Tips on JMP ing into Mixture Experimentation

Transcription

1 Tips on JMP ing into Mixture Experimentation Daniell. Obermiller, The Dow Chemical Company, Midland, MI Abstract Mixture experimentation has unique challenges due to the fact that the proportion of the ingredients making the total is important rather than the actual quantity of the ingredient. Although mixture designs and their analysis is quite different, JMP allows the user to handle all of these situations. This paper briefly reviews what a mixture experiment is and why it is different. This paper also delivers some useful techniques to allow users to avoid pitfalls in designing and analyzing such experiments with JMP. Background Traditional design of experiments (DOE) requires the experimenter to vary the controlled variables (factors) in a precise way so as to cause changes in a response variable. This technique is very effective and allows the experimenter to see if the level of a factor(s) influences the response. Typical designs of this type would be the full-factorial designs, fractional factorial designs, and central composite designs to name just a few. Mixture experimental design differs from the traditional approach because the controlled variables (ingredients or components) must add up to I or 100%. In other words, the amount of a controlled variable does not matter, but the proportion of the total is important. Some examples of mixture experiments would be making a blended fruit juice drink, a sausage recipe, or a mixed drink. However, as with classical designs, the researcher is often still interested in achieving the optimal response over the design space. Why ate Mixture Designs Different? Because the proportions of the ingredients are important, mixture experiments have a constraint that the sum of the ingredients must add to 1 or 100%. For a three component mixture, knowing the proportions for two components fixes the proportion of the third. This lack of complete independence between the ingredients causes the traditional experimental designs to breakdown. As an example, suppose we are interested in studying a 2 component mixture. We are interested in studying component A over the range between 30 and 70% and component B between 30 and 70%. A full-factorial design for this situation would be: Run Number Comp.A Comp.B From the above design, runs 1 and4 are not possible as they do not total 100%. Mixture Designs Since classical designs are not appropriate, special mixture designs are required. The most basic mixture design is the simplexcentroid design. This design is used in situations where there are no constraints on the components or the constraints form a smaller version of an unconstrained design space. For a three component study with no constraints the design space forms a triangle. One example where a simplex-centroid 799

2 design could be used is when all of the components are free to vary from 0 to 100%. See Figure 1 for a simplex-centroid design for three components. / \ ~. However, in most cases, the design points do not form a triangle or the components have physical restrictions on their ranges. In such a case, the design often contains the extreme vertices of the design, midpoints of long edges, and an overall centroid. Suppose component A ranges from 40-80%, B from 10-50%, and C from 0-70%. For this example, Figure 2 and Table 2 contain the design. \ : '\ 0.2 B Figure 1: 3 Component Simplex-Centroid Design \. 1 ' o Notice that the design points are at the vertices of the design space (allows estimation of main effects), midpoints of the edges (estimation of binary blending - 2- way interactions), and the overall centroid of the design space (estimation of ternary blending - 3-way interaction). This particular design is obtained from JMP using the Design Experiments option and choosing a Simplex -Centroid design from the Mixture design button with a degree of 3. The design points are (in standard order) in Table 1. A B C Table 1: Simplex-Centroid Design for 3 Ingredients o B Figure 2: 3 Component Extreme Vertices Design A B C Table 2: DeSign POints for Extreme Vertices DeSign Notice that none of these designs from JMP include replication. Tip #1: The user must manually choose which points to replicate. 800

3 This is a very subjective choice. In an ideal situation, the entire design would be replicated. However, in practice, only the extreme vertices and/or overall centroid are replicated. In JMP, select the rows to be replicated (do not have any column headings selected) and choose copy. Then deselect all rows and choose Paste at End. The user can then reorder the experiments in a random fashion. Analyzing Mixture Experiments The constraint on the sum of the components being 100% also impacts the analysis of mixture designs. For a 2 component case, if a full-factorial design could be completed, the linear model would be Y = A + 132B + I312AB. For a mixture design, the model must be modified. The 130 term can be rewritten as 130CA+B) since A+B=l. By simplifying, the model becomes Y = l3*la + l3*zb AB where 13*1 = and 13*2 = This is the Scheffe canonical form of the model Notice that one would not want to test the coefficients on the main effects against zero as in the normal case. Instead, one should test the main effect coefficients against the overall mean of the data. Also, the intercept is still estimated for the model, it is just combined with other parameter estimates. Thus, the Scheffe models are different from the usual no-intercept model in line.ar regression. Tip #2: To analyze a mixture design in JMP, use the Fit Model choice from the analysis menu. To develop the Scheffe mixture models in JMP, use the Mixture Response Surface model in the Effect Macros list. Note that this will check the no-intercept box, but this model will be different from the standard no-intercept regression model. Since the Scheffe mixture models are different from a no-intercept regression model, JMP must handle the analysis differently. Tip #3: Due to the Scheffe mixture model being fit, the leverage plots seen from the Fit Model Standard Least Squares Personality appear to be incorrect. Choosing a mixture response surface model causes JMP to generate a standard response surface model. The standard approach assumes 3-way and higher interactions are not important. However, many mixture scenarios benefit from estimating the 3-way term. Tip #4: One can manually add a 3- way term to the mixture model. JMP will give warning messages, but all analyses are correct. Although JMP gives the appearance of doing a no-intercept regression model, JMP is actually testing a Scheffe model. Tip #5: To make sure JMP is fitting a Scheffe model and not a no-intercept model, check the overall ANaVA table in the analysis. It should say "Tested against reduced model: Y=mean". If JMP is fitting a no-intercept model it will have Y=O instead of mean. Sample output is shown in Figure 3. I AnaJysis 01 Variance I Source OF Sum of Squares Mean Square Model Error C Total Tested against reduced model: Y =mean F RatiO Prob>F Figure 3: Sample output from MIXture Model. Earlier, in the derivation of the Scheffe models, it was noted that the parameter estimates for the main components should not be tested against O. The parameter estimates instead should be tested against the overall mean of the response. This must be done only for the main components. All tests concerning the interactions are still 801

4 against O. Unfortunately, IMP still tests these parameters against O. Typically, these tests are just ignored as one does not usually remove a main component term from a Scheffe mixture model. Tip #6: When examining the parameter estimates of a Scheffe model, the t-tests on the main component parameter estimates from JMP are not correct and should be ignored. Visualizing the Mixture Model While fitting models, JMP offers several great visual tools to enhance your understanding of the model. The Prediction Profile plot from the Screening model personality is one such tool. However, in a mixture scenario one must be careful in using the interactive prediction tools. Tip #7: Most of the interactive prediction tools in JMP do not restrict you to the mixture design space. Therefore, predictions for impossible mixture scenarios are possible. This is demonstrated in Figure 4. 1,nem i! ~ t-~- 7,.'.,. >,,,,'j] '/, o / " 1.,.1.0 O.B 0.5 _ <,2D (. 5.0 )<... \: Q.( 0.2 <,6,0 I!1IIIIIlII <.)0 > /0 Figure 5: Sample Ternary Plot Many times, as indicated in Figure 5, if the design space is constrained, the resulting ternary plot may be too small. Tip #8: Use the magnifying glass to zoom in on the ternary plot (Figure 6). The crosshair tool is also helpful in determining component levels to achieve a desired response. D.S, t Flgufe 4: Impossible mixture from prediction profile. Another great visual tool is the contour plot. IMP allows one to plot contours over the mixture design space. This option is found under the Graph menu and is called Ternary plot. One simply selects the components to be plotted. After the graph appears that shows the design points, one can use the checkmark (,I') to choose Add Contours. A sample contour plot is shown in Figure _'01,0 _"2.0 _ <=5.0 _'.7D.>/.0 OJ 0,2 l[t&;g ; (::: 3,0 _ <=6.0 6: Zoomed in area Plot

5 More than 3 Components. For showing examples, 3 component mixture designs are often used since the results can be seen visually. In practice, more than 3 components are often needed to make the product. However, the steps for the analysis and the tips given above still apply. However, visualizing 4 components or more on a temary plot gets more difficult. When more than three components are present, the ternary plot must be done for various levels of the components not present on the plot. This may cause the shape of the design space to change as "slices" are heing taken through the (n-l) dimensional design space (where n = the number of components). Beyond the possible changing shape of the design space, one needs to be aware how JMP handles the situation. For any ternary plot, JMP will rescale the components that are on the graph so that those components add up to 100%. For example, a 4 component design was completed with the following component ranges: A B C D 40 to 80% 10 to 50% Ot050% Ot020% The plot in Figure 7 has component D fixed at 10%. Notice that it appears as though component A can achieve almost 90%. This is due to JMP's rescaling of the axes. Tip #9: When more than three components are in a mixture, the axis on the ternary plot represents the proportion of the total of the three components visible on the ternary plot. For this example, since D=1O% of the mixmre, the axis scales represent the proportion of the 90% remaining in the mixture. 0' OJ OJ <= 0.0.'6.0 Figure 7: Ternary plot with 4 components. Conclusion JMP offers many tools for creating and analyzing mixture designs. These tools allow a greater understanding of the mixture model. However, there are several tips that can aid somebody in going through a mixture design. The tips presented in this paper are Tip #1: The user must manually choose which points to replicate. This is a very subjective choice. In an ideal situation, the entire design would be replicated. However, in practice, only the extreme vertices andlor overall centroid are replicated. In JMP, select the rows to be replicated (do not have any colurun headings selected) and choose copy. Then deselect all rows and choose Paste at End. The user can then reorder the experiments in a random fashion. Tip #2: To analyze a mixture design in JMP, use the Fit Model choice from the analysis menu. To develop the Scheffe mixture models in JMP, use the Mixture Response 803

6 Surface model in the Effect Macros list. Note that this will check the no-intercept box, but this model will be different from the standard no-intercept regression model. Tip #3: Due to the Scheffe mixture model being fit, the leverage plots seen from the Fit Model Standard Least Squares Personality appear to be incorrect. Tip #4: One can manually add a 3-way term to the mixture model. JMP will give waming messages, but all analyses are correct. Tip #5: To make sure JMP is fitting a Scheffe model and not a no-intercept model, check the overall ANOVA table in the analysis. It should say "Tested against reduced model: Y=m.ean". If JMP is fitting a no-intercept model it will have Y =0 instead of mean. Tip #6: When exanurung the parameter estimates of a Scheffe model, the t -tests on the main component parameter estimates from JMP are not correct and. should be ignored. Tip #7: Most of the interactive prediction tools in JMP do not restrict you to the mixture design space. Therefore, predictions from impossible mixture scenarios are possible. Tip #8: Use the magnifying glass to zoom in on the ternary plot. The crosshair tool can also be very helpful in determining component levels to achieve a desired response. Tip #9: When more than three components are in a mixture, the axis on the temary plot represents the proportion of the total of the three components visible on the ternary plot. References Cornell, J. A. (1990). Experiments with Mixtures: Designs, Models, and the Analysis of Mixture Data, 2 nd ed. John Wiley & Sons, New York, NY. Cox, D. R. (1971). "A Note on Polynomial Response Functions for Mixtures". Biometrika 58, pp Piepel, G. F. (1985). Models and Designs for Generalizations of Mixture Experiments Where the Response Depends on the Total Amount. University Microfilm International, Ann Arbor, MI. Piepel, G. F. and Cornell, J. A. (1994). "Mixture Experiment Approaches: Examples, Discussion, and Recommendations". Journal of Quality Technology, Vol 26, No.3, pp Piepel, G. F. and Cornell, J. A. (1985). "Models for Mixture Experiments When the Response Depends on the Total Amount". Technometrics 27, pp SAS Institute, Inc. (1995). Guide, Cary, NC. JMP User's SAS Institute, Inc. (1995). JMP Statistics and Graphics Guide, Cary, NC. Scheffe, H. (1958). "Experiments With Mixtures". Journal of the Royal Statistical Society B 20, pp Acknowledgetuents JMP software is a registered trademark of SAS Institute Inc. in the USA and other countries. indicates USA registration. 804