Response Surface Designs ... 71 The Response Surface Design Dialog ... 71 The Design Table ... 72 Axial Scaling Options ... 73 A Central Composite Design ... 74 Fitting the Model ... 75 A Box-Behnken Design: The Tennis Ball Example ... 76 Geometry of a Box-Behnken Design ... 78 Analysis of Response Surface Models ... 78
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Response Surface Designs
The Response Surface Design Dialog
The Response Surface Design command on the DOE main menu (or DOE JMP Starter tab page) displays the dialog, shown to the left in Figure 5.1, for you to enter factors and responses. When you click Continue the list of design selections shown on the right appears. The response surface design list has a Box- Behnken design and two types of central composite design, called uniform precision and orthogonal. These properties of central composite designs relate to the
number of center points in the design and to the axial values:
❿ Uniform precision means that the number of center points is chosen so that the prediction variance at the center is approximately the same as at the design vertices.
❿ For orthogonal designs, the number of center points is chosen so that the second order parameter estimates are minimally correlated with the other parameter estimates. Figure 5.1 Design Dialogs to Specify Factors and Choose Design Type
To complete the dialog, enter the number of factors (up to eight) and click Continue. In the table shown to the right in Figure 5.1, the 15- run Box-Behnken design is selected. Click Continue to use this design.
The left panel in Figure 5.2 shows the next step of the dialog. To reproduce the right panel of Figure 5.2 specify 1 replicate with 2 center points per replicate, and change the run order popup choice to Randomize. When you finish specifying the output options you want, click Make Table.
Figure 5.2 Design Dialog to Modify Order of Runs and Simulate Responses
The Design Table
The JMP data table (Figure 5.3) lists the design runs specified in Figure 5.2. Note that the design table also has a column called Y for recording experimental results.
Figure 5.3 The JMP Design Facility Automatically Generates a JMP Data Table
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Axial Scaling Options
When you select a central composite design and then click Continue, you see the dialog on the right in Figure 5.4. The dialog supplies default axial scaling information but you can use the options described next and enter the values you want.
Figure 5.4 CCD Design With a Specified Type of Axial Scaling
The axial scaling options control how far out the axial points are: Rotatable
makes the variance of prediction depend only on the scaled distance from the center of the design.
Orthogonal
makes the effects orthogonal in the analysis.
In both previous cases the axial points are more extreme than the –1 or 1 representing the range of the factor. If this factor range cannot be practically achieved, then you can choose either of the following options:
On Face
is the default. These designs leave the axial points at the end of the -1 and 1 ranges. User Defined
uses the value entered by the user, which can be any value greater than zero. Inscribe
rescales the whole design so that the axial points are at the low and high ends of the range (the axials are –1 and 1 and the factorials are shrunken in from that).
A Central Composite Design
The generated design, shown in the JMP data table in Figure 5.3, lists the runs for the design specified in Figure 5.2. Note that the design table also has a column called Y for recording experimental results.
Figure 5.5 shows the specification and design table for a 20-run 6-block Central Composite design with simulated responses.
Figure 5.5 Central Composite Response Surface Design
The column called Pattern identifies the coding of the factors. The Pattern column shows all the factor codings with “+” for high, “–” for low, “a” and “A” for low and high axial
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values, and “0” for midrange. If the Pattern variable is a label column, then when you click on a point in a plot of the factors, the pattern value shows the factor coding of the point. Note: The resulting data table has a Table Variable called Design
that contains the design type. This variable appears as a note at the top of the Tables panel to the left of the data grid. In this example,
Design says CCD-Orthogonal Blocks. The table also contains a model script stored as a Table Property, and displayed as a menu icon labeled Model.
.
Fitting the Model
When you click the Table Property icon for the model(in the Tables panel to the left of the data grid), a popup menu appears with the Run Script command. The Run Scriptcommand opens the Model Specification dialog window and lists the appropriate effects for the model you selected. This example has the main effects and interactions as seen in Figure 5.6. When you collect data, you can key or paste them into the design table and run this model. The model is permanently stored with the data table.
Figure 5.6 Model Specification dialog for Response Surface Design
&RS &RS &RS
A Box-Behnken Design: The Tennis Ball Example
The Bounce Data.jmp sampledata file has the response surface data inspired by the tire tread data described in Derringer and Suich (1980). The objective is to match a standardized target value, given as 450, of tennis ball bounciness. The bounciness varies with amounts of
Silica, Silane, and Sulfur used to manufacture the tennis balls. The experimenter wants to collect data over a wide range of values for these variables to see if a response surface can find a combination of factors that matches a specified bounce target.
To begin, select Response Surface Design from the DOE menu. The responses and factors information is in existing JMP files found in the Design Experiment Sample Data folder. Use the Load Responses and Load Factors commands in the popup menu on the RSM Design title bar to load the response file called Bounce Response.jmp and the factor file called Bounce Factor.jmp. Figure 5.7 shows the completed Response panel and Factors panel.
Figure 5.7 Response and Factors For Bounce Data
After the response data and factors data loads, the Response Surface Design Choice dialog lists the designs in Figure5.8.
5 Surface Figure 5.8 Response Surface Design Selection
The Box-Behnken design selected for three effects generates the design table of 15 runs shown in Figure5.9. The data are in the Bounce Data.jmp sample data table. The Table Variable (Model) runs a script to launch the Model Specification dialog.
After the experiment is conducted, the responses are entered into the JMP table.
Figure 5.9
JMP Table for a Three-Factor Box- Behnken Design
Geometry of a Box-Behnken Design
The geometric structure of a design with three effects is seen by using the Spinning Plot platform. The spinning plot shown in Figure 5.10 illustrates the three Box-Behnken design columns. Options available on the spin platform draw rays from the center to each point, inscribe the points in a box, and suppress the x, y, and z axes. You can clearly see the 12 points midway between the vertices, leaving three points in the center.
Figure 5.10 Spinning Plot of a Box- Behnken Design for Three Effects
Analysis of Response Surface Models
To analyze response surface designs, select the Fit Model command from the Analyze menu and designate the surface effects in the Model Specification dialog. To do this, select the surface effects in the dialog variable selection list and add them to the Effects in Model list. Then select Response Surface from the Effect Attributes popup menu (see
Figure 5.6).
However, if the table to be analyzed was generated by the DOE Response Surface designer, then the Run Model table variable script automatically assigns the response surface attribute to the factors, as previously illustrated in Figure 5.6.
Analysis Reports
The standard analysis results appear in tables shown in Figure 5.11, with parameter estimates for all surface and crossed effects in the model.
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The prediction model is highly significant with no evidence of lack of fit. All main effect terms are significant as well as the two interaction effects involving Sulfur.
Figure 5.11 JMP Statistical Reports for a Response Surface Analysis of Bounce Data
See Chapter 9, “Standard Least Squares: Introduction“ in the JMP Statistics and Graphics
Guide for more information about interpretation of the tables in Figure 5.11.
The Response Surface report also has the tables shown in Figure 5.12:
❿ The Response Surface table is a summary of the parameter estimates.
❿ The Solution table lists the critical values of the surface factors and tells the kind of solution (maximum, minimum, or saddlepoint).
❿ The Canonical Curvature table shows eigenvalues and eigenvectors of the effects. Note that the solution for the Bounce example is a saddlepoint. The Solution table also warns that the critical values given by the solution are outside the range of data values. See Chapter 11, “Standard Least Squares: Exploring the Prediction Equation“ in the JMP
Statistics and Graphics Guide for details about the response surface analysis tables in
Figure 5.12 Statistical Reports for a Response Surface Analysis
The eigenvector values show that the dominant negative curvature (yielding a maximum) is mostly in the Sulfur direction. The dominant positive curvature (yielding a minimum) is mostly in the Silica direction. This is confirmed by the prediction profiler in Figure 5.13. The Prediction Profiler
The response Prediction Profiler gives you a closer look at the response surface to find the best settings that produce the response target. It is a way of changing one variable at a time and looking at the effects on the predicted response.
Open the Prediction Profiler with the Profiler command from the Factor Profiling popup menu on the Response title bar. The Profiler displays prediction
traces for each X variable. A prediction
trace is the predicted response as one variable is changed while the others are held constant at the current values (Jones 1991).
The first profile in Figure 5.13 show initial settings for the factors Silica, Silane, and
Sulfur, which result in a value for Stretch of 396, which is close to the specified target of 450. However, you can adjust the prediction traces of the factors and find a Stretch value that is closer to the target.
The next step is to choose DesirabilityFunctions from the popup menu on the Profiler title bar. This command appends a new row of plots to the bottom of the plot matrix, which graph
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desirability on a scale from 0 to 1. The row has a plot for each factor, showing its desirability
trace, as illustrated by the second profiler in Figure 5.13. The Desirability Functions command also adds a column that has an adjustable desirability function for each Y variable. The overall desirability measure appears to the left of the row of desirability traces.
The response goal for Stretch is a target value of 450, as illustrated by the desirability function in Figure 5.13. If needed, you can drag the middle handle on the desirability function vertically to change the target value. The range of acceptable values is determined by the positions of the upper and lower handles. See Chapter 11, “Standard Least Squares: Exploring the Prediction Equation“ in the JMP Statistics and Graphics Guide for further discussion of the Prediction Profiler.
The overall desirability shows to the left of the row of desirability traces. However, note in this example that the desirability function is set to 450, the target value. The current
predicted value of Stretch, 396, is based on the default factor setting. It is represented by the horizontal dotted line that shows slightly below the desirability function target value. Figure 5.13 Prediction Profiler for a Response Surface Analysis
You can adjust the factor traces by hand to change the predicted value of Stretch. Another convenient way to find good factor settings is to select Maximize Desirability from the Prediction Profiler popup menu. This command adjusts the profile traces to produce the response value closest to the specified target (the target given by the desirability function).
Figure 5.14 shows the result of the most desirable settings. Changing the settings of Silica
from 1.2 to 0.94512, Silane from 50 to 50.0038, and Sulfur from 2.3 to 2.11515 raised the predicted response from 396 to the target value of 450.
Figure 5.14 Prediction Profiler for a Response Surface Analysis
A Response Surface Plot
Another way to look at the response surface is to use the Contour Profiler. The Contour Profiler command in the Factor Profiling menu brings up the interactive contour profiling facility as shown in Figure 5.15. It is useful for optimizing response surfaces graphically, especially when there are multiple responses. This example shows the profile to Silica and
Silane for a fixed value of Sulphur.
Options on the Contour Profiler title bar can be used to set the grid density, request a surface plot (mesh plot), and add contours at specified intervals, as shown in the contour plot in Figure 5.15.
The sliders for each factor set values for Current X and Current Y. The surface plots (mesh plots) at the bottom of the report illustrate the effect on the response surface when you set
Sulphur to its minimum (40) and then to its maximum (60). This change in the surface shape clearly shows that there is interaction between Sulfur and the other factors .
5 Surface Figure 5.15 Prediction Profiler for a Response Surface Analysis
Silane=40 Silane=60
Figure 5.16 shows the Contour profile when the Current X values have the most desirable settings as shown at the bottom in Figure 5.14 .
Figure 5.16 Prediction Profiler with High and Low Limits
The Prediction Profiler and the Contour Profiler are discussed in more detail in Chapter 11 of the Statistics and Graphics Guide, “Standard Least Squares: Exploring the Prediction Equation.”
6 Factorial
Chapter 6
Full Factorial Designs
In full factorial designs you perform an experimental run at every combination of the factor levels. The sample size is the product of the numbers of levels of the factors. For example, a factorial experiment with a two-level factor, a three-level factor, and a four-level factor has 2•3•4=24 runs.Factorial designs with only two-level factors have a sample size that is a power of two (specifically 2f where f is the number of factors.) When there are three factors, the factorial design points are at the vertices of a cube as shown in the diagram on the left. For more factors, the design point lie on a hypercube.
Full factorial designs are the most conservative of all design types. There is little scope for ambiguity when you are willing to try all combinations of the factor settings.
Unfortunately, the sample size grows exponentially in the number of factors, so full factorial designs are too expensive to run for most practical purposes.
Chapter 6
Contents
The Factorial Dialog ... 87 The Five-Factor Reactor Example... 886 Factorial
The Factorial Dialog
To start, select Full Factorial Design in the DOE main menu, or click the Full Factorial Design button on the JMP Starter DOE tab page. The popup menu on the right in Figure 6.1 illustrates the way to specify categorical factors with 2 to 9 levels. Add a continuous factor and two categorical factors with three and four levels respectively. Change the levels to those shown at the left in Figure 6.1.
Figure 6.1 Full Factorial Factor Panel
When you finish adding factors, click Continue. to see a panel of output options (as shown to the right). When you click Make Table, the table shown in Figure 6.2 appears. Note that the values in the
Pattern column describe the run each row
represents. For continuous variables, plus or minus signs represent high and low levels. Level numbers represent values of of categorical variables.
Figure 6.2 2x3x4 Full Factorial Design Table
inus sign for ow level of
ontinuous factor lus sign for igh level of ontinuous actor
evel number for ategorical ariable
The Five-Factor Reactor Example
Results from the reactor experiment described in Chapter 4, “Screening Designs” can be found in the Reactor 32 Runs.jmp sample data folder, (Box, Hunter, and Hunter 1978, pp 374-390). The variables have the same names: Feed Rate, Catalyst, Stir Rate,
Temperature, and Concentration. These are all two-level continuous factors.
To create the design yourself, select Full Factorial Design from the DOE main menu (or toolbar), or click Full Factorial Design on theDOE tab page of the JMP Starter window. Do the following to complete the Response panel and the Factors panel:
❿ Use the Load Responses command from the popup menu on the Full Factorial Design title bar and open the Reactor Response.jmp file to get the response specifications.
❿ Likewise, use the Load Factors command and open the Reactor Factors.jmp file to get the Factors panel.
6 Factorial Figure 6.3
Full-Factorial Example Response and Factors Panels
A full factorial design includes runs for all combinations of high and low factors for the five variables, giving 32 runs. Click Continue to see Output Options panel shown to the right. When you click Make Table, the JMP Table in Figure 6.4 is constructed with a run for every combination of high and low values for the five variables, and an empty Y column for
entering response values when the experiment is complete. The table has 32 rows, which cover all combinations of a five factors with two levels each. The Reactor 32 Runs.jmp
sample data file has these experimental runs and the results from the Box, Hunter, and Hunter study. Figure 6.4 shows the runs and the response data.
6 Factorial
Begin the analysis with a quick look at the data before fitting the factorial model. The plot on the right shows a distribution of the response, Percent Reacted, using the Normal Quantile plot option on the Distribution command on the Analyze menu. Start the formal analysis with a stepwise regression. The data table has a script stored with it that automatically defines an
analysis of the model that includes main effects and all two factor interactions, and brings up the Stepwise control panel. To do this, choose Run Script from the Fit Model popup menu on the title bar of the Reactor 32 Run.jmp table.
The Stepwise Regression Control Panel appears with a
preliminary Current Estimates report. The probability to enter a factor into the model is 0.05 (the default is 0.25), and the probability to remove a factor is 0.1.
A useful way to use Stepwise is to check all the main effects in the Current Estimates table, and then use Mixed as the Direction for the stepwise process, which can both include or exclude factors in the model.
Change from default settings: Prob to Enter Factor is .05 Prob to Leave factor is .10 Mixed direction instead of Forward or Backward
To do this, click the check boxes for the main effects of the factors as shown in Figure 6.5, and click Go on the Stepwise control panel.
Figure 6.5 Starting Model For Stepwise Process
The Mixed stepwise procedure removes insignificant main effects and adds important interactions. The end result is shown in Figure 6.6. Note that the Feed Rate and Stir Rate
factors are no longer in the model.
Figure 6.6 Model After Mixed Stepwise Regression
Click the Make Model button to generate a new model dialog. The Model Specification