Overview of Response Surface Designs
The Response Surface Design platform provides the classical central composite and Box-Behnken designs, including blocked versions of these designs. For central composite designs, you can control the placement of axial points and other aspects of the design. Response surface designs are available for continuous factors only and are provided for up to eight factors.
Tip: You can use DOE > Custom Design to construct optimal response surface designs that accommodate your specific experimental situation. Custom Design constructs response surface designs that are much more flexible than classical response surface designs. In particular, you can use the Custom Design platform to create response surface designs that involve categorical factors or more than eight continuous factors. You can also specify the number of runs and restrictions on the design space. For examples, see “Response Surface Experiments” in the “Examples of Custom Designs” chapter.
A central composite design (Figure 11.2) combines a two-level fractional factorial design and two other types of points:
• Center points, where all the factor values are set to the midrange value.
• Axial points, where one factor is set to a high or low value (an axial value) and all other factors are set to the midrange value.
Depending on your selections relative to axial points, a central composite design can have as many as five distinct settings for each factor and the axial points can extend beyond the specified range of the factors.
Figure 11.2 Central Composite Design for Three Factors
A Box-Behnken design (Figure 11.3) has only three levels per factor and has no design points at the vertices of the cube defined by the ranges of the factors. This type of design can be useful when you must avoid these points due to engineering considerations. But, the lack of design points at the vertices of the cube means that a Box-Behnken design has higher prediction variance, and so less precision, near the vertices compared to a central composite design.
Figure 11.3 Box-Behnken Design for Three Factors
In JMP, you can construct a response surface design in two ways:
• Using the Response Surface Design platform (for up to eight continuous factors)
• Using the Custom Design platform (and clicking the RSM button in the Model outline)
In both cases, the design table contains a Model script that you can run to fit a model. The Model script applies the Response Surface Effect attribute to each main effect, so that the main effects appear with a &RS suffix in the Fit Model window. This attribute ensures that the Fit Least Squares report contains a Response Surface report. For details about this report, see the Standard Least Squares Report and Options chapter in the Fitting Linear Models book.
Note: The Response Surface outline in the Standard Least Squares report is not shown for response surface designs that contain more than 20 continuous factors.
Example of a Response Surface Design
In this example, you construct a Box-Behnken design for a tire tread experiment. Your objective is to match a target value of 450 for a measure of elongation (Stretch). The stretch varies as a function of the amounts of Silica, Silane, and Sulfur used to manufacture the tire tread compound. You want to experiment over a wide range of factor settings to find the settings that achieve the target.
Construct a Box-Behnken Design
In this example, for convenience, you load the responses and factors from existing tables. When designing a new experiment on your own, enter the responses and factors manually. See “Responses” and “Factors”.
1. Select DOE > Classical > Response Surface Design.
2. Select Help > Sample Data Library and open Design Experiment/Bounce Response.jmp.
3. Click the Response Surface Design red triangle menu and select Load Responses.
4. Select Help > Sample Data Library and open Design Experiment/Bounce Factors.jmp.
5. Click the Response Surface Design red triangle menu and select Load Factors.
Figure 11.4 Responses and Factors Outlines for Tire Tread Design
In the Responses outline, notice that the Goal for Stretch is set to Match Target.
In the Choose a Design panel, possible designs appear.
Note: Setting the Random Seed in step 6 reproduces the exact results shown in this example. In constructing a design on your own, this step is not necessary.
6. (Optional) Click the Response Surface Design red triangle menu and select Set Random Seed. Type 12345 and click OK.
7. Click Continue to retain the Box-Behnken design selection.
8. Click Make Table.
Figure 11.5 Box-Behnken Design Table
At this point, conduct the experiment and enter the responses into the data table.
Analyze the Experimental Data
1. Select Help > Sample Data Library and open Design Experiment/Bounce Data.jmp.
The file Bounce Data.jmp contains your experiment results.
2. Run the Model script.
Notice that the main effects in the Construct Model Effects list are followed by the & RS suffix. This suffix indicates that these are response surface effects, which produce a Response Surface report in the Standard Least Squares report.
3. Click Run.
Figure 11.6 Lack of Fit and Effect Tests Reports
There is no indication of lack of fit and the Effect Tests report indicates that all but two higher-order terms (Silica*Silane and Silane*Silane) have p-values below 0.0001. See the Standard Least Squares chapter in the Fitting Linear Models book for more information about interpretation of the tables in Figure 11.6.
4. Click the disclosure icon next to Response Surface to open the report.
5. Click the disclosure icon next to Canonical Curvature.
Figure 11.7 Response Surface Report
The Coef table shown as the first part of the report gives a concise summary of the estimated model parameters. The first three columns give the coefficients of the second-order terms. The last column gives the coefficients of the linear terms. To see the prediction expression in its entirely, select Estimates > Show Prediction Expression from the Response Stretch red triangle menu.
The Solution report gives the coordinates of the point where the single critical value occurs. In this instance, that point is a saddle point (a point where neither a maximum nor a minimum occurs) and falls outside the range of the design space.
The Canonical Curvature report shows eigenvalues and eigenvectors of the effects. These give information about the nature and direction of the surface’s curvature. The large positive eigenvalue of 62.9095 indicates positive curvature and the eigenvector values indicate that the curvature is primarily in the Silica direction. The large negative eigenvalue of -74.9584 indicates negative curvature and the eigenvector values indicate that the curvature is primarily in the Sulfur direction.
See the Standard Least Squares chapter in the Fitting Linear Models book for details about the response surface analysis tables in Figure 11.7.
Next, use the prediction profiler and the contour profiler to find optimal settings.
Explore Optimal Settings
1. Click the Prediction Profiler red triangle menu and select Optimization and Desirability > Maximize Desirability.
Figure 11.8 Prediction Profiler for Bounce Data.jmp with Desirability Maximized
Note: Your optimal settings might differ. This is because there are many points for which the predicted Stretch is 450.
When you specified the response, the goal was set to match a target of 450, with lower and upper limits of 350 and 550. This goal was carried over to the design table and these limits were put in a Response Limits column property for Stretch. A desirability function is constructed from these response limits (top right cell in Figure 11.8). For details, see “Response Limits” in the “Column Properties” appendix.
When you maximize the desirability function, JMP identifies one combination of factor level settings (usually out of many possible combinations) that results in a predicted Stretch of 450. Figure 11.8 shows these settings as Silica = 1.069, Sulfur = 1.972, and Silane = 40.000. Next, you use the Contour Profiler to identify other points that maximize the desirability function.
For more information about the Prediction Profiler, see the Profiler chapter in the Profilers book.
2. Click the Response Stretch red triangle menu and select Factor Profiling > Contour Profiler. Suppose that you want to achieve your target while setting Sulfur to the value 2.0. Also, you want to make sure that the settings that you choose for Silane and Silica maintain predicted Stretch within 5 units of 450.
3. In the plot controls area above the plot, click the radio button under Vert to the left of Silane.
Figure 11.9 Contour Profiler for Bounce Data.jmp
The plot shows the contour of values of Silane and Silica for Stretch at 425 and Sulfur at 2.3.
4. Set the Current X for Sulfur to 2.
5. Set the Contour for Stretch to 450.
6. Set the Lo Limit and Hi Limit for Stretch to 445 and 455, respectively. Press Enter.
Figure 11.10 Contour Profiler Showing Optimal Settings for Silica and Silane
The unshaded band of Silica and Silane values gives predicted Stretch between 445 and 455 when Sulfur is set at 2.0. The values on the solid red curve give predicted Stretch of 450.
7. Drag the crosshairs that appear in the plot to the unshaded band to find settings for Silica and Silane that are best for your process from a practical perspective.
Suppose that your process is known to be more robust at low levels of Silane than at high levels. Then you might consider the settings in Figure 11.11.
Figure 11.11 Contour Profiler Showing Specific Settings for Silica and Silane
For Sulfur = 2.0, the factor settings identified by the crosshairs are Silica = 1.045 and Silane = 41.71. These settings are shown under Current X. At these settings, the predicted Stretch is 449.62071, shown next to Current Y.
For further information about the Contour Profiler, see the Contour Profiler chapter in the Profilers book.
Response Surface Design Window
The Response Surface Design window walks you through the steps to construct a design for modeling a quadratic surface. You can select a central composite design, a Box-Behnken design, or a blocked version of one of these design types. If you select a central composite design, you can adjust the axial points.
The Response Surface Design window is updated as you work through the design steps. The outlines, separated by buttons that update the outlines, follow the flow in Figure 11.12.
Figure 11.12 Response Surface Design Flow
The following sections describe the steps in creating a response surface design:
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 11.13 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
Factors in a response surface design can only be continuous.
Tip: Use DOE > Custom Design to create response surface designs that involve categorical factors.
The initial Factors panel for a response surface design appears with two continuous factors.
Figure 11.14 Factors Outline
The factors outline contains the following buttons.
Add
Enters the number of continuous factors specified.
Remove Selected
Removes the selected factors.
Tip: When you have completed your Factors panel, select Save Factors from the red triangle menu. This saves the factor names and values in a data table that you can later reload. See “Response Surface Design Options”.
The Factors outline 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 as Continuous. 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.
Factor Column Properties
For each factor, various column properties are saved to the data table for the completed design.
Design Role
Each factor is assigned the Design Role column property. The Role that you specify in defining the factor determines the value of its Design Role column property. When you select a design with a block, that factor is assigned the Blocking value. The Design Role property reflects how the factor is intended to be used in modeling the experimental data. Design Role values are used in the Augment Design platform.
Factor Changes
Each factor is assigned the Factor Changes column property with a setting of Easy. In the Response Surface Design platform, it is assumed that factor levels can be changed for each experimental run. Factor Changes values are used in the Evaluate Design and Augment Design platforms.
Coding
If the Design Role is Continuous, the Coding column property for the factor is saved. This property transforms the factor values so that the low and high values correspond to –1 and +1, respectively. The estimates and tests in the Fit Least Squares report are based on the transformed values.
RunsPerBlock
Indicates the number of runs in each block. When you select a design with a block and then click Make Table, a factor with the default name Block is added to the Factors list. The RunsPerBlock column property is saved for that factor.
Choose a Design
After you enter your responses and factors and click Continue, you select from a list of designs. The designs include two types:
Select the design that you want to use and click Continue.
Figure 11.15 Choose a Design Panel for Four Factors
Box-Behnken Designs
Box-Behnken designs have only three levels for each factor and have no design points at the vertices of the cube defined by the ranges of the factors. These designs can be useful when it is desirable to avoid extreme settings for engineering considerations. However, these designs result in higher prediction variance near the vertices than do central composite designs.
Central Composite Designs
Central composite designs have center points and axial points. An axial point is a point where one factor is set to a high or low value (an axial value) and all other factors are set to the midrange, or center, value.
A central composite design can have axial points that fall beyond the faces of the hypercube defined by the specified factor ranges. This means that each factor might require five distinct settings, including two that fall beyond the range of values specified in the Factors outline. However, JMP enables you to place design points on the face.
The following types of central composite designs are available:
Central Composite Design
The usual central composite design for the specified number of factors.
CCD-Uniform Precision
The number of center points is chosen so that the prediction variance near the center of the design space is very flat.
CCD-Orthogonal
The number of center points and the axial values are chosen so that the second-order parameter estimates are minimally correlated with the other parameter estimates.
CCD-Orthogonal Blocks
The second-order parameter estimates and block effects are minimally correlated with the other parameter estimates.
Axial Value
When you select a central composite design and then click Continue, you have the option to provide axial scaling information. In placing axial values, the values shown are used to multiply half of the specified range of a factor. If you specify a value of 1.0 next to Axial Value, then axial points in the resulting design are placed on the faces of the cube defined by the factors. You can set the axial value according to the following options:
Figure 11.16 Axial Value Panel
Rotatable
The prediction variance depends only on the scaled distance from the center of the design. The axial points are more extreme than the factor ranges. If this factor range cannot be practically achieved, select On Face or specify your own value.
Orthogonal
The effects are orthogonal. The axial points are more extreme than the factor ranges. If this factor range cannot be practically achieved, select On Face or specify your own value.
On Face
Places the axial points at the extremes of the specified factor ranges.
User Specified
Places the axial points at a distance specified by the value that you enter in the Axial Value text box.
Inscribe
Rescales the design so that the axial points are at the low and high ends of the factor range. The factorial design points are shrunken based on that scaling.
Specify Output Options
You can specify details for the output data table in the Output Options panel. When you finish, click Make Table to construct the data table for the design.
Run Order
The Run Order options determine the order of the runs in the design table. Choices include the following:
Keep the Same
Rows in the design table are in the same order as in the Design and Anticipated Coefficients outline.
Sort Left to Right
Columns in the design table are sorted from left to right.
Randomize
Rows in the design table are in random order.
Sort Right to Left
Columns in the design table are sorted from right to left.
Randomize within Blocks
Rows in the design table are in random order within the blocks.
Center Points and Replicates
Number of Center Points
The number of center points that appear in the design. A center point is a run where every continuous factor is set at the center of the factor’s range. The initial value shown is the number of center points in the design that you selected.
Number of Replicates
The number of times to replicate the entire design, including center points. One replicate doubles the number of runs.
Make Table
Click Make Table to create a design table that contains the runs for your experiment.
Figure 11.17 Orthogonal Central Composite Design for Bounce Factors and Response
The Design note in the Table panel at the upper left gives the design type that generated the table (Central Composite Design). This information can be helpful if you are comparing multiple designs.
Pattern Column
A Pattern column gives a symbolic description of the run in each row in terms of the factor values.
Tip: Pattern can be a useful label variable in plots.
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-
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Low value
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+
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High value
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0
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Midrange (center) value
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a
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Low axial value
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A
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High axial value
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Design Table Scripts
The design table includes the following scripts:
Model
Runs the Analyze > Fit Model platform.
Evaluate Design
Runs the DOE > Design Diagnostics > Evaluate Design platform.
DOE Dialog
Re-creates the Response Surface Design window that you used to generate the design table. The script also contains the random seed used to generate your design.
Response Surface Design Options
The red triangle menu in the Response Surface Design platform contains these options:
Save Responses
Saves the information in the Responses panel to a new data table. You can then quickly load the responses and their associated information into most DOE windows. This option is helpful if you anticipate re-using the responses.
Load Responses
Loads responses that you saved using the Save Responses option.
Save Factors
Saves the information in the Factors panel to a new data table. Each factor’s column contains its levels. Other information is stored as column properties. You can then quickly load the factors and their associated information into most DOE windows.
Note: It is possible to create a factors table by entering data into an empty table, but remember to assign each column an appropriate Design Role. Do this by right-clicking on the column name in the data grid and selecting Column Properties > Design Role. In the Design Role area, select the appropriate role.
Load Factors
Loads factors that you saved using the Save Factors option.
Save Constraints
(Unavailable for some platforms) Saves factor constraints that you defined in the Define Factor Constraints or Linear Constraints outline into a data table, with a column for each constraint. You can then quickly load the constraints into most DOE windows.
In the constraint table, the first rows contain the coefficients for each factor. The last row contains the inequality bound. Each constraint’s column contains a column property called ConstraintState that identifies the constraint as a “less than” or a “greater than” constraint. See “ConstraintState” in the “Column Properties” appendix.
Load Constraints
(Unavailable for some platforms) Loads factor constraints that you saved using the Save Constraints option.
Set Random Seed
Sets the random seed that JMP uses to control certain actions that have a random component. These actions include the following:
‒ simulating responses using the Simulate Responses option
‒ randomizing Run Order for design construction
‒ selecting a starting design for designs based on random starts
To reproduce a design or simulated responses, enter the random seed that generated them. For designs using random starts, set the seed before clicking Make Design. To control simulated responses or run order, set the seed before clicking Make Table.
Note: The random seed associated with a design is included in the DOE Dialog script that is saved to the design data table.
Simulate Responses
Adds response values and a column containing a simulation formula to the design table. Select this option before you click Make Table.
When you click Make Table, the following occur:
‒ A set of simulated response values is added to each response column.
‒ For each response, a new a column that contains a simulation model formula is added to the design table. The formula and values are based on the model that is specified in the design window.
‒ A Model window appears where you can set the values of coefficients for model effects and specify one of three distributions: Normal, Binomial, or Poisson.
‒ A script called DOE Simulate is saved to the design table. This script re-opens the Model window, enabling you to re-simulate values or to make changes to the simulated response distribution.
Make selections in the Model window to control the distribution of simulated response values. When you click Apply, a formula for the simulated response values is saved in a new column called <Y> Simulated, where Y is the name of the response. Clicking Apply again updates the formula and values in <Y> Simulated.
For additional details, see “Simulate Responses” in the “Custom Designs” chapter.
Note: 
You can use Simulate Responses to conduct simulation analyses using the JMP Pro Simulate feature. For information about Simulate and some DOE examples, see the Simulate chapter in the Basic Analysis book.
You can use Simulate Responses to conduct simulation analyses using the JMP Pro Simulate feature. For information about Simulate and some DOE examples, see the Simulate chapter in the Basic Analysis book.Advanced Options > Set Delta for Power
Specifies the difference in the mean response that you want to detect for model effects. See “Evaluate Design Options” in the “Evaluate Designs” chapter.
Save Script to Script Window
Creates the script for the design that you specified in the Response Surface Design window and saves it in an open script window.
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