
(2.15)
He also noted that when s^. or t is negative, hij or h123
takes on an extremely large value near the boundary
x = 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 = Z 6.x. + E E 8.. (x.+x.)h. .(x.,x .) +
1 1
ij
1 J ij 1 J
i=1 * i< j
+ 8* j (x. + + x )h.^ (x1 .. ., x ) ,
12...q 1 q 12...q 1 q
(2.16)
where the functions hj_j(xi, x^) jk^xi xj xk^ e1:c* are
homogeneous of degree zero. This model can be simplified by
deleting higher order terms. Becker (1978) gave some


17
suggestions for the h functions, and noted chat the H2 and
H3 models of (2.7) and (2.8) are of the form (2.16).
Aitchison and Bacon-Shone (1984) presented the
polynomial models
q-1
" 8 *
(2.17)
q-1
q-1
" Vi ll TV3
.2.18;
where z^ = logix^/xq), i=1,2, . ,q-1 Rewriting (2.17)
and (2.13) in terms of the component proportions xi gives
the symmetric model forms
<4 Q
n = S~ + Z 8.log x. ( E 8. a 0) (2.19)
0 i=1 1 1 i=1
q Q 2
n = 8n + Z 6.log x. + Z Z 8..(log x.-log x,) (2.20)
U i=i 1 L J
where the 8-¡_j are functions of the Note that the above
models are not directly applicable when the component pro
portions take on zero values, since log x -* - as x^ -* 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 zero-valued boundaries. Substituting x^ or
- x for x in (2.19) or (2.20) yields models useful for


18
constrained mixture experiments where f(x) -* as x^ -* Lj_
or .
Aitchison and Bacon-Shone pointed out that
3. = 0 (i=1,...,c); S. = 0 (1 < i < j < c) (2.21)
indicates the inactivity of components 1, 2, . c.
They also noted that
3.. = 0 (i=1,2,...,c; j=c+1,...,q)
S
(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
3.. 0 (i=1,2,...,c; ji+1,...,q)
^ vJ
(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} simplex-lattice
designs for exploring the full q-component simplex region in
an unconstrained mixture experiment. The {q,m}


19
simplex-lattice design (m=1,2, . .) consists of the
(q+J51) points in the simplex (1.11) that represent all
possible mixtures obtainable when the proportion of each
component can take on the values 0, 1/m, 2/m, . 1.
Examples of some {q,m} simplex-lattices are given in Figure
2.1 .
The simplex-lattice gives an equally spaced distribu-
*
tion of points over the simplex (1.11) and enables a Scheffe
canonical polynomial of degree m in the Xj_ to be fitted
exactly. For example, a {q,2} simplex-lattice supports the
fitting of the Scheffe canonical polynomial model (2.4),
while a {q,3} simplex-lattice supports the fitting of model
(2.5).
Scheffe (1963) presented the simplex-centroid designs
and associated "special" canonical polynomial models for
unconstrained mixture experiments. The simplex-centroid
design consists of 2q 1 points: the q pure components,
q
the (2) two-component blends with equal proportions of 1/2
q
for each of the proportions present, the (3) three-component
blends with equal proportions of 1/3 for each of the compo
nents present, . and the q-component blend with equal
proportions of 1/q for all components. The simplex-centroid
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 simplex-centroid design has an associated "special"
canonical polynomial model


20
{4,2}
(4,3}
Figure 2.1 Some {q,m} Simplex-Lattice Designs for Three and
Four Components
(a)
Cb)
Figure 2.2 Simplex-Centroid Designs for (a) Three
Components and (b) Four Components


21
q q q
n = Z SiX. + e E B .X.X. + z I E Siikx.x x
i-1 11 i + * + ^12...qX1X2 **Xq
(2.24)
This model contains 2^ 1 terms and hence provides an exact
fit to data collected at the points of the corresponding
simplex-centroid design. The special-cubic model
" = ilSlX* i<5 S123X1X2X3 (2-25)
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 in-which 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 (x^ = 1, x^ = 0, j^i)
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


22
component i is the imaginary line extending from the vertex
Xj_ 1 x. a 0, j^i, to the point Xj_ = 0, x^ = 1/(q-1),
on the opposite boundary. The points of an axial design lie
only on the component axes. Cornell and Gorman (1973)
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 D^j-optimal
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 Dj,j-optimal 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 x^ > 0.05, i=1,2,3 to avoid the
problems that occur in Xj_ when Xj_ = 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 r^ = x^/x-j. Hackler, Kriegel, and
Hader (1956), Donelson and Wilson (I960), 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 r^ =
x./(x.+x.+ . .).
-L X J


23
(a)
(.475 .05 ,.475)
(.14,.14,.72)
( .13 .05 .82)
( .05, .05, .90)
(.05,.13,.82)
Figure 2.3* Points of Supoort of D^-Optimal Designs for
Models 2.9 (a) and 2.10 (b)


24
Aitchison and Bacon-Shone (1984) did not discuss
designs for their log-ratio 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 Zj_ = log(Xj_/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 (Lj.) and upper (U^) bounds for the component
proportions. For fitting the Scheffe second-degree
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 first-degree
canonical polynomial model (2.3), by selecting only a subset


25
of all extreme vertices. The approach of 3nee and Marquardt
includes the use of the D^j, Gj^, and AN-optimality criteria
in choosing efficient designs.
Nigam, Gupta, and Gupta (1985) 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 (D^, G^, V^,
and Aj^-optimality) in building a design. Resulting designs
compare well with those developed by XVERT with respect to
measures such as G-efficiency, det(X X), and tr[(X X) ].
Snee (1975) discussed the development of mixture
designs for fitting the second-degree 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
computer-aided 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 computer-aided
design approach of Snee (1975).


26
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 Mixture-Process Variable Experiments
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 simplex-centroid design, he naturally
suggested a simplex-centroid x (k^ x k£ x . x kn)
factorial arrangement for mixture-process variable
experiments (where kj represents the number of levels of the
jth process variable). Such an arrangement can be thought
of as a simplex-centroid design at each of the k^ x k^ x
. . x k factorial points or alternately as an (k^ x k^ x


27
. . x kn) factorial at each of the 2q 1 simplex-centroid
points.
Scheffe also discussed an associated model for the
simplex-centroid x (k^ x k-2 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 z1 ,
, . 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 6x1 + SjXj 8x3 62X1X2 S13X1X3 a23x2x3
8?23X1X2X3 + S2
0
2X2 *
+ B23x2x3
+ 123x1X2X3^zj + ^1 x^ + xo +
1 T p 2 2
b12x x
B23 2X3
* 6i23X1X2X3lz1Z2 *
(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


23
or interactions containing only process variables). This is
due to the identity x^ + x2 + + xq = 1 .
The number of points in the simplex-centroid 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
simplex-centroid 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 (2^ 1)2n points of the simplex-centroid x 2n design
and the points of a 2Ci+n design (after removing the 2n
points corresponding to those combinations where all of the
q mixture components are absent). A fraccin of the sim
plex-centroid x 2n design is obtained by taking the points
corresponding to those in a fraction of the 2^+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 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 2J design is the mixture composition


29
(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 mixture-process 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 mixture-process
variable models containing fewer than the 2^+n 2n terms in
the complete model.
It was noted earlier that there are no terms in
Scheffe's mixture-process 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 two-component mixture experiment with one
process variable (at two levels), they considered the
canonical polynomial model
0 0 0
n = 81x1 + S2x2 + 8 -j2xi x2
"1
+ (B^x^ + 82x2 812X1 X2^Z1
(2.27)
The effects of the process variable z^ are contained in the
coefficients S^j, 8^, and If z-j has the same (constant)
11 1
effect on all compositions, then s-j = S£ and B^ =
yielding the reduced model


30
n = s1x1 + S2X2 + &12xix2 + 80Z1
(2.23)
where 8^=8^= 8^. Note that the terms in (2.23) are not a
subset of the terms in (2.27), specifically 3^ 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
z-j by considering (2.27).
To arrive at a reduced form of the combined model in
the Xj's and Zj_'s, Gorman and Cornell suggested repara
metrizing the general form of the mixture-process variable
q
canonical polynomial model by first substituting 1 £ x.
j=2 0
for x-| in all crossproduct terms involving x-j (alone) with
the process variables and then rewriting the terms in the
model. For the above example, substituting 1 x2 or x^ in
the terra S^jx^z^ in (2.27) yields
= 8x1 + 8x2 + S12x1x2 + B1(1 x2)z1
+ S2X2Z1 + 812X1X2Z1
81X1 + S2X2 + 012X1X2 + 52X2Z1
+ si2X1X2Z1 + 80Z1
(2.29)
111 11 1
where 2 3 B1 and 30 = S1* Hence 52 represents the
difference between the effect of on the linear blending
of x2 and x^ while 8^ represents the effect of z^ on the


31
linear blending of x^. Note that when the terms 2x2z1 and
S^2xix2z1 are 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 3^ of z^
represents the effect of z^ on the linear blending of ,
not an overall main effect of z-j (unless both 6 and S12 are
zero, in which case sq is a measure of the overall main
effect of z-j) .
In closing this section, it should be noted that
Scheffe's mixture-process variable models are still
applicable for constrained mixture-process 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 MIXTURE-AMOUNT 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 1.5)*
3.1 An Introduction to Mixture-Amount 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 mixture-amount experiment. An
example of a mixture-amount 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


33
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
mixture-amount 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. Mixture-amount 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 mixture-amount experiment with q = 2 components and
a total amount variable A at two levels (say A^ < A^)
Suppose the two components blend linearly at both amounts.


34
Several possible situations are shown in Figure 3.1, where
the pure component proportions are denoted by (x^,x9) =
(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 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 che
blending properties of the two components; at A-| 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


response
35
Figure 3.1. Plots of Several Two-Component Blending Systems
at Two Total Amounts A^ and A2


fitting data from a mixture-amount experiment. A technique
for doing so is suggested by recognizing the similarity of a
mixture-amount experiment to a mixture experiment with one
process variable. Likewise, a mixture-amount 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
mixture-amount experiments with or without process
variables. Mixture-amount experiments without process
variables are discussed in this chapter. The extension to
mixture-amount experiments with process variables is
discussed in Appendix G.
3.3 Mixture-Amount Models Based on Scheffe
Canonical Polynomials
Scheff's canonical polynomials (see Section 2.1) have
been shown to be a versatile class of equations for modeling
mixture response surfaces. Since a mixture-amount experi
ment is just a series of mixture experiments run at each of
several amounts A^, A£, ... 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


37
amount Ai. We consider the fitting of a Scneffe canonical
polynomial model such as (2.3) or (2.4) at each amount.
To begin the development, suppose a particular Scneffe
canonical polynomial model form, denoted by n^, 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 Iq vary as A varies, i.a., the parameters
of Hq depend on A. This dependence can be modeled (for each
parameter Bm in nc) using the standard polynomial form
(3.1)
where Bm(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 (r-1)th degree polynomial can be
used if desired, a second-degree polynomial will often
suffice in practice. Nonpolynomial functions of A that
might also be appropriate for certain applications are
(3.2)
or
(3.3)
In practice, A-1 and log A in the above equations would
usually be coded, as was A in (3-1).


38
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 mixture-amount model where q = 2 and r = 3* Then,
nc is of the form (2.4) and the Sm's are of the form (3-1),
which yields
n = S1(A)x1 + 82(A)x2 + 812(A)x1x2
. [s e]a 8^(a')2]x1 U2 sA' + s(A')]¡c2
o
1 '
2/i'v2-
+ [8^2 + 8 -} 2A' + 8^2(a')2]x^x
1 2
0 0 0
81 X-] + 8 2X2 + 012X1X2
2
+ Z
k=1
r a k
[ex
1A1
6 2x2 ^
(3.4)
Note that the subscript of a 6 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.


39
When the levels A1 A?, A^ 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) + 62x2 + S12X1X2 represents the linear and
nonlinear blending properties of the mixture
components at the average level of total amount,
ii)[s!jx1 &2x2 + S1 Zx'1 x21A' represents the linear
effect of total amount on the linear and nonlinear
blending properties of the mixture components,
iii)[B^x^ + B2xz + si2x1x2^A'^2 represents the
quadratic effect of total amount on the linear and
nonlinear blending properties of the mixture
components.
Thus the coefficients B^ and of the terms x1(a')ic and
x1x(a')c, 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 )^ 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 second-degree
orthogonal polynomials P^A) and P^(A)], the interpretations
of the coefficients change somewhat. Under this coding, the


40
coefficients 3, 32, and S2 measure the linear and
nonlinear blending properties of the components averaged
over the levels of total amount. The coefficients 3^ and
^ij ^=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 3 for
an example that illustrates the above interpretations.
For general q and r, a model of the form (3*4) is
written as
q .0
- 'Vi
i=1
q 0
* 14 Wa
+
r-1
Z
k=1
[ E S^x E E 8^ x x ](A'
i-1 11 i (3.5)
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
11 11
all compositions (which forces ^ ^ ~ = anc* s-]2
1 1
= s' = . = 8 =0); that A has a constant quadratic
15 i j q


41
9 2
effect for all compositions (which forces 8^ = 82 = * =
Sq and S^2 = 3^ = . = 3q-1 ,q = 0); . ; and that A
has a constant (r-1)th degree effecx for all compositions
(which forces 8^"^ = = . = 8^^ and S^2^ si3^ -
. . = 8£"1 = 0). In this case, the model of (3-5) takes
4. 1 > 4
the reduced form
n
? 3
£ 3.x
i=1
1 1
E E
4 0? .X. X .
1J 1 3
r_1 k < k
+ 2 s£(a >* ,
k=1 u
(3.6)
where the 3^ (= = . = 8k) k=1,2, . ,r-1,
represent the linear, quadratic, . (r-1)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
I
terms of (3.5); specifically the terms with A alone
(B§(A')k, k=1 ,2, . ,r-1) are-not 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 repararaetrization involves
q~1 k k
replacing x with 1 £ x. in the terms 8 x (A ) k=1,2,
q 1 4 4
. . ,r-1, of (3.5) and simplifying. For the q = 2, r = 3
example considered earlier (3.4) is reparametrized as


42
.0.
0
ti S1x1 + S2x2 + S12X1X2
+ Z [ 3 x+ 8p("1 x. ) + 3^pX^Xp](A. )'
k=1
= Sx1 + 32x2 s12X1X2
k=1
[2X1 + B12X1X2
k. 1 n k
+ 8 n ] (A ) j
(3.7)
where 6^ = 3^ 3^ and Bq = BX, k=1,2. Henee, 5^ rsPre
sents the difference between the linear effects of total
amount on the linear blending properties of x^ and x2, and
52 represents the difference between the quadratic effects
of total amount on the linear blending properties of x^ and
x2. Although (3*7) now contains the terms 8qA and 8q(A )
with a' and (a')^ alone, note that Bq (= S2) and 3q (= S2)
measure the linear and quadratic effects of total amount,
respectively, on the linear blending of x2, and do not
measure the overall linear and quadratic effects of total
amount [unless we find that 52 = <52 = 0 and S^]2 = S^2 = 0 in
(3.7)].
The Scheffe canonical polynomial mixture-amount 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


43
also of interest. Mixture-amount model forms for these
situations will now be discussed.
Let us suppose one of the Scneffe 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 mixture-amount model is derived
using this "most complicated" canonical polynomial, it will
be an adequate (but overparametrized) form for fitting data
from the mixture-amount experiment. The appropriate mix
ture-amount model form is a reduced form of the "adequate"
mixture-amount model. The nature of these model reductions
are determined for several situations in Appendix B.
Several canonical polynomial mixture-amount models of
practical interest have been discussed in this section and
are also discussed in Appendices A and 3. 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,
F*
where SSE
for error
(SSE
reduced
- SSE
full
)/(
e -
ef)
SSEfullTef
(3.8)
,, SSE,. e and e~ are the sum of squares
reduced full r f
and the error degrees of freedom for the reduced


44
and full models, respectively. As an example, suppose in a
mixture-amount 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
4 0 4 n 1 i
Model 1: n = E 3 x, + E E S-.x.x. + 3nA
i-1 11 i ,0
.0
q 1
. E 3 x A
11 w, ij l J i = 1 1 1
Model 2: n = E Six. + E E BT .x.x, +
11 1 i< j
Model 3: n = E 8?x. + E E S?.x.x. + E six,A
i-1 11 i q 1
+ E E 3..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(q-1 )/2 terms S^x^a', 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 = N-q(q+1) and er =
N-q(q+3)/2, respectively. If the test is significant, modal
3 is selected. If the test is not significant, then models
2 and 1 are compared, treating them as "full" and "reduced,"
respectively. The full vs. reduced model test (3*3) can


45
also be used to compare the models discussed in Appendix B
to the corresponding models without parameter restrictions.
Another model selection approach is ~o fit the repara
metrized form of a 'full1' mixture-amount model (such as
model 3 in the above example) and use variable selection
techniques such as all-possible-subsets 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 Mixture-Amount Models Based on Other
Mixture Model Forms
In the previous section, mixture-amount models were
developed by writing the parameters of Scheff 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), (5-2),
(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, A^ and Using Sm's
of the form (3.1) yields the mixture-amount model


46
q q .i
na 2 8.(A)x. +23 (A)x.
ii 1 1 i=1
q n -i i q o
= 2 (s'? + SU ) x. + 2 (3U +
i=1 1 1 1 i-1
1 N -1
s-iA )xi
q
z
i=1
B?X. +
1 1
q
2 6
i-1
+ 2 31x.A' + 2 31.xT1A1 (3-9)
i-1 1 1 i-1 ^ 1
Recall that a' denotes a coded form of the total amount
variable A. When the levels A-j and A2 are coded to have
mean zero, the terms in (3*9) have the following interpreta
tions:
(i) 2 8?x. and 2 B^x?1 respectively represent the
i=1 11 i=1 x 1
linear and nonlinear blending properties of the
mixture components at the average level of total
amount,
q i q i _*i i
(ii) 2 S.x.A and 2 S .x. A respectively represent
i-1 1 1 i-1 -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


47
refers to an extreme increase or decrease in the response
value as the value of x^ approaches zero.
As another example, assume that the second-order 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
= E 2.(A)x. + E E 8 (A)(x.x )
i-1 1 1 i 1/2
= Z [8^+S ] (log A) ] x, H- E E [8 +81 (log A)'](X x )1/2
i=1 1 1 1 i< j J
= E 8iXi Z Z 0?j(xiXj}
i-1 1 i 1/2
+ E six.(log A)' + E E si ,(x.x.)^2( log A) (3.10)
i-1 11 i The notation (log A)' above denotes a coded form of log A.
When the three levels log A-j log A£, and log A^ are coded
to have mean zero, the terms in (3.10) have the following
interpretations:
P o P n /p
(i) Z S:x. and Z Z 8^ (x.x.)'^ respectively repre-
i-1 11 i sent the linear and nonlinear blending properties
of the mixture components at the average level of
log (total amount),


48
Q 1 Q 1 1/2 '
(ii) Z six,(log A) and z Z s' (x.x.) (log A)
i=1 i< j J J
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 mixture-araounr models are
applicable for any of the other types of mixture-amount
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 mixture-amount models) may be needed. For
example, the reduced form of (3-9),
Q pv Q O A Q A |
n = Z S Tx + E8.X + IS.X.A (3.11)
i=1 1 1 i=1 -1 1 i=1 1 1
is appropriate if the total amount has a linear effect on


49
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
n
q
Z
i=1
0
S x. +
ii
q
z 8
i=1
1 f
(3.12)
and
n = Z s?x. + z Z S?^(x,xJ1^2 + sl(log 4)' (3.13)
i=1 1 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-
q-1 1 ,
sing x as 1 I x. in the terms 8 x A of (3.9) and
1 q i=1 1 q
6qxq(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
Scheff canonical polynomial mixture-amount models in
Appendix B. As an illustration, consider the situation
specified by the mixture models


50
V=-1 ax1 + bX2
"A'=+1 =X1 dx2 SX11 <3'U)
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 mixture-amount model for this situation is of
the form
n = 8x1 Sx2 * S^A' 8¡x2a'
* S^x^a' (3-15)
with as yet unknown parameter restrictions. Substituting
the data
X1
X2
a'
n
1
0
-1
a
in
o

.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
.0 e
01
' 2
s2 -
2
0 -1 2
1
01
c-a
_ 2
1
s2 -
d-b
2
1 e
0 -1 = 2
(3.1


51
The appropriate parameter restriction for this situation is
thus seen to be si-] = If the inverse nonlinear
blending occurred at the low level of A instead ox the high
level, the parameter restriction would be s_>] =
3-5 Mixture-Amount ModelsA Summary
A mixture-amount 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 log-ratio model, etc.) can
be used, and the parameters may be written as any function
of A. The simplest application of this mixture-amount
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 mixture-amount models
obtained by this technique provide for many of these
situations (see Appendices A and B).
The considerable flexibility of the mixture-amount
modeling technique and the resultant vast number of models


52
to be considered raises questions about the practical
aspects of selecting an appropriate mixture-amount model. A
natural model selection approach is suggested when data from
a complete (not fractional) mixture-amount experiment is
available. Since a mixture-amount 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, inverse-term
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 mixture-amount model. If only two levels of
A are considered in the mixture-amount 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 (5.2) or (5.3) rather than the linear form (3.1).
If the available data are from a fractional mixture-
amount experiment or are not from a mixture-amount
experiment at all, the above "natural" approach to model


selection nay 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 nay also be of help in
such situations. The practical aspects of selecting a
mixture-amount model will be considered further in Chapter 7
where several examples will be presented.
Finally, note that mixture-amount 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
mixture-amount model is appropriate for modeling the
response. By coding the levels of A (or A1, log A, etc.)
to have mean zero, mixture-amount models provide a descrip
tion of the blending properties of the components at the
average level of A (or A'1, log A, etc.) and explain how the
total amount affects these component blending properties.
By using orthogonal polynomial functions of A (or A- log
A, etc.) in mixture-amount models, descriptions of the
component blending properties averaged over the levels of A
(or A-1, 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 MIXTURE-AMOUNT EXPERIMENTS
Designs for both unconstrained and constrained mixture-
amount experiments are presented in this chapter. An
unconstrained mixture-amount experiment is one in which the
component proportions xH vary between 0 and 1. A
constrained mixture-amount 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 mixture-amount experiments is presented and
guidelines for selecting the levels of total amount to be
investigated are given. Techniques for fractionating
mixture-amount designs are discussed in Section 4.2.
4.1 Developing; Designs for Mixture-Amount Experiments
Since a mixture-amount experiment is defined as a
series of mixture experiments at several levels of total
amount, it is natural to propose as mixture-amount 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.
54


55
Defining a mixture-amount 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 affect 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 mixture-amount
experiment where the experimenter does not anticipate having
additive or inactive components, nor does he expect extreme
response behavior as component proportions approach zero.


56
Based on this knowledge, the experimenter selects the
special-cubic canonical polynomial as being an appropriate
model for describing component blending at each of the three
levels of total amount. An appropriate mixture-amount
design is then a three-component simplex-centroid 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}
simplex-lattice 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 simplex-centroid designs
at the high and low levels of total amount (see Figure 4.2).
As a second illustration, consider a constrained
mixture-amount experiment with three components and two
levels of total amount, where the component proportions are
constrained by .1 < x^ < .4, .1 < X£ .3, and .35 < *3 i
.75. An appropriate design for the special-cubic by linear
mixture-amount model
13 3
n = E [ Z B^x + E E fJ^.x.X + S ^ x X X ] (A' )h (4.1)
h=0 i=1 i consists of the vertices, centroids of the longest edges,
and the overall centroid of the constraint region at each of


57
A
1
A
2
A
3
Figure 4.1. Mixture-Amount Design Consisting of a Three
Component Simplex-Centroid Design at Each of
Three Amounts
X
1
X
2
X
3
A
1
A
2
A
3
Figure 4.2. Mixture-Amount Design Consisting of a {3,3}
Simplex-Lattice Design at the Middle Level and
a Simplex-Centroid Design at the Low and High
Levels of Total Amount


53
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 4.5*
Another important aspect of developing designs for
mixture-amount 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 higher-than-linear 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
A should be equally spaced on a log A or A" 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.


59
Table 4.1. Design Points for Fitting a Special-Cubic Model
in a Three-Component Constrained Mixture-Amount
Experiment at Two Levels of Amount
Pt<*>
X1
X2
X3
T
A
1
.10
.30
.60
-1
2
.10
.15
.75
-1
3
.15
.10
.75
-1
4
.40
.10
.50
-1
5
.40
.25
.35
-1
s
o
.35
.30
35
-1
7
.225
.30
.475
-1
a
.10
.225
.675
-1
9
.275
.10
.625
-1
10
.40
.175
.425
-1
11
.25
.20
.55
-1
12
.10
.30
.60
1
13
.10
.15
.75
1
14
.15
.10
.75
1
15
.40
.10
.50
1
16
.40
.25
.35
1
17
.35
.30
.35
1
18
.225
.30
.475
1
19
.10
.225
.675
1
20
.275
.10
.625
1
21
.40
.175
.425
1
22
.25
.20
.55
1
(a)
These point numbers are used in Figure 4.3


Figure 4.3. Mix ture-Amount Design for Fitting a Specia1-Cubic Model in a Constrained
Mixture-Amount Experiment


61
4.2 Fractionating; Designs for Mixture-Amount
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 mixture-amount 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 mixture-amount 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 mixture-amount designs are now
discussed.
Fractionating mixture-amount 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 log-ratio 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 mixture-amount design. If the q-1 mathematically
independent ratio or log-ratio variables are each
investigated at two levels and A is also investigated at two
levels, an appropriate mixture-amount design is a x 2 =
2^ factorial design. Similarly, a 3^ factorial mixture-
amount design is appropriate if the q-1 ratio or log-ratio
variables and the total amount variable A are each
investigated at 3 levels. Fractionation methods for 2^ and


62
3q designs are well known and many such fractional designs
have been tabled (e.g., see Cochran and Cox 1957).
Fractionation methods for the ZK3m series of factorial
designs are discussed briefly in Appendix G with respect to
mixture-araount-process variable experiments, but the
techniques are applicable here also.
The second fractionation method of Scneffe (1963),
discussed in Section 2.3, can be used to reduce the number
of points in simplex-centroid x 2 mixture-amount designs
(designs in which a q component simplex-centroid design is
set up at each of two levels of total amount). In general,
a simplex-centroid x 2 mixture-amount design supports
fitting a mixture-amount model of the general form
n =
[ 2 S?x
h=0 i=1
i i
+ 2 2 b^x.-x..
i ij 1 0
+2228?
i< j ijkxixjxk
+ 0? XX. ..x ](a' )h
12...q 1 2 q
(4.2)
However, fractions of a simplex-centroid x 2 mixture-amount
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 one-half fraction of the three
component simplex-centroid x 2 mixture-amount design is
listed in Table 4.2 and is pictured in Figure 4.4. This
seven-point design supports fitting either the special-cubic


Table 4.2. One-Half Fraction^ of a Simplex-Centroid x 2
Mixture-Amount Design for Three Components
i
X1
x2
X3
A
1/2
1/2
0
-1
1/2
0
1/2
-1
0
1 /2
1/2
-1
1
0
0
1
0
1
0
1
0
0
1
1
1/3
1 /3
1/3
1
(a) Fraction obtained using I = +ABCD as the defining
contrast. Switching the levels of A' yields the
I = -ABCD fraction.
Figure 4.4. Graphical Display of Design in Table 4.2


64
mixture model, or, the seven-term mixture-amount model
(4.3)
n
Fitting the special-cubic 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 one-half 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 one-half fraction of
the four-component simplex-centroid x 2 mixture-amount
design which is listed in Table 4.3 and is pictured in
Figure 4.5. This 15-point design supports fitting the 15-
term mixture model
n
4
+ I E z 3
i< j 0
. X X x,
ijk i j k
+ 31234X1X2X3X4
xx.x,x
(4.4)
or the 15-term mixture-amount models


65
n
4
+ Z £ E S
i< j or
n
(4.6)
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
simplex-centroid design for the purpose of estimating the
special cubic and quartic blending properties, it seems
apparent that the one-half 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


56
Table 4.3.
One-Half
Mixture-
Fraction^-3' of a
Amount Design for
Simplex-Centroid
Four Components
X
X1
X2
x3
x4
r
A
1/2
1/2
0
0
-1
1/2
0
1/2
0
-1
1/2
0
0
1 / 2
-1
0
1/2
1/2
0
-1
0
1/2
0
1/2
-1
0
0
1/2
1/2
-1
1/4
1/4
1/4
1/4
-1
1
0
0
0
0
1
0
0
1
0
0
1
0
1
0
0
0
1
1
1/3
1/3
1/3
0
1
1/3
1/3
0
1/3
1
1/3
0
1/3
1/3
1
0
1/3
1 /3
1/3
1
(a) Fraction obtained using I = +ABCDE as the defining
contrast. Switching the levels of A' yields the
I = -ABODE fraction.
Figure 4-5. Graphical Display of Design in Table 4-3


67
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 simplex-centroid x 2 mixture-amount 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 computer-aided
design approach. One such approach based on D^-optimality
is discussed below.
The fractionation methods discussed so far are appli
cable only for certain types of mixture-amount designs.
However, the computer-aided design approach, introduced in


58
Section 1.1, provides a method for fractionating any
mixture-amount design for both unconstrained and constrained
mixture-amount experiments. Recall thax the computer-aided
design approach involves choosing a criterion of interest
(e.g., Dn, Gn, Vn, or AN-optimality) 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
mixture-amount design to be fractionated. The Dj,j-optimality
criterion (which seeks to maximize det(x'x), where X is the
N-point 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 Djj-
optimal design, it often does so; when it does not, the
resulting design is near Dj^-optimal.
We discuss the development of D^-optimal designs for
canonical polynomial mixture-amount models. The development
for other families of mixture-amount models proceeds in much
the same way.
The candidate points for a given design/model are
usually the points of the associated complete mixture-amount
design. Several examples are given below.
0 The candidate points for the models (A1) (A5) in
Appendix A are (assuming an unconstrained mixture-
amount experiment) the simplex vertices at each


69
level of total amount. Since there are no
nonlinear blending terms in these models, the Djp
optimal design will not contain binary, ternary,
. . etc. mixtures even if included in the
candidate list. For a constrained mixture-amount
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 mixture-amount
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 mixture-amount model under consideration
contains special cubic terms. For a constrained
mixture-amount experiment, the candidate points
would consist of the constraint region vertices and
edge centroids at each level of total amount. The
two-dimensional face centroids would be included if
the mixture-amount model contains special cubic
terms.
0 The candidate points for a full cubic canonical
polynomial mixture-amount model in an unconstrained
mixture-amount experiment are the points of a {q,3}
simplex-lattice at each level of total amount.
The D^-optimal (or near D^-optimal) designs for several
of the canonical polynomial mixture-amount models of
Appendix A were obtained using the DETMAX program for three
component unconstrained mixture-amount experiments with two
and three levels of total amount. Some of the many possible
Dj^-optimal designs for the models considered are given in
Appendix C. The results for q = 3 suggest procedures for
developing Djpoptimal designs (without the need of a
computer program such as DETMAX) for unconstrained


70
mixture-amount 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.
0 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 D^j-optimal 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 D^-
optimal designs for any value of N > p.


71
Table 4-4. Sequential DN-Optimal 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 =
I 6X.
1=1 1 1
q 0 1.1
+ E E 3. .X.x + S-A
Kj ^ 1 J 0
N
Procedure*
P
alaill +
2
p+1 to C
C+1 to C+p
The smallest possible D^-optimal 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*


72
Table 4.5. Sequential DN-Qptimal 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
n
q o q o q 1
Z 8.x. + Z Z s._x x + Z 3ix.A
i=1 1 1 i< j 1 J i=1
N
Procedure*
q (q+1) + The smallest possible D^-optimal design
2 + q for this model contains1the 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.
p+1 to C Add the remaining edge centroid points
until a full set of candidate points is
obtained.
C+1 to C+2p 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.
C+2q+1 to C+p 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 = 5.


73
Table 4.6. Sequential DN-Optimal 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
n =
q
z
i=1
Sx.
1 1
q
+ z z
i 8 .x.x .
ij i 3
^ 1 f
X S x A
i=i 1 x
H t
z z a. x. x. a
i N
Procedure*
p = C The smallest possible DN-optimal design
for this model consists of a full set of
candidate points.
C+1 to 2C 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 D^-Optimal Design Development
Procedure for the Special-Cubic by Constant
Mixture-Amount 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
n =
q
E
i=1
Sx.
1 1
q
E E
i + E
q
E E
i 8? x. x .x,
ljk i j k
1 f
3A
N
Procedure*
p = C/2 + 1
p+1 to C
C+1 to C+p
The smallest possible D^-optimal 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 fuil 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.


75
Table 4.8. Sequential DN-Optimal Design Development
Procedure for the Special-Cubic by Linear
Mixture-Amount 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
n = Z
i-1
6X. + Z Z 6 .x
1 1 i . X .
1 0
q
+ z z z
i< j 6? x x x,
Ijk 1 a k
Z 31 x A '
i = 1 1 1
N
Procedure*
p = C/2 + q
P+1 to p+(|)
p+(|)+1 to C
C+1 to C+2q
C+2q+1 to
C+2q+(
C+2q+(^)+1 to
C + p
The smallest possible D^-optimal design
for thi3 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*


76
Table 4.9. Sequential DN-Optimal Design Development
Procedure for the Special-Cubic by Linear
Mixture-Amount 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
n = Z [ Z 8.x.
h=0 i=1 1 1
9 u 9 h i h
+ Z Z 8 .x. x + Z Z Z Sn x. x .X, ] (A )n
Kj 1 J l N
Procedure*
P c
The smallest possible D^-optimal design
for this model consists of a full sec of
candidate points.
C+1 to 2C
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.


77
Table 4.10- Sequential (Near) D,\j-0ptimal 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
= 2 8 x. +
i1 1 1
£ £ 8 ; .x. X +
i 1 ^2 '
2 3 x A + £ sfx.(A )
i=1 11 i=1 1 1
N
Procedure*
q(q + 1 ) The smallest possible DN-optimal design
2 for this model contains^the vertices at
all three levels of A and covers the edge
centroid positions once.
p+1 to p+(5) 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.
p+(^)+1 to C
Add the remaining edge centroids to
complete a full set of candidate points.
C+1 to C+3d 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
(all vertices). Variances of the 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 A1 =
and A' = +1.
-1


73
Table 4.10.-continued.
N Procedure*
C+3q+1 to C+p
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 G for examples of designs
generated by this procedure for the case q = 3-


79
Table 4.11. Sequential (Near)D^-Optimal 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 o q o q 1
n = £ 3.x. + £ £ S-.x.x. + £ 3.X A
i-1 11 i q 1 < q 2 >2
+ 2 £ 8iix x-A + £ Six.(A )
i N Procedure*
p = q2 + 2q The smallest possible Djq-optimal 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.
p+1 to C
Add edge centroids on the A' = 0 simplex
to obtain a full set of candidate points.
C+1 to C+2q 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.
(3q2+7q+2)/2 to
6q+4(2)~1
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-


ao
Table 4.11.-continued.
N Procedure*
6q + 4(g)
This unique design consists of the
vertices at all three levels of A twice,
and the edge centroids on rhe A1 = -1 and
A' = +1 simplexes twice. Note that it
does not contain a full set of candidate
points.
6q+4(^) +1 to
C+p
Add edge centroids on the A1 =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.


81
Table 4.12. Sequential DN-Optimal 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 o q o q i i q 1 t
n = E 3.x. + Z Z 3..X.X. -t- E 0.x.A + E Z S..X.X.A
i=1 11 i q ? i o q p t p
+ 2 S|Xj(A + E Z S^ixixi(A )
i=i i N Procedure*
p = C The smallest possible DN-optimal design
for this model consists of one full set of
candidate points.
C+1 to C+p 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' = *1 simplexes lowers
the variances of the s' while
concentrating points on the A' = 0 simplex
lowers the variances of the sU and 3 .
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.


32
A potential criticism for the use of D^-optimal 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 mixture-amount model having x^A and x^x^A terms
corresponding to the x.¡_ and x^x^ 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 Dj^-optimal for their corresponding models, but
are also D^-optimal 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 D^-optimal for the model of
Table 4.5 (and vice versa), although it may be possible to


33
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 D^-
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 D^-optimal
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,
0 the procedures of Tables 4.7 and 4.8 yield designs
that are also DN-optimal for the model of Table 4.9
0 the procedures of Tables 4.10 and 4.11 yield
designs that are also Dj^-optimal 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 D^-optiraal 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 D^-optimal 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
\ A
be criterion based measures such as tr[(X X) ],
max[x'(x'x)1x], or avg[x'(X'X)-1x], 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 D^-optimal designs are given in
Appendix D for some of the three-component designs from
Appendix C.
The D^-optimal computer-aided 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 mixture-amount experiments. One
must have and use a computer program (such as DETMAX) for
each particular application. As an example, consider the
three component constrained mixture-amount 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 special-cubic by linear
mixture-amount model
n = l
i=1
3
E E
i 0 0 J 1 A '
hjVj + Bi23xiV3 J/iV
(4.7;
To summarize, in this section we have discussed several
techniques for fractionating mixture-amount designs for


N
Design 1
Design 2
Figure 4.6. DETMAX Designs for the Constrained Mixture-Amount Experiment Listed in
Table 4.1
CD
VJ1


Design 1
Figure 4.6.-continued.
Design 2
The pattern of the design to
the left is unique, apd yields
det(X'X) = 9-7 x 10~15. The
full set of 22 candidate points
yields det(X'X) = 9.1 x 10_1i?.
J
oa


37
unconstrained mixture-amount experiments, including a
computer-aided design approach. This approach was also used
to fractionate a constrained mixture-amount design. It is
clear that the computer-aided design approach is quite
powerful and can be used to fractionate any mixture-amount
design (including the ratio or log-ratio variable
designs). The DN-optimality 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
Mixture-amount 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. Mixture-amount 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.
as


89
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 aif 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 second-degree polynomial model of the form
q
n = aQ + z a a. (5.1)
u i=1 1 1
q q 2 q
n = a0 + Z a.a. + Z a,.a. + Z Z a.-a.a, (5.2)
u i=1 1 1 i-1 i< j J
Similar models are obtained by substituting log aj_ (or other
functions of the a) for the 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 (1985).
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 (a^) may be written as the


90
product of the proportion of the component in the mixture
(Xj_) and the total amount of the mixture (A): a^ = x^A,
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 a^ = x^A
allows us to view the response as a function of the
component proportions and the total amount of the mixture,
i.e.,
n = f(a^,a2,...,aq)
= f ( x^ A,x^A,,XqA)
= g(x.| ,X£,... ,Xq,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 a-¡_ 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 Mixture-Like 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 a. The approach is introduced with the
following hypothetical two-component example.
A soft-drink 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 (a^,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 = a1a1 + a>,a2 + a12a1a2 (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 mixture-amount and component amount approaches,


92
first consider the plot of the sweetener combinations given
in Figure 5.1. The combinations all lie on the line
a2 = (-4/3)a1 + 12 (5.5)
The company has chosen a desired sweetness level based on
the amounts a1 = 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
a,,/ 9 + a2/12 = 1 (5-6)
q
is reminiscent of the restriction ex. =1 in a mixture or
i=1 1
mixture-amount experiment. Since the total amounts of the
five combinations chosen for the experiment are different,
it is natural to compare this approach to the mixture-amount
approach. There is a clear difference between the two
approaches; with a mixture-amount 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.


Full Text

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