2. EXPERIMENTS WITH MIXTURES


WHAT IS A MIXTURE

A mixture is a blend of materials that make up some total formula amount. Examples of mixtures include the gases in a semiconductor wafer deposition chamber; the resins, solvents, and pigments in a house paint; or the liquid hydrocarbons and additives in gasoline. A mixture consists of q+1 linearly independent variables that are constrained to a q-dimensional dependent space in which the levels of the q+1 variables always sum to a constant. The constant in mixtures equals the total sample amount. The constant is equal to one for standardized variables and 100% for unstandardized variables. A mixture is therefore a special type of multiple constraint (a restriction on the allowable level setting combinations of two or more experiment variables).

To illustrate the mixture constraint, consider three linearly independent variables; X1, X2, and X3, with standardized ranges of zero to plus one. Independent variables can be set at any level setting combination within their allowable ranges. Geometrically this independent variable space is represented as a cube in the figure below.

However, mixture variables are constrained to a two-dimensional (q = 2) space in which X1 + X2 + X3 always equals a constant. Within the standardized independent space, the q-dimensional constrained space is represented by the triangle within the cube. The triangle space is called a simplex space. To more easily see the simplex space within the cube, the X3 axis (the Y axis) in the figure above has been reversed from that normally used. The contraction of the independent variable space into the q-dimensional dependent space results from the mixture multiple constraint. The mixture constraint is presented in equation form below.

The figure below also illustrates the q-dimensional simplex space for three experiment variables. All points on the surface or in the interior of the simplex sum to one. The figure below illustrates the typical simplex coordinates of the three vertices; (1, 0, 0), (0, 1, 0), and (0, 0, 1), and the center point; with coordinates (1/3, 1/3, 1/3). In addition the figure below also shows the three edge midpoints; with coordinates (1/2, 1/2, 0), (1/2, 0, 1/2), and (0, 1/2, 1/2), and three interior points; with coordinates (2/3, 1/6, 1/6), (1/6, 2/3, 1/6), and (1/6, 1/6, 2/3).

As with other design geometries, the simplex geometry can be extended to any number of dimensions. For example, given four mixture variables, the simplex geometry is that of a three-dimensional pyramid having four identical sides, each an equilateral triangle. For dimensions above four, the simplex geometry becomes a hyperpolyhedron.


FACTORIAL DESIGNS AND MIXTURE EXPERIMENTS

Prior to true mixture design software like CARD, factorial designs were the only real Design of Experiments (D.o.E.) option for mixture experiments. However, factorial designs are intended for "independent" experiment variables. An independent variable is simply one for which the level setting can be adjusted independently of other variables in the same experiment system. The effect of an independent variable on a given response results directly from its level setting. The same is not true for mixture variables. A mixture variable can not be adjusted independently of the other variables in the mixture. Most importantly, mixture variables are "proportional" effectors. The effect of a mixture variable on a given response results from its proportional amount in the mixture, i.e., its amount relative to the amounts of the other mixture variables.

The value of a factorial design lies in its ability to uncorrelate the experiment variable linear-effects terms from each other. This uncorrelation can extend to variable interaction-effects terms and simple nonlinear-effects terms. However, the ability of a factorial design to uncorrelate experiment variable terms requires that the variables be independent. Trying to adapt factorial designs to mixture variables can cause profound information problems in the experiment design. The problems are compounded by the mixture multiple constraint, which compromises standard correlation analysis, and therefore the ability to identify any problems resulting from the adaptation of the factorial design to the mixture experiment.

As stated, the mixture multiple constraint results in a perfect multiple correlation between the variable linear terms and the constant (). This is why you must eliminate the constant in a regression analysis of mixture experiment data. The mixture multiple constraint also requires a different standardization of the experiment variable terms prior to carrying out a correlation analysis of a mixture experiment. As a result, you can not use standard correlation analysis to determine the amount of correlation among the mixture variable terms in your design matrix. For mixture designs a standard correlation analysis will not correctly define whether your analysis of the experiment results will yield accurate estimates of the variable effects. Also, standard correlation analysis is limited in its ability to diagnose problems in a design matrix. This is why D.o.E. FUSION uses a more correct analysis of a design matrix called eigen analysis. D.o.E. FUSION automatically executes an eigen analysis of every design matrix and translates the results into easily-understood good/bad measures. These measures are even color coded: Red for bad and Blue for good.


ILLUSTRATING THE FACTORIAL DESIGN PROBLEM

Example Design 1A, presented below, is a two-level full factorial design for three variables. The minimum (R-min) and maximum (R-max) values of each variable's experimental range are presented in the variable columns below the eight runs of the design. This factorial design uncorrelates the experiment variables from each other. The uncorrelation extends to the variable pairwise interaction effects terms, designated .


EXAMPLE DESIGN 1A: FACTORIAL DESIGN


Run #
Variable A (X1)
(grams)
Variable B (X2)
(grams)
Variable C (X3)
(grams)
TSA
(grams)
1 0 0 80 80
2 20 0 80 100
3 0 20 80 100
4 20 20 80 120
5 0 0 100 100
6 20 0 100 120
7 0 20 100 120
8 20 20 100 140
R-min 0 0 80 -
R-max 20 20 100 -



In the full factorial design above, the total sample amount (TSA) changes from run to run. The TSA varies from a low of 80 grams in Run #1 to a high of 140 grams in Run #8. In addition to making the design awkward to carry out, varying the TSA changes the relative proportions of the variable level settings between experiment runs. You can see this by comparing Example Design 1A above with Example Design 1B below. Example Design 1B was created by translating the variable level settings in each run of Example Design 1A into their actual mixture proportions, given that the three experiment variables make up the entire mixture. To do this we simply converted the level setting of each variable in a given run to its corresponding percent of the total sample amount (TSA) for that run. For example, in Example Design 1A the level setting of Variable C in Run #1 is 80 grams and the TSA is 80 grams. Therefore, Variable C is present in Run #1 at 100% [(80 grams/80 grams)100%], which is the Variable C level setting in Run #1 of Example Design 1B.

You can easily see the how the change in the TSA changes the relative proportions of the variable level settings between experiment runs by comparing Example Designs 1A and 1B. In Runs #7 and #8 of Example Design 1A, the amounts of Variables B and C are constant. However, in Example Design 1B the relative proportion of Variable C to Variable B changes from 4.88:1 to 5.14:1. The variation in the TSA also changes the ranges of the experiment variables. You can see these changes by comparing the corresponding R-min and R-max values for Variable C in Example Design 1A with the corresponding values in Example Design 1B.


EXAMPLE DESIGN 1B: RUNS ADJUSTED TO A 100 PERCENT MIXTURE


Run #
Variable A (X1)
(%)
Variable B (X2)
(%)
Variable C (X3)
(%)
TSA
(%)
1 0 0 100 100
2 20 0 80 100
3 0 20 80 100
4 17 17 66 100
5 0 0 100 100
6 17 0 83 100
7 0 17 83 100
8 14 14 72 100
R-min 0 0 66 -
R-max 20 20 100 -



The changes in (1) the relative proportions of the variable level settings between experiment runs, and (2) the variable ranges are due to the changing TSA between runs. These changes may seem trivial. However, they can profoundly affect correlation among the mixture variable effects terms. The first question we must ask is whether or not the translation of Example Design 1A into Example Design 1B (its correct mixture representation) has in any way compromised the uncorrelation of the variable effects terms achieved by the full factorial design. The second question we must ask is how do we answer the first question, since we can not use standard correlation analysis to identify correlation problems in the mixture experiment design.


EXAMPLE DESIGN 2A: FACTORIAL DESIGN


Run #
Variable A (X1)
(grams)
Variable B (X2)
(grams)
Variable C (X3)
(grams)
1 15 5 30
2 40 5 30
3 15 40 30
4 40 40 30
5 15 5 70
6 40 5 70
7 15 40 70
8 40 40 70



To illustrate the problem, lets first compare Example Design 1A with Example Design 2A. Both designs are two-level full factorial designs for three variables. Given independent variables, both designs uncorrelate the experiment variable main effects terms and the interaction effects terms. However, lets now compare Example Design 1B with Example Design 2B below. As before, Example Design 2B represents the translation of Example Design 2A into its correct mixture design representation, again given that the three variables make up the entire mixture. Can you tell that the translation of Example Design 1A into its correct mixture representation (Example Design 1B) has severely compromised the uncorrelation of the variable effects terms? Can you tell that the translation of Example Design 2A into Example Design 2B has not compromised the uncorrelation of the variable effects terms?


EXAMPLE DESIGN 2B: RUNS ADJUSTED TO A 100 PERCENT MIXTURE


Run #
Variable A (X1)
(%)
Variable B (X2)
(%)
Variable C (X3)
(%)
1 30 10 60
2 53 7 40
3 18 47 35
4 36 36 28
5 17 5 78
6 35 4 61
7 12 32 56
8 27 27 46



D.o.E. FUSION's eigen analysis of an experiment design matrix generates a statistic called a variance inflation factor (VIF) for each variable-effect term. These VIF's range from a low of one, for a term that has no correlation to any other term or terms, to a high of infinity, for a term that is perfectly correlated to one or more other terms. On the scale of one to infinity, we will use the convention that a term has no correlation problems if its corresponding VIF is less than or equal to 10.

In Example Design 1A all experiment variable linear terms and pairwise interaction terms have VIF's equal to 1.0. This means that, as expected, there is no correlation at all among these terms in this design. However, in Example Design 1B the VIF's range from a low of 1.4 for Variable C to a high of 120 for both Variable A and Variable B. For the interaction-effects terms, the VIF's range from a low of 41 to a high of 75. This means that the translation of the factorial design into its true mixture representation has profoundly compromised the uncorrelation of the variable-effects terms in this design. You should understand that no analysis of data, no matter how sophisticated, can eliminate the problem correlations from this design and yield accurate estimates of the variable effects. Therefore, Example Design 1 is a fatally-flawed experiment.

In Example Design 2A all experiment variable linear terms and pairwise interaction terms also have VIF's equal to 1.0. As stated, Example Design 2B represents the translation of the full factorial design into its true mixture representation. In this design the VIF's do increase somewhat. However, the largest VIF in this design is 9.7, corresponding to the X1X3 term (the interaction between Variable A and Variable C). This means that the translation of the factorial design into its true mixture representation has not significantly compromised the uncorrelation of the variable-effects terms in this design.

The comparisons of Example Designs 1 and 2 discussed above clearly show the risk in using factorial designs as a basis for mixture experiments. Depending on our variable ranges, one factorial mixture experiment design may yield good data while another may be fatally flawed. At the very least we need a product like D.o.E. FUSION to tell us which is which. However, as we shall see later, D.o.E. FUSION can generate true mixture designs which do not have any of the inherent risk in factorial mixture experiment designs.

The next problem that we must address is the translation of the factorial design into its mixture representation when the mixture contains more than the number of variables in our experiment. For example, suppose we have a mixture consisting of four or more variables: A, B, C, D, E, .... Suppose further that we want to study variables A, B, and C, and that we want to use Example Design 2A as the factorial design on which to base our experiment. In this case we would have an additional variable called BASE. In a given design run, the BASE level setting represents the total amount of variables D, E, ... , and is equal to the TSA minus the amounts of variables A, B, and C. However, since we must include variables D and E in each run, we must define a final sample amount (FSA) that is larger than the largest TSA in the original factorial design. The BASE variable level settings in a given run will then equal the FSA minus the TSA for that run. To illustrate, Example Design 3 below represents a translation of Example Design 2A into a mixture experiment that includes the BASE variable. In the original design the largest TSA was 150 grams, corresponding to Run #8. However, for this translation we used an FSA of 200 grams. This sets the BASE variable level setting in Run #8 to 50 grams (200 grams - 150 grams), which translates to 25% ([50 grams/200 grams]*100%).


EXAMPLE DESIGN 3: RUNS ADJUSTED TO A 100 PERCENT MIXTURE WITH BASE


Run #
Variable A (X1)
(%)
Variable B (X2)
(%)
Variable C (X3)
(%)
BASE
(%)
1 7.5 2.5 15 75.0
2 20.0 2.5 15 62.5
3 7.5 20.0 15 57.5
4 20.0 20.0 15 45.0
5 7.5 2.5 35 55.0
6 20.0 2.5 35 42.5
7 7.5 20.0 35 37.5
8 20.0 20.0 35 25.0



In Example Design 2B the translation of the factorial design into its true mixture representation did not compromise the uncorrelation of the variable terms. However, Example Design 3 contains profound correlations between the variable linear-effects terms and the pairwise interaction effects terms involving the BASE variable (the Xi*X4 terms, i 4).. Therefore, whether or not Example Design 3 above is a good experiment design or a fatally flawed experiment design depends entirely on the nature of the BASE variable. If the BASE variable is only water or a true diluent, then the Xi*X4 interaction effects are probably nonexistent. In this case Example Design 3 is a good experiment design. However, if the BASE variable consists of an additional component or components that are themselves effectors of the responses of interest, then these variables may very well interact with our three experiment variables. In this case, as in the case of Example Design 1B, the translation of the factorial design into its true mixture representation has profoundly compromised the uncorrelation of the variable-effects terms. Again, no analysis of data, no matter how sophisticated, can eliminate the problem correlations from this design and yield accurate estimates of the variable effects. Therefore, Example Design 3 is a fatally-flawed experiment.


D.o.E. FUSION: TRUE MIXTURE EXPERIMENT DESIGNS

D.o.E. FUSION generates true mixture experiment designs using the most advanced algorithm design technology in software today. D.o.E. FUSION is the only PC software product that can design experiments with both mixture and process (non-mixture) variables, and only D.o.E. FUSION can address so many real-world restrictions on how mixture variables can be combined. These constraints include Boundary constraints (upper and/or lower limit level setting combinations), Ratio constraints (stoichiometry), and Mixture Within Mixture constraints (constraints on subset concentrations like total solvent concentration).

D.o.E. FUSION can easily design an experiment that overcomes the problems of factorial mixture designs. For example, D.o.E. FUSION designed a mixture experiment using Variables A, B, and C and the ranges from Example Design 1A. This is Example Design 4 below. Recall that the translation of Example Design 1A into its correct mixture representation profoundly compromised the uncorrelation of the variable-effects terms. However, Example Design 4 is a true mixture design, and has no such correlation problems among the variable-effects terms. In fact, the largest VIF in this design is 2.0. This clearly demonstrates the superiority of a true mixture design capability like D.o.E. FUSION. It is simply the best way to design correct and effective experiments for studying mixtures.


EXAMPLE DESIGN 4: D.o.E. FUSION TRUE MIXTURE DESIGN


Run #
Variable A (X1)
(grams)
Variable B (X2)
(grams)
Variable C (X3)
(grams)
1 7 7 86
2 0 20 80
3 0 0 100
4 20 0 80
5 0 10 90
6 10 10 80
7 10 0 90
8 3 13 84
9 3 3 94
10 13 3 84

 

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