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Comparison of Slack Variable and Mixture Experiment Approaches Samantha M. Landmesser1 and Greg F. Piepel2 Department of Statistics, University of Tennessee, Knoxville, TN 2 Statistical Sciences, Pacific Northwest National Laboratory, Richland, WA 1

Abstract In a mixture experiment, the response variable depends on the proportions of the components, which must sum to one. Because of this constraint, standard polynomial models cannot be used to analyze mixture experiment data. To get around this, some researchers ignore one of the components and use standard polynomial models in the remaining components. Because the component proportions must sum to one, the ignored component (referred to as the slack variable (SV)) makes up the remaining proportion of the mixture. In the literature, there have been many examples of researchers using the SV approach instead of a mixture approach for modeling. We have analyzed data from several of these examples using both approaches. For examples whose goal was to screen the mixture components (screening examples), we fit full linear models and identified which components were important using both approaches. In the screening examples, the mixture modeling approach revealed that the SV had a significant effect on the response. For examples that had sufficient data to fit quadratic models to the data (quadratic examples), we used stepwise regression to develop reduced quadratic models for the SV approach, and partial quadratic mixture (PQM) models for the mixture approach. In the quadratic examples, the PQM models identified the SV and/or one of its quadratic blending terms as having a significant effect on the response variable. Hence, by completely ignoring a component’s effect on the response, SV analysis carries a greater risk of wrong conclusions than the mixture approach. There are fewer possible reduced quadratic SV models than possible PQM models because the reduced quadratic SV models are a subset of the class of PQM models. As a result, PQM models will always fit the data as well as, or better than, the best reduced quadratic SV model. We conclude that it is better to analyze mixture experiments using methods specifically developed for them instead of using standard methods with the SV approach. Keywords: Mixture Components, Model Reduction, Partial Quadratic Mixture Models, Variable Selection. 1. Introduction In a mixture experiment, the response variable depends on the proportions of the components, which must sum to one. Many researchers use the slack variable (SV) approach when designing and analyzing mixture

experiments. Using an approach specifically for mixture experiments provides several advantages over the SV approach. Using mixture experiment examples from the literature, we illustrate that the mixture approach for modeling and data analysis provides results that are the same or better without the risk of incorrect conclusions that can occur with the SV approach. A SV design involves setting the values of all but one of the components, often using a standard statistical design such as a fractional factorial, central composite, or BoxBehnken design. The remaining component, called the slack variable, has a proportion equal to one minus the sum of the proportions of the other components. When modeling the data from these types of experiments, it is often assumed that the SV doesn’t have a significant effect on the response. An inactive “filler” component is an example of a component having no effect. Practitioners also often assume the SV has no effect if it makes up the vast majority of the mixture. In some cases, practitioners assume that the SV has at most an additive linear effect. A diluent component (i.e., a component that dilutes the mixture) would be an example of a component with a linear additive effect. The mixture experiment approach takes into account the unique restrictions of mixture experiments. Specifically, the proportions of the components (xi) must fall between zero and one ( 0 ≤ x i ≤ 1 , i = 1, 2, … , q) and must sum to q

one ( ∑ x i = 1 ). The mixture approach uses every i =1

component in the design of the experiment and the analysis of the data. This approach does not make any assumptions about the SV or require any specific situation to be applicable. Previous literature contains several examples of mixture experiments analyzed using the SV approach. Table 1 briefly summarizes some aspects of several mixture experiment examples from the literature that used the SV approach. In each of the examples, the researchers used the SV approach to analyze the data and make conclusions about how the variables affected the response. We analyzed data from several of these examples using the mixture and SV approaches. The examples fall into two groups, screening examples and quadratic examples. In the screening examples, the goal was to identify the components having significant effects on a response variable. In quadratic examples, the goal

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Table 1: Summary of Slack Variable Experiments from the Literature Slack Variable (SV) Example MacDonald and Bly (1966) Fonner et al. (1970) Cain and Price (1986) Goh and Roy (1989) Takayama and Nagai (1991) Chan & Kavanagh (1992) Abdullah et al. (1993) Guillou & Floros (1993) Levison et al. (1994) Piepel and Cornell (1994) Setz et al. (1997) Cornell (2000) Example #1(e) Cornell (2000) Example #2 Cornell (2000) Example #3(e) Cornell & Gorman (2003)

q

(a)

5 3 4 11 5 8 3 4 6 4 5 3 3 4 4

n/d

(a)

27/25 9/9 19/19 16/16 12/9 12/12 9/9 15/13 36/27 20/15 24/21 9/7 13/9 20/10 14/12

#RVs

(a)

SV Component

8 4 3 5 3 7 6 5 3 4 3 1 1 1 1

Shortening Dicalphos Base Oil SiO2 Water Water Water Brine Base Water Water TPE Varied Varied Varied x4

(c)

SV Linear Additive?

No No No No No No No No No No No No No No No

No No No No No No No No No No No No No No No

SV Inactive?

(b)

SV Range

(0.030, 0.160) (0.0171, 0.9314) (0.845, 0.925) (0.364, 0.495) (0.4011, 0.7005) (0.7926, 0.9352) (0.3086, 0.5110) (0.894, 0.998) (0.355, 0.775) (0.85, 1.00) (0.120, 0.750) (0, 1.00) (0, 1.00) (0, 1.00) (0.949, 0.975)

(d)

(a) q = the number of mixture components that were varied, n = the total number of data points, d = the number of distinct data points, # RVs = the number of response variables. (b) The component proportion range of the SV. If the experiment had some components constant, the range shown is for proportion of the SV component relative to the components that were varied. (c) This column indicates whether the SV component is inactive for one or more of the response variables. (d) This column indicates whether the SV component has a linear additive effect on one or more response variables. (e) This example was also investigated by Khuri (2005). was to develop the best quadratic model that adequately approximates the relationship between the response variable and the mixture components. Section 2 presents the mixture and SV models used to fit data from the literature examples and the summary statistics used to assess and compare the models. Section 3 presents the results of fitting mixture and SV models for selected examples. We also discuss some examples for which the mixture approach resulted in better models and explain why that occurred. Section 4 summarizes the work and explains why the mixture approach is more appropriate in all situations than the SV approach.

2.1 Slack Variable Models The SV approach is sometimes used by practitioners for screening, that is, discovering which mixture components are important for a given response variable. In this case, the full linear SV model is used q −1

E ( y ) = α 0 + ∑ α i xi i =1

More often, researchers are interested in obtaining an adequate approximation of the true relationship between the response variable and mixture compositions. In that case, the full quadratic SV model q −1

q −1

q − 2 q −1

i =1

i =1

i =1 j =i +1

E ( y ) = α 0 + ∑ α i xi + ∑ α ii xi2 + ∑ ∑ α ij xi x j

2. Methods Both mixture and SV models were used to analyze the data from the literature examples. For each screening example, we fit the full linear models (mixture and SV) and identified significant terms. For each quadratic example, we fit the full quadratic models (mixture and SV) and then reduced them using stepwise regression. We determined the best-fitting reduced model based on regression summary statistics. Each of the models and summary statistics used is discussed in this section.

(1)

(2)

is often used. Removing non-significant terms from the full quadratic SV model in Equation (2) results in a reduced quadratic SV model. 2.2 Mixture Experiment Models

Mixture models include every component in the model, and don’t have an intercept (constant term). The most commonly used mixture models are the Scheffé models

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(Scheffé 1958, Cornell 2002, Smith 2005). The full linear Scheffé model is q

E ( y ) = ∑ β i xi

(3)

i =1

When a researcher wants to investigate the quadratic blending effects of mixture components on the response variable, the full quadratic Scheffé model q −1 q

q

E ( y ) = ∑ α i x i + ∑ ∑ β ij x i x j i =1

i =1 j = i +1

(4)

is appropriate. Notice that linear as well as nonlinear blending terms involving the component treated as the SV (that is, xq) are included in model (4). This allows researchers to investigate the effects of the SV that are ignored in the SV models (1) and (2). The linear mixture model (3) is not useful for directly testing the significance of component effects on a response variable (see Cornell 2002 and Smith 2005). Instead, the component slope linear mixture (CSLM) model discussed by Piepel (2006, 2007) can be used for this purpose. The CSLM model is given by q

q

i =1

i =1

E( y ) = γ 0 + ∑ γ i (1 − si ) xi = γ 0 + ∑ γ i xi'

(5a)

where xi’ = (1 − si)xi and the coefficients γi (i = 1, 2, … , q) are subject to the restriction q

(

)

∑γ j 1 − s j s j = 0

j =1

(5b)

with s = (s1, …, sq) being a reference mixture. In (5a), γ0 is the response value at the reference mixture and γi represents the slope of the response surface along the Cox effect direction (Cornell 2002, Section 5.9) for the ith component. The CSLM model (5) provides the same fit to data as the linear mixture model (3). Because the γi coefficients are the slopes of the response along component effect directions, standard t-tests can be used to assess whether the mixture components have significant effects on the response variable.

Piepel, Szychowski, and Loeppky (2002) discuss partial quadratic mixture (PQM) models that contain a subset of all quadratic (squared and crossproduct) terms including those involving the SV. A PQM model has the general form q

E ( y) = ∑ α i xi i =1

(6) q −1 q ⎧q ⎫ 2 + Selected ⎨ ∑ β ii x i + ∑ ∑ β ij x i x j ⎬ i =1 j = i +1 ⎩i =1 ⎭ where “Selected” means that only a subset of the squared and crossproduct terms can be chosen. The main limitation is that if a squared term for a component is selected, all of the crossproduct terms involving that component cannot also be selected. Doing so would result in an exact collinearity in the model. Variable selection methods (such as stepwise regression) would never select terms forming an exact collinearity, so this is not an issue of practical concern. Note that the class of reduced quadratic SV models and the class of reduced quadratic Scheffé models are both subclasses of the class of PQM models. That is, the class of PQM models contains models that are equivalent to all reduced quadratic SV models and all reduced quadratic Scheffé models. The class of PQM models also contains models not in the classes of reduced quadratic SV models and reduced quadratic Scheffé models. Hence, it is guaranteed that the best-fitting PQM model will always fit as well or better than any reduced quadratic SV model or reduced quadratic Scheffé model. 2.4 Statistics Used to Evaluate and Compare Models

Three different R2 statistics were used to evaluate mixture and SV models. These are denoted as R 2 , R A2 , and R P2 , which are defined as follows. The (ordinary) R2 statistic is given by n

2 ∑ ( ˆyi − yi )

2.3 Reduced Quadratic Models

Because we are interested in identifying which quadratic terms have a significant effect on the response, the full quadratic models [Equations (2) and (4)] are reduced by removing nonsignificant terms. When the full quadratic SV model in Equation (2) is reduced to contain a subset of the quadratic terms not involving the SV, the resulting model is called a reduced quadratic SV model. When the full quadratic Scheffé model in Equation (4) is reduced to contain a subset of the crossproduct terms (including the crossproduct terms involving the SV), the resulting model is called a reduced quadratic mixture model.

R 2 = 1 − i =n1 2 ∑ ( yi − y )

(7)

i =1

and is interpreted as the fraction of variability in the data accounted for by the fitted model. The adjusted R2 statistic is given by n

R A2 = 1 −

2 ∑ ( ˆyi − yi ) ( n − p )

i =1 n

2 ∑ ( yi − y ) ( n − 1)

(8)

i =1

and is interpreted as the adjusted fraction of variability in the data accounted for by the fitted model. The adjustment is for the number of parameters (p) and number of data

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points (n) used in fitting the model. The predicted R2 statistic is given by n

RP2

∑ ( ˆy(i ) − yi )

2

= 1 − i =1n 2 ∑ ( yi − y )

(9)

i =1

and is calculated by a method equivalent to leaving each data point out of the model fit, and then evaluating how well the model predicts the property for that data point. R P2 estimates the fraction of variability that would be explained in predicting new observations drawn from the same mixture composition space. The Root Mean Square Error is given by n

RMSE =

2 ∑ ( ˆyi − yi )

i =1

(10)

n− p

If the fitted model is adequate and does not have a statistically significant lack-of-fit, this statistic provides an estimate of the standard deviation representing the experimental and measurement uncertainty associated with the processes of conducting the experiment and measuring the response variable.

mixture components on a response variable of interest when using a fitted mixture model. The model is used to predict, for each component, the response for a series of compositions lying along an effect direction (e.g., the Cox effect direction) for that component. Along such a direction, the component of interest is varied within the allowable composition region of interest. The remaining components change proportionally to offset the changes in the component of interest. The predicted response values are plotted on the y-axis and changes in each component from its reference mixture value are plotted on the x-axis. The predicted response values along the effect direction for a given component form a component response trace. The response traces for the components varied in a mixture experiment plotted together form a component response trace plot. Components with steeper response traces have stronger effects on the response. A response trace that is nearly horizontal indicates the corresponding component has little or no effect on the response. Components whose response traces are very close may have similar effects on the response. Thus, component response trace plots can be used to guide the reduction of components in a mixture experiment model (e.g., see Piepel and Redgate 1997). 3. Results

2.5 Component Response Trace Plots

Component response trace plots (Cornell 2002, Section 5.9) provide for graphically assessing the effects of

Each literature example listed in Table 1 was analyzed using both the SV and mixture approaches. Table 2 summarizes the results of fitting linear models for two

Table 2: Important Components Identified in Two Screening Examples Using Linear Models Example Response Variable Flow Mixture TiO2, Approach Na2O (CSLM Model)

Slack Variable Approach (Reduced Linear Model by Stepwise Regression) Important Components Identified by the Original Authors(a)

Goh & Roy (1989), SV = SiO2 Tension Opacity TiO2 SiO2

Chan & Kavanagh (1992), SV = Water

Compression Na2O, SiO2

Acidity SiO2, Li2O, Na2O

TiO2, Na2O

TiO2, Al2O3, Na2O, Li2O, Na2SiF6

TiO2, Al2O3

Na2O, TiO2, Na2SiF6, CaO, Al2O3, ZnO, CaF2, PbO

Na2O, Li2O, Al2O3, Na2SiF6

TiO2, Na2O

TiO2

TiO2

Na2O

Na2O, Li2O, Al2O3, Na2SiF6

RMFH AEO, AEOS, NaLAS, DEALAS, TEALAS, Water AEO, AEOS, NaLAS, DEALAS, TEALAS,

GSTT AEO, AEOS, CDEA, water

lnVisc lnCLPT NaCl, No NaLAS, components DEALAS, significant water

AEOS, CDEA

NaCl, No NaLAS, components DEALAS, significant CDEA

AEO, AEOS, NaLAS, DEALAS, TEALAS, CDEA

AEO, N/A AEOS, NaLAS, DEALAS, CDEA

N/A

(a) In some cases authors only identified the component(s) with the largest effects rather than components with significant (i.e., non-negligible) effects.

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Table 3: Summary Statistics of Best-Fitting Reduced Quadratic Models for Selected Literature Examples Example

Cornell and Gorman (2003) SV = x4

Response Dissolution Variable Mixture Statistic and SV R2 0.978 R2A 0.960 R2P 0.910 RMSE 0.465

Fonner et al. (1970) SV = Dicalphos

Takayama and Nagai (1991) SV = Water

Y1 Mixture 0.979 0.977 0.973 0.550

tL SV

0.979 0.977 0.973 0.541 Starch2, Quadratic Stearic2, stearic Terms in 2 2 x2 , x3 , x1x3 dicalphos acid2, Best ×starch starch× Model stearic

Guillou and Floros (1993) SV = Brine Base

ln(Rp)

TIS

Tmin

Mixture and SV 0.932 0.892 0.720 1.858

Mixture and SV 0.918 0.871 0.616 0.463

Mixture and SV 0.734 0.586 0.293 10.803

dLim× dLim Ethanol2, Water, ×Eth dLim2 Ethanol2

Ethanol2, dLim2

Mixture

SV

0.886 0.821 0.656 0.128

0.816 0.746 0.674 0.152

screening examples. Table 3 summarizes the results of fitting quadratic models when there were sufficient data to do so for selected quadratic examples. The models in Table 3 contain all linear terms appropriate for mixture or SV models. The last row of Table 3 summarizes the (i) quadratic terms in reduced quadratic SV models, and (ii) the selected quadratic terms in PQM models with the mixture approach. 3.1 Screening Examples

The mixture and SV approaches reached conflicting conclusions as to which components were significant in the screening examples, as illustrated in Table 2. For some of the responses in the Goh and Roy (1989) example, the CSLM model identified the SV to be an important component. The response trace plot in Figure 1 also shows that the SV (SiO2) has a negative effect on the

ln(Ymax) Mixture

SV

0.936 0.888 0.761 1.758

0.932 0.880 0.745 1.858

CaCl2× K-sorbate2, K-sorbate2, K-sorbate, brine2, NaCl2, CaCl22 K-sorbate2 CaCl2

compression response. The CSLM model identified Na2O as the only other important component, while the SV approach identified several other components as important. For the Chan and Kavanagh (1992) example, the SV (water) has a negative effect on each of four response variables, as shown in the response trace plots in Figure 2. For the responses RMFH and lnVisc, the SV approach doesn’t identify all of the important components that the mixture approach (using the CSLM model) identifies. These screening examples show that the SV approach can make incorrect conclusions such as not recognizing the SV as an important component and including components in the SV linear model that have negligible effects. 3.2 Quadratic Examples

There are also several discrepancies between the best reduced quadratic models that were fit to the quadratic examples from the literature. For the examples in Table 3, the mixture approach identified quadratic terms involving the SV to have significant effects on three responses from different examples. The SV and mixture experiment approaches resulted in equivalent models for four of the responses.

Figure 1: Response Trace Plot of Compression Response from Goh and Roy (1989) Example

For the tL response in the Takayama and Nagai (1991) example, the PQM model contains a crossproduct term involving the SV (water). The SV approach only identified one significant quadratic term and it didn’t match the term identified by the PQM approach. The PQM model was able to achieve a better fit than the SV model, as evidenced by the R 2 , R A2 , R P2 , and RMSE values shown in Table 3.

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In the Cornell and Gorman (2003) example, the SV (x4) does not have a quadratic effect on the Dissolution response, so the PQM and reduced quadratic SV models identified the same quadratic terms and have the same fit.

lnCLPT 1.25

C omponent nNaC l nA EO nA EOS nNaLA S nDEALAS nTEA LA S nC DEA nW ater

Fitted lnCLPT

1.00 0.75

4. Conclusions

0.50

The SV approach makes the assumption that the SV either has no effect on the response (e.g., an inactive “filler” component) or that it only has a linear additive effect on the response (e.g., a diluent component). In the literature SV examples analyzed (15 examples with 55 responses, only some of which are discussed in this paper), we were unable to find an example of the SV having no effect or an additive linear effect (see Table 1). Hence, these assumptions appear to be rarely satisfied in practice. The mixture experiment approach does not make the same questionable assumptions, and is applicable whether or not a particular component has no effect or a linear additive effect.

0.25 0.00

-0.050

-0.025 0.000 0.025 0.050 0.075 deviation from reference blend in proportion

GSTT 120

C omponent nNaC l nA EO nA EOS nNaLA S nDEALAS nTEA LA S nC DEA nW ater

110

Fitted GSTT

100 90

The full linear and full quadratic mixture experiment models are equivalent to the SV full linear and full quadratic models, respectively. When using the full models, the fits of the SV and mixture models (linear or quadratic of each) will be the same but the coefficients and their levels of significance will not. The advantage to using the mixture approach comes when using (i) CSLM models (or Scheffé linear mixture models and response trace plots) to assess component linear blending effects and (ii) PQM models to understand quadratic blending effects of components. The SV approach can lead to incorrect conclusions about the significance of component linear and quadratic blending effects. Also, the SV models are not capable of including quadratic blending terms involving the SV. This limits the class of quadratic models possible with the SV approach, which may result in an inaccurate representation of the SV (and other components) effects.

80 70 60 -0.050

-0.025 0.000 0.025 0.050 0.075 deviation from reference blend in proportion

lnVISC 7

C o mpo nent nNaC l nA EO

6

nA EO S nNaLA S nDEA LA S nTEA LA S nC DEA nW ater

Fitted lnVISC

5 4 3 2 1 0 -0.050

-0.025 0.000 0.025 0.050 0.075 deviation from reference blend in proportion

RMFH 14

C o m p o n en t n N aC l nA EO

Fitted RMFH

13

nA EO S n N aLA S n D E A LA S n TE A LA S nC DEA n W ater

12

11

10

9 -0.050

-0.025 0.000 0.025 0.050 0.075 d e v ia t io n fr o m r e fe r e nc e b le nd in p r o po r t io n

Figure 2: Response Trace Plots of Each Response Variable in Chan and Kavanagh (1992) Example

The SV and mixture experiment modeling approaches often identify different components with significant effects, both with linear models and quadratic models. However, the SV and mixture experiment modeling approaches can also result in equivalent models. This should not be taken as evidence of the superiority of the SV approach. It is always possible to obtain a quadratic mixture model equivalent to a given quadratic SV model by using the PQM modeling methods of the mixture experiment approach. However, it not always possible to obtain a given mixture model by using the SV approach. This is because the SV approach leaves out the linear SV component and all of the quadratic terms involving the SV. The mixture approach is more efficient and thorough

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for obtaining an accurate model relating a response variable to the component proportions. It is important that researchers be aware of the advantages that mixture models present for the analysis of data from mixture experiments. The SV approach ignores the effect of the SV on the response, and has fewer potential reduced models compared to the class of PQM models. The mixture models were specifically designed for mixture experiments and provide for obtaining the correct analysis and conclusions about linear and quadratic blending effects of the mixture components. When analyzing mixture experiments, it is recommended to use the mixture approach rather than the SV approach. Acknowledgements

Ms. Landmesser’s work was performed as part of a 2007 summer internship at the Pacific Northwest National Laboratory (PNNL). It was supported by the Department of Energy, Office of Science, Scientific Undergraduate Laboratory Internship Program. We thank Scott Cooley of the Statistical Sciences group at PNNL for reviewing a draft of this paper. References

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