Biplot Displays for Looking at Multiple Response Data in Mixture Experiments

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Wendell F. SMITH, Jr. John A. CORNELL

Research Laboratories Department of Statistics-Agricultural Experiment Station

Eastman Kodak Company University of Florida

Rochester, NY 14650-2103 Gainesville, FL 3261 I-0560

In the analysis of mixture experiments it is not unusual to have many response variables (or physical characteristics of the end product) under investigation simultaneously. Generally these responses are correlated with one another, and the analyst must compromise by con- centrating on what he or she considers the most important characteristics, at the expense of others, in selecting the optimum formulation. Furthermore, in those situations in which some responses are positively correlated while others are negatively correlated, it can be very difficult to understand the relative importance of the different components. In such situations, a biplot display can help the analyst not only to understand the underlying structure of the data better but also to understand the roles played by the different components. Biplots are so named because both row (mixture formulation) and column (response) information are displayed in a single plot. This graphical technique has been applied successfully to mixture experiments in the photographic development area and in plastics formulation, as our examples will show.

KEY WORDS: Component effects; Cox directions; Singular value decomposition

1. INTRODUCTION

Mixture experiments consist of varying the proportions of two or more ingredients and studying the changes that occur in the measured response(s) that is(are) assumed to be functionally related to ingredient composition. The controllable variables are nonnegative proportionate amounts of the mixture in which the proportions are by volume, weight, or mole fraction. In a q-ingredient (or q-component) mixture, if xi represents the proportion of the ith component present in the mixture, then

OIXiI 1, i = 1,2, . . . , q, (1) and

glxi = l.

The composition or factor space of the q components, by virtue of the restrictions (1) and (2), takes the form of a (q - 1)-dimensional simplex, a straight line if q = 2, an equilateral triangle for q = 3, and a tetrahedron when q = 4. When additional constraints are imposed on the proportions in the form of lower and upper bounds

O<LiIXi51Ji<1, i = 1, 2, . . . ) q, (3) these additional constraints (3) alter the shape of the experimental region from that of a simplex to one of an irregularly shaped convex polyhedron inside the simplex.

Mixture model forms most commonly used in fitting mixture data are the canonical polynomials introduced by Scheffk (1958, 1963). The first-degree or linear blending model is

Y = i piXi + E. (4)

i=l

In (4) Y represents the observed value of the response, the coefficient pi is the expected response to component i (i.e., at xi = l), and E is the random error in the observed response value. Higher-degree models contain nonlinear blending terms of the type

Pijxixj) PijkXiXjXk Y and yijXiXj(x, - xi). In this article,

only linear blending models of the form (4) will be considered, however. A complete source devoted en- tirely to mixture designs and model forms for both simplex-shaped and polyhedral regions is the book by Cornell (1990).

Nearly all mixture experiments involve multiple (m 2 2) response variables Y,, Y,, . . . , Y, that are of interest simultaneously. Some examples are as follows:

1. Solubility Y,, tint strength Y,, porosity Y3, and sheen Y4 of a house paint consisting of pigment, ve- hicle, solvent, and additives

2. Viscosity YI, chemical durability Y,, and elec- trical conductivity Y3 of borosilicate glass, which is a solid, stable medium used for the disposal of high- level radioactive nuclear wastes and which comprises up to nine major oxide components (Piepel 1990)

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3. Tensile modulus Yl, tensile strength Y,, warp Y3, flexural modulus Y,, and flexural strength Y, of a plastic compound consisting of a mixture of resin, glass fibers, and microspheres (Bohl 1987)

Usually the measured responses are correlated with one another (some positively and some negatively) so that finding a feasible mixture blend for which the values of all the response variables or characteristics fall within their specific limits simultaneously is not an easy task. In general, when each response is studied separately, as has typically been the case in the literature, trade-offs have to be made by sacrificing the least important characteristics or responses while trying to optimize the most important characteristics.

We shall see how these trade-offs can be understood or visualized more easily through the use of biplots.

In addition to finding a feasible blend or a set of blends for the multiple responses, generally one as- pires to a better understanding of the overall mixture system by studying the effects of the components.

Until now, component effects have been discussed only for a single response, but we shall generalize the univariate approach to cover multiple responses.

The remainder of this article is organized as follows. In Section 2, we provide motivation for a biplot. To do this, a method for computing effects is reviewed and a component-effects plot and a biplot are compared for a photographic-film-coating experiment. In Section 3, a method for constructing a biplot for studying a set of mixture data is outlined.

In Section 4, a numerical example of a three- component plastics-compounding experiment in which the usefulness of a biplot is illustrated on five responses is presented. In Section 5, the article con- cludes with a discussion and summary.

2. A COMPONENT-EFFECTS PLOT AND A BIPLOT FOR A PHOTOGRAPHIC-FILM-

COATING EXPERIMENT

The concept of a component effect in a mdtiCOm-

ponent mixture system is quite simple. For a particular response Y, in a system in which the linear blending model (4) has been fitted and assessed as being satisfactory, the effect of component i is simply the difference in the estimated response for blends in which component i is at its high and low proportion settings, respectively-that is, P(xi,high) - Y(x,,r,,).

Piepel(1982) referred to this type of an effect as the total effect of component i. Thus to compute this difference requires only finding the coordinates of the blends x,,high and Xi,row and substituting these values into the fitted model.

The following three-component example is pro-

vided to illustrate the method used to compute the

total effects of three components whose proportions are denoted by x1, x2, and xj. By limiting ourselves to only three components, we are able to illustrate the meaning of an effect and of effects directions more easily than with four and more components. A constrained subregion for a three-component mixture system is shown in Figure 1. The subregion is defined by the constraints .28 : x1 I .60, .08 5 x2 5 52, .12 5 x3 % .48. Generally, for fitting the three- term linear blending model, data are collected at the six extreme vertices of the subregion and also at the centroid of the subregion, the latter point serving as a meaningful location for which to check the “lack of fit due to nonlinear blending of the components”

of the fitted linear blending model when a measure of the experimental error variance is available. In most applications, the coordinates of the centroid blend are found by averaging the coordinates of the extreme vertices or, alternatively, the centroid may be the center of mass; see Piepel (1983). In this article, the average of the coordinates of the extreme vertices is used to define the coordinates of the centroid.

Dashed lines that correspond to Cox’s (1971) directions (see also Cornell 1990, chap. 5) for components one, two, and three are also drawn in Figure 1. Each of the dashed lines is drawn from the centroid blend, whose coordinates are denoted by c = (c,, c2, c3) = (.4266, .2867, .2867), to a vertex of the triangle. The dashed line drawn from c to the xi = 1 vertex is the Cox direction for component i. The dashed lines are also extended from c away from each vertex to intersect a boundary of the subregion. The locations where the dashed lines meet the boundaries of the subregion are the high and low proportion blends for each component, respectively. The dashed lines in Figure 1 will be called COX directional rays.

The method for calculating the coordinates of the

Xi,high and xi,low blends was described by Cornell (1990,

p. 264), and by Piepel (1982). The result of applying this method to the present example leads to the blends given in Figure 1. When the fitted linear blending model Y(x) = b,xr + b2x2 + b,x, is used, the magnitudes of the estimated total effects of components one, two, and three, expressed as functions of the coefficient estimates 6, , b,, and b,, are

E, = Y(.60,.20,.20) - ?(.28,.36,.36)

= .32b, - .16b2 - .16b3

E2 = p(.29,.52,.19) - p(.55,.08,.37)

= - .26b, + .44b, - .lSb, E, = ?(.31,.21,.48) - Y(.53,.35,.12)

= .22b, - .14b, + .36b,, (5)

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=3

=2 high=

low =

c29,

/ / I \

I .

I

=2 I

I x3

x1 low = (28, .36, 36)

Figure 1. A Three-Component Constrained Region Defined by the Bounds 28 s x, 5 .60, .08 5 x2 5 .52, and. 12 I x3 5 The dashed lines going through the centroid blend c correspond to Cox directions. The mixture blends are at the high and proportions of the respective components on the Cox directional rays.

.48.

low

which, when expressed in matrix notation, simplifies to

E _3x1 = &x3&x1. (6) The elements in the matrix D, X 3 are the coefficients of the b, in (5) and b,, i = (b, , b2, b,)=, where T represents the transpose.

2.1 Component-Effects Plot

Effects can be presented graphically in an effects plot (Piepel 1982), which is also called a response trace (Cornell 1990, chap. 5; Hare 1985). The real utility of an effects plot is that it can be drawn for systems involving many components; see, for example, the work of Piepel (1982), who plotted the effects of 11 chemical components on leachability of glass used for waste vitrification. In effects plots, values of P(x) for those blends that lie on the Cox directional rays inside and on the boundaries of the experimental region (either a constrained subregion or the simplex itself) are plotted on the ordinate, whereas the abscissa displays the difference in the proportion of the ith component for each of the blends taken along the ith Cox directional ray relative to a reference mixture, where the latter is often chosen to be the centroid of the design region. Effects plots can be generated with any type of mixture-model form (linear or nonlinear), but, as we mentioned in Section 1, the effects plots and biplots discussed later in this article were generated using only linear blending terms in the mixture component proportions. An effects plot is illustrated in Figure 2 for a four-

component mixture experiment that was carried out for the purpose of optimizing a paper-based photographic coating. In this experiment, one of the responses of interest was the resistance to photochemical dye fading of the photographic dye; the effects plot in Figure 2 shows how the resistance to fading (or light stability) was affected when changes were made in the proportions of each of the components along the Cox directional rays.

Figure 3 displays the makeup of a paper-based photographic coating of the type that was used in this experiment. In this coating, the emulsion layer consists of an oil phase that has been dispersed in a mixture of water, gelatin, and silver halide. The oil phase comprises five components-a dye precursor (called a coupler), two coupler solvents (SOLVl and SOLV2), and two stabilizers (STAB1 and STAB2).

In the actual mixture experiment, the proportion of the coupler in the oil phase was held fixed in all blends, resulting in a four-component mixture system in which the proportions of the two coupler solvents and the two stabilizers were varied over a constrained subregion of the mixture space. The ultraviolet (UV) layer located above the emulsion layer attenuates the photochemical fading of the dye by short wavelength light. Two levels of UV absorber were used and thus

“UV” was considered a process variable. The effect of the UV layer is omitted in Figure 2 but will be included in the biplot of Figure 4, Section 2.2.

After exposure and processing, a colored image dye was formed from the coupler in the emulsion layer. Thirteen processed film samples were submit- ted to an accelerated light-fading test for a fixed

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- -I -

m i “‘\

“1.

SOLVP ,/”

,’ ,’

,’ I’

,’ ,’

,’ I’

.’ 8’

I’ ,’

,’ ,’

.’ I’

a I’

l- 0.0;

u-l b- 2

-‘* soL”,

-O.l- ,’ I’

-I ,’ ,’ “\

,’ ,’ “1

,’ .’ “\.-

,’ STAB2

/’ ,’

,’ I’

-0.2 ,’

I I I I I I I r

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4

CHANGE FROM REFERENCE MIXTURE

Figure 2. An Effects Plot of a Four-Component Mixture Experiment. The component effects are measured as departures in the estimated response at points taken on the rays relative to the estimate of the response at a reference mixture.

amount of time. Following the test, the percent of color image remaining was recorded. The percents were then standardized, with a positive value for the standardized response corresponding to a higher-than- average percent of color image remaining, and fit using a linear Scheffe mixture model with an additional term for the process variable. The effects plot in Figure 2 was constructed for the four mixture variables.

) UV absottkg layer

Dye precursor (coupler) Sohrents (2)

Stabilizers (2)

Figure 3. A Single-Layer Coating of a Photographic Paper.

The emulsion layer contains the mixture components.

For this response, some conjectures one could make from the plot are as follows:

1. Individually increasing the proportions of SOLV2 or STAB1 will cause the response to increase, but an increase in the proportions of SOLVl and STAB2 will cause the response to decrease. For this particular dye-fading response, an increase in the response value corresponds to an increase in light stability or a lesser degree of color fading.

2. The response is least sensitive to changes in composition resulting from a change in the proportion of STAB1 and, in fact, STAB1 may be thought of as having only a negligible effect on the response.

In the photographic experiment described here, light stability was only one of 14 responses being studied. In all there were four sensitometric responses (which influence color reproduction), four thermal yellowing responses, three photochemical yellowing responses, and three photochemical dye- fading responses. The four thermal responses differed in temperature and humidity, but the three photochemical responses differed in light intensity and composition of the light. To assess the effects of the mixture components on all of the responses, 14

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effects plots like the one in Figure 2 were required.

Trying to decipher the information contained in multiple effects plots motivated us to seek an alter- native method for displaying the component effects in multiple-response mixture experiments. Our so- lution to this dilemma is the biplot.

2.2 The Biplot

The biplot (Gabriel 1971, 1981) is a graphical display of a q x m data matrix by means of two- dimensional markers r-r, r2, . . . , r9 for its rows and two-dimensional markers cr , cz, . . . , c, for its columns. In the context of the photographic experiment, the data matrix would be 5 x 14; the five rows represent the effects of each of the five factors (four mixture components and one process variable) and the 14 columns, the multiple responses. At this point, in Figure 4 we present a biplot for the photographic experiment without explaining how it was constructed other than to say the linear blending model plus UV term was fitted to all 14 responses. Our

purpose in presenting the biplot before going into the details of how to construct one is to illustrate how useful the biplot can be for assessing graphically the effects of many factors on a set of multiple responses. In Section 3, a method for constructing an effects biplot is reviewed. Further details can be found in the articles by Gabriel (1971, 1981) and in Jack- son’s (1991) book.

In Figure 4 the row markers, rr, r2, . . . , r5 representing the five factors are denoted by asterisk symbols and the column markers, cr, c2, . . . , cl4 representing the 14 responses by vectors emanating from the origin. The arrow drawn from the SOLV2 row marker to one of the three photochemical dye- fading vectors will be explained. It is arbitrary which markers are represented by symbols and which ones by vectors (Jackson 1991). In this type of biplot, however, the magnitude of the correlation between any two responses is approximated by the cosine of the angle between the vectors, and therefore we suggest plotting the column markers as vectors. Without yet considering the row markers, we infer the fol-

1

tz 0 I- : IL

-1 -1

STAB2 *

Thermal yellowing (4)

SOLVl *

Photochemical

yellowing (3)

* uv

,STABl

Sensitometry (4)

0 FACTOR 2

1

Figure 4. A Plot of the Column Markers (response variables) in the Photographic Film Experiment. A projection is made from the SOL V2 component onto one of the photochemical dye-fading response vectors.

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lowing from the disposition of the column markers (or response vectors) in Figure 4:

1. Thermal yellowing responses and photochemical yellowing responses are highly correlated with one another. The correlation is positive, suggesting that if a treatment combination (mixture blend and/

or UV setting) improves thermal yellowing it will also improve photochemical yellowing.

2. The sensitometric responses are highly correlated with one another and are negatively correlated with the thermal and photochemical responses. When yellowing is improved, the sensitometric responses will degrade.

3. Photochemical dye-fading responses are correlated with one another and are nearly orthogonal to or uncorrelated with the yellowing and sensitometric responses.

To visually estimate the effect of a specific factor on a particular response, a perpendicular projection is made from the symbol representing the factor onto the response vector of interest, extended if necessary. The effect is then estimated mentally by taking the product of the distance from the origin to the point of intersection of the projection (hereafter called the d length of the projection) made onto the response vector and the length of the response vector projected upon. As an example, the row marker for SOLV2 has been projected onto the vector for one of the three photochemical dye-fading responses (compare the arrow in Fig. 4). This response is the same one used in the effects plot in Figure 2. The projection is made onto the positive side of the vector, indicating that SOLV2 affects this response in a positive manner. The d length of the projection is the distance from the origin to the tip of the arrow.

All of the responses in this experiment were signed such that more positive is better, and consequently an increase in the proportion of SOLV2 in the mixture improved or affected positively this particular photochemical dye-fading response. An improve- ment in dye fading means that there was less fading;

that is, the image was more stable to light. The projections for the other three mixture components and UV are not shown, but it is easy to see that increasing the proportion of STAB2 decreased dye stability and that STAB1 and SOLVl had little or no effect on this response. These conclusions are in agreement with the effects plot in Figure 2. As an aside, increasing the level of UV also improved photochemical dye stability.

The biplot in Figure 4 summarizes the effects of the mixture components and the UV variable on the separate groups of responses in this experiment. From Figure 4, we can infer the following:

1. The yellowing responses, both thermal and photochemical, are affected positively by STAB2.

Because this is in the desired direction, yellowing actually decreased. To a lesser extent, SOLVl and SOLV2 have undesirable effects on the yellowing responses-the responses were affected negatively (meaning yellowing increased) with increasing proportions of the solvents. Increasing the level of UV had a greater positive effect on the photochemical yellowing responses than on three of the four thermal yellowing responses, we think. The exception would be the rightmost thermal yellowing response in quadrant I (upper right quadrant). Because of the angle of this latter vector, the d length of the projection of UV onto it is only slightly shorter than the d length of the projection of UV onto the photochemical yellowing response vectors. This thermal yellowing vector is longer than the photochemical yellowing vectors, however, and therefore the product of the UV projection d length and the length of this vector is about the same as the products in the case of the photochemical yellowing responses. Note that the projections of UV onto the other three thermal yellowing responses, particularly the two in quadrant II (upper left quadrant), are shorter in A length than the UV projection onto the rightmost thermal yellowing response.

2. Photochemical dye fading is improved (dye stability increases) by UV and SOLV2. UV has a greater effect than SOLV2 on two of the three dye-fading responses. In the case of the lowermost response of the three dye-fading responses in quadrant IV (lower right quadrant), the effects of UV and SOLV2 are about the same because the d lengths of the projections onto this response vector are approximately equal. STAB2 has a deleterious effect on photochemical dye stability, because it projects onto the negative extensions of these vectors. Increasing the proportion of STAB2 has a particularly deleterious effect on the lowermost of the three dye-fading responses in quadrant IV. SOLVl and STAB1 have virtually no effect on the dye-fading responses.

3. The sensitometric responses are improved by SOLVl and SOLV2 and are significantly degraded by STAB2. UV and STAB1 have virtually no effect on these responses.

When these inferences were compared with the data in the original 5 X 14 effects matrix, all but one was in agreement with the data. Increasing the level of UV actually had a more positive effect on the photochemical yellowing responses than on all of the thermal yellowing responses and not just three out of the four as previously thought. The reason for our earlier misconception involving the rightmost thermal yellowing response is that the biplot is not,

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in most cases, a perfect representation of a data matrix (effects matrix in our case) but instead is only an approximate representation of the data matrix if the rank of the data matrix is greater than 2. Gabriel (1971) presented a measure of goodness of fit for a biplot (the formula is presented in Sec. 4) that gave a value of .925 when applied to the present example.

This means that (1 - .925) x 100% = 7.5% of the information in the original 5 x 14 effects matrix is unaccounted for by the biplot.

Summarizing, then, we see from the biplot that we have a fundamental trade-off in this system. Those factors that improve thermal and photochemical yellowing (more positive response) will degrade the sensitometric responses, and vice versa. In the absence of other factors, we would have to make a choice about which groups of responses to optimize. The photochemical dye-fading responses could be improved more or less independently of the other responses by increasing the level of UV.

Could we have arrived at the same conclusions by constructing separate effects plots, like that in Figure 2, for the 14 responses? It is possible but highly im- probable. The biplot in Figure 4 not only provides information about how each of the components affects each of the responses, but, just as importantly, it graphically displays the correlational structure among the response variables. A method for constructing a biplot of multiple effects in a mixture experiment is now outlined.

3. MATHEMATICAL DEVELOPMENT OF

BIPLOTS

For discussion purposes, the following notation will be used: Let x, represent the ith observed value of the jth response variable, i = 1, 2, . . . , II, j = 1, 2 . . 3 ^m, and let the number of mixture components and process variables combined be q. The start- ing point in the construction of a biplot is thus an n x m matrix of response variables, qj, which we shall standardize. The standardization removes the different metrics or units of scale of the response variables and is accomplished by defining

xj - yj zij =

v& (Kj - y,)?’ (7) where 7, = Z:=i x,/n so that zi, is the ith standardized value for the jth response. The matrix of standardized responses, Z, xm, is said to be in correlation form because the elements of the matrix Z’Z are the pairwise correlations among the m responses.

Let us write the linear blending model for the jth standardized response in matrix notation as

zj = x pi + E,’ j = 1, 2, . . . , m. (8)

In (S), Z, is the jth column of Z,,,, X is an n x q matrix in which the ith row contains the settings of the mixture components and process variables corresponding to the ith trial, pj is a q x 1 vector of linear blending coefficients and process variable effects, and E/ is an II x 1 vector of random errors.

By keeping the form of the linear blending model the same for all responses, a q x m matrix of un- weighted least squares estimates of the coefficients in the m separate models of (8) is

B qxm = [b,, b,, . . . , b,]

= (x=x)~~qxqTx.z,x,. (9) Furthermore, by generalizing Equation (6) to a multiple response setting, a q X m matrix of effects can be calculated from the q x m matrix of coefficient estimates

E qxm = DqxqBqxm. (10)

It is the matrix Eqxm that we now seek to biplot.

The biplot is a method of displaying a lower rank (usually rank 2) approximation of a data matrix (Ga- briel 1971, 1981). Because data matrices are usually of rank greater than 2 and we choose the display to be two-dimensional, the first step we shall take is to approximate the data matrix (hereafter, when we refer to the data matrix we shall be talking about the effects matrix, Eqxm) by a matrix E,*,,, where the rank of E,*,, is 2.

To define the matrix EGx m, first we write the matrix Eqxm as a product of three matrices. Let the q x m matrix of effects Eqxm be of rank 7, where 7 I min(q - 1, m). It is possible to write Eqxm, using the singular value decomposition (SVD) (e.g., see Good 1969; Jackson 1991, chap. 10; Mandel 1982), as

E qxm = PqxAx~Q7TXm, 01) where the columns of P are orthonormal eigenvectors of EE’ and the columns of Q are orthonormal eigenvectors of ETE; that is, P’P = QTQ = I,,,. The matrix A,, T is a diagonal matrix of ordered singular values 6, 2 & 2 . . . z e, > 0. These singular values are the positive square roots of the eigenvalues A,, A . . . > A, of the square matrix ETE as well as EET.

C?n a per-element basis, each element ejj of E, xm can be expressed as

7

eij = C ekPikqjk,

k=l (12)

where pik and qjk are the kth element in the ith row of P and the jth column of QT, respectively.

The rank-two matrix E,*,, , which approximates the matrix Eqxm, is now defined by letting r = 2 in

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Equation (12), as

= [P1,Pzl [$ ;J [$I. (13) In (13), pr and p2 are the first two (leftmost two) columns of P in (11) and qT and qT are the top two rows of QT in (11). Equation (13) is the best rank- two approximation of E, X m in the sense of minimiz- ing the Euclidean norm of the matrix of residuals

IIE qxm - JGnll. (14)

We are now ready to biplot the elements of the matrix EGX m. To do this, Gabriel (1981) and Jackson (1991) discussed two types of factorizations that are possible with E,*,, based on (13). Either factorization can be used for constructing a biplot, but, in this article, we shall use only one to construct a biplot and use the other to construct a different type of plot, called a j plot. The first, referred to as the GHT factorization, is E,*,, = GHT, where

G _9x2 H mx2

The second, referred E’ 9xm = JKT, where J 9x2 K _mx2

= [PI> P21

= [Qll, e2q21. (15) to as the JKT factorization, is

= [&PlT e2p21

= [Sl, q21. (16)

[There are, of course, many other factorizations possible for E*, depending on how the diagonal matrix in Eq. (13) is partitioned.] In the GHT factorization, the row markers rr, r2, . . . , r4 for the biplot are the q rows of Gqx2, and the column markers cl, c2, . . . ) c, are the m rows of H, x2. In the JKT factorization, the row markers for the biplot are the q rows of Jqx2, and the column markers are the m rows of Kmx2. The biplot of the photographic film- coating experiment, Figure 4, is a GHT biplot in which the row markers defined the factor symbols and the column markers represented the response vectors.

From the standpoint of reproducing the elements of E&n by taking inner products of vectors, it does not matter which factorization is used. In the case of the GHT factorization, each element ei*, of E,*,, is equal to the inner product g’hj, where gT is the ith row of G and hj is the jth column of HT. Similarly, in the JKT factorization, each element ec of EGx ,,, is equal to the inner product jTk,, where jf- is the ith row of J and kj is thejth column of KT. Furthermore, an inner product such as g,Th, (or j,Tk,) takes on a geometric meaning if it is written in the form

g:h, = h,Tgi = I$/ ) hjl cos ( fl,), (17)

where ) g,T] and ) h,/ are the lengths of the vectors and f3, is the angle between gr and hj. Because the inner product is a scalar and owing to the symmetry of the product of two vectors, any element e$ of E,*,, is reproduced by multiplying together the d length of the projection of either vector onto the other and the length of the vector projected upon. Recall our discussion in Section 2 for visually estimating the effect of a factor on a response when looking at the biplot of Figure 4. We said the effect was estimated by mentally multiplying the d length of the projection from the factor’s symbol and the length of the response vector projected upon. This mental exercise, although expressed in (17), is the method used when estimating inner products that are equal to the e,*l of E,*,, and estimate eij of E, xm.

So, which factorization do we use to construct the biplot? We use the GHT factorization in which the columns of G are orthonormal, and therefore fXxqGqx2 = ^I, (Gabriel 1971, 1981). As a conse- quence we have

E mxq ^*T E* qxm = Knxz GLqGqxd-G-xm

= HmxJJZTxm. (18)

When the columns of Eqxm are mean corrected, then HHT is n - 1 times the variance-covariance matrix for E,*,,, and the cosines between the ^hvectors (hi of HT) are equal to the correlations between the columns of E,*, m. If the rank-two approximation of E qxm W&n is good, then the correlations between the columns of E,*,, will closely approximate the correlations between the columns of E, xm. In the photographic film example of Section 2 and in the example discussed in Section 4, the responses and not the effects were mean corrected using Equation (7). As a result, the means of the columns of E, xm and E,*,,,, are approximately, although not exactly, equal to 0. The cosines between the h vectors therefore provide only approximate estimates of the correlations between the responses.

To estimate and plot the effects of the components on the multiple responses, we use the JK’ factorization. In the JKT factorization, the columns of K are orthonormal, and therefore KlxmKmx2 = I, (Gabriel 1971, 1981), so that

E,*x,EZ& = J,xKx9KqxzJ2Txq

= J,xzJ& (19) The q diagonal elements of E,*,,E~~, will approximate the q diagonal elements of EqxmEzxqr where the latter are equal to the squared lengths of the effects vectors for the mixture components and process variables. If we take the length of an effect vector to be a relative measure of the magnitude of the

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effect of that component and/or process variable on the multiple responses and call it a composite effect, then the lengths of the row markers in a JKT biplot approximate the absolute magnitudes of the composite effects. Moreover, since the composite effect of a component is the square root of the sum of squares of the separate response effects, where the separate effects could be negative or positive, the composite effect is nonnegative.

Rather than construct a JKT biplot, we shall construct a j plot in which only the row markers in J are plotted as vectors. [Our terminology “j plot” is bor- rowed from Gabriel (1981), who introduced the term

“h-plot” in the GHT factorization.] From the j plot, inferences can then be made about the relative directions as well as the approximate relative magnitudes of the composite effects. For the photographic- film-coating experiment discussed earlier in Section 2, Figure 5 is one such j plot. When all 14 responses are considered, it is clear from Figure 5 that among the five factors being studied STAB2 had the greatest effect on the responses and STAB1 had the smallest effect, but the effects of SOLVl, SOLV2, and UV

2.5

1.5

0.5 g L Q L -0.5

-1.5

-2.5

are approximately equal in magnitude and are between those of the stabilizers. Moreover, the composite effect of SOLVl is opposite in sign to the composite effect of STAB2, although the others are between these two.

The following example, taken from the literature, is presented to illustrate the usefulness of a GHT biplot and a j plot for a three-component plastics- compounding-mixture experiment in which five separate responses or characteristics of the plastic were measured.

4. A PLASTICS-COMPOUNDING

EXPERIMENT-BOHL’S (1987) RESIN DATA Virtually all plastics are complex formulations of polymer and additives, the latter added to improve key characteristics or properties of the plastic compound. In the following three-component experiment (q = 3), a virgin resin was studied alone and with each of two additives (glass fibers and microspheres). A lower bound of 80% of the mixture was set on the resin, allowing the two additives, individually and combined, to make up at most 20% of any

STABi

-2.5 - 1 . 5 - 0.5 0.5 ^1.5 2.5

FACTOR 2

Figure 5. A j Plot of the Composite-Effects Vectors for the Photographic-Film Experiment.

(10)

Figure 6.

The dashed

glass fibers (x 2)

glass fibers (x 2) microspheres (x .$ microspheres (x .$

The Experimental Region and Coordinate Settings of the Seven Blends in the Plastics-Compounding The Experimental Region and Coordinate Settings of the Seven Blends in the Plastics-Compounding lines correspond to Cox directions.

lines correspond to Cox directions.

mixture. These bounds, when expressed in terms of the component proportions x1 (resin), x2 (glass fibers), and xj (microspheres), are

.80 5 xl, 0 5 x2 5 .20, 0 5 x3 5 .20. (20) The triangular factor space of feasible mixture formulations is shown in Figure 6. The seven distinct mixture blends that were run in the experiment are also shown in Figure 6. The centroid blend was rep- licated twice, making n = 8.

Five properties of the plastic compound were measured-tensile strength Yr , tensile modulus Y,, flexural strength Y3, flexural modulus Y,, and warp Ys.

Experiment.

Table 1 lists the values of the properties given as percent of that of virgin resin alone along with the proportions of resin, glass fibers, and microspheres in each of the seven blends. The limiting values for the five responses that were considered necessary for the compound to meet specifications are also listed.

The first step in the analysis is to define the matrix of centered and standardized responses Z,, 5.

Defining

zij = (yj - yj) $I1 ^(Kj - ^Tj12]1'*~

j = 1, 2, 3, 4, 5,

Table 1. Plastics-Compounding Experimental Data

Component proportions Properties IV, z specification limit)

Glass Tensile Tensile Flexural Flexural

Resin fibers Microspheres strength modulus strength modulus Warp X? X2 X3 (Y, 2 120) (Y, 2 200) (Y, 2 140) IV, 2 2001 (Y, 5 100)

1.0 0 0 100 100 100 100 100

.9 .l 0 177 264 171 216 194

.9 0 .I 86 141 100 124 a3

.867 ,067 ,067 139 264 145 199 106

,867 ,067 .067 137 248 141 176 111

.a .2 0 217 521 211 421 372

.a .1 .I 148 366 153 286 161

.8 0 .2 79 213 86 163 67

(11)

Table 2. Coefficient Estimates of the Fitted Linear Blending Models in the Standardized Response Values

Coefficient estimate

Response

Tensile strength Tensile modulus Flexural strength Flexural modulus Warp

Resin (x,)

-.24 -.53 - .28 - .49 .27

Glass fibers (x,)

4.52 5.59 4.69 5.44 - 4.48

Microspheres (x,) RZ

- I .38 .saia

1.35 .9845

-1.10 .9807

.95 .9582

1.03 .8899

we have

The positive and negative signs in the W column of

Z 8x 5 have been reversed so that a high positive value (or low warp value) is preferred.

Calculating the component effects is the next order of business. First, however, we shall fit the linear blending model to each standardized response and check the adequacy of fit of the model to each response. The proportions for resin, glass fibers, and microspheres in Table 1 are used to define the matrix x 8x3 in the first-order model Zj = Xp, + F (j = 1, 2,3,4,5); the estimated coefficients in the five fitted linear blending models are listed in Table 2. The R2 values for each fitted model are also listed in Table 2. A check on the adequacy of fit of each of the five fitted models did not suggest the need to include any of four candidate nonlinear blending terms, so the linear blending model was considered adequate. The estimated Cox effects [elements of the matrix E in (23) rounded to two decimal places], which were calculated using Equation (lo), where the matrix Dxx3 is of the form

[

.2000 - .lOOO - .lOOO

D= - .1286 .1385 - .0099 ) (22) - .1286 - .0099 .1385 I

are listed in Table 3. The elements in the matrix D were obtained by taking the differences in the coordinates of the blends xi high and xi ,0w along each of the three Cox directional rays. For resin, x1 high = (l,W) and x1 low = (.8,.1,.1); for glass fibers, X2high

= (.8,.1385,.0615) and x~,~,,, = (.9286,0,.0714), and

for microspheres, xg high = (.8,.0615,.1385) and

x3 low = (.9286,.0714,0). The absolute values of the effects entries in Table 3 suggest that the order of importance among the components, when taken over all five responses, is glass fibers > resin > microspheres. We shall confirm the preceding conjecture with a j plot of the composite effects.

The effects matrix E3xs, which we shall approximate with E;x5 for constructing the biplot, is of the form

TS TM FS FM W

[

-.3623 -.8008 -.4134 - .7381 .3982 E 3x5 = ^.6713 ^.8298 ^.6955 A078 - .6643

- .2054 .2001 - .1636 .1414 .1523 1

(23) The rank of the matrix E is 2 since the rank of the D matrix in (22) is 2. To find the elements e; of E;x5, we use the matrices P3x2, A2x2r and Qc,, in the SVD of Equation (11). These matrices are

QT = .3632 .5519 .3874 .5252 -.3710

-.5173 .5269 - .4103 .3757 ^.38101 ⁽²⁴⁾

The rank-two approximation matrix E:x 5 is identical in form to E, x 5 in (23), as it must be since the rank

of ^E3x5 is 2, and therefore our biplot will be of the

elements of E, x 5 in (23).

For the GH’factorization, the form of the matrices G and HT are obtained from the matrices P, A, and

Table 3. Estimated Component Effects of the Standardized Responses in the Plastics-Compounding Experiment

Tensile Tensile Flexural Flexural Component strength modulus strength modulus Warp

Resin - .36 - .a0 - .41 - .74 .40

Glass fibers .67 .83 .70 .ai - .66

Microspheres -.21 .20 -.I6 .I4 .I5

(12)

Q’ in (24) as

- .6115 - .4164 resin G _3x2 =p= .7912 - .3167 I glass fibers

- .0048 X522 microspheres and

TS TM FS FM W

H:,, = AQr= .7537 1.1453 .8038 1.0899 -.7698 -.2368 .2412 - .1878 .I720 ^.I7441

The biplot, shown in Figure 7, was constructed by plotting the rows of G and using an asterisk to signify the location of each of the three components in the plot, along with plotting the columns of Hr in the form of vectors. The biplot clearly shows the following:

1. Tensile strength, flexural strength, flexural modulus, and tensile modulus are all positively correlated with one another, but warp is negatively correlated with the strength and modulus responses.

1.2

0.6

-0.6

-1.2

2. Increasing the proportion of resin increases warp and decreases the strength and modulus responses.

3. Increasing the proportion of glass fibers is very favorable for increasing the strength and modulus responses and also decreases warp.

4. Tensile and flexural moduli and warp increase with an increase in the proportion of microspheres, but tensile and flexural strength decrease.

What is the relative ordering of the components in terms of their composite effects on the five responses? This question can best be answered by constructing a j plot that uses as vectors the row markers of the J matrix in the JKT factorization. From (16) and (24), we have

c

- 1.2690 - .1906 resin J 3x2 = 1.6419 - .1450 glass fibers

- .0099 .3900 microspheres I (25) and the j plot of these vectors is shown in Figure 8.

The lengths of the vectors or relative measures of

RESIN *

,* __-_._-- __-_.

SPHERES

-1.2 -0.6 0.0 0.6 1.2

FACTOR 2

FLgure 7. The GHT Factorization &plot of the Plastics-Compounding Experiment.

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2

1

g 0

L 6 LL

-1

-2

FIBERS

RES I N

SPHERES ---__---.

I I

-2 -1 0 1 2

FACTOR 2

Figure 8. A j Plot of the Three Composite-Effects Vectors in the Plastics-Compounding Experiment.

the composite effects of resin, glass fibers, and microspheres, respectively, are the square roots of the diagonal elements of JJT( = EET). With the J matrix in (25) or E matrix in (23), these lengths or relative measures are VQZU = 1.28 for resin, V?YZU = 1.65 for glass fibers, and m = .39 for microspheres. These values support our earlier contention on looking at the effects entries in Table 3, where we inferred that the order of importance among the three components was glass fibers > resin > microspheres. From the j plot, we also see that the composite effect of resin is opposite in sign to that of glass fibers. From this we suspected that resin affects strength and modulus responses negatively, whereas glass fibers have a positive effect on strength and modulus responses. The composite effect of microspheres on these particular responses is some- where in between those of resins and of glass fibers.

Before leaving this example we would be remiss if we did not address the obvious question of how much information on the response variables and the effects of the components is lost in the rank-two approximation of the data matrix. (In this article, the

data matrix was a matrix of component effects on a group of measured responses.) An absolute measure of goodness of fit, given by Gabriel (1971), in using EGx m in (13) in place of E, X m, where E, X m is of rank r = min(q - 1, m), is

pp = 1 - IIE - E* 11’

llEll*

e: + e:

=-

c;=, e:*

For the plastics-compounding experiment, this value is p$*) = 1, meaning only (1 - pi*)) X 100% = 0%

of the information in E, X 5 was unexplained in using E* 3X5. This, of course, was noted earlier when we stated that the rank of the matrix E, X 5 was 2, which it must be when q = 3 and m 2 2, and therefore

E* _3x5 ₌ E3x5. When, however, q > 3 and m 2 3 so

that r > 2, then the value of pp is greater than 0, in which case one should calculate the percentage (1 - &)) x 100% of information in Eqxm that is lost or unexplained when using the rank-two matrix E* qxm. And although Gabriel (1971) stated only that