• TABLE OF CONTENTS
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 Front Cover
 Front Matter
 Table of Contents
 Introduction
 Mixture experiments
 Summary and conclusions
 Back Cover














Group Title: Bulletin - Agricultural Experiment Station, University of Florida ; 879 (technical)
Title: Models and designs for experiments with mixtures
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 Material Information
Title: Models and designs for experiments with mixtures
Series Title: Bulletin Agricultural Experiment Station, University of Florida
Physical Description: v. : ill. ; 23 cm.
Language: English
Creator: Cornell, John A., 1941-
Linda, Steve B
Publisher: Agricultural Experiment Station, Institute of Food and Agricultural Sciences, University of Florida
Place of Publication: Gainesville
Publication Date: 1991-
 Subjects
Subject: Mixtures -- Statistical methods   ( lcsh )
Mixtures -- Experiments   ( lcsh )
Genre: government publication (state, provincial, terriorial, dependent)   ( marcgt )
bibliography   ( marcgt )
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 Notes
Bibliography: Includes bibliographical references (Pt. I, p. 45).
Statement of Responsibility: John A. Cornell and Steve B. Linda.
General Note: Cover title.
Funding: Bulletin (University of Florida. Agricultural Experiment Station) ;
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Table of Contents
    Front Cover
        Front Cover
    Front Matter
        Front Matter
    Table of Contents
        Table of Contents
    Introduction
        Page 1
    Mixture experiments
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Some remarks about modeling a response surface
            Page 6
            Page 7
        The simplex-lattice designs for exploring the whole simplex region
            Page 8
            Page 9
        The canonical form of mixture polynomials
            Page 10
            Page 11
            Page 12
            Page 13
        Example no. 1: the Vendex-Kelthane pesticide experiment
            Page 14
            Page 15
            Page 16
        Example no. 2: fruit punch experiment
            Page 17
            Page 18
            Page 19
            Page 20
            Page 21
            Page 22
            Page 23
            Page 24
            Page 25
            Page 26
            Page 27
            Page 28
            Page 29
            Page 30
            Page 31
        Example no. 3: adding sweeteners to a popular sports drink
            Page 32
            Page 33
            Page 34
            Page 35
            Page 36
            Page 37
            Page 38
            Page 39
            Page 40
            Page 41
    Summary and conclusions
        Page 42
        Page 43
        Page 44
        Page 45
        Page 46
        Page 47
        Page 48
        Page 49
    Back Cover
        Back Cover
Full Text



March 1991


Models and Designs for
Experiments With Mixtures
Part I: Exploring the Whole
Simplex Region

John A. Cornell and Steve B. Linda


Agricultural Experimert Station CE;.
Institute of Food and Agricultural Sciences
University of Florida, Gainesville
J. M. Davidson, Dean [' o'

Ir


Bulletin 879 (Technical)








Models and Designs for

Experiments With Mixtures



John A. Cornell and Steve B. Linda



Authors


John A. Cornell, Professor, and Stephen B. Linda, Senior Statisti-
cian, Department of Statistics, Institute of Food and Agricultural
Sciences, University of Florida, Gainesville 32611







Part I: Exploring the whole simplex region.

This work is Part I of a two-part series that discusses mixture
experiments. In Part I, we introduce and define a mixture experiment
and present some of the more frequently used statistical designs
and models. Included also are techniques used in analyzing mixture
data where the analysis is performed using SAS version 6.03. Data
from three separate mixture experiments provide the setting for the
different analyses.
In Part II, the discussion centers on placing additional constraints
in the form of lower and upper bounds, 0 < Li -_ xi Ui < 1, on the
component proportions. The effect of the additional constraints forces
the experimental region to become a subregion of the simplex-shaped
region in Part I. Constructing designs and fitting equations to data
collected from the subregion (or constrained region) both are more
complicated than when the region is a simplex. Several numerical
examples are provided in order to illustrate the strategy behind
constructing designs and the interpretation of fitted models in highly
constrained mixture problems.








Contents

Introduction ............................................ 1
Mixture Experiments ................ .................. 1
Some Remarks about Modeling a Response Surface ......... 6
The Simplex-Lattice Designs for Exploring Whole
Simplex Regions ..................................... 8
The Canonical Form of Mixture Polynomials .............. 10
Example No. 1: Vendex-Kelthane Pesticide Experiment ... 14
Example No. 2: Fruit Punch Experiment ............... 17
Example No. 3: Adding Sweeteners to a Popular Sport
Drink ............................... 32
Summary and Conclusions ............................. 42








Introduction


Experiments are carried out in order to answer questions. A statis-
tically designed experiment is a collection of experimental trials that
are patterned in such a way as to provide clear answers to unambigu-
ous questions. Statistically designed experiments (SDE) allow the
experimenter to measure the influence of one or more factors, or
experimental variables, on a measured response where response is
used here to mean the outcome of an experimental trial. SDE may
also help in eliminating or reducing the influence of unwanted vari-
ables and allow the estimation of the magnitude of experimental
error. A further advantage of statistically designed experiments is
that they require the experimenter to think about what it is that he
or she is trying to accomplish before resources are committed to
experimentation. Often this step alone is the major contributor to
the success of SDE's.

Mixture Experiments

Mixture experiments are performed in many areas of agricultural
research. In a mixture experiment two or more ingredients are mixed
or blended together to form an end product. Some examples are,
a) Blending chemicals to form pesticides or herbicides.
b) Mixing three types of saltwater fish to make sandwich patties.
c) Combining protein-rich feed supplements in diets for chicks.
d) Blending juices from watermelon, orange, and pineapple to
make a fruit punch.
In each of the mixtures above, one or more characteristics of the
end product is/are of interest to the experimenter. For example, in
a), one might be interested in the efficacy of the pesticide in killing
fire ants. In b), one is interested in the texture as well as the flavor
of the patties. In c), the characteristic or response of interest is
reduction in the fat gain in weight of the chicks during a three-month
period of time; and d), the fruitiness flavor of the punch. If the
experimentation can be controlled in terms of varying the ingredient
proportions so that these characteristics can be made to depend only
on the relative proportions (or percentages) of the ingredients present
in the blend and not on the total amount of the mixture, then we
have a mixture experiment.
A feature of a mixture experiment is that the variables controlled
by the experimenter represent proportionate amounts of the mixture
where the proportions are by volume, by weight, or by mole fraction.
When expressed as fractions of a mixture, the proportions sum to
unity. Thus, if the number of ingredients (or components) in the








Introduction


Experiments are carried out in order to answer questions. A statis-
tically designed experiment is a collection of experimental trials that
are patterned in such a way as to provide clear answers to unambigu-
ous questions. Statistically designed experiments (SDE) allow the
experimenter to measure the influence of one or more factors, or
experimental variables, on a measured response where response is
used here to mean the outcome of an experimental trial. SDE may
also help in eliminating or reducing the influence of unwanted vari-
ables and allow the estimation of the magnitude of experimental
error. A further advantage of statistically designed experiments is
that they require the experimenter to think about what it is that he
or she is trying to accomplish before resources are committed to
experimentation. Often this step alone is the major contributor to
the success of SDE's.

Mixture Experiments

Mixture experiments are performed in many areas of agricultural
research. In a mixture experiment two or more ingredients are mixed
or blended together to form an end product. Some examples are,
a) Blending chemicals to form pesticides or herbicides.
b) Mixing three types of saltwater fish to make sandwich patties.
c) Combining protein-rich feed supplements in diets for chicks.
d) Blending juices from watermelon, orange, and pineapple to
make a fruit punch.
In each of the mixtures above, one or more characteristics of the
end product is/are of interest to the experimenter. For example, in
a), one might be interested in the efficacy of the pesticide in killing
fire ants. In b), one is interested in the texture as well as the flavor
of the patties. In c), the characteristic or response of interest is
reduction in the fat gain in weight of the chicks during a three-month
period of time; and d), the fruitiness flavor of the punch. If the
experimentation can be controlled in terms of varying the ingredient
proportions so that these characteristics can be made to depend only
on the relative proportions (or percentages) of the ingredients present
in the blend and not on the total amount of the mixture, then we
have a mixture experiment.
A feature of a mixture experiment is that the variables controlled
by the experimenter represent proportionate amounts of the mixture
where the proportions are by volume, by weight, or by mole fraction.
When expressed as fractions of a mixture, the proportions sum to
unity. Thus, if the number of ingredients (or components) in the








system under investigation is denoted by q and if the proportion of
the ith component in the mixture is represented by xi, then
x 0, i = 1, 2, ..., q (1)
and
q
2 xi = X1 + x2 + ... + xq = 1.0 (2)
i=l
As an example, in formulating the fruit punch, we might let water-
melon comprise 60% of the punch (orx1 = 0.60), orange juice comprise
15% of the punch (or x2 = 0.15), and pineapple 25% of the punch (or
x3 = 0.25). Note that while the xi could represent nonnegative percen-
tages of the mixtures or the fruit punch that was just mentioned,
when they are divided by 100%, they would be fractions as in Equ-
ation (2).
The following is an example of a simple mixture experiment. Two
chemicals, Vendex and Kelthane, are to be mixed with three other
ingredients in a 30%:70% ratio of chemical to the other ingredients.
The purpose is to develop a liquid pesticide to be used for killing
mites on strawberry plants. Each chemical can be mixed alone with
the other ingredients and any combination of the two chemicals can
be mixed with the other ingredients. Tolerance to the pesticide by
the mites after a period of time is of concern to the experimenter.
It is suspected that if the two chemicals are blended together along
with the other ingredients, the efficacy of the pesticide will be longer
lasting than if each chemical appears by itself in the pesticide and
the single-chemical pesticides are applied sequentially.
An experiment is set up consisting of five different combinations
of the two chemicals mixed with the other ingredients to form five
different pesticide blends. [This particular design will be discussed
in the next section.] Each of the five pesticide blends is to be sprayed
on a group of ten strawberry plants where each group of ten plants
was uniformly infested with mites prior to applying the pesticides.
Six replications of the five blends are planned in order to obtain an
estimate of the experimental error variation (group to group variabil-
ity within each particular blend). The same amount of pesticide is
sprayed on each group so as to remove any effect the amount may
have. The objective of the experiment is to find the blend of Vendex
and Kelthane that produced the highest percent mortality of the
mites after a specified period of time.
In terms of percentages of Vendex and Kelthane with the other
ingredients, one blend contains Vendex (V) only with the other ingre-
dients in a 30%:70% ratio while another blend contains only Kelthane
(K) with the other ingredients in a 30%:70% ratio. The single chem-








ical blends are called single-component mixtures. The remaining
three pesticide blends consist of 22.5%:7.5%:70% of V:K:other ingre-
dients, 15%:15%:70% of V:K:other ingredients, and 7.5%:22.5%:70%
of V:K:other ingredients. In the 70% of each blend that is contributed
by the other three ingredients, the relative proportions of the three
ingredients remain fixed. Thus we have only a two component mix-
ture experiment here because only the percentages of the two chem-
icals vary from one blend to the next. When expressed in terms of
the proportions of the two chemicals in each of the five pesticide
blends, the ratios of V:K are 1.0:0, 0:1.0, 0.75:0.25, 0.50:0.50 and
0.25:0.75, respectively. The average percent mortality, when taken
over 60 strawberry plants, of each of the five blends and the chemical
proportion of Vendex and Kelthane in each blend are presented in
Table 1, while a plot of the percent mortalities is drawn in Figure 1.

Tabel 1. Pesticide experimental data.

Vendex (V) Kelthane (K) Average Percent Mortality (%)
Blend % (x1) % (x2) Over the Six Replications

1. 30 1.00 0 0 67
2. 22.5 .75 7.5 .25 75
3. 15 .50 15 .50 79
4. 7.5 .25 22.5 .75 58
5. 0 0 30 1.00 35


According to the data values in Table 1, the observed average
mortality of mites from using Vendex alone (with the other ingre-
dients) is 67%, while when using Kelthane alone the average mortal-
ity is 35%. If the chemicals are strictly additive in their effectiveness,
the average mortality of any blend consisting of both V and K would
be the volumetric average of the single-chemical blend averages. For
example, if additive blending is present, the (0.50, 0.50) blend would
have an aveYage mortality that is calculated to be (67% x 0.50)
+ (35% x 0.50) = 51%. Similarly, the average mortality of the (0.25,
0.75) blend would be (67% x 0.25) + (35% x 0.75) = 44.5%. However,
since from Table 1 the average percent mortality of the (0.50, 0.50)
blend is 79% and this value exceeds the volumetric average of 51%,
the chemicals Vendex and Kelthane are said to be synergistic (ben-
eficial) in their blending with each other. If the average percent
mortality of the (0.50, 0.50) blend had been lower than 51%, then
the chemicals are said to be antagonistic in their blending with each
other. The average mortalities of the remaining blends, (0.75, 0.25)
and (0.25, 0.75), exceed their respective volumetric averages, 59%










90 -


75- 075--- 79
067
Average 60 058
Percent
Mortality
45 -

035
30 -


15 -

I I I I
(1.0,0) (.75,.25) (.50,.50) (.25,.75) (0,1.0)
Proportion of Chemical (Vendex, Kelthane: V,K)
Figure 1. Percent mortality of mites for each of the five chemical blends.

and 44.5%, respectively, further supporting the synergistic blending
properties of the two chemicals. We shall illustrate how to test for
synergism or antagonism between a pair of components later in the
example presented in the section on models for mixture experiments.
Let us return to Equations (1) and (2) which represent the principal
restrictions on the component proportions in mixture experiments.
As a result of the restrictions (1) and (2) on the values of xi, the
experimental region or composition space of interest is a regular
(q-1)-dimensional simplex. For q=2 components, the factor space
is a straight line, as shown in Figure 1, where the horizontal axis
lists the proportions associated with the two chemicals that make
up the five blends. For q=3, the factor space is an equilateral
triangle, and for four components the factor space is a tetrahedron,
see Figure 2. Note that since the proportions sum to unity as shown
in Equation (2), the xi are constrained variables and altering the
proportion of one component in a mixture will produce a change in
the proportions of at least one other component in the experimental
region.
The coordinate system for the values of the mixture component
proportions is a simplex coordinate system written as (x1, X2, ..., Xq).




xi=1


(a)
S1+x2+X3=




( 0,1,0)

V X2





(b)



2=1 4=



x3=1

Figure 2. The three-component and four-component simplex factor spaces.
(a) With three components, the simplex is an equilateral triangle.
(b) For four components, the simplex is a tetrahedron.

With three components for example, the vertices of the triangle
represent single-component mixtures xi = 1, xj = xk = 0 for i,j,k
= 1,2 and 3, i 4 j # k, and are denoted by (1,0,0), (0,1,0), and
(0,0,1). The interior points of the triangle represent mixtures in
which none of the components are absent, that is, x, > 0, x2 > 0
and x3 > 0. The centroid of the triangle corresponds to the mixture
with equal proportions (1/3, 1/3, 1/3) from each of the components.
Shown in Figure 3 is the three-component composition space where
the coordinates can be plotted on triangular graph paper which has
lines that are parallel to the three sides of the equilateral triangle.
In mixture experiments, the experimental data are defined on a
quantitative scale and are said to be the yield or some other physical
property of a product formed from the blend. The purpose of the
experimental program will be to model the blending surface with
some form of mathematical equation so that
a) predictions of the response for any mixture or combination of
the ingredients can be made empirically, or,










(1,0,0)
x =1




1 I 1 1
















x2 X 1
X" 1I I (0,0,1)
(0,1,0) (0,2,.'


Figure 3. Triangular coordinate graph paper for plotting three component
coordinates.

b) some measure of the influence on the response of each com-
ponent singly and in combination with the other components
(joint blending of the components) can be obtained.
We shall refer to these experimental objectives throughout this
bulletin.

Some Remarks About Modeling a Response Surface

A convenient way to evaluate the performance of a mathematical
equation in representing the mixture system is through the concept
of a response surface. Initially it is assumed that there exists some
functional relationship

=q = (X, x2, ..., Xq)
which defines the dependence of the response f, such as the mortality
of the mites, on the proportions x1, x2, ..., xq of the components. The
function 4 is a continuous function in the xi and ( is represented
usually by a first-degree polynomial








S= P1X1 + P2X2 + + PqXq, (3)
or a second-degree polynomial (say for q = 3).
1 = Pi1X1 + P2X2 + P3x3 + 12x1X2 + P13x1x3 + f23X2x3. (4)
In (3) and (4) the Pi's are unknown parameters in the models and
the xi's represent the component proportions. On some occasions a
third-degree polynomial (a cubic or reduced form of a cubic equation
with certain terms omitted from the complete cubic equation) may
be necessary to represent the surface.
In an experimental program consisting of N trials, the observed
value of the response in the uth trial, denoted by Yu, is assumed to
vary about a mean of q with common variance a2 for all u = 1,2,...,N.
The observed value contains additive experimental error E
Yu = 'q + Eu, 1 < u < N (5)
where the errors Eu are assumed to be uncorrelated and identically
distributed with zero mean and common variance cr2. As an example,
in the pesticide experiment that was discussed earlier, the average
mortality for the x = 0.50 and x2= 0.50 blend was observed to be
79%. Now the true mortality for the 50%:50% blend might be some
value in the range of 75% to 85% and for our experiments the differ-
ences between what was observed and the true but unknown r is
known as experimental error.
After the N observations are collected, the unknown parameters
or coefficients in the model are estimated by the method of least
squares. Once the coefficient estimates are calculated they are sub-
stituted into the model for use in predicting response values. To
illustrate, let us assume there are only two components in the system
and their proportions are denoted by x, and x2. Let the equation to
be fitted be written as the first-degree polynomial
Yu = PXl + P2x2 + .u (6)
[The absence of the parameter pg in Equation (6) is due to the
restriction that x, + x2 = 1. We shall discuss the derivation of the
form of Equation (6) in the section on statistical models.] With N >
2 observations collected on Yu, we can obtain the estimates bl, and
b2 of the parameters p1, and p2, respectively. If it is decided that
the estimates are not zero, then the unknown parameters in Equa-
tion (6) are replaced by their respective estimates to give the equa-
tion, called the fitted model,
Y(x) = blx1 + b2x2 (7)
where Y(x), read "Y hat", denotes the predicted or estimated value
of 9 for given values of x, and x2 that could be substituted in (7).








In the sections that follow, we shall discuss some statistical
methods that have proven to be very useful when performing mixture
experiments. These methods include (1) selecting the type of model
equation to be fitted to the experimental data, (2) choosing the design
program that defines which blends to run for the purpose of collecting
the data, and (3) using the appropriate techniques in the analysis
of such data. We briefly discuss these methods for experiments where
all combinations of the ingredients are possible as well as for exper-
iments in which only certain combinations are feasible. Some of the
expository papers written on methods for analyzing data from mix-
ture experiments are by Cornell (1973, 1979), Gorman and Hinman
(1962), Hare (1974), and Snee (1971, 1973, 1979). Mixture designs,
models, and techniques used in the analysis of data are discussed
in considerable detail in Cornell (1990).

The Simplex-Lattice Designs for Exploring
the Whole Simplex Region
For investigating the response surface over the entire simplex
region, a natural choice for a design would be one with points that
are positioned uniformly over the simplex factor space. Such a design
is the {q,m} simplex-lattice introduced by Schefft (1958) where the
points are defined by the following coordinate settings: the propor-
tions assumed by each component take the m + 1 equally spaced
values from 0 to 1,
xi = 0, _' .' 1 (8)
and all possible combinations (mixtures) of the components are con-
sidered, using the proportions in (8) for each component.
For a q = 3 component system, suppose each component is to be
set at the three proportions xi = 0, 1/2, and 1, for i= 1, 2, and 3,
which is the same as setting m = 2 in (8). The {q = 3, m = 2} simplex-
lattice consists of the six points on the boundary of the triangular
factor space (x1,,x 3) = (1,0,0), (0,1,0), (0,0,1), (1/2,2,0), (/2,0,/2),
(0,12,/2). The three vertices (1,0,0), (0,1,0) and (0,0,1) represent the
individual components while the points (/2,/2,0), (1/2,0,V2), and
(0,Y2,V2) represent the binary blends or two-component mixtures and
are located at the midpoints of the three sides of the triangle. The
{3,2}, {3,3}, and {4,2} simplex-lattices are shown in Figure 4.
An alternative arrangement to the {q,m} simplex-lattice is the
simplex-centroid design introduced by Scheff6 (1963). In a q-
component simplex-centroid design, the number of points is 2q-1.
The design points correspond to the q permutations of (1,0,0,...,0),
the (q) permutations of (V2,/2,0,0,...,0), the (q) permutations of
(1/3,1/3,1/3,0,...,0), ..., and the centroid point (1/q,1/q,...,1/q). A three-














(1,0,0)


, ,0, )


- (0,0,0,1)
(0,1,0) i ) (0,0,1) (0,1,0) (0, ( 1 (0,0,1)(0,1,0,0)
(,'2'2 3 3' 1 I o,0,,)
(0,, 2,0) (0,0,2
3,2 (3,3 2 (0,0,1,0)
3,2 ,33


Figure 4. The {3,2}, {3,3} and {4,2} simplex-lattice arrangements and the coordinate settings of the design points.








component simplex-centroid design is the {3,2} simplex-lattice with
an additional point at the centroid of the triangle. A 4-component
simplex-centroid design consisting of 24-1 = 15 points is the {4,2}
simplex-lattice augmented with points at the centroids of the four
faces and at the centroid of the tetrahedron itself. The centroid of
the tetrahedron has the coordinates (X1,X2, X3, 4) = (1/4, 1/4, 1/4, 1/4).
Besides experimental regions, mixture experiments also differ
from the ordinary regression problems in the form of the polynomial
model to be fitted. ScheffM (1958, 1963) introduced the canonical
polynomials for use with the simplex-lattices and simplex-centroid
designs.

The Canonical Form of Mixture Polynomials
The canonical form of the mixture polynomial is derived by apply-
ing the restriction x1 + x2 + ... + xq = 1 to the terms in the standard
polynomial and then simplifying. For example, with two components
whose proportions are denoted by x1 and x2, the standard first-
degree polynomial is written as
I = o + P31x + P2x2-
However, since x1 + x2 = 1, we can replace P0 by P0(xl + x2 = 1)
in n to get
S= (po + P1)X1 + (Po + P2)2
= Pxl + P2x2
so that the constant term, Po, is removed from the model as previously
displayed in equation (3). For the second-degree polynomial the quad-
ratic terms pp1x2 and P22x2 are also removed from the model along
with the constant term Bo, giving the form (4) for q = 3. Thus, the
mixture models have fewer terms than the standard polynominals.
Hereafter, we shall periodically refer to the canonical form of the
polynomial equations as Scheffl-type models.
In general, the canonical forms of the mixture models or the
Scheff6-type models (with the primes removed from the pi's) are:
q
Linear I = Pixi (9)
i=l
q q-lq
Quadratic 11 = Pixi + I Ipijxixj (10)
i=l i







Full cubic (11)
q q-1q q-1 q q-2q-lq
= i+ X iijXi+ ixx + ~yiXiXj(Xi-Xj) + I 2 ijk3XixjXk
i=1 i
Special cubic (12)
q q-1q q-2q-1q
'9 = 2 Pixi + E 23ijxixj + E I2 EPjkixjXk
i=1 i With q = 3 components for example, the quadratic and special-cubic
models respectively, are
T1 = P 1X + 22x2 + P3X3 + 1212 + P13x1x3 + 23x2x3
T1 = PX 2 + P + 3 + P12x1X2 + P13x1x3 + P23X2X3 + P123x1X2X3
Generally, the linear (9), quadratic (10), and full cubic models (11)
are associated with the {q,1}, { q,2}, and {q,3} simplex-lattices, respec-
tively. The special cubic equation is a reduced form of third-degree
polynomial that provides measures of the ternary blends of the three
components, i, j, and k. It represents the lowest form of polynomial
of degree higher than two and contains q(q2+5)/6 terms while the
full cubic model contains q(q+1)(q+2)/6 terms.
The canonical form of the polynomial in q components that is to
be fitted to data collected at the points of the simplex-centroid design
is
(13)
q q q
7 = P i + 2. ijXiXj + X Y2 kxiXjXk + ... + 12...qxx2...xq
i=1 i The model of Equation (13) contains 2q-1 terms and this number
corresponds exactly to the number of points in the simplex-centroid
design.
The terms in the canonical polynomial models have simple in-
terpretations. In Equation (10) for example, if xi = 1 and therefore
xj = 0 forj j i, then = pi, that is, Pi is the expected response to the
pure com onent i. When the blending is strictly additive the model
is = i = Pixi which is the equation of a planar surface. With a
quadratic model, the second-degree terms (the Pixixjy) describe quad-
ratic departure of the response surface from the planar surface.
Higher-order terms such as P123x1x2x3 and y xixj(xi xj) describe
additional departures in the shape of the response surface from that
of a plane beyond those described by the second-degree terms.








To illustrate the shape of a mixture surface that was generated
by the second-degree mixture model

I = Plxl + P2X2 + P3x3 + P12x1X2 + P13x1X3 + P23X2x3 (14)
let us refer to the picture in Figure 5 of a second-degree response
surface directly above the three-component triangle. At each of the
vertices of the triangle representing the three single-component mix-
tures, the three heights of the surface are denoted by p1, p2, and P3,
respectively. These heights are represented by the coefficients of the
terms, pixi, i= 1,2,3 in the second-degree model and by themselves
define the plane T = Plxl + P2x2 + 3x3. The remaining three
terms, P3xixxj, i = 1,2, i from the plane along the edges of the triangle. Shown in Figure 5
is the deviation, p12x1x2, of the surface from the plane along the
x1-x2 edge at the particular point x1=x2 1/2 in which case the
magnitude of the departure is P12/4.




n B x1+ 02x2+ 03x3








012











X2= 1
Figure 5. The departure p1/4 of the surface along the xl x2 edge of
the triangle from the plane defined by r = PfIx + P2X2
+ P3X3.








When data are collected at the points of the {q,m} simplex-lattice,
the formulas for the estimates of the coefficients in the canonical
polynomials are expressed as simple functions of the observed values
of the response collected at the points of the design. For instance,
suppose ni observations are taken on the single component i (xi =
1, xj = 0,jk i) and the average of the n; observations is denoted by
Yi. Further suppose that n, observations are taken on the 50%:50%
binary mixture (xi = 1/2,x- = 1/2,xk = 0 for all i i and j and the average of the ni observations is denoted by Yi.
Then the least squares formulas for calculating the coefficient esti-
mates bi and bj in the second-degree model of Equation (10), are
bi = Yi, i= 1,2,...,q
(15)
bij = 4Yj 2(Yi + Yj), ij= 1,2,...,q, i
Moreover, if with the simplex-centroid design nik observations are
taken on the x. = x. = xk = 1/3 blend and the average is denoted
by Yijk, then the formula for calculating the estimate of Pijk in (12) is
bijk = 27Yijk 12(Yij + ik + Yjk) + 3(Yi + Yj + Yk) (16)
Note that the scalar quantities 4 and 2 in the formula for b,. and
the quantities 27, 12 and 3 in the formula for bijk do not depend on
the values of ni, ni, and nik but rather come from the values ofxi,
xj and xk. As a reminder, the simple formulas in (15) and (16) occur
only when fitting the models (9), (10) and (12) to their respective
simplex-lattice and simplex-centroid arrangements. In cases where
additional blends are added to these designs, the estimation formulas
for the coefficients are more complicated. Formulas for estimating
the parameters in models up to degree four are presented in Chapter
2 of Cornell (1990).
We shall return now to the five-blend pesticide experiment that
was introduced previously where the two chemicals Vendex and
Kelthane were mixed with three other ingredients in a 30%:70%
ratio. Our purpose here is to illustrate the fitting of a Scheff&-type
second-degree model to data that was taken from the three points
of a {q = 2, m = 2} simplex-lattice. Then the data from all five blends
will be fitted. The objective of the experiment is to see if some or
all of the blends consisting of both Vendex and Kelthane are more
effective in killing mites than could be expected by applying Vendex
or Kelthane separately (with the other ingredients.).








Example No. 1: Vendex-Kelthane Pesticide Experiment
Let us recall the two-component pesticide experiment involving
the chemicals Vendex and Kelthane. The five blends that were used
in this experiment are listed in Table 1 and comprise the {2,2}
simplex-lattice (blends 1, 3, and 5) plus two extra blends (2 and 4).
[Actually the five blends comprise a {2,4} simplex-lattice.]
To illustrate the use of the formulas in (15) for calculating the
estimates of the coefficients in the second-degree model

Yu = PlXl + 2x2 + P121X2 + u
we shall first use only the data (average percentage mortality) from
the {2,2} simplex-lattice blends 1, 3, and 5. The data at the two extra
blends, blends 2 and 4, can be used to check the appropriateness of
the second-degree model that is fitted to the three blends 1, 3, and
5. One way of checking for model adequacy is by comparing the
observed average percent mortality values at the blends 2 and 4
with the average percent mortality values predicted at these blends
with the fitted model. If the differences between the observed and
predicted percent mortality values are large, then the data at blends
2 and 4 need to be fitted along with the data at blends 1, 3, and 5,
to obtain the final fitted model form or, a higher-degree model such
as a cubic model would have to be fitted to the data at the five blends.
Generally when data are available from more blends than there are
terms in the model, the complete set of data is fitted to provide the
final prediction equation.
Using the data from blends 1, 3, and 5 only and designating Vendex
to be component one and Kelthane to be component two, the esti-
mates of the coefficients p1, P2, and 012 in the model above are
and b1 = 67 (blend 1), b2 = 35 (blend 5),
and
b12 = 4 (blend 3) 2 (blend 1 + blend 5)
= 4(79) 2(67 + 35) = 316 204 = 112.
The fitted model obtained by substituting the estimates b1, b2, and
b12 into the equation above for the parameters P1, P2, and P12 respec-
tively, is A
Y(x) = 67x, + 35x2 + 112xlx2
or
Percent Mortality = 67V + 35K + 112VxK (17)
where in (17) we have replaced x, with V and x2 with K.
Predicted average percent mortality values at the five blends using
the fitted model (17) are







A
blend 1, Y(1.0,0) = 67(1.0) + 35(0) + 112(1.0)(0) = 67
A
blend 3, Y(0.5,0.5) = 67(0.5) + 35(0.5) + 112(0.5)(0.5) = 79
A
blend 5, Y(0,1.0) = 67(0) + 35(1.0) + 112(0)(1.0) = 35
blend 2, Y(0.75,0.25) = 80
blend 4, 9(0.25,0.75) = 64
Note that at blends 1, 3, and 5, the predicted percent mortality
values are equal to the observed percent mortality values. This will
always happen when the number of distinct terms in the fitted model
is equal to the number of distinct points in the design. At blends 2
and 4 on the other hand, the predicted values 80 and 64 are slightly
different from their respective observed percent mortality values,
75 and 58. The differences in the percent 80 75 = 5% at blend
2 and 64 58 = 6% at blend 4 are not large enough however to
cause us to be concerned that the fitted model is inadequate and to
suspect that a higher-than-second-degree model is required for this
data set.
Fitted to the complete set of five data values, the second-degree
model is
Percent Mortality = 66.6V + 34.2K + 99.4VxK. (18)
The values of the coefficient estimates in this latter fitted model (18)
are very close to the values of the estimates (particularly b1 and b2)
in (17) that were obtained using the data from the {2,2} lattice. This
means the data from the extra two blends (2 and 4 in Table 1) follows
pretty closely the quadratic nature of the estimated percent mortality
curve generated by the data from blends 1, 3, and 5. Shown in Figure
6 is a plot of equation (18) taken over the entire range of the data.
This simple two-component example could easily be expressed in
terms of the proportion of the single chemical Vendex (V) since it is
known that in all of the blends making up the chemical proportion
of the pesticide, K= 1-V. Thus, if one substitutes 1-Vfor K in (18),
the resulting fitted model becomes
Percent Mortality = 66.6V + 34.2(1-V) + 99.4V(1-V)
= 34.2 + 131.8V 99.4V2 (19)
Equation (19) is of the second degree in Vendex. It could also be
expressed as the second-degree equation, Y(K) = 66.6 + 67.0K
- 99.4K2, in Kelthane. A drawback to expressing the equation (18)
in terms of only V (or only K) is that the system we are studying is










904


75- ---


Average 60 -
Percent
Mortality
(%) 45-


30-


15-

I I I I I
(1,0) (.75,.25) (.50,.50) (.25,.75) (0,1)
Proportion of Chemical (Vendex, Kelthane: V,K)
Figure 6. The percent mortality curve, Y(V,K) = 66.6V + 34.2K
+ 99.4VxK, over the range of values of the five blends.
really a two-chemical mixture system and by expressing the model
in one chemical only the effect of the other chemical is obscured in
the expression.
We shall now present the analysis of a three-juice fruit punch
experiment. In this experiment, ten different blends of watermelon
juice, pineapple juice, and orange juice are to be evaluated by a
sensory panel in terms of overall general acceptance to see if
watermelon juice can be considered an acceptable ingredient in the
fruit punch. In each of the blends to be studied, watermelon juice is
required to comprise at least 30% of the punch. A Scheffe-type second-
degree model will be fitted in order to determine if the juices, when
blended together, produce higher general acceptance scores than
those particular punch blends that define the vertices of the 3-com-
ponent triangle.









Example No. 2: Fruit Punch Experiment (Huor et al., 1980)

Watermelon (xl), pineapple (x2), and orange (x3) juice concentrates
were used as primary ingredients of a fruit punch. In each blend,
watermelon juice was forced to make up at least 30% of the fruit
punch so that in place of pure pineapple juice or pure orange juice
or pineapple and orange juice combined, these blends were replaced
by 30% watermelon with 70% pineapple juice, 30% watermelon with
70% orange juice, and 30% watermelon with 35% pineapple and 35%
orange juice, respectively. The experimental design consisting of six
blends comprising the {3,2} simplex-lattice plus four blends inside
the triangle is shown in Figure 7.
The ingredient proportions and the average acceptance values
that were scored on a scale of 1 (extremely poorer than reference)
100% Watermelon
A %1=1
1. (1,0,0)


65% Watermelon
35% Orange
(1/2.0,1/2) /


65% Watermelon
35% Pineapple
\ (1/2.1/2,0)


4 10.


Watermelon


30% Watermelon
35% Pineapple
35% Orange
(0,1/2,1/2)


100% Pineapple 100% Orange

Figure 7. The ten blends of the three-component fruit punch experiment.


2= 1
(0,1,0)








to 9 (extremely better than reference) for three replications of each
blend are listed in Table 2. Each general acceptance value in Table
2 represents an average of eight panelists' scores. Blends 1 to 6
comprise the {3,2} simplex lattice design while blends 7 to 10 are
the interior blends.
Table 2. Fruit punch general acceptance ratings.

Watermelon Pineapple Orange General
Acceptance Average
Blend %W (X1) %P (x2) %0 (x3) (Yu) Y1

1. 100 1.0 0 0 0 0 4.3,4.7,4.8 4.60
2. 65 0.5 35 0.5 0 0 6.3,5.8,6.1 6.07
3. 30 0 70 1.0 0 0 6.5,6.2,6.3 6.33
4. 30 0 35 0.5 35 0.5 6.2,6.2,6.1 6.17
5. 30 0 0 0 70 1.0 6.9, 7.0, 7.4 7.10
6. 65 0.5 0 0 35 0.5 6.1, 6.5, 5.9 6.17
7. 54 0.34 23 0.33 23 0.33 6.0,5.8,6.4 6.07
8. 80 0.72 10 0.14 10 0.14 5.4,5.8,6.6 5.93
9. 40 0.14 40 0.57 20 0.29 5.7,5.0,5.6 5.43
10. 40 0.14 20 0.29 40 0.57 5.2,6.4,6.4 6.00


%W 30
z x 70 x2


%P %0
70 x3 70


To illustrate the calculations of the values of the estimates of the
coefficients in the Scheff6-type second-degree model

Yu = Pl 22 + + P33 +12X1X2 +P13x13 + P23x2x3 + u,
when the data are collected only from the {3,2} simplex-lattice design,
we shall use only the data for the blends 1 to 6 in Table 2. According
to Equation (15), the estimates are
bl = 4.60 (blend 1), b2= 6.33 (blend 3), b3= 7.10 (blend 5)
b12 = 4(blend 2) 2(blend 1 + blend 3)
= 4(6.07) 2(4.60 + 6.33) = 24.28 21.86 = 2.42
b13 = 4(blend 6) 2(blend 1 + blend 5)
= 4(6.17) 2(4.60 + 7.10) = 24.68 23.40 = 1.28
b23 = 4(blend 4) 2 (blend 3 + blend 5)
= 4(6.17) 2(6.33 + 7.10) = 24.68 26.86 = -2.18.
Substituting the estimates into the second-degree equation, the fitted
model for the data from blends 1 to 6 is









(20)
Y(x) = 4.60xI + 6.33x2 + 7.10x3 + 2.42XlX2 + 1.28xlx3 2.18x2x3.
The values of the coefficient estimates in (20) were calculated above
using the averages of the three replicates of each blend and thus
reflect a slight amount of roundoff error. If the 18 individual values
from blends 1 to 6 are used in the calculations, the estimates of the
binary blending coefficients are b12 = 2.40, b13 = 1.27, and b23
= -2.20.
For this particular experiment, it was of interest to investigate
three-juice blends inside of the triangle as well as the blends at the
vertices and midpoints of the edges of the triangle. Blend 7 is the
three-juice blend at the centroid of the fruit-punch triangle while
blends 8, 9, and 10 are positioned between the centroid and the three
vertices of the triangle (see Figure 7). These four locations along
with blend 4 are fruit punch compositions consisting of all three
juice concentrates so that with this experiment half of the blends
consist of single- and two-component mixtures and half of the blends
are complete mixtures.
Having illustrated the calculation formulas (15) for the estimates
of the coefficients in the second-degree model when data are collected
only at the six points of the {3,2} simplex-lattice, let us now fit the
second-degree model to the entire set of 30 acceptance values that
are listed in Table 2. By fitting the model to the entire set of data
from the ten blends, the resulting fitted equation will reflect the
shape of the surface over the entire triangle better than equation
(20) does where the latter was fitted only to data collected from the
boundaries (the vertices and midpoint of the edges) of the triangle.
The second-degree model fitted to the entire set of 30 acceptance
values in Table 2 is
(21)
Y(x) = 4.77x1 + 6.27x2 + 7.11x3 + 2.15x1x2 + 1.10x1x3 3.54x2x3.
(0.24) (0.25) (0.25) (1.13) (1.13) (1.02)
The numbers in parentheses directly below the coefficient estimates
are the estimated standard errors (positive square root of the vari-
ance) of the coefficient estimates. These standard error values were
taken from the computer printout of the analysis. We shall discuss
the printout shortly.
The coefficient estimates in the fitted model (21) describe the type
of blending that occurred among the three juice concentrates in
terms of the acceptance ratings of the fruit punch blends. Before we
attempt to interpret the meanings of the coefficient estimates, let









us discuss the analysis of the complete set of 30 data values. The
programming statements for using SAS (1985) that provided the
analysis of the data, are listed in Appendix A.
In this experiment, ten different blends were each scored three
times by a panel of 8 to 9 persons resulting in a total of 30 observa-
tions. The persons serving on the panels scoring the general accept-
ance of the juice blends changed during the course of the experiment
so that the 30 general acceptance scores were assumed to reflect a
representative sample of the general public. The design thus was
viewed as being a completely random design so that the initial statis-
tical model for this completely random set of 30 acceptance scores is
Ylu= 1p + B + Elu u = 1,2,3; 1 = 1, 2,..., 10. (22)
In (22), YIu represents the acceptance score of the uth replication of
the lth blend, [i is the overall mean of the ten blends, B is the
difference between the true mean acceptance score of the lth blend
and the overall mean, and Elu is the random error associated with
the observed acceptance score for the uth replication of the Ith blend.
For purposes of testing hypotheses (that is, performing statistical
tests) we assume the Elu are independently distributed normal ran-
dom variables with a zero mean and variance a2.
The total variation in the 30 acceptance scores, when taken about
the overall average acceptance score, is
10 3
Corrected Total S.S. = 12 (Ylu )2
1=1u=1
10 3
where Y = Y E Ylu/30 = 179.6/30
=l1u=1 = 5.9867
10 3 2

10 3 1= 1 u=l
1 2 Y2_
2 lu ~------
1=lu=l 30
= 1089.04 1072.2053
= 13.8347.
The quantity 2 = 1Y u= 1089.04 used in calculating the Cor-
rected Total sum of squares is called the Uncorrected Total sum of
squares.
In an experiment such as this one, one of the first questions we
would ask ourselves is, "Are the average acceptance scores for the








ten different blends different from each other?" Estimates of the
average acceptance scores for the ten blends are listed in the right-
most column (YI) of Table 2. To answer this question, we compute
the variation among the means of the ten different blends as well
as the variation within the ten different blends and compare these
two variance quantities with a statistical test. The variation among
the ten blend averages as well as within the ten blends are computed
as
10
Among Blends S.S. = I nl(Y y)2
1=1
= 3[(4.60 5.99)2 + (6.07 5.99)2 + ... + (6.00
5.99)2]
= 11.0080 with 10 1 = 9 degrees of freedom
10 3
Within Blends S.S. = I y (YIu y)2
l=lu=l
= (4.3 4.60)2 + (4.7 4.60)2 + ... + (6.4
6.00)2
= 2.8267 with 20 degrees of freedom.
The sum of the Among Blends sum of squares and the Within Blends
sum of squares equals the Corrected Total sum of squares.
To test the hypothesis of the equality among the ten blends (or
their means), we set up the F-statistic

F Among Blends S.S./9
Within Blends S.S./20
11.0080/9
= 8.65
2.8267/20
and compare the calculated value of F against a tabled value of F.
At the 0.01 level of significance, the tabled F value is F(9,2o,0.o1)
= 3.46 and since 8.65 > 3.46, we reject the equality of the mean
acceptance scores of the ten blends and conclude that some blends
have a more desirable (higher) acceptance rating than do others.
Let us now return to the fitted model (21) and discuss the type of
blending that exists among the three juice concentrates.
Shown in Figures 8a and 8b are pictures or plots of the fruit punch
general acceptance surface that were generated using the G3D pro-
cedure in SAS/GRAPH, (see Cornell et al., 1983). The vertical heights








of the surface above the vertices of the triangle in Figures 8a and
8b represent the acceptance scores for the juice combinations 100%
watermelon (x1 = 1.0), 30% watermelon with 70% pineapple (x2
= 1.0), and 30% watermelon with 70% orange (x3 = 1.0), respec-
tively. These heights are represented by the coefficients b1 = 4.77,
b2 = 6.27, and b3 = 7.11 of the linear blending terms in the second-
degree model

A (23)
Y(x) = 4.77x1 + 6.27x2 + 7.11x3 + 2.15xlx2 + 1.10lOx1 3.54x2x3.
Since b3 > b2 > bl, the combination of 30% watermelon with 70%
orange juice appears to be more acceptable than the watermelon-
pineapple juice combination, which is more acceptable than the 100%
watermelon juice. .


YHAT

7.09




6.06




5.03



4.00


I -U."d =1 -0.80
-0.80
Figure 8a. Fruit punch experiment.
Y = 4.77xi + 6.27x2 + 7.11x3 + 2.15x1x2 + 1.10lX3 3.54x2x3.

The quadratic shape of the punch acceptance surface is a result of
the joint blending of pairs of juice concentrates. In Figure 8a along
the x, x2 edge of the triangle, the surface bends upwards (owing
to the positive sign of b12 = 2.15) indicating that as the proportion
of watermelon in the blend decreases down to approximately xI







YHAT


7.08 / tl













0.33

= -0.80


Figure 8b. Fruit punch experiment.
Y = 4.770x + 6.27x2 + 7.11x3 + 2.15xlX2 + 1.10xx3 3.54x2x3
4.00 5 /. /,' 0.23
0.900










the x1 x3 edge in Figure 8b, the surface bends slightly upwards
indicating slight synergistic (non-additive) blending between orange
juice and watermelon. Along the x2 x3 edge in Figure 8b, the
surface bends downwards reflecting the antagonistic (nonbeneficial)
blending effect of pineapple and orange when watermelon is fixed
at 30% of the blend. This latter antagonistic blending property be-
tween pineapple and orange is represented by the negative sign of
b23 = -3.54 in the second-degree fitted model (23).
The computer printout of the analysis of the 30 general acceptance
values is listed in Table 3. Several entries on the printout are worth
mentioning at this point. Specifically, since the second-degree model
that was fitted and is shown as Equation (23) does not contain a
constant term po, the NOINT (or NO INTercept) option was used
in the model statement which is the first model statement listed in
Appendix A. As a result, with this model form the degrees of freedom
(DF) and the Sum of Squares for the Model and Uncorrected Total
entries in Table 3 are not corrected for the overall mean of the 30
data values. Consequently, the DF for both the Model and Uncor-








Table 3. Fruit punch data. Computer output from analysis of the 30 general acceptance values. Model equivalent to equation
(23) fitted, with NOINT option used in model statement.

General Linear Models Procedure

Dependent Variable: Y
Sum of Mean
Source DF Squares Square F Value Pr> F

Model 6 1084.49825 180.74971 955.14 0.0001
Error 24 4.54175 0.18924

Uncorrected Total 30 1089.04000


R-Square C.V. Root MSE YMean

0.995830 7.266430 0.43502 5.9866667

Source DF Type I SS Mean Square F Value Pr> F

X1 1 480.255527 480.255527 2537.82 0.0001
X2 1 364.276843 364.276843 1924.95 0.0001
X3 1 236.866735 236.866735 1251.68 0.0001
X1X2 1 0.650804 0.650804 3.44 0.0760
X1X3 1 0.169700 0.169700 0.90 0.3531
X2X3 1 2.278635 2.278635 12.04 0.0020












Table 3. (Continued)

Source DF Type III SS Mean Square F Value Pr> F
Xl 1 76.171495 76.171495 402.51 0.0001
X2 1 121.609914 121.609914 642.62 0.0001
X3 1 156.473002 156.473002 826.85 0.0001
X1X2 1 0.679641 0.679641 3.59 0.0702
X1X3 1 0.177607 0.177607 0.94 0.3423
X2X3 1 2.278635 2.278635 12.04 0.0020
T for HO: Pr > TI Std Error of
Parameter Estimate Parameter = 0 Estimate
Xl 4.773601512 20.06 0.0001 0.23793379
X2 6.266368008 25.35 0.0001 0.24719381
X3 7.108060450 28.76 0.0001 0.24719381
X1X2 2.148058023 1.90 0.0702 1.13347590
X1X3 1.098086662 0.97 0.3423 1.13347590
X2X3 -3.536609727 -3.47 0.0020 1.01919177











Table 4. Fruit punch data. Computer output from analysis of the 30 general acceptance values. Model equivalent to equation
(25) fitted. Coefficients for x1, x2, x3 in a Scheff6-type model obtained by ESTIMATE statements.


General Linear Models Procedure
Dependent Variable: Y
Sum of Mean
Source DF Squares Square F Value Pr>F
Model 5 9.29291245 1.85858249 9.82 0.0001
Error 24 4.54175421 0.18923976
Corrected Total 29 13.83466667


R-Square C.V. Root MSE YMean
0.671712 7.266430 0.43502 5.9866667
Source DF Type I SS Mean Square FValue Pr> F
X1 1 5.11961353 5.11961353 27.05 0.0001
X2 1 1.07415889 1.07415889 5.68 0.0255
X1X2 1 0.65080425 0.65080425 3.44 0.0760
X1X3 1 0.16970035 0.16970035 0.90 0.3531
X2X3 1 2.27863544 2.27863544 12.04 0.0020









Table 4. (Continued)


Source

X1
X2
X1X2
X1X3
X2X3


Parameter


betal
beta2
beta3


Parameter


INTERCEPT
XI
X2
X1X2
X1X3
X2X3


Estimate
4.77360151
6.26636801
7.10806045


Estimate

7.108060450
-2.334458938
-0.841692441
2.148058023
1.098086662
-3.536609727


Tfor HO:
Parameter = 0

20.06
25.35
28.76
T for HO:
Parameter = 0


28.76
-6.85
-2.42
1.90
0.97
-3.47


Type IIISS
8.88687702
1.10482246
0.67964140
0.17760739
2.27863544


F Value


Mean Square
8.88687702
1.10482246
0.67964140
0.17760739
2.27863544


46.96
5.84
3.59
0.94
12.04


Pr>F
0.0001
0.0237
0.0702
0.3423
0.0020


Pr > ITI


0.0001
0.0001
0.0001

Pr> |TI


0.0001
0.0001
0.0237
0.0702
0.3423
0.0020


Std Error of
Estimate
0.23793379
0.24719381
0.24719381
Std Error of
Estimate

0.24719381
0.34065742
0.34834804
1.13347590
1.13347590
1.01919177








rected Total in Table 3 are inflated by 1 and the Sum of Squares for
the Model and Uncorrected Total are also inflated. The correct DF
and Sum of Squares for the Model and Corrected Total entries are
listed in Table 4.
Since the Sum of Squares entries for the Model and Uncorrected
Total are inflated in Table 3, so also are the values of the F test
(F Value = 955.14) and of R2 (R-Square = 0.995830). The correct
values of the F test and of R2 are listed in Table 4.
The printout for Table 4 was generated by deleting one of the
linear blending terms from the second-degree model and allowing
SAS to include a constant term (or intercept) in the fitted model.
For our example p3x3 was deleted. The deletion of one of the Pixi
terms in the model is necessary because SAS automatically inserts
a constant term (po) in the model unless one specifies NOINT in the
model statement and it is not possible to estimate the intercept po
and all of the linear coefficients pi, i = 1, 2, ..., q owing to the
restriction x1 + x2 + ... + Xq = 1. Consequently with the addition
of the constant term (or INTERCEPT) in the model and the deletion
of one of the linear terms, the DF and Sum of Squares for the Model
and Corrected Total entries in Table 4 are adjusted for the overall
mean.
The estimates of the coefficients in the second-degree model of
Equation (23) are listed at the bottom of Table 3. Listed also are the
estimated standard errors of the coefficient estimates and the values
of the corresponding t-test statistic
Estimate
t = (24)
Std. Error of Estimate
The level of significance of the t test in (24) is given under Pr > |TI
where we see from Table 3 that all three linear blending coefficient
estimates are significantly greater than zero and the nonlinear
blending ofx2 with x3 is significantly different from zero. Generally
when an indicated level of significance is less than 0.05 we declare
the coefficient estimate to be different from zero.
The model that was fitted to provide the printout in Table 4 is of
the form
Y(x) = bo + bx1 + b2x2 + bl2X1x2 + bl3x1X3 + b23x2x3 (25)
= 7.11 2.33x1 0.84x2 + 2.15x1x2 + 1.0lxx3 3.54x2x3.
(0.25) (0.34) (0.35) (1.13) (1.13) (1.02)
The quantities in parentheses below the coefficient estimates are
the estimated standard errors of the coefficient estimates. These
estimated standard errors are listed in Table 4 in the bottom right-
most column.








The difference between the fitted models of Equations (23) and
(25) exists only with the first three terms in the two models. In
Equation (25) above, the intercept bo is equal to b3 in Equation (23).
The b*,i = 1, 2 in (25) represent differences between the bi and b3
in Equation (23). More specifically,
in Eq. (25) Eq. (23)
b* = -2.33 equals b1 b3 = 4.77 7.11
b2 = -0.84 equals b b3 = 6.27 7.11.
Thus if one fits the model of Equation (25) in order to obtain the
correct analysis of variance table as shown at the top of Table 4 but
wishes to have a Scheff6-type model of the form shown in Equation
(23), it is easy to rewrite Equation (25) as
Y(x) = (b*+bo)x1 + (b2+bo)x2+ box3 + bl2x1x2 + bl3x1x3 + b23x2x3
= (-2.33+7.11)x1 + (-0.84+7.11)x2 + 7.11x3 + 2.15x1x2
+ 1.10x1X3 3.54x2x3
= 4.77x1 + 6.27x2 + 7.11x3 + 2.15xlx2 + 1.10lxx3 3.54x2x3
which is exactly the model of Equation (23). The estimates of the
coefficients for the first three terms (linear blending terms) are listed
in Table 4 in the PARAMETER column and are labeled betal, beta2,
and beta3, respectively.
Summarizing the results of the fruit punch data set, the fitted
models (23) and (25) and the tests of significance on the model co-
efficient estimates in Tables 3 and 4 tell us that
bl2 = 2.14, Pr > ITI = 0.0702
The average overall acceptance of the x1 = x2 = 1/2 blend
or the 65%W 35%P juice blend is almost significantly
(P = 0.07) higher than the average of the acceptance values
of the 100%W (xI = 1) and the 30%W 70%P juice (x2
= 1) blends.
b23 = -3.47, Pr > ITI = 0.0020
The average overall acceptance of the x2 = xg = 1/2 blend
or the 30%W 35%P 35%0 blend is significantly lower
than the average of the acceptance values of the 30%W
70%P (x2 = 1) and the 30%W 70%0 (x3 = 1) blends.
model (25)
b' = -2.33 (=b, b3) Pr > ITI = 0.0001
The difference b3 b1 is significantly different from zero
meaning the estimate b3 = 7.11 is significantly higher than








the estimate b1 = 4.77. This says that the average overall
acceptance of the 30%W 70%0 (x3 = 1) blend is signifi-
cantly higher than the average overall acceptance to 100%W
(xI = 1).
2 = -0.84 (=b2 b3) Pr > IT = 0.0237
The estimate b3 = 7.11 is significantly higher than the esti-
mate b2 = 6.27, or the average overall acceptance of the
30%W 70%0 (x3 = 1) blend is significantly higher than
the average overall acceptance of the 30%W 70%P (x2
= 1) blend.
The next course of action would be to plot the contours of the
estimated fruit punch surface in an attempt to determine the blend
of watermelon, pineapple, and orange juices that produced the high-
est average overall acceptance value. A contour plot of the estimated
fruit punch surface is presented in Figure 9. According to the contour
curves, the highest average overall acceptance score appears at the
x3 = 1 vertex of the triangle which corresponds to the 30% water-
melon 70% orange juice blend. Concluding this, some confirma-
tory experiments using such blends as 35%W 65%0, 30%W -
5%P 65%0, and 30%W 70%0 might be tried next. The program
statements for generating the contour plot in Figure 9 are listed in
Appendix B. x=1


5.3






5. 8 /
S\ / //
/ \.


/ / /




-/ ---- \ \
x2=l X3=l
Figure 9. Contour plot of the fruit punch overall acceptance surface.








We shall now present another three-component model fitting exer-
cise where in this example the shape of the surface will require the
fitting of a higher-than-second-degree equation. As with Example 2,
in Example 3 the purpose of the experiment is the approximation
of a response (flavor) surface so that the best blend of the three
ingredients can be determined by studying the contour plot and
three-dimensional graph of the estimated surface.








Example No. 3: Adding Sweeteners to a Popular Sport Drink
Three sweeteners, glycine, saccharin, and an enhancer were con-
sidered individually and in combination with one another as addi-
tives for sweetening a popular athletic drink. The amount of
sweetener was held fixed in all blends at 4% of the total volume.
Ten different combinations of the three sweeteners were selected
as the blends to be studied and the sweetener proportions are listed
in Table 5. The design used is a simplex-centroid design (blends 1-7)
augmented with three interior blends (8, 9, and 10). This experimen-
tal plan or design provides a good coverage of the triangle and a
nice balance in the types of mixtures to be studied in that it consists
of three single-component mixtures, three two-component mixtures,
and four three-component or complete mixtures. The data values,
also listed in Table 5, represent an intensity of aftertaste score for
each blend where each data point is an average of 50 replies from
a large consumer population. Each respondent was asked to rate
the intensity of aftertaste of the drink using a scale of 1 (positively
no aftertaste) to 30 (very extreme aftertaste or very sweet). A high
score (greater than 20) is considered undesirable, while a score in
the range of 5 to 10 is considered desirable.
Initially a second-degree model of the form (10) was fitted with
the hope that the intensity of aftertaste scores could be modeled with
such a simple equation. The fitted model is
(26)
Y(x) = 12.43x1 + 5.20x2 + 3.84x3 + 18.41x1X2 + 15.69xlx3
(4.81) (4.81) (4.81) (22.19) (22.19)
+ 13.23x2x3
(22.19)
and the quantities in parentheses are the estimated standard errors
of the coefficient estimates. Shown in Table 6 is the analysis of
variance (ANOVA) table for the fitted second-degree model
(27)
Y(x) = 3.84 + 8.59x1 + 1.36x2 + 18.41x1x2 + 15.69xlX3 + 13.23x2x3
which is equivalent in fit to model (26) but which contains an inter-
cept term (bo = 3.84). In (27) the P3X3 term in the second-degree
model (26) was deleted, exactly as was done with equations (23) and
(25) of Example 2, so as to produce the correct sums of squares
quantities in the ANOVA Table 6.












Table 5. Data from the artificial sweetener experiment.

Average
Intensity of
Percent of Total Drink (%) Component Proportions Aftertaste
Blend Glycine Saccharin Enhancer x = G/4% 2 = S/4% 3 = E/4% Score (Y)

1. 4 0 0 1 0 0 10
2. 0 4 0 0 1 0 6
o 3. 0 0 4 0 0 1 4
S4. 2 2 0 1/2 1/2 0 15
5. 2 0 2 1/2 0 1/2 13
6. 0 2 2 0 1/2 1/2 12
7. 4/3 4/3 4/3 1/3 1/3 1/3 8
8. 8/3 2/3 2/3 2/3 1/6 1/6 19.5
9. 2/3 8/3 2/3 1/6 2/3 1/6 6
10. 2/3 2/3 8/3 1/6 1/6 2/3 7











Table 6. Sweetener additives data. Computer output from analysis of the average intensity of aftertaste scores. Model of the
form of equation (27) fitted. Coefficients for xl, x2, x3 in a Scheff6-type model obtained by ESTIMATE statements.


General Linear Models Procedure

Dependent Variable: Y
Sum of Mean
Source DF Squares Square F Value Pr> F
Model 5 109.5528620 21.9105724 0.88 0.5653

Error 4 99.6721380 24.9180345

Corrected Total 9 209.2250000


R-Square C.V. Root MSE YMean

0.523613 49.66962 4.991797 10.0500000

Source DF Type ISS Mean Square F Value Pr> F

X1 1 69.44444445 69.44444445 2.79 0.1704
X2 1 2.08333333 2.08333333 0.08 0.7868
X1X2 1 16.83619580 16.83619580 0.68 0.4573
X1X3 1 12.32674961 12.32674961 0.49 0.5206
X2X3 1 8.86213877 8.86213877 0.36 0.5830










Table 6. (Continued)


Source

X1
X2
X1X2
X1X3
X2X3


Parameter


betal
beta2
beta3


Parameter

INTERCEPT
X1
X2
X1X2
X1X3
X2X3


DF Type IIISS

1 40.59204545
1 1.02272727
1 17.16205511
1 12.45486047
1 8.86213877


Estimate


Mean Square

40.59204545
1.02272727
17.16205511
12.45486047
8.86213877


TforHO:
Parameter = 0


12.43013468
5.20286195
3.83922559


Estimate


3.83922559
8.59090909
1.36363636
18.41414142
15.68686869
13.23232323


T for HO:
Parameter = 0


0.80
1.28
0.20
0.83
0.71
0.60


F Value

1.63
0.04
0.69
0.50
0.36

Pr > |T


0.0612
0.3406
0.4698

Pr > ITI


0.4698
0.2709
0.8493
0.4533
0.5186
0.5830


Pr> F

0.2709
0.8493
0.4533
0.5186
0.5830

Std Error of
Estimate

4.81426298
4.81426298
4.81426298

Std Error of
Estimate

4.81426298
6.73093733
6.73093733
22.18828417
22.18828417
.22.18828417








The fit of the models (26) or (27) is not particularly satisfactory
as indicated by the low value ofR2 = 0.5236 in Table 6 and the fact
that none of the crossproduct coefficient estimates are signifi-
cantly different from zero at the 0.05 level. Yet the taste score values
possess considerable variation and are for the most part higher in
value with the 2- and 3-sweetener blends than with the single-
sweetener blends. This suggests there is curvature in the taste score
surface. Hence the next step in the analysis is to fit the second-degree
model (26) with the extra term b123x1x2x3 added to it and see if the
special-cubic model improves the fit.
The fitted special-cubic model is
(28)
A
Y(x) = 12.05x1 + 4.82x2 + 3.46x3 + 29.75x1x2 + 27.03x3X3
(4.56) (4.56) (4.56) (22.97) (22.97)
+ 24.57x2x3- 183.71x1X2X3.
(22.97) (151.47)
This model possesses an R2 value of 0.6803 which still is quite low.
Furthermore, upon dividing each of the crossproduct coefficient
estimates by their respective estimated standard errors, the largest
quotient is t = 29.75/22.97 = 1.30 which clearly is not significantly
different from zero. Thus the special-cubic model (28) does not seem
to fit the data values close enough for us to feel comfortable in using
it to generate surface contour plots.
One technique that is used often to check the fit of a model is to
calculate the values of the residuals, Y Y(x), and look for clues
from the residual values that indicate why the fit is poor and possibly
to suggest ways in which the model form can be improved upon.
Also, since residuals are imitators of the random errors in the
observed values, studying residual patterns is helpful in checking
the constant variance assumption of the random errors. Generally
however, unless the number of observations is at least 30 (a rough
rule of thumb), it is usually difficult to deduce anything about the
constant variance property of the errors from studying the pattern
of the residual values. Nevertheless we shall look at the residual
values at the ten design points or blends used in this experiment to
see if there are any indications or clues that might suggest a different
model than (28) ought to be used in fitting our data.
Table 7 lists the residuals from fitting the special-cubic model
(28). Two residual values stand out as being larger than the rest.
These residuals are at the interior blends (x1, x2, X3) = (2/3, 1/6, 1/6)
and (1/6, 2/3, 1/6) and have the values 6.5 and -3.2, respectively.
This seems to indicate that the highest (19.5) and lowest (6) taste








score values inside the triangle are not being fitted as close as the
other taste score values are with the special-cubic model (28) and
an adjustment in the model form needs to be made in order to fit
these values closer.

Table 7. Residuals obtained from the fit of the special-cubic model (28).
A A
Blend Y(observed) Y(estimated) residual = Y-Y

1. 10 12.1 -2.1
2. 6 4.8 1.2
3. 4 3.5 0.5
4. 15 15.9 -0.9
5. 13 14.5 -1.5
6. 12 10.3 1.7
7. 8 9.0 -1.0
8. 19.5 13.0 6.5
9. 6 .9.2 -3.2
10. 7 8.3 -1.3


An adjustment to the special-cubic model is made by dropping the
P123x1x2x3 term and adding the three terms P1123x2x2x3, P1223x1x2x3,
and P1233xlX2x2. The resulting expression is called the special-
quartic model, Cornell (1986), and is of the form
(29)

1 = P1 + P2x2 + P3x3 + 12xlX2+ P13x1x3 + 23X2X3 + P1123xx2x3
+ P1223X1X2X3 + P1233x1x2x3.
The quartic terms P1123x"x2x3, 1223x1x2X3, and P1233xlx2x in (29)
provide extra support at the noncentroid interior blends (xi, x2, x3)
= (2/3, 1/6, 1/6), (1/6, 2/3, 1/6), and (1/6, 1/6, 2/3), respectively.
Fitted to the ten taste score values in Table 5, the special-quartic
model is
(30)
A
Y(x) = 10.11x1 + 6.11x2+4.11x3+28.45x1x2 + 24.45x1x3+28.45x2x3
(1.61) (1.61) (1.61) (7.88) (7.88) (7.88)
+ 584.29xYx2x3 693.71x1x2x3 441.71x1x2X3.
(165.27) (165.27) (165.27)









Table 8. Sweetener additives data. Computer output from analysis of the average intensity of aftertaste scores. Model equivalent
to equation (29), but with an intercept term fitted. Coefficients for xl, x2, x3 in a Scheff6-type model obtained by ESTIMATE
statements.


General Linear Models Procedure


Dependent Variable: Y


Source

Model
Error

Corrected Total


Sum of
F Squares

8 206.6318627
1 2.5931373
9 209.2250000


R-Square
0.987606


Source


X1
X2
X1X2
X1X3
X2X3
X1X1X2X3
X1X2X2X3
X1X2X3X3


C.V.
16.02311


Type I SS
69.44444445
2.08333333
16.83619580
12.32674961
8.86213877
0.20466009
78.35152498
18.52281570


F Value

9.96


F Value


Pr>F
0.2405


YMean

10.0500000
Pr> F


Mean
Square
25.8289828
2.5931373



RootMSE

1.610322

Mean Square
69.44444445
2.08333333
16.83619580
12.32674961
8.86213877
0.20466009
78.35152498
18.52281570


26.78
0.80
6.49
4.75
3.42
0.08
30.21
7.14


0.1215
0.5348
0.2381
0.2738
0.3157
0.8256
0.1146
0.2279





Table 8. (Continued)


Source

X1
X2
X1X2
X1X3
X2X3
X1X1X2X3
X1X2X2X3
X1X2X3X3


Parameter


Estimate

4.1127451
6.0000000
2.0000000
28.4509804
24.4509804
28.4509804
584.2941173
-693.7058821
-441.7058822


Parameter
INTERCEPT
X1
X2
X1X2
X1X3
X2X3
X1X1X2X3
X1X2X2X3
X1X2X3X3


DF Type IIISS

1 18.00000000
1 2.00000000
1 33.83801028
1 24.99210865
1 33.83801029
1 32.41183861
1 45.68686878
1 18.52281570


Estimate P

10.11274510
6.11274510
4.11274510


TforHO:
Parameter = 0
2.56
2.63
0.88
3.61
3.10
3.61
3.54
-4.20
-2.67


Mean Square

18.00000000
2.00000000
33.83801028
24.99210865
33.83801029
32.41183861
45.68686878
18.52281570

Tfor HO;
arameter = 0

6.30
3.81
2.56


betal
beta2
beta3


F Value


6.94
0.77
13.05
9.64
13.05
12.50
17.62
7.14


0.1003
0.1636
0.2371


0.2371
0.2309
0.5412
0.1719
0.1984
0.1719
0.1755
0.1489
0.2279


Pr> F

0.2309
0.5412
0.1719
0.1984
0.1719
0.1755
0.1489
0.2279
Std Error of
Estimate

1.60637038
1.60637038
1.60637038
Std Error of
Estimate

1.6063704
2.2773393
2.2773393
7.8760340
7.8760340
7.8760340
165.2693931
165.2693931
165.2693931








The computer-generated ANOVA table for the fitted model (30) is
provided in Table 8 and three-dimensional plots of the estimated
intensity of aftertaste surface, using the fitted model (30), are shown
in Figures 10a and 10b. The program statements for generating the
plots of the three-dimensional surface are listed in Appendix C.


Y_HAT

18.85



12 57



6.28



0.00 X









X2=


Figure 10a. Surface plot of the intensity of aftertaste surface.


The fit of the special-quartic model (30) is considerably better than
the fit of the special-cubic model (28) as evidenced by the value of
R2 = 0.9876 from Table 8 (compared to R2 = 0.6803 with the fitted
special-cubic model). The increase in the value ofR2 with model (30)
over that with model (28) is due partially to the fact that model (30)
contains two more terms than model (28). Nevertheless, the largest
residual value that was calculated using model (30) is 0.7, and this
value occurs at the three interior blends 8, 9, and 10. Thus, because
of the smaller values of the residuals we are confident that the model
(30) provides a better description of the kind of blending that takes








YIIAT


18.85



12.57



6.28



0.00













Figure 10b. Surface plot of the intensity of aftertaste surface.

place among the three sweeteners in explaining the variation in the
aftertaste scores than does model (28). The estimated intensity of
aftertaste score surface is depicted in Figures 10a and 10b.
Upon viewing the estimated intensity of aftertaste surfaces in
Figure 10, one notices the mound shaped surface at the ternary
blend (xl, x2, x3) = (2/3, 1/6, 1/6) or (G, S, E) = (8/3%, 2/3%, 2/3%)
which is emphasized by the term 584.29 x x2x3 in model (30). Com-
plete blends of the three sweeteners in the neighborhood of the top
of the mound exhibit antagonistic blending among the three sweeten-
ers that in turn produce average intensity values exceeding the
average of the intensity values of the three pure or single-
components which is (10.0 + 6.0 + 4.0)/3 = 6.67. Furthermore, the
depressions in the surface at the two blends (x1, x2, xa) = (1/6, 2/3,
1/6) or (G, S, E) = (2/3%, 8/3%, 2/3%) and (x1, x2, x3) = (1/6, 1/6,
2/3) or (G, S, E) = (2/3%, 2/3%, 8/3%), which are the results of the
terms -693.71xx22x3 and -441.7x1x2x2 in model (30), indicate that
there are three-sweetener blends that have lower average intensity
of aftertaste scores than many of the two-sweetener combinations.







This phenomenon is illustrated in the contour plot of the surface,
shown in Figure 11, where the shaded region includes those blends
with an average intensity of aftertaste score between 5 and 10.


xl 1


15- 1' .

p/ ,-'----:-~---'- 1----\,

12





-----------

X2= 1 X3 1
Figure 11. Contour plot of the intensity of aftertaste surface.

Summary and Conclusions

Mixture experiments are performed in many areas of agricultural
research. These experiments differ from standard factorial-type
experiments in that with mixture experiments the controllable vari-
ables represent proportions of the total mixture and therefore the
values of the xi are constrained (xi- 0, x1 + x2 + ... + X = 1). As
an example, suppose we have two types of saltwater fish (mullet
and sheepshead) and the fish are to be blended together to form
sandwich patties. If the amount of fish in each patty made is held








constant, then as the mullet proportion of the fish blend is increased,
the sheepshead proportion of the patty must decrease.
In a 2q factorial experiment on the other hand, each of the factors
has two levels (low and high, absence or presence, etc.) and the
experimental program generally consists of performing all combina-
tions (2q of them) of the low and high levels of the q factors. Once
again if we think of making the two-fish patties only this time we
choose to use a factorial arrangement where each fish type has a
low and a high level FiL and FiH, i = 1, 2, respectively, then the 22
factorial consists of the four combinations (FIL, F2L), (FL, F2H),
(FlH, F2L) and (FZH, F2H). Now the two extreme combinations (FiL,
F2L) and (FlH, F2H) will contain quite different amounts of fish in
the patties. Thus we must either change (increase) the size of the
patties with the (FlH, F2H) combination in order to keep the propor-
tion of fish in the patty constant, or we must relax the assumption
of requiring a constant proportion of fish be present in the patties.
In mixture experiments, since the component proportions are
constrained in value (that is, to lie between zero and one), the experi-
mental region or factor space of all possible compositions is a (q-1)-
dimensional simplex. With q = 2, the factor space is a straight line;
for q=3, the factor space is an equilateral triangle; and for q=4,
the factor space is a tetrahedron. In order to explore the entire
simplex region, special types of designs called simplex-lattice
arrangements are used. These designs consist of the individual com-
ponents, pairs of components, triplets, etc. With q= 3 for example,
the individual or single-component mixtures are the vertices of the
triangle. The two-component or binary blends are located on the
edges or sides of the triangle, and the three-component blends are
points in the interior of the triangle.
Besides the difference in experimental regions, mixture experi-
ments differ from the ordinary types of experiments in the form of
statistical model that is fitted. Mixture models are simpler in form
than the standard regression equations because mixture models
possess fewer numbers of terms. Yet mixture models are capable of
describing the type of blending (linear blending or nonlinear blend-
ing) that is present among the mixture components.
In this bulletin, three examples of mixture experiments are pre-
sented. They are
(i) blending two chemicals (Vendex and Kelthane) for killing
mites on strawberry plants,
(ii) blending juices from watermelon, pineapple, and orange to
form a fruit punch, and
(iii) adding sweeteners (glycine, saccharin, and an enhancer) to
a popular athletic drink.








In each of these experiments, a statistical model is fitted to the data
collected and the resulting model is used to generate pictures (in
the form of three-dimensional plots and contour curves) of the
response surface. The pictures are then used to elucidate information
about the type of blending that occurred in the experiment.
In this bulletin we have only touched on the tip of the mixture
experiment iceberg. In many mixture experiments, the component
proportions cannot range from zero to one because all of the ingre-
dients must be present to form a valid blend. This is accomplished
by placing constraints on the component proportions of the form
0 U< 1, i = 1, 2, ..., q (31)

where Li is a lower bound for xi and Ui is an upper bound for xi.
What the constraint in (31) means is that component i must be
present in all blends in at least a proportion Li and cannot be present
in a proportion greater than Ui in any blend. In the artificial
sweetener experiment for example, suppose we insist that among
the three sweeteners glycine, saccharin, and enhancer, that glycine
(x1) makes up at least 20% but not more than 80%, saccharin (x2)
comprises at least 10% but not more than 70%, and the enhancer
(x3) makes up least 5% but not more than 65%. Then in addition to
the standard constraints xi 0 and x1 + x2 + x3 = 1, we have the
following
0.20 < x, s 0.80
0.10 < x2 < 0.70 (32)
0.05 < x3 < 0.65.
The constraints (32) are called lower and upper bound constraints
and their presence will complicate the design and model fitting
strategies slightly in comparison to the techniques discussed in Part
I. We shall address the lower and upper bound constraints experi-
ment in Part II.








References


Cornell, J.A. (1973). Experiments with mixtures: A review.
Technometrics 15:437-455.
Cornell, J.A. (1979). Experiments with mixtures: An update and
bibliography. Technometrics 21:95-106.
Cornell, J.A. (1990). Experiments With Mixtures: Designs, Models,
and the Analysis of Mixture Data. 2nd Edition, John Wiley, New
York.
Cornell, J.A. (1986). A comparison between two ten-point designs
for studying three-component mixture systems. J. of Quality
Technology 18:1-15.
Cornell, J.A., J.T. Shelton, R. Lynch and G.F. Piepel (1983). Plotting
three-dimensional response surfaces for three-component mix-
tures or two-factor systems. Bulletin 836, Agricultural Expt. Stat.,
Inst. of Food and Agric. Sci., Univ. of Florida, Gainesville, FL.
Gorman, J.W. and J.E. Hinman (1962). Simplex-lattice designs for
multicomponent systems. Technometrics 4:463-487.
Hare, L.B. (1974). Mixture designs applied to food formulation. Food
Technology 28:50-62.
Huor, S.S., E.M. Ahmed, P.V. Rao, and J.A. Cornell (1980). Formu-
lation and sensory evaluation of a fruit punch containing water-
melon juice. J. of Food Sci. 45:809-813.
SAS Institute (1980). SAS/GRAPH User's Guide. Version 6.03. SAS
Institute, Raleigh, North Carolina.
Scheffe, H. (1958). Experiments with mixtures. J. Royal Statist.
Soc. B, 20:344-360.
Scheff6, H. (1963). The simplex-centroid design for experiments with
mixtures. J. Royal Statist. Soc. B, 25:235-263.
Snee, R.D. (1971). Design and analysis of mixture experiments.
J. of Quality Technology 3:159-169.
Snee, R.D. (1973). Techniques for the analysis of mixture data.
Technometrics 15:517-528.
Snee, R.D. (1979). Experimenting with mixtures. CHEMTECH
9:702-710.








Appendix A. SAS program statements used to produce Tables 3 and 4.
options s = 76 ps = 62 nodate nonumber;
data;
input point xl x2 x3 yl y2 y3;
array ys{3} yl-y3;
do i=1 to 3;
y=ys{i};
output;
end;
keep xl x2 x3 y;
cards;
1 10 0 4.3 4.7 4.8
2 .5 .5 0 6.3 5.8 6.1
3 0 10 6.5 6.2 6.3
4 0 .5 .5 6.2 6.2 6.1
5 0016.9 7 7.4
6 .5 0 .5 6.1 6.5 5.9
7 .34.33 .33 6 5.8 6.4
8 .72 .14 .14 5.4 5.8 6.6
9 .14.57 .29 5.7 5 5.6
10 .14.29 .57 5.2 6.4 6.4
run;
proc glm;
model y=xl x2 x3 xl*x2 xl*x3 x2*x3 / noint;
proc gim;
y= xl x2 xl*x2 xl*x3 x2*x3;
estimate 'betal' intercept 1 xl 1;
estimate 'beta2' intercept 1 x2 1;
estimate 'beta3' intercept 1;
run;








Appendix B. SAS/PC program statements used to produce Figure 9.
/*
The following DATA step creates points on the fitted response
surface.
*/
data;
do vl =-.6 to 1 by .02;
do v2 = -.8 to .9 by .02;
xl=(sqrt(6)*vl + 1)/3;
x2=(1 xl sqrt(2)*v2)/2;
x3=1 xl x2;
y_hat= 0;
if (0 le xl le 1) and (0 le x2 le 1) and (0 le x3 le 1) then do;
yhat=4.77*xl + 6.27*x2 + 7.11*x3 + 2.15*xl*x2
+ 1.1*xl*x3 3.54*x2*x3;
end;
output;
end;
end;
run;
/*
The following two statements cause the graphics output to be
written to the file 'c:Vmixtures\gsfs\fruit9.gsf', and specify the
device driver for a Hewlett-Packard Laserjet printer.
The binary file 'c:\mixtures\gsfs\fruit9.gsf', can be copied to the
device, i.e., if the device is the printer, then a hard copy of Figure
9 can be obtained by the following DOS command:
copy c:\mixtures\gsfs\fruit9.gsf prn /b
*/
filename grafout 'c.\mixtures\gsfs\fruit9.gsf';
options gsfname = grafout gsfmode = replace dev = hplj5p2
handshake = none nodisplay;
proc gcontour;
title 'Figure 9. Fruit punch experiment.';
plot vl*v2=y.hat / levels = 4.9 5.3 5.8 6 6.3 6.5 6.8 7.0;
run;








Appendix C. SAS/PC program statements used to produce Figures
10a and 10b.
/*
The following DATA step creates points on the fitted response
surface for Figure 10a.
*/
data one;
do vl= -.6 to 1 by .02;
do v2= -.8 to .9 by .02;
xl= (sqrt(6)*vl + 1)/3;
x2=(1 xl sqrt(2)*v2)/2;
x3=1 xl x2;
y-hat= 0;
if (0 le xl le 1) and (0 le x2 le 1) and (0 le x3 le 1) then do;
yhat = 10.11*xl + 6.11*x2 + 4.11*x3 + 28.45*xl*x2
+ 24.45*xl*x3 + 28.45*x2*x3
+ 584.29*xl*xl*x2*x3 693.71*xl*x2*x2*x3
441.71*xl*x2*x3*x3;
end;
output;
end;
end;
run;
/*
The following two statements cause the graphics output to be
written to the file 'c.\mixtures\gsfs\sweetl0a.gsf', and specify the
device driver for a Hewlett-Packard Laserjet printer.
The binary file 'c:\mixtures\gsfs\sweet10a.gsf' can be copied to
the device, i.e., if the device is the printer, then a hard copy of
Figure 10a can be obtained by the following DOS command;
copy c:\mixtures\gsfs\sweetl0a.gsf prn/ b
*/

filename grafout 'c:\mixtures\gsfs\sweetl0a.gsf';
options gsfname = grafout gsfmode = replace dev= hplj5p2
handshake = none nodisplay;
proc g3d data= one;
title 'Figure 10a. Artificial sweetener experiment.';
plot vl*v2 = y-hat / rotate = 45 tilt = 45;
run;








Appendix C (continued.). SAS/PC program statements used to
produce Figures 10a and 10b.
/*
The following DATA step creates points on the fitted response
surface for Figure 10b.
*/
data two;
do ul= -.6 to 1 by .02;
do u2= -.8 to .9 by .02;
x2=(sqrt(6)*ul + 1)/3;
x3= (1 x2 sqrt(2)*u2)/2;
xl=l x2 x3;
y-hat= 0;
if (0 le xl le 1) and (0 le x2 le 1) and (0 le x3 le 1) then do;
yhat = 10.11*xl + 6.11*x2 + 4.11*x3 + 28.45*xl*x2
+ 24.45*xl*x3 + 28.45*x2*x3
+ 584.29*xl*xl*x2*x3 693.71*xl*x2*x2*x3
441.71*xl*x2*x3*x3;
end;
output;
end;
end;
run;
/*
The following two statements cause the graphics output to be
written to the file 'c:\mixtures\gsfs\sweetl0b.gsf', and specify the
device driver for a Hewlett-Packard Laserjet printer.
*/
filename grafout 'c:\mixtures\gsfs\sweetl0b.gsf';
options gsfname = grafout gsfmode = replace dev = hplj5p2
handshake = none nodisplay;
proc g3d data=two;
title 'Figure 10b. Artificial sweetener experiment.';
plot ul*u2 = y.hat / rotate = 45 tilt = 35;
run;































































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