Title: Models and designs for generalizations of mixture experiments where the response depends on the total amount
CITATION THUMBNAILS PAGE IMAGE ZOOMABLE
Full Citation
STANDARD VIEW MARC VIEW
Permanent Link: http://ufdc.ufl.edu/UF00102790/00001
 Material Information
Title: Models and designs for generalizations of mixture experiments where the response depends on the total amount
Physical Description: Book
Language: English
Creator: Piepel, Gregory Frank, 1954-
Copyright Date: 1985
 Record Information
Bibliographic ID: UF00102790
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: ltuf - AEH8277
oclc - 14972002

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. Andr4 Khuri, Dr. Frank Martin, and

Dr. Esam Ahmed for serving on my committee. I would also

like to thank Dr. Randy Carter, who served on my Part C and

oral defense examining committees. Words of thanks also go

to Cynthia Zimmerman for her expert job of typing this

manuscript and to Joe Branch for his work in preparing the

artwork for the figures.

Finally, I would like to thank my wife, Polly, and

daughter, Erin, for their support and encouragement over the

course of my studies and research.


iii














TABLE OF CONTENTS


Page

ACKNOWLEDGMENTS......................... .. ............... iii

ABSTRACT .................... ............ .......... .vii

CHAPTER

ONE INTRODUCTION........ .................. ... ..... 1

1.1 The Response Surface Problem...............1
1.2 Mixture Experiments--An Introduction........6
1.3 The Subject of This Research--
Generalizations and Extensions of
Mixture Experiments........................9

TWO LITERATURE REVIEW--MIXTURE 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............ 2

3.1 An Introduction to Mixture-Amount
Experiments............................ 52
3.2 Including the Total Amount in Mixture
Models................................... 34
3.3 Mixture-Amount Models Based on
Scheffe Canonical Polynomials..............36
3.4 Mixture-Amount Models Based on
Other Mixture Model Forms.................. 45
3.5 Mixture-Amount Models--A 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








FIVE MODELS AND DESIGNS BASED ON THE COMPONENT
AMOUNTS ...................................... .88

5.1 Standard Designs and Polynomial Models
Based on the Component Amounts.............89
5.2 Models and Designs for Experiments
Where the Component Amounts Have a
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........................167

8.1 Summary ............. ... .......... ....... 168
8.2 Recommendations........................ 174

APPENDICES

A SCHEFFE CANONICAL POLYNOMIAL MIXTURE-AMOUNT
MODELS........................... .... ......... 178

A.1 Models in Which the Components Blend
Linearly ....... ................... .....178
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 Blending at Two
Amounts .............. .................. . 183
B.2 Linear, Quadratic, and Quadratic
Blending at Three Amounts..................185








B.3 Linear, Linear, and Quadratic
Blending at Three Amounts..................189
B.4 Special-Cubic Blending....................190

C THREE COMPONENT DN-OPTIMAL 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-OPTIMAL 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














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


vii









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 xl, x2, xq. The predictor

variables are quantitative and continuous and their values

are assumed to be controlled by the experimenter. The

response y is quantitative and continuous. The functional

relationship between the predictor variables and the

response may be expressed as


Yk = f(xkl'xk2',.. Xkq) + k k=1,..., N, (1.1)

where yk is the kth of N observations of the response in an

experiment, xki is the value of the ith predictor variable

for the kth observation, and sk is the experimental error

contained in the kth observation.

The form of the function f in (1.1) is usually not

known and may be quite complex. In practice, an

approximating function is identified with as simple a form

as possible; often first or second-degree polynomials in the

predictor variables xl, x2, . are adequate. The

selection of an appropriate approximating function (often




2



referred to as model selection) is the first step in solving

a response surface problem. Usually a model linear in the

parameters is chosen.

A linear response surface model may be written in

matrix notation as


y = X8 + (1.2)


where y is an Nxl vector of observed response values, X is

an Nxp matrix of known constants (N 2 p 2 q), 8 is a pxl

vector of unknown parameters, and E is an Nxl vector of

random errors. It is usually assumed that E(e) = Q and

Var(E) = a2V, where V is a diagonal matrix. Most often in

practice V = IN (the NxN identity matrix containing ones on

the main diagonal and zeros elsewhere). Since E(E) = Q, the

model (1.2) can alternately be expressed as


n = E(y) = XB (1.3)


Another step in solving the response surface problem is

to estimate the parameters 8 and refine the model form if

necessary. If we assume Var(E) = c2IN, then the ordinary

least squares estimator of 8 is given by


8 = (X'X)'X'y (1.4)


and has variance








-12
Var(8) = (X X) a (1.5)


The portion of response surface analysis involving

model selection, parameter estimation, and model refinement

is known as regression analysis. Hence, the model

parameters are called regression coefficients and the

response surface model is called the regression model.

Once a fitted regression model y = X8 is obtained, the

next step is to test it for adequacy of fit. If it is found

to be adequate, it can then be used to make predictions of

expected response values for any set of predictor variable

values x-, x2, . xq within the experimental region.

If we let XO represent this set of values expanded to

resemble the terms in the model, the predicted value and its

variance for the expected mean response at x0 are



y(x ) = X 0 (1.6)


and

Var[y(x )] = Var(x 8)
0, -1


= x(X X)-1 2 (1.7)


Under the assumption Var(s) = a2V (where V is a known

diagonal matrix, not necessarily the identity matrix), s is

estimated by weighted least squares, yielding









8 = (XV' X)- X'V (1.8)



Var(B) = (X V-1 X) 2 (1.9)


and

Varly(x )] = x (X V' X)- 0 (1.10)
~ -00


For more details on these formulas and regression analysis

in general, see Draper and Smith (1981) or Montgomery and

Peck (1982).

Note that the parameter estimators, parameter estimator

variances, and prediction variances in (1.4) through (1.10)

all depend on the Nxp matrix X, which is referred to as the

(expanded) design matrix. Clearly the experimental design

chosen is of great importance in determining the fitted

model and its properties. Box and Draper (1975) gave 14

criteria to consider in choosing a response surface

design. Myers (1971) presented several classes of response

surface designs which support the fitting of first and

second-degree polynomial models in the predictor

variables. Among the designs discussed by Myers are the 2q

and 3q factorials, the 2q-k 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








the design from a candidate list so as to optimize the

design criterion selected. Several design criteria of

interest are:



1. D-optimality seeks to maximize det(X'X) or
equivalently minimize det[(X X)- '.

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-optimality seeks to minimize trace [(X'X)-1].


Designs consisting of N points obtained by using these

optimality criteria are referred to as DN, GN, VN, and AN-

optimal designs. These design criteria and computer

programs for implementing them are discussed by St. John and

Draper (1975), Mitchell (1974), and Welch (1984).

In summary, the major parts of a response surface

analysis are:



1. Selection of an appropriate model to approximate
the response surface over the region of interest.

2. Development of a design which supports the fitting
of the selected model form and provides for testing
the adequacy of fit of the model.

3. Fitting the chosen model, testing it for adequacy
of fit, and revising the model if necessary.

4. Determination of the levels or ranges of the
predictor variables that yield the optimum response
value.









In the following chapters, we will be concerned mainly with

the first two items listed above, that is, model selection

and design development.



1.2 Mixture Experiments--An Introduction

A mixture experiment involves mixing two or more

components (ingredients) together to form some end product,

and then measuring or observing one or more properties of

the resulting mixture or end product. In the usual

definition of a mixture experiment (Cornell 1981, Scheffe

1958), the properties of the mixture are assumed to depend

on the proportions of the components present and not on the

total amount of the mixture. Some examples of mixture

experiments are:



1. Sandwich fish patties made using mullet,
sheepshead, and croaker (Cornell and Deng 1982).
The texture of the fish patties was one of several
responses of interest.

2. Coatings (paints) made from blending a prime
pigment, vehicle, and two extender pigments (Hesler
and Lofstrom 1981). Hiding power and scrubbability
were the properties of interest.

3. Waste glasses obtained by mixing SiO2, B203, Al20,
CaO, MgO, Na20, ZnO, Ti02, Cr203, FepO and NiO
(Chick, Piepel, Mellinger et al. 198T). Leach
rates, viscosity, conductivity, and crystallinity
were several of the glass properties investigated.


A mixture experiment problem is clearly a response surface








problem, with the proportions of the components in a mixture

being the predictor variables.

In a mixture experiment (as defined above), the

response to a mixture of q components is a function of the

proportions xj, x2, . xq of components in the

mixture. Since xi represents the proportion of the ith

component in the mixture, the following constraints hold:


q
0 < x. < 1 (i=1,2,. ..,q); x. = 1 (1.11)
i=1 1


Mixture experiments having only these constraints are

referred to as unconstrained mixture experiments. Physical,

theoretical, or economic considerations often impose

additional constraints in the form of lower and upper bounds

on the levels of components


0 < Li < x < U 1 (i=1,2,...,q) (1.12)


Experiments where these additional constraints are imposed

on the xi are referred to as constrained mixture experi-

ments.

The region of mixture component combinations defined by

constraints (1.11) and (1.12) is referred to as the

constraint region. Geometrically, restriction (1.11)

defines the constraint region as a regular (q-1)-dimensional

simplex. In general, restrictions (1.12) reduce the

constraint region given by (1.11) to an irregular








(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 pseudocomponents. If at least one component

has a nonzero lower bound, then the pseudocomponent values

xi may be obtained from the original component values xi by


x. L
xi = i = 1,2,...,q (1.13)
1 E L
j=1

q
where E L < 1. Crosier (1984) referred to this as the
j=1
L-pseudocomponent transformation. If at least one variable

has a nonunity upper bound, then pseudocomponent values may

be obtained by the U-pseudocomponent transformation,


U. x.
xi = q i = 1,2,...,q (1.14)
Z U 1
j=1

q
where E U. > 1. Crosier (1984) presented additional
j=1 3
discussion on the use of these two pseudocomponent trans-

formations and gave guidelines for choosing between them.








Models and designs for mixture experiments are reviewed

in Chapter 2. Before proceeding to that material, however,

the purpose and subject of this research is presented.



1.3 The Subject of This Research--Generalizations
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 REVIEW--MIXTURE 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

Scheffe (1958) developed canonical forms of polynomial

models for mixture experiments by substituting the mixture
q
constraint E xi = 1 into certain terms in the standard
i=1
polynomial models and then simplifying. For example, with

q = 2 mixture components, the standard second-degree

polynomial model is


n = 0 x + a2x2 + ,12x1x2

2 2
+ a11x1 222 .(2.1)

Multiplying the constant term by unity and applying the
2
mixture restriction x + x2 = 1 to the x. terms yields
1 2 1
2 2
10 = 0(x1 + x2), = x1(1 x2) and x2 = x2(1 x1).
Hence, (2.1) can be reduced to the form

n = S 1x1 + 622 + 12x1x2 (2.2)








where a1 = o0 + a1 + al1' 2 = a0 + 2 + a22, and

312 = a12 all a22
The general forms of the first, second, and third-

degree canonical polynomial models in q mixture components

are


q
n = E Bixi (2.5)
i=1

q q
n = E Six + Z i 8 xx. (2.4)
i=1 i
and

q q q
n = Sixi + EE s jx + r ij i.x .j(xi xj)
i=1 i
q
+ E Bij kx x xk (2.5)
i
The special-cubic canonical polynomial model is a reduced

form of the full cubic model (2.5) obtained by deleting the

5ijxixj(xi xj) terms.

In each of the above model forms, the first q terms,

81x1 + . + 8qxq, represent the linear blending of the

components while the remaining terms represent nonlinear

blending of the components. We shall refer to these

phenomena throughout, as the linear and nonlinear blending

properties of the components.

Scheffe's canonical polynomial models are widely used

and have been shown to adequately approximate many types of








mixture response surfaces. However, there are certain types

of mixture surfaces for which the canonical polynomial

models are not adequate. For example, when one or more of

the mixture components have an additive effect, Becker

(1968) recommended that homogeneous models of degree one be

used. [A function f(x,y, . ,z) is homogeneous of degree

n if f(tx,ty, . ,tz) = tnf(x,y, . ,z) for every

t > 0.] Three such models, which Becker referred to as H1,

H2, and H3, are given by


q
H1: n = E .x +
i= 1


q
Z 8ij min(xi,x ) +
i

(2.6)


q
H2: n = B ix +
i=1 1


+ 812...q


q
H3: n = B ixi +
i=i1


+ 812...q


If the denominator

defined to be zero.

H1, H2, and H3 (the

are often adequate.


q 2-1
Z 8 ij xix./(x+x) + . .
i

x1x2... Xq/(x1+X2+...+x )q-1


S1/2
E Z 8 (x.x.)
ij (1 x
i

(x x2...x )/q .


(2.7)


+ . .


(2.8)


of a term in H2 is zero, that term is

In practice, the second-order forms of

first two sets of terms in each model)

Snee (1973) discussed the types of


+ 812...q min(x x2, ...,x q) ,








curvature generated by these second-order Becker models over

the region 0 < xi < 1.

Becker further noted that the forms of H1, H2, and H3

implicitly assume the response surface attains its maximum

(or minimum) at the centroid of the simplex. He suggested

alternate forms of the models for situations where this is

not the case (see Becker 1968 or Cornell 1981).

Draper and St. John (1977a) proposed several mixture

models which consist of Scheffe canonical polynomial models

plus inverse terms of the form xi1. For example, the first

and second-degree models with inverse terms are


q q 1
--1
n = Bixi + Z 8_ix (2.9)
i=1 i=1

q q q
-1
n = i + E 8ij.xix + 8 ix (2.10)
i=1 i
Inverse terms may be added to any Scheffe polynomial model

in a similar manner.
An -1
An inverse term xi proves helpful in situations where

an extreme change in the response (f(x) ") occurs as the

proportion of a component tends to its lower bound of

zero. For similar situations where a component has a non-

zero lower bound Li, inverse terms of the form (xi Li)-1

may be used. If f(x) + - as xi + Ui (where Ui represents

the component's upper bound such that 0 < L.< U. < 1), then
1 1-
inverse terms of the form (Ui xi-1 are appropriate.
inverse terms of the form (U. x.) are appropriate.








It is assumed when using any of the above models with

inverse terms that the experimental region itself does not

include the boundary of any component i that causes f(x) +

. as xi Li or Ui. See Draper and St. John (1977a) for

further discussion of this assumption.

In some mixture experiments, interest centers around

ratios of component proportions and how the response depends

on these ratios (see e.g., Hackler, Kriegel, and Hader 1956

or Kenworthy 1963). Snee (1973) noted that models based on

ratios are useful alternatives to the Scheffe and Becker

models for particular types of surfaces because the ratio

models describe a different type of curvature than do the

other models.

A ratio model is developed by replacing the set of

component proportions xi, i=1,2, . ,q, with an

equivalent set of ratio variables rj, j=1,2, . ,q-1. In

general, there are many possibilities. For example, with

three components the following equivalent transformations

(among others) are possible:


Transformation rC r2

R1 x1/x2 x2/x3

R2 x1/x2 x3/x2

R3 x1 x2/x3

R4 x1/(x2 + x ) x2/x3


It is seen that ratios may not be defined if certain








component proportions take on zero values in the

denominator. In such cases, Snee (1973) suggested adding a

small positive quantity c to each xi so that the denominator

is always greater than zero.

The above example illustrates that only q-1 ratio

variables are needed to replace the q component proportions
q
(owing to the mixture restriction Z x. = 1). Because of
i=1 2
this reduction in the number of variables, the ratio

variables are mathematically independent. Hence, standard

polynomial models in the ratio variables, such as

q-1
n = a0 + Z ajrj (2.11)
j=1

and

q-1 q-1 q-1 2
n = 0 + rr + E ajk rk j la r (2.12)
j=1 j
may be used. Further discussion of ratio models may be

found in Snee (1973) and Cornell (1981).

Becker (1978) presented additional models for mixture

experiments with additive or inactive components. For

q = 3, Becker suggested the following model form when at

least one component is inactive:


n = O + 1x1/(x1+x2) + '2x2/(x2+x3) + B3x3/(x3+x1)

q
+ E z ijhij(xi,x ) + 8123hl23(x1,x2,x3) (2.15)
i







Here hij and h123 are specified functions which are

homogeneous of degree zero. Inactivity of a component xi is

suspected when Bi = 0, Sij = 0 for jPi, and 8123= 0. The

hij and h123 functions suggested by Becker are


xs / x s
hi(xi,xj) i (2.14)
ij i j/ \i j/
1, 1 13


h123(x1,x2,x ) = x1 x2 txx2 ( (2.15)
(77 1 2 x1+2 ) 2+x5) x x

He also noted that when si or ti is negative, hij or h123

takes on an extremely large value near the boundary

xi = 0. Models of the form (2.13) are then alternatives to

the inverse term models of Draper and St. John (1977a)

discussed earlier.
Becker (1978) also made a general model suggestion for

mixture experiments with additive components, extending his

earlier work (Becker 1968). This suggestion is to consider

the model


q q
n 8.x. + Z E 8.. (x +x )hi (x ,x ) + . .
i=1' i
+ 812...q(x1+...+xq)h12...q(x1 ...,x q) (2.16)

where the functions hij(xi,xj), hijk(xixj,xk), etc. are

homogeneous of degree zero. This model can be simplified by

deleting higher order terms. Becker (1978) gave some








suggestions for the h functions, and noted that the H2 and

H3 models of (2.7) and (2.8) are of the form (2.16).

Aitchison and Bacon-Shone (1984) presented the

polynomial models


q-1
n = 8 + Bizi (2.17)
i=1

q-1 q-1
S= + E 8.z + Z i z (2.18)
i=1 i
where zi = log(xi/xq), i=1,2, . ,q-1. Rewriting (2.17)

and (2.18) in terms of the component proportions xi gives

the symmetric model forms


q q
n = +r iilog xi ( 0i = 0) (2.19)
i=1 i=1

q q 2
S= + og x + E i6 (log xi-log xj) (2.20)
i=1 i
where the 8ij are functions of the yij. Note that the above

models are not directly applicable when the component pro-

portions take on zero values, since log xi + - as xi + 0.

This behavior suggests models (2.19) and (2.20) as alterna-

tives to the inverse term models of Draper and St. John

(1977a) when the component proportions approach but do not
equal zero-valued boundaries. Substituting xi Li or

Ui xi for xi in (2.19) or (2.20) yields models useful for








constrained mixture experiments where f(x) + as xi Li

or U.i

Aitchison and Bacon-Shone pointed out that


s. = 0 (i=1,...,c); Bi = 0 (1 < i < j < c) (2.21)


indicates the inactivity of components 1, 2, . c.

They also noted that


.. = 0 (i=1,2,...,c; j=c+1,...,q) (2.22)


indicates that components 1, 2, . c are additive with

respect to components c+1, c+2, . q. This is a more

general concept of additivity than that considered by Becker

(1968, 1978), where he implied


8i = 0 (i=1,2,...,c; j=i+1,...,q) (2.23)


indicates that components 1, 2, . c are additive.


2.2 Mixture Experiment Designs

As with any response surface problem, choosing an

experimental design is an important part of a mixture

experiment. Designs for both constrained and unconstrained

mixture experiments are reviewed.

Scheffe (1958) proposed the {q,m} simplex-lattice

designs for exploring the full q-component simplex region in

an unconstrained mixture experiment. The fq,m}








simplex-lattice design (m=1,2, . .) consists of the
(q+m-1) points in the simplex (1.11) that represent all

possible mixtures obtainable when the proportion of each

component can take on the values O, 1/m, 2/m, . 1.

Examples of some {q,m} 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 xi 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 (1965) 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











(2/3 o, 1/3)


(0,1,0) /
X2


(1/2 o, 02)


(0/3 0,2/3)


(0,0,1)


{3,2}


{4,2}


(3,3}


{4,3}


Figure 2.1 Some {q,m} Simplex-Lattice Designs for Three and
Four Components


X3 X2


Figure 2.2 Simplex-Centroid Designs for (a) Three
Components and (b) Four Components








q q q
n = E S.xX + E s ij.x x + xxijkXix
i=1 i
+. . + 812...qX1X2...Xq (2.24)


This model contains 2q 1 terms and hence provides an exact

fit to data collected at the points of the corresponding

simplex-centroid design. The special-cubic model


3 3
S= E ixi + 8 ijxixj + 123x1 x2x3 (2.25)
i=1 i
is an example of (2.24) for the case q = 3.

Becker (1978) proposed radial designs for mixture

experiments for the purpose of detecting inactive components

or components with additive effects. He defined a radial

design as one 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 (xi = 1, xj = 0, jWi)

corresponding to those components thought to have additive

effects or thought to be inactive components. For

constrained mixture problems, focal points might be the

simplex vertices, the vertices of the pseudocomponent

simplex, or other points depending on how the concept of a

component effect is to be defined (see Piepel 1982).

The radial designs of Becker (1978) are an extension of

the axial designs proposed by Cornell (1975). The axis of








component i is the imaginary line extending from the vertex

xi = 1, xj = 0, j i, to the point xi = 0, x, = 1/(q-1), j i,

on the opposite boundary. The points of an axial design lie

only on the component axes. Cornell and Gorman (1978)

illustrated the use of an axial design for detecting an

additive blending component. Other uses for axial designs

have been discussed by Cornell (1975, 1977).

Draper and St. John (1977b) presented DN-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 DN-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 xi 1 0.05, i=1,2,3 to avoid the

problems that occur in xj1 when xi = 0.
Standard response surface designs such as factorials

and central composite designs are appropriate for fitting

polynomial models in ratio variables. Kenworthy (1963)

discussed factorial designs for situations where each ratio

variable is of the form rk = xi/xj. Hackler, Kriegel, and

Hader (1956), Donelson and Wilson (1960), Kissell and

Marshall (1962), and Kissell (1967) all used central

composite designs in situations where their ratio variables

were defined using ratio functions of the form ri =

x./(xi+x j + .).













(.90,.05,.05)

(.78,.17,.05)


(.2, .6,.2)
(.17,.78,.05)
(.05,.90,.05).

x2


(.2,.2,.6)
(.17,.05,.78)
.(.05,.05,.90)


(.05,.78,.17)


(.82,.05,.13)

(.72,.14,.14)


(.475,.05,.475)


(.14,.14,.72)
(.13,.05,.82)
(.05,.05,.90)


(.05,.82,.13)


Figure 2.3.


(.05, .13,.82)


Points of Support of DN-Optimal Designs for
Models 2.9 (a) and 2.10 (b)


(.14,
(.13,
(.05,








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 zi = log(xi/xj) are applicable.

Most of the designs discussed above were originally

proposed for unconstrained mixture experiments. For certain

constrained mixture experiments where the constraint region

is again a regular simplex, the designs discussed so far can

be adapted by using a pseudocomponent transformation.

However, most constrained mixture experiments have an

irregular hyperpolyhedron as a constraint region. We now

discuss designs for this type of constrained mixture

experiment.

McLean and Anderson (1966) presented an algorithm for

generating the vertices of the constraint region given the

lower (Li) and upper (Ui) bounds for the component

proportions. For fitting the Scheffe 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








of all extreme vertices. The approach of Snee and Marquardt

includes the use of the DN, GN, and AN-optimality criteria

in choosing efficient designs.

Nigam, Gupta, and Gupta (1983) presented the XVERT1

algorithm for generating designs for fitting the first-

degree canonical polynomial model (2.3). The XVERT1

algorithm is considerably faster than XVERT since it does

not depend on any of the optimality criteria (DN, GN, VN,

and AN-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)-1].
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).








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 ExDeriments

In mixture experimentation, it may be of interest to

observe changes in the response values caused by varying the

levels of n process variables in addition to the q mixture

component proportions. Scheffe (1963) gave an example where

the response is "road octane number" of a blend of

gasolines, and the make and speed of the car might be varied

as well as the proportions of the gasolines.

Since the main purpose of Scheffe's article was to

introduce the simplex-centroid design, he naturally

suggested a simplex-centroid x (k1 x k2 x . x kn)

factorial arrangement for mixture-process variable

experiments (where k- 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 k1 x k2 x

. x k factorial points or alternately as an (kI x k2 x








. . x kn) factorial at each of the 2q 1 simplex-centroid

points.

Scheffe also discussed an associated model for the

simplex-centroid x (k1 x k2 x .. x kn) design. The

notation gets rather messy if the process variables are

considered as classification variables, and there are more

than two levels of each factor. For process variables zl,

z2, . zn measured on a continuous scale, the model can

be represented somewhat easier. As an illustration, the

model for q = 3 components and n = 2 process variables,

where each process variable is set at two levels, is given

by



n ax x + a 0x3 + 8 2xxx2 + B xx + B8 x2x
1X1 1 2 42 3 3 812X12 13 1 3 23 2 3

2 .
+ 8 3xqx2x + [8x + 8 + 2 832


+8123 x1x2x z 2 3 j= x1 x2 2 23 2x


12
+ 8123xx2x3]zlz2 (2.26)


In general, when there are n process variables each at two

levels, the complete canonical polynomial contains 2q+n 2n

terms and is of degree q+n in the x's and z's.

Note in (2.26) that there are no terms involving only

the process variables (main effects of the process variables








or interactions containing only process variables). This is

due to the identity x + x2 + + xq =1.

The number of points in the 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 (2q 1)2n points of the simplex-centroid x 2n design

and the points of a 2q+n design (after removing the 2n

points corresponding to those combinations where all of the

q mixture components are absent). A fraction of the sim-

plex-centroid x 2n design is obtained by taking the points

corresponding to those in a fraction of the 2q+n design.

Points in the resulting design that have one mixture com-

ponent present correspond to pure mixtures, points with two

mixture components present correspond to binary mixtures

with each component proportion equaling 1/2, . and so

on. The process variable combinations of high and low

levels of the zi at these mixture points are interpreted as

usual. As an example, for q = 5 and n = 2 let A, B, C, D,

and E represent the mixture components and let F and G

represent the two process variables. Then the point "acdg"

in the 27 design is the mixture composition








(1/3,0,1/3,1/3,0) run at the low level of process variable F

and at the high level of process variable G.

Cornell and Gorman (1984) presented various fractional

design plans for 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 2q+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
= 81x + 1 2x2 + 812x1x2


+ (B11 x 2 12x2)z1 (2.27)


The effects of the process variable z1 are contained in the

coefficients 61, B2, and 812. If zI has the same (constant)
11 21
effect on all compositions, then S1 = a2 and 812 = O,

yielding the reduced model









0X + X + x x + 1(2.28)
n = 81 + 8x2 1212 O+ 1 (2.28)

1 1 1
where = 81 = Note that the terms in (2.28) are not a
0 1 2
subset of the terms in (2.27), specifically S8z is not con-

tained in (2.27). Gorman and Cornell also noted that one

can get a distorted view of the effects and significance of

zi by considering (2.27).
To arrive at a reduced form of the combined model in

the xi's and zi's, Gorman and Cornell suggested repara-

metrizing the general form of the mixture-process variable
q
canonical polynomial model by first substituting 1 Z x.
j=2 J
for xl in all crossproduct terms involving x, (alone) with

the process variables and then rewriting the terms in the

model. For the above example, substituting 1 x2 for x1 in

the term 8xizi1 in (2.27) yields


o 0 o 1
n = 8 X1 + 82x2 + 2X1X2 + 81(1 x2)z1

1 1



= sBx1 + S x + 2X1X2 + 62x2zx


+ 812x1x2z1 + z1 (2.29)

1 1 1 1 1 1
where 8 8 and = B Hence, 6 represents the

difference between the effect of z1 on the linear blending

of x and x while 81 represents the effect of z on the
2 1 0 1








linear blending of xj. Note that when the terms 6x2z1 and

12xlx2z1 are omitted from (2.29), we obtain (2.28). Hence,
the reparametrized model is suitable for obtaining reduced

model forms through subset selection procedures. Also, note

that the reparametrized form (2.29) now has a term with z,

alone. But, as noted above, the coefficient 81 of zi

represents the effect of z, on the linear blending of xj,

not an overall main effect of z1 (unless both 61 and 812 are

zero, in which case 80 is a measure of the overall main

effect of zl).

In closing this section, it should be noted that

Scheffe's 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 1.3).



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








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 A1 < A2).

Suppose the two components blend linearly at both amounts.








Several possible situations are shown in Figure 3.1, where

the pure component proportions are denoted by (xl,x2) =

(1,0) and (0,1). Figure 3.1(a) illustrates the case where

changing the level of the total amount has no effect on the

response (the lines are coincident), while in Figure 3.1(b)

increasing the amount from A1 to A2 increases the response

at all mixtures by a constant amount. Note that Figure

3.1(b) illustrates a situation where the total amount does

not affect the component blending properties but does affect

the response. Figures 3.1(c) and (d) illustrate cases where

the total amount does affect the component blending proper-

ties. In Figure 3.1(c), we see that an increase in the

value of the response results from raising the level of the

amount of the mixture, and the effect of raising the amount

becomes larger as the proportion of component 2 in the

mixture increases. Figure 3.1(d) represents a situation

where changing the amount has a considerable effect on the

blending properties of the two components; at A1, increasing

the proportion x2 produces an increase in the response

value, while at A2 it results in a decrease in the response

value.


3.2 Including the Total Amount in Mixture Models

Since the response in a typical mixture experiment does

not depend on the total amount, the usual mixture model

forms must be modified to incorporate amount effects for











AA2 A
Al


c(d)
on .
I AI




(1,0) (0,1) (1,0) (0,1)
(a) (b)








A2
A1

A2



Al



(1,0) (0,1) (1,0) (0,1)
(c) (d)



Figure 3.1. Plots of Several Two-Component Blending Systems
at Two Total Amounts A1 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

Scheff6's canonical polynomials (see Section 2.1) have

been shown to be a versatile class of equations for modeling

mixture response surfaces. Since a mixture-amount experi-

ment is just a series of mixture experiments run at each of

several amounts A1, A2, . Ar, r > 2, it is natural to

envision fitting the entire experimental data set as a

series of smaller experiments which are performed at each








amount Ai. We consider the fitting of a Scheffe canonical

polynomial model such as (2.3) or (2.4) at each amount.

To begin the development, suppose a particular Scheffe

canonical polynomial model form, denoted by nC, adequately

describes the component blending at each of the r levels of

A. If the total amount of the mixture affects the response,

the parameters of nC vary as A varies, i.e., the parameters

of nC depend on A. This dependence can be modeled (for each

parameter 8m in nC) using the standard polynomial form


r-1
Sm(A) = + z k(A, (3.1)
k=1

where Sm(A) denotes that the parameter is considered to be a

function of A, and A denotes a coded version of A.

Although (3.1) implies an (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


r-1
B (A) = 0 + Z A-k (3.2)
Sm ,k=1m

or

r-1
S(A) = O + k (log A)k (3.3)
k=1

In practice, A-1 and log A in the above equations would

usually be coded, as was A in (3.1).








By writing the parameters in a Scheffe canonical

polynomial model as functions of A, a new model is obtained

that enables us to measure the effects of total amount on

the blending properties of the components. This model

derivation technique may be applied to any of the mixture

models reviewed in Section 2.1 (see Section 3.4). Any model

obtained in this manner will be referred to as a mixture-

amount model.

As an example, let us derive the form of a quadratic by

quadratic mixture-amount model where q = 2 and r = 3. Then,

nC is of the form (2.4) and the 8m's are of the form (3.1),
which yields


n = B1(A)x1 + B2(A)x2 + B12(A)x1x2





12 12 12 10 2
= [% + 0 A + 1(1 ) x [82 + 82A + 82(A ) ]x2





= Bxi + 2x2 + B2x1lx2


2 k k k k
+ E [x + 2 + B12x1x2](A ) (5.4)
k=1

Note that the subscript of a a parameter in (3.4) refers to

the components that are present in the associated term while

the superscript refers to the power of the A variable for

that term.








When the levels A,, A2, A3 are coded to have zero mean

(usually -1, 0, +1 if the levels are equally spaced), the

terms in the combined model (3.4) have the following

interpretation:


i) soX1 + BOX + 8 2X1x2 represents the linear and

nonlinear blending properties of the mixture
components at the average level of total amount,

ii) [BLx1 + 8 x2 1 l2X12]A' represents the linear
effect of total amount on the linear and nonlinear

blending properties of the mixture components,

iii) [Szx1 + 2x2 2xlx2](A')2 represents the

quadratic effect of total amount on the linear and
nonlinear blending properties of the mixture

components.


Thus the coefficients 8 and 8j of the terms xi(A )k and

xixj(A')k, k=1,2, in (3.4) are measures of the effects of
changing the amount of the mixture on the linear and

nonlinear blending properties of the mixture components (at
the average level of total amount).
When the levels of A' and (A')2 are coded to be the

coefficients of orthogonal polynomials [i.e., when A and

(A')2 in (3.4) are replaced by the first and second-degree
orthogonal polynomials P1(A) and P2(A)], the interpretations
of the coefficients change somewhat. Under this coding, the








coefficients 8$, B2, and B12 measure the linear and

nonlinear blending properties of the components averaged

over the levels of total amount. The coefficients 8 and

8j, k=1,2, are measures of the effects of changing the

amount of the mixture on the linear and nonlinear blending

properties of the mixture components (averaged over the

levels of total amount). See Section B.2 of Appendix B for

an example that illustrates the above interpretations.

For general q and r, a model of the form (3.4) is

written as

q q 0
n = E 8.x + r Z 8 ijxixj
i=1 i
r-1 q k
+ E [ E x + E 8 .x x ](A ) (3.5)
k=1 i= i
Depending on the way in which changing the total amount

affects the component blending for a particular application,

all of the terms in (3.5) may or may not be needed. Several

reduced forms of (3.5) that may be appropriate for various

applications are listed and discussed in Appendix A.

Suppose now the amount of the mixture does not affect

the blending properties of the components but does have an

effect on the value of the response. For the model of

(3.5), this implies that A has a constant linear effect for
1 1 1 1
all compositions (which forces 81 = = .. = 1 and 12
1 1
= 81 = 1 = 0); that A has a constant quadratic
13 =q-1,q








effect for all compositions (which forces 82 = 2 =

82 and 82 = _2 1 = 0); . ; and that A
S 12 q-1,q
has a constant (r-1)th degree effect for all compositions

(which forces 8r-1 = B-1 = 1 and 8 =1 = r1
1 2 q 12 13
= Sr q = O). In this case, the model of (3.5) takes

the reduced form


q 0q 0 r-1 k k
n = SOx. + E B .x.x. + S (A ) (3.6)
i=1 1 1 i

where the 8k (= = ), k=1,2, . ,r-
0 1 q
represent the linear, quadratic, . (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

terms of (3.5); specifically the terms with A alone

(80(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 reparametrization involves
q-1 k k
replacing xq with 1 xi in the terms 8 x (A), k=1,2,
4 i=1 'qq
. ,r-1, of (3.5) and simplifying. For the q = 2, r = 3

example considered earlier (3.4) is reparametrized as








0 0 0
n = 1 x1 + 2 + 812X1X2

2 k k kk
+ z [11X1 + 2(1 x1) + 12XX2(A
k=1

2

2 k k k k


k k k k k 1








x2. Although (3.7) now contains the terms +A' and 8((A')2
where 6A and ()2 and e, note 2, k=1,2. Hence, ( repre-






measure the linear and quadraticnear effects of total amount,

amounrespectively, on the linear blending properties of x, and do not

measurements the overall difference between the quadratic effects of total
of total amount unlesson the linear blending properties of x and
x(. Although (7)7) now contains the terms OA. and 8O(A











The Scheffe canonical polynomial mixture-amount models
considered Ad thus far were all developed under the assumption

measure the linesame canonical polynomial form id quadratic effects of appropriate forunt,

despctively, onbing the component blending at each total amount.
measure the overall linear and quadratic effects of total

amount [unless we find that 61 =

The Scheffe canonical polynomial 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








also of interest. Mixture-amount model forms for these

situations will now be discussed.

Let us suppose one of the Scheffe canonical polynomial

forms (linear, quadratic, cubic, etc.) is appropriate for

describing component blending at each level of A, and that

the appropriate forms are not the same for all levels of

A. Further, consider the most complicated form (i.e.,

highest degree) that is needed at one of the levels of A.

Then, if the form of the 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 B. To determine if

one model is better than another, or if one model is most

appropriate for a particular application, one can perform a

series of full vs. reduced model tests,


(SSEduced SSEful)/(er e f)
F* = SSEu/e (5.8)


where SSE reduced, SSEfull, er and ef are the sum of squares

for error and the error degrees of freedom for the reduced









and full models, respectively. As an example, suppose in a

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


q 0q 0 1'
Model 1: n = S S.x. + z 8 .x.x. + $ A
i=l1 I i
q 0 q 0 q '
Model 2: n = E B.x. + r Z 8B .x.x. + EZ .x.A
i=1 i

q 0 q0 q 1
Model 3: n = Sixi + E E ijxix + S xi A
i=1 i
q 1
+ Z S ij.x.x A ,
i

which are models (A6), (A7), and (A8) in Appendix A. We

begin by fitting models 3 and 2, treating them as the "full"

and "reduced" models respectively, and performing the test

(3.8). The test is a measure of the significance of the

q(q-1)/2 terms 80xixjA', 1 i < j < q, in model 3 over and

above the contribution of the terms contained in model 2.

The error degrees of freedom are ef = N-q(q+1) and er =

N-q(q+3)/2, respectively. If the test is significant, model

3 is selected. If the test is not significant, then models

2 and I are compared, treating them as "full" and "reduced,"

respectively. The full vs. reduced model test (3.8) can








also be used to compare the models discussed in Appendix B

to the corresponding models without parameter restrictions.

Another model selection approach is to fit the repara-

metrized form of a "full" 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 Scheffe canonical

polynomial models as functions of the total amount A. This

technique may also be used with any of the other mixture

models taken from the literature, many of which were

presented in Section 2.1. Expressions such as (3.1), (3.2),

(3.3) or any other appropriate function of A may be used for

the parameters of the mixture model chosen. Any such model

obtained by this technique is referred to as a mixture-

amount model.

As an example, assume the inverse term model (2.9) is

appropriate at each of two amounts, Al and A2. Using am's

of the form (3.1) yields the mixture-amount model









q9 -1
n = S S.(A)x. + 8 (A)x


q Aq 0 x-1

i=.1 1 i=1 -

q0 q 0 -1
6 8.x. + a .ix
i=1 i=1


q1 q 1 1
+ E S.ixA + x A (3.9)
i=1 i=1 -i

Recall that A denotes a coded form of the total amount

variable A. When the levels Al and A2 are coded to have

mean zero, the terms in (3.9) have the following interpreta-

tions:


q 0 q 0 -1
(i) 2 80xi and Z B_0 respectively represent the
i=1 i=1
linear and nonlinear blending properties of the

mixture components at the average level of total

amount,


q q -1 I
(ii) E 1 xiA and a 8xi A respectively represent
i=1 i=1
the linear effects of total amount on the linear

and nonlinear blending properties of the

components.


The phrase "nonlinear blending" in the above interpretations








refers to an extreme increase or decrease in the response

value as the value of xi approaches zero.

As another example, assume that the 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


q q
n = Z S.(A)xi + E 8 .(A)(xi x )1/2
i=1 i
q 0 1 q
= + (log A)]xi + E E [8 +81 (log A) ](x.xj)
i=1 1 i i

0 q 0 1/2
Z Sixi + E Sij (xix )
i=1 i q 1 q 1 1/2'

+ 8x.(log A) + E 1 (x.x.)1/2(log A) (3.10)
i=1 1 i
The notation (log A) above denotes a coded form of log A.

When the three levels log A1, log AZ, and log A3 are coded

to have mean zero, the terms in (3.10) have the following

interpretations:


q 0 q 0 1/2
(i) x and Z 8 i(xxj)1/ respectively repre-
i=1 i sent the linear and nonlinear blending properties

of the mixture components at the average level of

log (total amount),











q q 1 1/2
(ii) E Bixi(log A) and S B ij(xix ) (log A)
i=1 i respectively represent the logarithmic effects of

total amount on the linear and nonlinear blending

properties of the mixture components.



All of the techniques discussed in Section 3.3.,

Appendix A, and Appendix B for deriving or reducing the

Scheffe canonical polynomial mixture-amount 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 0 q 0 -1 q
xn = E + Z X + Z-i ixiA (3.11)
i=1 i= -i=1

is appropriate if the total amount has a linear effect on








the linear component blending properties but does not affect

the nonlinear blending properties.

If the total amount does not affect the blending prop-

erties but does affect the response, the appropriate models

for the two examples are


q 0 q 0 -1 1
n = S x. + E 8 ,x. + 88 A(.12)
i=1 i=I -

and

q 0 q 0 1/2 1 A)
n = x. + E Z S (.x.x)12 + s(log A)' (3.15)
i=1 i
These models are reduced forms of (3.9) and (3.10),

respectively. However, the terms in (3.12) and (3.13) are

not subsets of the terms in (3.9) and (3.10). The Gorman

and Cornell (1982) reparametrization technique [reexpres-
q-1 1
sing x as 1 z x. in the terms 8 x A of (3.9) and
q i1 i=1
sqx (log A)' of (3.10)] discussed in Section 3.3 is

applicable here.

For each of (3.9) and (3.10), it is implicitly assumed

that the same mixture model is valid at each level of A.

For situations where this is not the case, the appropriate

parameter restrictions can be obtained as was done for the

Scheffe canonical polynomial mixture-amount models in

Appendix B. As an illustration, consider the situation

specified by the mixture models









"A =-l = ax1 + bx2

-1
nA' =' cx1 + dx2 + ex (3.14)

where the two levels of A are coded as -1 and +1 (A denotes

the coded version of the total amount variable A). The

appropriate mixture-amount model for this situation is of

the form


0 0 0 -1 1 xA' 1
n = 81x + + -1x1 1+ A + 82x2A

1 -1 '
+ 8 x1 A (3.15)


with as yet unknown parameter restrictions. Substituting

the data


X1 x2 A
1 0 -1 a
.05 .95 -1 .05a + .95b
.50 .50 -1 .50a + .50b
1 0 1 c+e
.05 .95 1 .05c + .95d + 20e
.50 .50 1 .50c + .50d +2e

into (3.15) and solving the resulting system of equations

yields the parameter estimates


0 a+c 0 b+d O e
81 a 2 2 = 2 -1 2

1 c-a 1 d-b 1 e (1
81 = 2 82 = 2 81 = (3.16)









The appropriate parameter restriction for this situation is

thus seen to be all = sI. If the inverse nonlinear

blending occurred at the low level of A instead of the high

level, the parameter restriction would be al8 = -O$1



3.5 Mixture-Amount Models--A 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








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 (3.2) or (3.3) rather than the linear form (3.1).

If the available data are from a fractional mixture-

amount experiment or are not from a mixture-amount

experiment at all, the above "natural" approach to model








selection may not be appropriate. In such situations, the

sequential "full vs. reduced" model procedure discussed at

the end of Section 3.3 is appropriate. The Gorman and

Cornell "reparametrization followed by variable selection"

technique discussed in Section 3.3 may also be of help in

such situations. The practical aspects of selecting a

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 A-1, 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-1, 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 xi 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.








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 effect total

amount has on these properties, the same mixture model is

usually considered at each level of total amount. Then, the

same mixture design (corresponding to the mixture model

under consideration) is constructed at each level of total

amount. However, in situations where it is known beforehand

that components blend differently at different amounts, or

where additional investigation into the component blending

at one amount is desired, one can choose to run different

mixture designs at different total amounts. Also,

fractionated designs (which we discuss in Section 4.2) can

be viewed as different mixture designs at each level of

total amount.

As an illustration of the design development process,

consider a q = 3, r = 3 unconstrained 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.








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,1 .4, .1 < x2 .3, and .35 x3 <

.75. An appropriate design for the special-cubic by linear

mixture-amount model


1 3 h h h ( h
n = z [ Z Bix. + zE 8 .ixix. + 8 x -x ](A ) (4.1)
h=O i=1 i
consists of the vertices, centroids of the longest edges,

and the overall centroid of the constraint region at each of



























Figure 4.1. Mixture-Amount Design Consisting of a Three
Component Simplex-Centroid Design at Each of
Three Amounts


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








the two levels of amount (based on the recommendation of

Snee 1975--see Section 2.2). The points of this design are

listed in Table 4.1 and are pictured in Figure 4.3.

Another important aspect of developing designs for

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 should be equally spaced on a log A or A-1 scale,

respectively. Regardless of the scale chosen, the equally

spaced levels should be spread far enough apart to yield

detectable differences in response as the level of A

changes.









Table 4.1.



Pt(a)

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22


Design Points for Fitting a Special-Cubic Model
in a Three-Component Constrained Mixture-Amount
Experiment at Two Levels of Amount


x1

.10
.10
.15
.40
.40
.35
.225
.10
.275
.40
.25
.10
.10
.15
.40
.40
.35
.225
.10
.275
.40
.25


.30
.15
.10
.10
.25
.30
.30
.225
.10
.175
.20
.30
.15
.10
.10
.25
.30
.30
.225
.10
.175
.20


_ 3
XL

.60
.75
.75
.50
.35
.35
.475
.675
.625
.425
.55
.60
.75
.75
.50
.35
.35
.475
.675
.625
.425
.55


A

-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
1
1
1
1
1
1
1


(a) These point numbers are used in Figure


4.3.














C

,-4



cl
0


C




*a
e-4



o
a,


0




c
o0



-4

1-


C)



cn






cli


0 C
CL
-4










0)
o4-









O4-

1-
00C










*ca)
03 &0






C C











< <
i i
(U (


'4








4.2 Fractionating Designs.for 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 2q-1 x 2 =

2q factorial design. Similarly, a 3q 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 2q and








3q designs are well known and many such fractional designs

have been tabled (e.g., see Cochran and Cox 1957).

Fractionation methods for the 2k3m series of factorial

designs are discussed briefly in Appendix G with respect to

mixture-amount-process variable experiments, but the

techniques are applicable here also.

The second fractionation method of Scheffe (1963),

discussed in Section 2.3, can be used to reduce the number

of points in 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


1 q h q h q h
S= C ix + Z xx + xx I+ 8ijkXiXjXk
h=O i=1 i
h h
++ x 2...qX1 ...x q](A )h( (4.2)


However, fractions of a 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.




x1

1/2
1/2
0
1
0
0
1/3


One-Half Fraction(a) of a Simplex-Centroid x 2
Mixture-Amount Design for Three Components


x2

1/2
0
1/2
0


1/3


x3

0
1/2
1/2
0
0
1
1/3


A

-1
-1
-1
1
1
1
1


(a) Fraction obtained using I = +ABCD as the defining
contrast. Switching the levels of A' yields the
I = -ABCD fraction.


A' = -L


A' = +1


Figure 4.4. Graphical Display of Design in Table 4.2








mixture model, or, the seven-term mixture-amount model


3 3 0 )
n = E Sx. + E Z B .x.x + 80A (4.3)
i=1 i
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


4 O 4 0 4 0
n E i .x. + E x x. + E E B xijk.x ix
i=1 i

+ 0 234x2xx4 (4.4)
1234xi 2 '


or the 15-term mixture-amount models









4 0 4 0
n = Si x + z Z 8 x.x.
i=1 i
4 0
+ EL 8ijkxixjxk + A (4.5)
i
or

4 4
n= Bx. + EE i. x x
i=1 i
4
+ 1 1 '
+ x 8Sx.A + {one 8 .x.x.A term} (4.6)
i=-1 J

These three models are appropriate under different

assumptions about component blending and how the total

amount affects the response, if at all. Model (4.6) is the

only one of the three that allows for the component blending

properties being affected by total amount, and does so at

the cost of assuming there is no special cubic or quartic

blending among the four components. Since the face

centroids (1/3,1/3,1/3,0), . (0,1/3,1/3,1/3) and the

overall centroid (1/4,1/4,1/4,1/4) are included in the

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








Table 4.3.


x1

1/2
1/2
1/2
0
0
0
1/4
1
0
0
0
1/3
1/3
1/3
0


One-Half Fraction(a) of a Simplex-Centroid x 2
Mixture-Amount Design for Four Components


x2


1/2
0
0
1/2
1/2
0
1/4
0
1
0
0
1/3
1/3
0
1/3


0
1/2
0
1/2
0
1/2
1/4
0
0
1
0
1/3
0
1/3
1/3


x


A


0
0
1/2
0
1/2
1/2
1/4
0
0
0
1
0
1/3
1/3
1/3


(a) Fraction obtained using I = +ABCDE
contrast. Switching the levels of
I = -ABCDE fraction.


A' = -1


as the defining
A' yields the


A' = +1


Figure 4.5. Graphical Display of Design in Table 4.3








vertex points and another edge centroid (see later in this

section for a discussion of such designs).

The above two examples illustrate that the second

fractionation method of Scheffe can be used to fractionate

the 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 DN-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




68


Section 1.1, provides a method for fractionating any

mixture-amount design for both unconstrained and constrained

mixture-amount experiments. Recall that 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 DN-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 DJ-

optimal design, it often does so; when it does not, the

resulting design is near DN-optimal.

We discuss the development of DN-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.


O The candidate points for the models (Al) (A5) in
Appendix A are (assuming an unconstrained mixture-
amount experiment) the simplex vertices at each








level of total amount. Since there are no
nonlinear blending terms in these models, the DN-
optimal design will not contain binary, ternary,
. etc. mixtures even if included in the
candidate list. For a constrained 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 DN-optimal (or near DN-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

DN-optimal designs for the models considered are given in

Appendix C. The results for q = 3 suggest procedures for

developing DN-optimal designs (without the need of a

computer program such as DETMAX) for unconstrained








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 positions--The possible geometric locations of the
design points regardless of the level of total
amount.


0 point--A specific candidate point chosen for the
design.


O full set--All 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 DN-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 DN-


optimal designs for any value of N 2 p.








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 = SOx. + E .x.x. + A
i=1 i

N

aq(q+1)
2





p+1 to C




C+1 to C+p


Procedure*

The smallest possible DN-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.








Table 4.5. Sequential DN-Optimal Design Development
Procedure for Model (A7) in Appendix A


Candidate Points
Simplex vertices and edge centroids at the two levels of A
(assumed coded as -1 and +1). There are C = q(q+1)
candidate points.


Model


q 0 q
n = Z .x. + E
i=1 i

N


2 + q





p+1 to C



C+1 to C+2p




C+2q+1 to C+p


0 q 1
Sijoxix + x 8ixiA


Procedure*

The smallest possible D -optimal design
for this model contains the vertices at
both levels of A and points which cover
the edge centroid positions once. The
edge centroids may be chosen at either
level of amount so long as each position
is covered.

Add the remaining edge centroid points
until a full set of candidate points is
obtained.

Add additional points to cover each vertex
position once, then twice (without repli-
cating among the additional points).
These points serve as second replicates of
the vertex positions at each level of A.

Add points to cover each edge centroid
position once. Note that an N = C+p
design is just a full set of candidate
points plus an N = p design. Hence, the
procedure cycles, continuing as above.


* See Figure C.2 in Appendix C for examples of designs
generated by this procedure for the case q = 3.








Table 4.6. Sequential 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

q q q0 q 1 q 1
1n = BZ x. + Z B ..x.x. + x.A + Z Z B .x.x.A
i=1 i

N


p = C


C+1 to 2C


Procedure*

The smallest possible DN-optimal design
for this model consists of a full set of
candidate points.

Add additional points until a second full
set is obtained. In choosing additional
points, it is not necessary to cover each
position once before covering a position
twice (once at each of the two levels of
A). However, points should not be
replicated within the additional points.
Note that the procedure cycles, continuing
as above.


* See Figure C.3 in Appendix C for examples of designs
generated by this procedure for the case q = 3.




74



Table 4.7. Sequential DN-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

q0 q0 q0 1
S= E 8 x + E ..x.x + EZ i kXiXX + 50A
i=1 i< i

N


p = C/2 + 1



p+1 to C




C+1 to C+p


Procedure*

The smallest possible DN-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 full set of candidate
points is obtained.

Add additional points to cover each
position once. Note that an N = C+p
design is just a full set plus an N = p
design. Hence, the procedure cycles,
continuing as above.


* See Figure C.4 in Appendix C for examples of designs
generated by this procedure for the case q = 3.








Table 4.8. Sequential 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

q 0q 0 q0 1q 1
n = x. + E x Z B .x.x. + E 8 x + E .x.A-
i=1 1 1 i

N

p = C/2 + q





p+1 to p+(2)

p+( )+1 to C


C+1 to C+2q


C+2q+1 to
C+2q+(q)

C+2q+(q)+1 to

C+p


Procedure*

The smallest possible DN-optimal design
for this model contains points chosen to
cover all positions once and the vertices
twice (once at each of the two levels of
A).

Add the remaining edge centroids.

Add the remaining face centroids until a
full set of candidate points is obtained.

Add additional points to cover the vertex
positions once, then twice.

Add additional points to cover the edge
centroid positions once.


Add additional points to cover the face
centroid positions once. Note that an
N = C+p design is just a full set of
candidate points plus an N = p design.
Hence, the procedure cycles, continuing
as above.


* See Figure C.5 in Appendix C for examples of designs
generated by this procedure for the case q = 3.








Table 4.9.


Sequential 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 q h q h h
n = 1 [ hx + E jxx. + E 8jk iXj x ]( )h
h=O i=1 i

N


p = C


C+1 to 2C


Procedure*

The smallest possible DN-optimal design
for this model consists of a full set of
candidate points.

Add additional points until a second full
set is obtained. In choosing additional
points, it is not necessary to cover each
position once before covering a position
twice (once at each of the two levels of
A). However, points should not be repli-
cated within the additional points. Note
that the procedure cycles, continuing as
above.


* See Figure C.6 in Appendix C for examples of designs
generated by this procedure for the case q = 3.








Table 4.10.


Sequential (Near)D1-Optimal Design Development
Procedure for Model (A12) in Appendix A


Candidate Points
Simplex vertices and edge centroids at the three levels of A
(assumed coded as -1, 0, and +1). There are C = 3q(q+1)/2
candidate points.


Model


q 0
n =r Z 8x
i=1


q 1 q 2 2
Si.x.A + x x(A )
i=1 i=1


N

S= q(q+) 2q




p+1 to p+(q)






p+(q)+1 to C


C+1 to C+3q


Procedure*

The smallest possible DN-optimal design
for this model contains the vertices at
all three levels of A and covers the edge
centroid positions once.


Add points to cover the edge centroid
positions twice. Two edge centroids at
each of the three levels of A gives the
smallest variances for parameter
estimates.


Add the remaining edge centroids to
complete a full set of candidate points.


Add vertex points until all vertices are
included again (replicated twice).
Slightly larger determinants are obtained
if one first covers the q vertex positions
once, then twice, and finally three times
(1ll vertices). Variances of the sO and
8 are smaller if the vertex points are
concentrated at A' = 0, while the
variances of the 8' are smaller if the
vertex points are concentrated at A' = -1
and A' = +1.


q B
+ ZE .x.x. +
i



78


Table 4.10.-continued.


N


C+3q+1 to C+p


Procedure*


Add points to cover the edge centroid
positions once. Note that an N = C+p
design is just a full set of candidate
points plus an N = p design. Hence, the
procedure cycles, continuing as above.


* See Figure C.7 in Appendix C for examples of designs
generated by this procedure for the case q = 3.




79


Table 4.11.


Sequential (Near)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 0
n = s.x. + s s
i=1 i i
q i
+ s ixi .xj.A
i

N

p = q2 + 2q






p+1 to C



C+1 to C+2q





(3q2+7q+2)/2 to
6q+4(q)-1
2


0
8 .x.x. +
lijx3i


q 2
+ E 8
i=1 1


q 1
E 8 .A
i=1 1 1


xi(A' )2


Procedure*


The smallest possible DN-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.


Add edge centroids on the A' = 0 simplex
to obtain a full set of candidate points.


Add vertices on the A' = -1 and A' = +1
simplexes. Slightly larger determinants
are obtained if one first covers the q
vertex positions once, then twice.


Designs for N in this range are not easy
to describe. They are not sequentially
obtainable from the above designs as they
do not contain a full set of candidate
points. See Figure C.8 in Appendix C for
some examples when q = 3.




80


Table 4.11.-continued.


N


Procedure*


6q + 4(q)







6q+4(q)+1 to
C+p


This unique design consists of the
vertices at all three levels of A twice,
and the edge centroids on the A' = -1 and
A' = +1 simplexes twice. Note that it
does not contain a full set of candidate
points.


Add edge centroids on the A' = 0 simplex.
Note that an N = C+p design is just a full
set of candidate points plus an N = p
design. Hence, the procedure cycles,
continuing as above.


* See Figure C.8 in Appendix C for examples of designs
generated by this procedure for the case q = 3.








Table 4.12. Sequential 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 0 q0 q 1 q 1
I = E X. + ijx.x. + Z i.xA + Z Z a xix.A
i=1 i

q 2 q 2 2
+ Six.(A ) + E B ijix (A )
i=1 i

N


p =C


C+1 to C+p


Procedure*

The smallest possible DN-optimal design
for this model consists of one full set of
candidate points.


Add additional points until a second full
set is obtained. It is not necessary to
cover all positions once (or twice) before
covering some positions twice (or three
times). Concentrating the new points on
the A' = -1 and A' = t1 simplexes lowers
the variances of the B while
concentrating points on the AO = 0 smplex
lowers the variances of the a and B.
For larger N, the procedure cycles,
continuing as above.


* See Figure C.9 in Appendix C for examples of designs
generated by this procedure for the case q = 3.








A potential criticism for the use of DN-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 xiA and xixjA terms

corresponding to the xi and xixj terms. The models in

Tables 4.4 and 4.5 are reduced forms of the model in Table

4.6 and do not have this "symmetry of terms" property. The
"nonsymmetry of terms" for the models in Tables 4.4 and 4.5

is why the corresponding procedures for developing DN-

optimal designs are not as straightforward as the procedure

of Table 4.6. The point of this discussion is that designs

developed by the more restrictive procedures in Tables 4.4

and 4.5 also satisfy the procedure of Table 4.6; that is,

the procedures of Tables 4.4 and 4.5 generate designs that

are not only DN-optimal for their corresponding models, but

are also DN-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 DN-optimal for the model of

Table 4.5 (and vice versa), although it may be possible to








construct such designs for certain values of N. As an

example, when q = 3 the designs for 9 N < 14 displayed in

the first design columns of Figures C.1 and C.2 are DN-

optimal for the models of both tables.

The above discussion uses the procedures and models of

Tables 4.4, 4.5, and 4.6 to illustrate that DN-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,


O 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 DN-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 DN-optimal 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 DN-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

be criterion based measures such as tr[(X'X)-1],

max[x'(X'X)-1x], or avg[x'(X'X)-lx], where the maximum or

average is computed over the candidate points. Examples of

how these characteristics and properties might be used to

choose among several DN-optimal designs are given in

Appendix D for some of the three-component designs from

Appendix C.

The DN-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


O 3 O 0 13
i = i i x + E E 8 x.x. + 8 x12x3 + E .x.A (4.7)
i=1 i
To summarize, in this section we have discussed several

techniques for fractionating mixture-amount designs for















4-

a





0
4-4







*M
.-4





a






















0
4-)












0
a,








o
0 *


C
E-



cO
02
Ct-











a E-

















0 *
0 4-3 s
Sa -l
br 0 () -r-



'-0 P *I
4-) C) Z C 01







C C .. 0 --
.P- E -4 C -
T [ Od >, rc,
TM aT


4-)


S0 a *0-
cu a)
4 XX C)

.0 ..C X 0




1 *= 1 T




E-4 4-4























tic
*-4
a 0


















C) (u ( /








CO N C -T








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.


88


1








5.1 Standard Designs and Polynomial Models Based
on the Component Amounts

Let us consider an experimental approach in which the

controllable variables are the amounts of the individual

components, denoted by ai, i=1,2, . ,q. Typically, an

experimenter wishing to model the response as a function of

the individual component amounts, would select either a

first or second-degree polynomial model of the form


q
n = a0 + Z aiai (5.1)
i=1l

q q 2 q
n = C0 + Z aa i+ E La ia + E aiaa (5.2)
i=1 i=-1 i
Similar models are obtained by substituting log ai (or other

functions of the ai) for the ai in (5.1) and (5.2). These

models, or those in (5.1) and (5.2), would be fitted to data

collected at the points of any standard response surface

design (as discussed in Section 1.1). Studies of this type

were performed by Hader et al. (1957), Moore et al. (1957),

Suich and Derringer (1977), and Valencia (1983).

An experiment conducted using the above standard

design, component amount model approach will be referred to

as a component amount (CA) experiment. A component amount

experiment is a type of general mixture experiment (as

defined in Section 1.3). To see this, first note that the

amount of an individual component (ai) may be written as the








product of the proportion of the component in the mixture

(xi) and the total amount of the mixture (A): ai = xiA,

i=1,2, . ,q. Although the response in a component

amount experiment is nominally assumed to be a function of

the component amount variables, the relationship ai = xiA

allows us to view the response as a function of the

component proportions and the total amount of the mixture,

i.e.,


n = f(al,a2,...,a )


= f(x1A,x2A,...,xqA)


= g(x ,x2,...,x ,A) (5.3)

Recalling the definition (in Section 1.3) of a general

mixture experiment as one in which the response is assumed

to be a function of the component proportions and possibly

the total amount of the mixture, (5.3) shows that a compo-

nent amount experiment is a type of general mixture experi-

ment.

It is of interest to note that the component amount

variables ai in a component amount experiment are mathemati-

cally independent. A type of experiment formulated in terms

of the component amounts where this is not the case is

discussed in the next section.




91


5.2 Models and Designs for Experiments Where the
Component Amounts Have a 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 ai. 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 (al,a2) = (9,0),

(6.75,3), (4.5,6), (2.25,9), and (0,12). The data collected

are to be used for fitting the model


n = a1 + 1 a2 + aa2 12ala2 (5.4)

which will in turn be used to determine the best combination

of sweeteners S1 and S2.

To see what makes this experimental approach different

from the mixture-amount and component amount approaches,








first consider the plot of the sweetener combinations given

in Figure 5.1. The combinations all lie on the line


a2 = (-4/3)al + 12 (5.5)


The company has chosen a desired sweetness level based on

the amounts al = 9 and a2 = 12, and the level is constant on

the line (5.5). They wish only to consider combinations of

the sweeteners along this line. The line (5.5) places a

restriction on the amounts a, and a2 of the two sweeteners

and thus they are not mathematically independent as is the

case with the component amount variables in the component

amount approach.

The restriction (5.5) on the component amount vari-

ables, rewritten as


a1/9 + a2/12 = 1 (5.6)

q
is reminiscent of the restriction Z x. = 1 in a mixture or
i=1 '
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.




University of Florida Home Page
© 2004 - 2010 University of Florida George A. Smathers Libraries.
All rights reserved.

Acceptable Use, Copyright, and Disclaimer Statement
Last updated October 10, 2010 - Version 2.9.9 - mvs