Overview of Mixture Designs
You can choose from the following types of mixture designs:
Optimal
invokes the custom designer with all the mixture variables already defined.
Simplex Centroid
Specify the degree to which the factor combinations are made.
Simplex Lattice
Specify how many levels you want on each edge of the grid.
Extreme Vertices
Specify linear constraints or restrict the upper and lower bounds to be within the 0 to 1 range.
ABCD Design
Creates a screening design for mixtures devised by Snee (1975).
Space Filling
Constructs a design that accommodates linear constraints. Design points are spread throughout the design space.
Linear Constraints
You can specify linear constraints in the following design types:
• Optimal
• Extreme Vertices
• Space Filling
• Design Workflow
Mixture Design Window
The mixture design window updates as you work through the design steps. For more information, see “The DOE Workflow: Describe, Specify, Design”. The workflow and outlines vary depending on design type.
Figure 13.2 Workflow for Optimal Mixture Designs
Responses
Use the Responses outline to specify one or more responses.
Tip: When you have completed the Responses outline, consider selecting Save Responses from the red triangle menu. This option saves the response names, goals, limits, and importance values in a data table that you can later reload in DOE platforms.
Figure 13.3 Responses Outline
Add Response
Enters a single response with a goal type of Maximize, Match Target, Minimize, or None. If you select Match Target, enter limits for your target value. If you select Maximize or Minimize, entering limits is not required but can be useful if you intend to use desirability functions.
Remove
Removes the selected responses.
Number of Responses
Enters additional responses so that the number that you enter is the total number of responses. If you have entered a response other than the default Y, the Goal for each of the additional responses is the Goal associated with the last response entered. Otherwise, the Goal defaults to Match Target. Click the Goal type in the table to change it.
The Responses outline contains the following columns:
Response Name
The name of the response. When added, a response is given a default name of Y, Y2, and so on. To change this name, double-click it and enter the desired name.
Goal, Lower Limit, Upper Limit
The Goal tells JMP whether you want to maximize your response, minimize your response, match a target, or that you have no response goal. JMP assigns a Response Limits column property, based on these specifications, to each response column in the design table. It uses this information to define a desirability function for each response. The Profiler and Contour Profiler use these desirability functions to find optimal factor settings. For further details, see the Profiler chapter in the Profilers book and “Response Limits” in the “Column Properties” appendix.
‒ A Goal of Maximize indicates that the best value is the largest possible. If there are natural lower or upper bounds, you can specify these as the Lower Limit or Upper Limit.
‒ A Goal of Minimize indicates that the best value is the smallest possible. If there are natural lower or upper bounds, you can specify these as the Lower Limit or Upper Limit.
‒ A Goal of Match Target indicates that the best value is a specific target value. The default target value is assumed to be midway between the Lower Limit and Upper Limit.
‒ A Goal of None indicates that there is no goal in terms of optimization. No desirability function is constructed.
Note: If your target response is not midway between the Lower Limit and the Upper Limit, you can change the target after you generate your design table. In the data table, open the Column Info window for the response column (Cols > Column Info) and enter the desired target value.
Importance
When you have several responses, the Importance values that you specify are used to compute an overall desirability function. These values are treated as weights for the responses. If there is only one response, then specifying the Importance is unnecessary because it is set to 1 by default.
Editing the Responses Outline
In the Responses outline, note the following:
• Double-click a response to edit the response name.
• Click the goal to change it.
• Click on a limit or importance weight to change it.
• For multiple responses, you might want to enter values for the importance weights.
Response Limits Column Property
The Goal, Lower Limit, Upper Limit, and Importance that you specify when you enter a response are used in finding optimal factor settings. For each response, the information is saved in the generated design data table as a Response Limits column property. JMP uses this information to define the desirability function. The desirability function is used in the Prediction Profiler to find optimal factor settings. For further details about the Response Limits column property and examples of its use, see “Response Limits” in the “Column Properties” appendix.
If you do not specify a Lower Limit and Upper Limit, JMP uses the range of the observed data for the response to define the limits for the desirability function. Specifying the Lower Limit and Upper Limit gives you control over the specification of the desirability function. For more details about the construction of the desirability function, see the Profiler chapter in the Profilers book.
Factors
Add factors in the Factors outline.
Tip: When you have completed the Factors outline, consider selecting Save Factors from the red triangle menu. This saves the factor names, roles, changes, and values in a data table that you can later reload.
Figure 13.4 Factors Outline
Add
Enter the number of factors to add and click Add.
Remove Selected
Removes the selected factors.
Factors List
The Factors list contains the following columns:
Name
The name of the factor. When added, a factor is given a default name of X1, X2, and so on. To change this name, double-click it and enter the desired name.
Role
Specifies the Design Role of the factor. The Design Role column property for the factor is saved to the data table. This property ensures that the factor type is modeled appropriately.
Values
The experimental settings for the factors. To insert Values, click on the default values and enter the desired values.
Editing the Factors List
In the Factors list, do the following:
• To edit a factor name, double-click the factor name.
• To edit a value, click the value in the Values column.
Linear Constraints
Click the Linear Constraint button to enter one or more linear inequality constraints. A template appears for a linear expression involving all the continuous factors in your design. Enter coefficient values for the factors and select the direction of the inequality to reflect your linear constraint. Specify the constraining value in the box to the right of the inequality. To add more constraints, click Linear Constraint again.
Note: When you save a script for a design that involves a linear constraint, the script expresses the linear constraint as a less than or equal to inequality (
).
).Examples of Mixture Design Types
This section presents examples of each of the different mixture design types:
Optimal Mixture Design
Optimal mixture designs use the Custom Design platform with the mixture variables entered into the response and factors panels.
To create an optimal mixture design:
1. Select DOE > Classical > Mixture Design.
2. Enter factors and responses. The steps for entering responses are outlined in “Responses” in the “Custom Designs” chapter.
3. After you enter responses and factors, click Continue.
4. Click Optimal on the Choose Mixture Design Type panel.
5. Open the Define Factor Constraints node and click Add Constraint to add linear constraints, if you have any.
6. Add effects to the model using the instructions below.
7. In the Design Generation panel, make selections relative to blocks, center points, replicates, and the number of runs.
8. Click Make Design to generate the Mixture Design report, which displays the design and Design Evaluation report.
9. Click Make Table in the Output Options panel of the Mixture Design report to generate the data table.
Adding Effects to the Model
Initially, the Model panel lists only the main effects corresponding to the factors that you entered, as shown in Figure 13.5.
Figure 13.5 The Model Panel
However, you can add factor interactions, specific crossed factor terms, powers, or Scheffé Cubic terms to the model.
• To add interaction terms to a model, click the Interactions button and select 2nd, 3rd, 4th, or 5th. For example, if you have factors X1 and X2, click Interactions > 2nd and X1*X2 is added to the list of model effects.
• To add crossed effects to a model, highlight the factors and effects that you want to cross and click the Cross button.
• To add powers of continuous factors to the model, click the Powers button and select 2nd, 3rd, 4th, or 5th.
• When you want a mixture model with third-degree polynomial terms, the Scheffé Cubic button provides a polynomial specification of the surface by adding terms of the form X1*X2*(X1-X2).
Simplex Centroid Design
A simplex centroid design of degree k with n factors consists of mixture runs with the following characteristics:
• all one factor
• all combinations of two factors at equal levels
• all combinations of three factors at equal levels
• and so on, up to k factors at a time combined at k equal levels.
A center point run with equal amounts of all the ingredients is always included.
Creating the Design
To create a simplex centroid design:
1. Select DOE > Classical > Mixture Design.
2. Enter factors and responses. The steps for entering responses are outlined in “Responses” in the “Custom Designs” chapter.
3. After you enter responses and factors, click Continue.
4. Enter the number of ingredients in the box under K. JMP creates runs for each ingredient without mixing. JMP also creates runs that mix equal proportions of K ingredients at a time to the specified limit.
5. Click the Simplex Centroid button.
6. View Design and Output Options, as illustrated in Figure 13.6.
Figure 13.6 Example of Factor Settings and Output Options
7. Specify Run Order, which is the order in which the runs that appear in the data table that you will create. Run order choices are:
Keep the Same: The rows (runs) in the output table appear as they do in the Factor Settings panel.
Sort Left to Right: The rows (runs) in the output table appear sorted from left to right.
Randomize: The rows (runs) in the output table appear in a random order.
Sort Right to Left: The rows (runs) in the output table appear sorted from right to left.
Randomize within Blocks: The rows (runs) in the output table appear in random order within the blocks that you set up.
8. Specify Number of Replicates. The number of replicates is the number of times to replicate the entire design, including center points. Enter the number of times you want to replicate the design in the associated text box. One replicate doubles the number of runs.
9. Click Make Table.
Simplex Centroid Design Examples
The table of runs for a design of degree 1 with three factors (left in Figure 13.7) shows runs for each single ingredient followed by the center point. The table of runs to the right is for three factors of degree 2. The first three runs are for each single ingredient, the second set shows each combination of two ingredients in equal parts, and the last run is the center point.
Figure 13.7 Three-Factor Simplex Centroid Designs of Degrees 1 and 2
To generate the two sets of runs in Figure 13.7:
1. Choose DOE > Classical > Mixture Design.
2. Enter three mixture factors.
3. Click Continue.
4. Enter 1 in the K box, and click Simplex Centroid to see the design on the left in Figure 13.8.
5. Click the Back button, click Continue, and then enter 2 in the K box. Then click Simplex Centroid to see the design on the right in Figure 13.8.
Figure 13.8 Create Simplex Centroid Designs of Degrees 1 and 2
As another example:
1. Choose DOE > Classical > Mixture Design.
2. Enter five factors and click Continue.
3. Use the default value, 4, in the K box.
4. Click Simplex Centroid.
5. Click Make Table.
Figure 13.9 shows part of the 31-run design. Note that your table might look different because the design was created with Run Order set to Randomize.
Figure 13.9 Partial Listing of Factor Settings for Five-Factor Simplex Centroid Design
Simplex Lattice Design
The simplex lattice design is a space filling design that creates a triangular grid of runs. The design is the set of all combinations where the factors’ values are i / m, where i is an integer that varies from 0 to m such that the sum of the factor values is 1.
To create simplex lattice designs:
1. Select DOE > Classical > Mixture Design.
2. Enter factors and responses. The steps for entering responses are outlined in “Responses” in the “Custom Designs” chapter.
3. Click Continue.
4. Specify the number of levels that you want in the Mixture Design Type panel and click Simplex Lattice.
Figure 13.10 shows the runs for three-factor simplex lattice designs of degrees 3, 4, and 5, with their corresponding geometric representations. In contrast to the simplex centroid design, the simplex lattice design does not necessarily include the centroid.
Figure 13.10 Three-Factor Simplex Lattice Designs for Factor Levels 3, 4, and 5
Figure 13.11 lists the runs for a simplex lattice of degree 3 for five effects. In the five-level example, the runs creep across the hyper-triangular region and fill the space in a grid-like manner.
Figure 13.11 JMP Design Table for Simplex Lattice with Five Variables, Order (Degree) 3
Extreme Vertices Design
The extreme vertices design can be selected only if you have modified the ranges on the factors in the Factors panel or if you have specified a linear constraint. The extreme vertices design accounts for factor limits and selects vertices and their averages (formed by factor limits) as design points. Additional limits are usually in the form of range constraints, upper bounds, and lower bounds on the factor values.
The extreme vertices design finds the corners (vertices) of a factor space constrained by limits specified for one or more of the factors. The property that the factors must be nonnegative and must add up to one is the basic mixture constraint that makes a triangular-shaped region.
Sometimes other ingredients need range constraints that confine their values to be greater than a lower bound or less than an upper bound. Range constraints chop off parts of the triangular-shaped (simplex) region to make additional vertices. It is also possible to have a linear constraint, which defines a linear combination of factors to be greater or smaller than some constant.
The geometric shape of a region bound by linear constraints is called a simplex. Because the vertices represent extreme conditions of the operating environment, they are often the best places to use as design points in an experiment.
You usually want to add points between the vertices. The average of points that share a constraint boundary is called a centroid point, and centroid points of various degrees can be added. The centroid point for two neighboring vertices joined by a line is a second degree centroid because a line is two dimensional. The centroid point for vertices sharing a plane is a third degree centroid because a plane is three dimensional, and so on.
Creating the Design
Follow these steps to create an extreme vertices design. The next sections show examples with specific constraints.
1. Select DOE > Classical > Mixture Design.
2. Enter factors and responses. These steps are outlined in “Custom Design Window” in the “Custom Designs” chapter. If your factor ranges are constrained, enter the limits as upper and lower limits in the Factors panel (see Figure 13.12).
3. Click Continue.
4. If you have linear constraints, click Linear Constraints and enter them.
Note: The extreme vertices design can be selected only if you have modified the ranges on the factors in the Factors panel or if you have specified a linear constraint.
5. In the Degree text box, enter the degree of the centroid point that you want to add. The centroid point is the average of points that share a constraint boundary.
6. If you have linear constraints, click the Linear Constraints button for each constraint that you want to add. Use the text boxes that appear to define a linear combination of factors to be greater or smaller than some constant.
7. Click Extreme Vertices to see the factor settings.
8. Specify the Run Order. This determines the order of the runs in the data table when it is created. Run order choices are:
Keep the Same: The rows (runs) in the output table appear as they do in the Design panel.
Sort Left to Right: The rows (runs) in the output table appear sorted from left to right.
Randomize: The rows (runs) in the output table appear in a random order.
Sort Right to Left: The rows (runs) in the output table appear sorted from right to left.
Randomize within Blocks: The rows (runs) in the output table appear in random order within the blocks that you set up.
9. Enter the sample size that you want in the Choose desired sample size text box.
10. (Optional) Click Find Subset to generate the optimal subset having the number of runs specified in sample size box described in Step 8. The Find Subset option uses the row exchange method (not coordinate exchange) to find the optimal subset of rows.
11. Click Make Table.
An Extreme Vertices Example with Range Constraints
The following example design table is for five factors with the range constraints shown in Figure 13.12, where the ranges are smaller than the default 0 to 1 range.
1. Select DOE > Classical > Mixture Design.
2. Add two additional factors (for a total of 5 factors) and give them the values shown in Figure 13.12.
3. Click Continue.
4. Enter 4 in the Degree text box (Figure 13.12).
Figure 13.12 Example of Five-factor Extreme Vertices
5. Click Extreme Vertices.
6. Select Sort Left to Right from the Run Order menu.
7. Click Make Table.
Figure 13.13 shows a partial listing of a resulting design. Note that the Rows panel in the data table shows that the table has the default 116 runs.
Figure 13.13 JMP Design Table for Extreme Vertices with Range Constraints
Suppose you want fewer runs. You can go back and enter a different sample size (number of runs).
8. Click Back and then click Continue.
9. Enter 4 in the Degree text box and click Extreme Vertices.
10. Choose desired sample size text box, enter 10.
11. Click Find Subset to generate an optimal subset having the number of runs specified.
The resulting design (Figure 13.14) is an optimal 10-run subset of the 116 current runs. This is useful when the extreme vertices design generates a large number of vertices. Your design might look different, because there are different subsets that achieve the same D-efficiency.
Figure 13.14 JMP Design Table for 10-Run Subset of the 116 Current Runs
Note: The Find Subset option uses the row exchange method (not coordinate exchange) to find the optimal subset of rows.
An Extreme Vertices Example with Linear Constraints
Consider the classic example presented by Snee (1979) and Piepel (1988). This example has three factors, X1, X2, and X3, with five individual factor bound constraints and three additional linear constraints:
|
X1 ≥ 0.1
X1 ≤ 0.5
X2 ≥ 0.1
X2 ≤ 0.7
X3 ≤ 0.7
|
90 ≤ 85*X1 + 90*X2 + 100*X3
85*X1 + 90*X2 + 100*X3 ≤ 95
.4 ≤ 0.7*X1 + X3
|
To enter these constraints:
1. Enter the upper and lower limits in the factors panel.
2. Click Continue.
3. Click the Linear Constraint button three times. Enter the constraints as shown in Figure 13.15.
4. Click the Extreme Vertices button.
5. Change the run order to Sort Right to Left, and keep the sample size at 13. See Figure 13.15 for the default Design and completed Output Options.
6. Click Make Table.
Figure 13.15 Constraints
This example is best understood by viewing the design as a ternary plot, as shown at the end of this chapter, in Figure 13.18. The ternary plot shows how close to one a given component is by how close it is to the vertex of that variable in the triangle. See “Creating Ternary Plots” for details.
Extreme Vertices Method: How It Works
If there are linear constraints, JMP uses the CONSIM algorithm developed by R.E. Wheeler, described in Snee (1979) and presented by Piepel (1988) as CONVRT. The method is also described in Cornell (1990, Appendix 10a). The method combines constraints and checks to see whether vertices violate them. If so, it drops the vertices and calculates new ones. The method named CONAEV for doing centroid points is by Piepel (1988).
If there are no linear constraints (only range constraints), the extreme vertices design is constructed using the XVERT method developed by Snee and Marquardt (1974) and Snee (1975). After the vertices are found, a simplex centroid method generates combinations of vertices up to a specified order.
The XVERT method first creates a full 2nf – 1 design using the given low and high values of the nf – 1 factors with smallest range. Then, it computes the value of the one factor left out based on the restriction that the factors’ values must sum to one. It keeps points that are not in factor’s range. If not, it increments or decrements the value to bring it within range, and decrements or increments each of the other factors in turn by the same amount. This method keeps the points that still satisfy the initial restrictions.
The above algorithm creates the vertices of the feasible region in the simplex defined by the factor constraints. However, Snee (1975) has shown that it can also be useful to have the centroids of the edges and faces of the feasible region. A generalized n-dimensional face of the feasible region is defined by nf – n of the boundaries and the centroid of a face defined to be the average of the vertices lying on it. The algorithm generates all possible combinations of the boundary conditions and then averages over the vertices generated on the first step.
ABCD Design
This approach by Snee (1975) generates a screening design for mixtures. To create an ABCD design:
1. Select DOE > Classical > Mixture Design.
2. Enter factors and responses. The steps for entering responses are outlined in “Responses” in the “Custom Designs” chapter.
3. After you enter responses and factors, click Continue.
4. Click the ABCD Design button.
5. View factor settings and Output Options.
6. Specify Run Order, which is the order you want the runs to appear in the data table when it is created. Run order choices are:
Keep the Same: The rows (runs) in the output table appear as they do in the Factor Settings panel.
Sort Left to Right: The rows (runs) in the output table appear sorted from left to right.
Randomize: The rows (runs) in the output table appear in a random order.
Sort Right to Left: The rows (runs) in the output table appear sorted from right to left.
Randomize within Blocks: The rows (runs) in the output table appear in random order within the blocks that you set up.
7. Specify Number of Replicates. The number of replicates is the number of times to replicate the entire design, including center points. Enter the number of times you want to replicate the design in the associated text box. One replicate doubles the number of runs.
8. Click Make Table.
Space Filling Design
The Space Filling mixture design type spreads design points fairly uniformly throughout the design region. It accommodates linear constraints. The design is generated in a fashion similar to the Fast Flexible Filling design method found under DOE > Special Purpose > Space Filling Design (“Fast Flexible Filling Designs”).
Two red triangle options relate specifically to Space Filling Designs:
FFF Optimality Criterion
For the Fast Flexible Filling mixture design type, enables you to select between the MaxPro criterion (the default) and the Centroid criterion. See “FFF Optimality Criterion”.
Advanced Options > Set Average Cluster Size
For the Fast Flexible Filling mixture design type, enables you to specify the average number of randomly generated points used to define each cluster or, equivalently, each design point. See “Set Average Cluster Size”.
FFF Optimality Criterion
The algorithms for Fast Flexible Filling designs begin by generating a large number of random points within the specified design region. These points are then clustered using a Fast Ward algorithm into a number of clusters that equals the Number of Runs that you specified.
The final design points can be obtained by using the default MaxPro (maximum projection) optimality criterion or by selecting the Centroid criterion. You can find these options under FFF Optimality Criterion in the report’s red triangle menu. MaxPro
For p factors and n equal to the specified Number of Runs, the MaxPro criterion strives to find points in the clusters that minimize the following criterion:
The MaxPro criterion maximizes the product of the distances between potential design points in a way that involves all factors. This supports the goal of providing good space-filling properties on projections of factors. See Joseph et al. (2015). The Max Pro option is the default.
Centroid
This method places a design point at the centroid of each cluster. It has the property that the average distance from an arbitrary point in the design space to its closest neighboring design point is smaller than for other designs.
Note: You can set a preference to always use a given optimality criterion. Select File > Preferences > Platforms > DOE. Select FFF Optimality Criterion and select your preferred criterion.
Set Average Cluster Size
The Set Average Cluster Size option is found under Advanced Options in the Mixture Design red triangle menu. This option enables you to specify the average number of uniformly generated points used to define each cluster or, equivalently, each design point.
By default, if the number of Runs for the Space Filling design type is 200 or smaller, a total of 10,000 random uniformly generated points are used as the basis for the clustering algorithm. When the number of Runs exceeds 200, the default value is 50. Increasing this value can be particularly useful in designs with a large number of factors.
Note: Depending on the number of factors and the specified value for Runs, you might want to increase the average number of initial points per design point by selecting Advanced Options > Set Average Cluster Size.
Linear Constraints
The design region can be restricted by selecting the Linear Constraint option in the Linear Constraints outline.
When you specify linear constraints, the random points that form the basis for the clustering algorithm are randomly distributed within the constrained design region. The clustering algorithm uses these points.
Space Filling Example
To create a space filling design:
1. Select DOE > Classical > Mixture Design.
2. Enter factors and responses.
3. Click Continue.
4. Add Linear Constraints, if you have any.
5. Specify the number of runs you want in the Runs box to the right of the Space Filling button in the Mixture Design Type panel.
6. Click Space Filling.
A Space Filling Example with a Linear Constraint
Consider a three-factor mixture design with the single linear constraint: 
. Figure 13.16 shows a ternary plot for a 30-run Space Filling design that satisfies this constraint. (For a discussion of ternary plots, see “Creating Ternary Plots”.) This design is constructed using the Centroid FFF Optimality Criterion. Note that the points fall in the constrained design region and are fairly well spread throughout this region.
. Figure 13.16 shows a ternary plot for a 30-run Space Filling design that satisfies this constraint. (For a discussion of ternary plots, see “Creating Ternary Plots”.) This design is constructed using the Centroid FFF Optimality Criterion. Note that the points fall in the constrained design region and are fairly well spread throughout this region.Figure 13.16 Space Filling Design with One Linear Constraint
Creating Ternary Plots
A mixture problem in three components can be represented in two dimensions because the third component is a linear function of the others. The ternary plot in Figure 13.18 shows how close to one (1) a given component is by how close it is to the vertex of that variable in the triangle. The plot in Figure 13.17 illustrates a ternary plot.
Figure 13.17 Ternary Plot for Mixture Design
The Piepel (1979) example referenced in “An Extreme Vertices Example with Linear Constraints” is best understood by the ternary plot shown in Figure 13.18.
To view a mixture design as a ternary plot:
1. Create the Piepel mixture data as shown previously, or open the table called Piepel.jmp, found in the Design Experiments folder of the Sample Data Library.
2. Choose Graph > Ternary Plot.
3. In the Ternary Plot launch window, specify the three mixture components and click OK.
The ternary plot platform recognizes the three factors as mixture factors, and also considers the upper and lower constraints entered into the Factors panel when the design was created. The ternary plot uses shading to exclude the unfeasible areas excluded by those constraints.
The Piepel data had additional constraints, entered as linear constraints for the extreme vertices design. There are six active constraints, six vertices, and six centroid points shown on the plot, as well as three inactive (redundant) constraints. The feasible area is the inner white polygon delimited by the design points and constraint lines.
Figure 13.18 Diagram of Ternary Plot Showing Piepel Example Constraints
Fitting Mixture Designs
When fitting a model for mixture designs, take into account that the factors sum to a constant; a traditional full linear model are not fully estimable.
The recommended response surface model is called the Scheffé polynomial (Scheffé 1958). See the discussion of Cox Mixtures and the Scheffé Cubic macro in the Standard Least Squares chapter in the Fitting Linear Models book. The Scheffé polynomial model does the following:
• suppresses the intercept
• includes all the linear main-effect terms
• excludes all the square terms (such as X1*X1)
• includes all the cross terms (such as X1*X2)
To fit a model:
1. Choose DOE > Classical > Mixture Design and make the design data table. To fit a model, the Y column in the data table must contain values. Either assign responses or select Simulate Responses from the red triangle menu before you click Make Table.
2. The design data table stores the model in the data table as a table property. This table property is a JSL script called Model, located in the left panel of the table.
3. Click the green triangle next to the Model script to open the Fit Model launch window, which is automatically filled with the saved model.
4. Click Run on the Fit Model window.
In this model, the parameters are easy to interpret (Cornell 1990). The coefficients on the linear terms are the fitted response at the extreme points where the mixture consists of a single factor. The coefficients on the cross terms indicate the curvature across each edge of the factor space.
The model report usually has several sections of interest, including the whole model tests, Analysis of Variance reports, and response surface reports, which are described below.
Whole Model Tests and Analysis of Variance Reports
In a whole-model Analysis of Variance table, JMP traditionally tests that all the parameters are zero except for the intercept. In a mixture model without an intercept, JMP looks for a hidden intercept, in the sense that a linear combination of effects is a constant. If it finds a hidden intercept, it does the whole model test with respect to the intercept model rather than a zero-intercept model. This test is equivalent to testing that all the parameters are zero except the linear parameters, and testing that they are equal.
The hidden-intercept property also causes the R2 to be reported with respect to the intercept model rather than reported as missing.
Understanding Response Surface Reports
When there are effects marked as response surface effects “&RS,” JMP creates additional reports that analyze the fitted response surface. These reports were originally designed for full response surfaces, not mixture models. However, JMP might encounter a no-intercept model and find a hidden intercept with linear response surface terms, but no square terms. Then it folds its calculations, collapsing on the last response surface term to calculate critical values for the optimum. This can be done for any combination that yields a constant and involves the last response surface term.
A Chemical Mixture Example
Three plasticizers (p1, p2, and p3) comprise 79.5% of the vinyl used for automobile seat covers (Cornell, 1990). Within this 79.5%, the individual plasticizers are restricted by the following constraints: 0.409 ≤ x1 ≤ 0.849, 0 ≤ x2 ≤ 0.252, and 0.151 ≤ x3 ≤ 0.274.
Create the Design
To create Cornell’s mixture design in JMP:
1. Select DOE > Classical > Mixture Design.
2. In the Factors panel, use the three default factors but name them p1, p2, and p3, and enter the high and low constraints as shown in Figure 13.19. Or, load the factors with the Load Factors command in the red triangle on the Mixture Design title bar. To import the factors, open Plastifactors.jmp, found in the Design Experiment sample data folder that was installed with JMP.
Figure 13.19 Factors and Factor Constraints for the Plasticizer Experiment
3. Click Continue.
4. Enter 3 in the Degree text box.
5. Click Extreme Vertices.
6. Click Make Table. JMP uses the 9 factor settings to generate a JMP table (Figure 13.20).
Figure 13.20 Extreme Vertices Mixture Design
Next, you add an extra five design runs by duplicating the vertex points and interior point, to give a total of 14 rows in the table.
Note: To identify the vertex points and the center (or interior) point, use the sample data script called LabelMixturePoints.jsl in the Sample Scripts folder installed with JMP.
7. Run the LabelMixturePoints.jsl to see the results in Figure 13.21, and highlight the vertex points and the interior point as shown.
Figure 13.21 Identify Vertices and Center Point with Sample Script
8. Select Edit > Copy, to copy the selected rows to the clipboard.
9. Select Rows > Add Rows and type 5 as the number of rows to add.
10. Click the At End radio button on the dialog and then click OK.
11. Highlight the new rows and select Edit > Paste to add the duplicate rows to the table.
The Plasticizer design with the results (Y values) that Cornell obtained are available in the sample data. Open Plasticizer.jmp in the sample data folder installed with JMP to see this table (Figure 13.22). You can check that, up to run order, the design that you created is identical to Cornell’s.
Figure 13.22 Plasticizer.jmp Data Table from the Sample Data Library
Analyze the Mixture Model
Use the Plasticizer.jmp data from the sample data library (Figure 13.22) to run the mixture model:
1. Click the green triangle next to the Model script in the upper left corner of the data table.
A completed Fit Model launch window appears.
2. Click Run to see the response surface analysis.
3. Plasticizer.jmp contains a column called Pred Formula Y. This column was added after the analysis by selecting Save Columns > Prediction Formula from the Response Y red triangle menu.
4. To see the prediction formula, right-click (press Ctrl and click on Macintosh) the column name and select Formula:
0–50.1465*p1 – 282.198*p2 – 911.648*p3 + p2*p1*317.363 + p2*p1*1464.330 + p3*p2*1846.218
Note: These results correct the coefficients reported in Cornell (1990).
The Response Surface Solution report (Figure 13.23) shows that a maximum predicted value of 19.570299 occurs at point (0.63505, 0.15568, 0.20927).
Figure 13.23 Mixture Response Surface Analysis
The Prediction Profiler
The report contains a prediction profiler.
1. If the profiler is not visible, click the red triangle in the Response Y title bar and select Factor Profiling > Profiler. You should see the initial profiler shown in Figure 13.24.
The crossed effects show as curvature in the prediction traces. Drag one of the vertical reference lines, and the other two lines move in the opposite direction maintaining their ratio.
Note: The axes of prediction profiler traces range from the upper and lower bounds of the factors, p1, p2, and p3, entered to create the design and the design table. When you experiment moving a variable trace, you see the other traces move such that their ratio is preserved. As a result, when the limit of a variable is reached, it cannot move further and only the third variable changes.
2. To limit the visible profile curves to bounds that use all three variables, select Profile at Boundary > Stop at Boundaries from Prediction Profiler red triangle menu.
3. If needed, select the Optimization and Desirability > Desirability Functions command to display the desirability function showing to the right of the prediction profile plots in Figure 13.25.
4. Select Optimization and Desirability > Maximize Desirability from the Prediction Profiler menu to see the best factor settings.
The profiler in Figure 13.25, displays optimal settings (rounded) of 0.6350 for p1, 0.1557 for p2, and 0.2093 for p3, which give an estimated response of 19.5703.
Figure 13.24 Initial Prediction Profiler
Figure 13.25 Maximum Desirability in Profiler for Mixture Analysis Example
The Mixture Profiler
The Fit Model report also has a Mixture Profiler that is useful for visualizing and optimizing response surfaces from mixture experiments.
Many of the features are the same as those of the Contour Profiler. However, some are unique to the Mixture Profiler:
• A ternary plot is used instead of a Cartesian plot, which enables you to view three mixture factors at a time.
• If you have more than three factors, radio buttons let you choose which factors to plot.
• If the factors have constraints, you can enter their low and high limits in the Lo Limit and Hi Limit columns. This setting shades non-feasible regions in the profiler.
Select Factor Profiling > Mixture Profiler from Response Y red triangle menu to see the mixture profiler for the plasticizer data, shown in Figure 13.26.
Figure 13.26 Mixture Profiler for Plasticizer Example
A Ternary Plot of the Mixture Response Surface
You can also plot the response surface of the plasticizer data as a ternary plot using the Ternary Plot platform and contour the plot with information from an additional variable:
1. Choose Graph > Ternary Plot.
2. Specify plot variables (p1, p2, and p3) and click X, Plotting, as shown in Figure 13.27.
3. To identify the contour variable (the prediction equation), select Pred Formula Y and click the Contour Formula button. The contour variable must have a prediction formula to form the contour lines, as shown by the ternary plots at the bottom in Figure 13.28. If there is no prediction formula, the ternary plot only shows points.
Figure 13.27 Ternary Plot Launch Window
4. Click OK and view the results, as shown in Figure 13.28.
By default, the ternary plot shows contour lines only. Add a fill by selecting Contour Fill from the Ternary Plot red triangle menu and then selecting Fill Above or Fill Below.
Figure 13.28 Ternary Plot of a Mixture Response Surface
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