Overview of Definitive Screening Design
Investigators use screening designs when they want to identify the factors that have the most substantial effects on a response. A screening design enables you to study a large number of factors in a fairly small experiment.
Many standard screening designs focus on estimating main effects. Definitive screening designs offer advantages over standard screening designs. They avoid confounding of effects and can identify factors having a nonlinear effect on the response. For details about the advantages and construction of definitive screening designs, see Jones and Nachtsheim (2011).
For designs containing only continuous factors, compare these properties of definitive screening designs versus standard screening designs:
Note: When quadratic effects are mentioned, the standard screening designs are assumed to have center points.
• Main effects are orthogonal to two-factor interactions.
‒ Definitive Screening Designs: Always
‒ Standard Screening Designs: Only for Resolution IV or higher
• No two-factor interaction is completely confounded with any other two-factor interaction.
‒ Definitive Screening Designs: Always
‒ Standard Screening Designs: Only for Resolution V or higher
• All quadratic effects are estimable in models containing only main and quadratic effects.
‒ Definitive Screening Designs: Always
‒ Standard Screening Designs: Never
These properties are described more fully in the remainder of this section.
Standard Screening Designs
Standard screening designs, such as fractional factorial or Plackett-Burman designs, attempt to study many factors with a relatively small allocation of resources. However, standard screening designs have several undesirable features:
• They can alias some main effects with two-factor interactions. In Plackett-Burman designs, for example, main effects are correlated with several two-factor interactions. If one or more two-factor interaction effects are substantial, then the experimenter must perform additional runs to resolve the ambiguities.
• They can also confound some two-factor interactions with each other. Consequently, if a two-factor interaction effect is substantial, then the experimenter must perform additional runs to resolve the remaining ambiguities.
• Continuous factors are usually set at only two levels (low and high). However, engineers and scientists often prefer designs where continuous factors are set at three levels (low, middle, and high). This is because two levels are not sufficient to detect nonlinearity, which is common in physical systems. You can use a traditional screening design with added center points to detect nonlinearity, but such a design does not identify the responsible factors.
Definitive Screening Designs
Using definitive screening designs, you can do the following:
• Avoid model ambiguity, enabling you to identify important factors more quickly and efficiently.
• Identify the cause of nonlinear effects while avoiding confounding any terms up to second order. So not only can you detect nonlinearity, as you might with center points in a traditional screening design, but you can identify the responsible factors.
Definitive screening designs offer the following advantages:
• Definitive screening designs require only a small number of runs. For six or more factors, the number of required runs is usually only a few more than twice the number of factors. For more detail on the number of runs, see “Conference Matrices and the Number of Runs”.
• Main effects are orthogonal to two-factor interactions. This means that estimates of main effects are not biased by the presence of active two-factor interactions, whether these interactions are included in the model or not. Note that resolution III screening designs confound some main and interaction effects. Also, Plackett-Burman designs produce biased main effect estimates if there are active two-factor interactions.
• No two-factor interaction is completely confounded with any other two-factor interaction. However, a two-factor interaction might be correlated with other two-factor interactions. Note that resolution IV screening designs completely confound some two-factor interaction effects.
• All quadratic effects are estimable in models comprised only of main effects and quadratic terms. This enables you to identify the factors that account for nonlinearity. Note that traditional screening designs with added center points do not allow estimation of all quadratic effects in models consisting of main and quadratic effects.
• Quadratic effects are orthogonal to main effects and not completely confounded with two-factor interactions. A quadratic effect might be correlated with interaction effects.
• For 6 through at least 30 factors, it is possible to estimate the parameters of any full quadratic model involving three or fewer factors with high precision.
• For 18 factors or more, they can fit full quadratic models in any 4 factors. For 24 factors or more, they can fit full quadratic models in any 5 factors.
Definitive Screening Design Platform
The Definitive Screening Design platform enables you to construct definitive screening designs for continuous factors and for two-level categorical factors. It also enables you to construct blocked designs. You can add extra non-center runs that enhance the ability of the design to reliably detect effects when many effects are active.
To view the absolute values of the correlations among effects, use the Color Map on Correlations provided as part of the Design Evaluation outline in the Definitive Screening Design window. You can compare the aliasing structure of definitive screening designs to that of other designs by comparing their color maps on correlations. For details, see “Color Map on Correlations” in the “Evaluate Designs” chapter.
For details, see “Structure of Definitive Screening Designs”. For information about definitive screening designs with blocks, see “Blocking in Definitive Screening Designs”. For suggestions on how to analyze data obtained using definitive screening designs, see “Analysis of Experimental Data”.
Fit Definitive Screening Platform
After you run a Definitive Screening Design (DSD), analyze your results using the Fit Definitive Screening platform. Standard model selection methods applied to DSDs can fail to identify active effects. To identify active main effects and second-order effects, the Fit Definitive Screening platform uses an algorithm called Effective Model Selection for DSDs. This algorithm leverages the special structure of DSDs. See Chapter 8, “The Fit Definitive Screening Platform”.
If you create your DSD in JMP, the design table contains a script called Fit Definitive Screening that automatically runs an analysis using the Effective Model Selection for DSDs methodology.
Examples of Definitive Screening Designs
This section contains the following examples:
Definitive Screening Design
Suppose that you need to determine which of six factors have an effect on the yield of an extraction process.
Create the Design
The factors and their settings are given in the data table Extraction Factors.jmp. You create a definitive screening design to investigate.
1. Select DOE > Definitive Screening > Definitive Screening Design.
2. Double-click Y under Response Name and type Yield.
3. From the red triangle menu, select Load Factors.
4. Open the Extraction Factors.jmp sample data table, located in the Design Experiment folder.
The factor names and ranges are added to the Factors outline.
Figure 7.2 Responses and Factors for Extraction Design
5. Click Continue.
The Design Options outline opens. Here you can specify a blocking structure. There is no need to block in this example, so you accept the default selection of No Blocks Required.
You can also choose to add Extra Runs, which greatly enhance your ability to detect second-order effects. A minimum of four Extra Runs is highly recommended and is the default.
6. Click Make Design.
The Definitive Screening Design window updates to show a Design outline and a Design Evaluation outline.
Figure 7.3 Design Outline
7. Open the Design Evaluation > Color Map on Correlations outline.
The Color Map on Correlations assigns a color intensity scale to the absolute values of correlations among all main effects and two-factor interactions.
Figure 7.4 Color Map on Correlations for Extraction Design
Note the following:
‒ The solid deep blue area shows that there is no correlation between main effects or between main effects and two-factor interactions.
‒ The lighter blue and gray areas indicate that the absolute correlations between two-factor interactions are small.
‒ The solid red squares indicate absolute correlations of 1. These all appear on the diagonal, reflecting the expected correlation of an effect with itself.
In the Output Options panel, note that the Run Order is set to Randomize.
8. Click Make Table to obtain the data table shown in Figure 7.5.
Note: The runs in your design might appear in a different order than the order shown in Figure 7.5.
Figure 7.5 Definitive Screening Design for Extraction Process
Comparison with a Fractional Factorial Design
Suppose that you had chosen a traditional screening design instead of the definitive screening design in “Definitive Screening Design”. This example compares the two designs in terms of confounding.
1. Select DOE > Classical > Screening Design.
2. Double-click Y under Response Name and type Yield.
3. Select Load Factors From the red triangle next to Definitive Screening Design.
4. Open the Extraction Factors.jmp sample data table, located in the Design Experiment folder.
The factor names and ranges are added to the Factors outline.
5. Click Continue.
6. Select Choose from a list of fractional factorial designs.
7. Click Continue.
Potential designs appear in the Design List.
Figure 7.6 Screening Design List for Six Continuous Factors
8. Select the sixteen-run fractional factorial design with no blocks, shown highlighted in Figure 7.6.
9. Click Continue.
10. Open the Display and Modify Design > Aliasing of Effects outline.
Figure 7.7 Aliasing of Effects for Fractional Factorial Design
The Aliasing of Effects outline for the 16-run fractional factorial design shows that every two-factor interaction is confounded with at least one other two-factor interaction. In this fractional factorial design, the Ethanol*Time interaction is confounded with Methanol*pH. To determine which interaction is active, you need to run additional trials. If the factors had been entered in a different order, the Ethanol*Time interaction might have been aliased with two other two-factor interactions.
In the section “Definitive Screening Design”, you constructed a 17-run definitive screening design. The Color Map on Correlations for this DSD (Figure 7.4) shows that no two-factor interactions are confounded with any other two-factor interactions. For the fractional factorial design, there are seven instances of confounded two-factor interactions. If you suspect that there are active two-factor effects, the DSD is the better choice.
You can conduct a more thorough comparison of the two designs using the Compare Designs platform (DOE > Design Diagnostics > Compare Designs). See Chapter 16, “Compare Designs”.
Definitive Screening Design with Blocking
Suppose that, due to raw material constraints, the extraction experiment requires that you run it using material from two separate lots. You can generate a definitive screening design with a blocking variable to account for the potential lot variation.
Create the Design
The extraction factors and their settings are given in the data table Extraction Factors.jmp. Generate a definitive screening design with a block as follows.
1. Select DOE > Definitive Screening > Definitive Screening Design.
2. Double-click Y under Response Name and type Yield.
3. From the red triangle menu, select Load Factors.
4. Open the Extraction Factors.jmp sample data table, located in the Design Experiment folder.
The factor names and ranges are added to the Factors outline.
5. Click Continue.
The Design Options outline opens. Here you can specify a blocking structure.
6. Select Add Blocks with Center Runs to Estimate Quadratic Effects.
Leave Number of Blocks set at 2.
You are recreating the design for the Extraction2 Data.jmp sample data table, which was created without Extra Runs. Although four Extra Runs are strongly recommended, you will not add Extra Runs in this example.
7. Next to Number of Extra Runs, select 0.
8. Click Make Design.
The Definitive Screening Design window updates to show a Design outline and a Design Evaluation outline.
Check that Block has been added to the Factors outline and to the Design.
9. In the Factors outline, Double-click Block and type Lot.
In the Output Options panel, note that the Run Order is set to Randomize within Blocks.
10. Click Make Table.
Note: The runs in your design might appear in a different order than the order shown in Figure 7.8.
Figure 7.8 Definitive Screening Design with Block for Extraction Process
Notice that run 1 is a center point run in Lot 1 and run 14 is a center point run in Lot 2.
Analyze the Experimental Data
At this point, you conduct your experiment and record your data in the Yield column of the design table (Figure 7.8). The Extraction2 Data.jmp sample data table contains your experimental results. The runs in the Extraction2 Data.jmp sample data table are in a different order than those in Figure 7.8.
To explore all second-order effects, one option is to use All Possible Models regression. Another option is to use forward stepwise regression. However, these standard methods often fail to identify active effects. For this reason, you use the Fit Definitive Screening platform.
1. Select Help > Sample Data Library and open Design Experiment/Extraction2 Data.jmp.
2. In the Table panel of the design table, click the green triangle next to the Fit Definitive Screening script.
Figure 7.9 Fit Definitive Screening Report
The effects identified by Fit Definitive Screening as potentially active are listed in the Combined Model Parameter Estimates report.
3. Click the Run Model button at the bottom of the Combined Model Parameter Estimates report.
This fits a standard least squares model to the effects identified as potentially active.
Figure 7.10 Standard Least Squares Report
The Actual by Predicted Plot shows no evidence of lack of fit. The Effect Summary report shows that Methanol*Methanol is not significant. You decide to remove this effect from the model.
4. Select Methanol*Methanol in the Effect Summary report and click Remove.
Figure 7.11 Final Set of Active Effects
The remaining effects are all significant. You conclude that these are the active effects.
Comparison of a Definitive Screening Design with a Plackett-Burman Design
Plackett-Burman designs are an alternative to fractional factorials for screening. However, Plackett-Burman designs have complex aliasing of the main effects by two-factor interactions.
This example shows how to compare a definitive screening with a Plackett-Burman design using the Evaluate Design platform. For an extensive example using the Compare Designs platform, see “Designs of Same Run Size” in the “Compare Designs” chapter.
The Definitive Screening Design
1. Select DOE > Definitive Screening > Definitive Screening Design.
2. Type 4 in the Add N Factors box and click Continuous.
3. Type 2 in the Add N Factors box and click Categorical.
Your window should appear as shown in Figure 7.12.
Figure 7.12 Definitive Screening Dialog with 4 Continuous and 2 Categorical Factors
4. Click Continue.
This example does not require a block. Under the Design Options Outline, check that the No Blocks Required option is selected.
In order to compare designs of approximately equal sizes. do not add Extra Runs.
5. Next to Number of Extra Runs, select 0.
6. Click Make Design.
The design that is generated has 14 runs.
7. Open the Design Evaluation > Color Map On Correlations outline.
Figure 7.13 Color Map for Definitive Screening Design
Notice that the categorical main effects have small correlations with each other and with the continuous factors’ main effects. These correlations lead to a small reduction in the precision of the estimates.
8. Do not close your Definitive Design Screening window until you compare the color map with that of the Plackett-Burman design, below.
The Plackett-Burman Design
Now create a Plackett-Burman design using the same factor structure.
1. Select DOE > Classical > Screening Design.
2. Type 4 in the Add N Factors box and click Continuous.
3. Type 2 in the Add N Factors box and click Categorical > 2 Level.
4. Click Continue.
5. Select Choose from a list of fractional factorial designs and click Continue.
Potential designs appear in the Design List, shown in Figure 7.14.
Figure 7.14 Plackett-Burman Design
6. Select the 12 run Plackett-Burman design. See Figure 7.14.
7. Click Continue.
8. Open the Design Evaluation > Color Map On Correlations outline.
Compare the color map for the 12-run Plackett-Burman design to the color map for the 14-run definitive screening design.
Figure 7.15 Plackett-Burman Correlations (left) and Definitive Screening Correlations (right)
Figure 7.15 shows both color maps, but shows only the portion of the Plackett-Burman color map that involves main effects and two-way interactions. (To construct the color map for the Plackett-Burman design without the three-way interactions, construct the design. Then obtain the color map using Evaluate Design.)
In the color map for the Plackett-Burman design on the left, you see that most two-factor interactions are correlated with main effects. This means that any non-negligible two-factor interaction will bias several main effects. This can lead to a failure to identify an active main effect or the false conclusion that an inactive main effect is active.
Contrast this with the color map for the definitive screening design on the right. With only two additional runs, the definitive screening design trades off a small increase in the variance of the main effects for complete independence of main effects and two-factor interactions.
Definitive Screening Design Window
The definitive screening design window updates as you work through the design steps. For more information, see “The DOE Workflow: Describe, Specify, Design” in the “Starting Out with DOE” chapter. The outlines, separated by buttons that update the outlines, follow the flow in Figure 7.16.
Figure 7.16 Definitive Screening Design Flow
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 7.17 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.
Figure 7.18 Factors Outline
The factors outline contains the following buttons.
Continuous
Enters the number of continuous factors specified in Add N Factors.
Categorical
Enters the number of nominal factors specified in Add N Factors.
Remove
Removes the selected factors.
Add N Factors
Adds multiple factors of a given type. Enter the number of factors to add and click Continuous or Categorical. Repeat Add N Factors to add multiple factors of different types.
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 “Definitive Screening 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. 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 type the desired values.
Editing the Factors Outline
In the Factors outline, note the following:
• To edit a factor name, double-click the factor name.
• To edit a value, click the value in the Values column.
Factor Types
Continuous
Numeric data types only. A continuous factor is a factor that you can conceptually set to any value between the lower and upper limits you supply, given the limitations of your measurement system.
Categorical
Either numeric or character data types with two levels. For a categorical factor, the value ordering is the order of the values, as entered from left to right. This ordering is saved in the Value Ordering column property after the design data table is created.
Factor Column Properties
For each factor, various column properties are saved to the data table.
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 add a block under Design Options, 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 definitive screening designs, 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.
Value Ordering
If the Design Role is Categorical or Blocking, the Value Ordering column property for the factor is saved. This property determines the order in which levels of the factor appear.
RunsPerBlock
Indicates the number of runs in each block. When a Block is selected in the Design Options outline and you then click Make Design, a factor with the default name Block is added to the Factors list. The RunsPerBlock column property is saved for that factor.
Design Options
The Design Options outline enables you to specify the blocking structure, the number of blocks, and the number of extra runs. Block effects are orthogonal to the main effects. Block sizes need not be equal.
The outline contains the following options:
No Blocks Required
Indicates that the design will not contain a blocking factor. This is the default selection.
Add Blocks with Center Runs to Estimate Quadratic Effects
Adds the number of blocks specified in the Number of Blocks text box. Constructs a design where block effects are orthogonal to main effects and where the model consisting of all main and quadratic effects is estimable. For details, see “Add Blocks with Center Runs to Estimate Quadratic Effects”
Add Blocks without Extra Center Runs
Adds the number of blocks specified in the Number of Blocks text box. Adds only as many center runs as required by the design structure. Constructs a design where block effects are orthogonal to main effects, but the model consisting of all main effects and quadratic effects might not be estimable. For details, see “Add Blocks without Extra Center Runs”.
Note: Use the Add Blocks without Extra Center Runs option only if you can assume that not all quadratic effects are important.
Number of Blocks
Indicates the number of blocks to add. The number of blocks that you can add ranges from two to the number of factors.
Number of Extra Runs
Adds non-center runs that enable you to conduct effective model selection. See “Extra Runs” and “Effective Model Selection for DSDs” in the “The Fit Definitive Screening Platform” chapter.
Tip: Adding runs to your design with the Extra Runs option enhances your ability to detect effects in the presence of many active effects. The recommended number of Extra Runs is four, which dramatically improves the power of the design to identify active second-order effects.
Make Design
Generates the design, presents it in the Design outline, and provides evaluation information in the Design Evaluation outline. The Output Options panel also appears, enabling you to create the design table.
Blocking in Definitive Screening Designs
This section describes the two blocking options:
If a design contains a categorical factor, a center run is a run where all continuous factors are set at their middle values. If all factors are categorical, the two blocking options are available. Both options produce designs whose blocks are orthogonal to main effects.
Add Blocks with Center Runs to Estimate Quadratic Effects
Note: For details about the construction and properties of blocked designs that estimate quadratic effects, see Jones and Nachtsheim (2016). The paper also contains information about treating the blocks as random effects.
The Add Blocks with Center Runs to Estimate Quadratic Effects option constructs a design with these properties:
• Block effects are orthogonal to main effects.
• The model consisting of all main and quadratic effects is estimable.
If a design contains only continuous factors, a blocked design for k factors having these properties can be constructed as follows:
• Remove the center run from the DSD design for k factors.
• Assign conference matrix foldover pairs to the same block.
• Add one center run to each block.
When some factors are categorical, the Add Blocks with Center Runs to Estimate Quadratic Effects option adds pairs of center runs within certain blocks. This structure ensures orthogonality and the ability to estimate all main and quadratic effects.
Because the only requirement on block size is that a block contains a foldover pair, the number of blocks can range from 2 to k, if k is even and from 2 to k+1, if k is odd. See “Conference Matrices and the Number of Runs”. JMP attempts to construct blocks of equal size.
Add Blocks without Extra Center Runs
The Add Blocks without Extra Center Runs option constructs a design that has a single center run when all factors are continuous and two center runs when some factors are categorical. The resulting design has these properties:
• Block effects are orthogonal to main effects.
• Each block effect is confounded with a linear combination of quadratic effects. This implies that the model consisting of all main and quadratic effects is not estimable.
For this reason, use this option only if you can assume that some quadratic effects are negligible.
A blocked design for k factors without extra center runs is constructed as follows:
• Assign conference matrix foldover pairs to the same block.
• If all factors are continuous, assign the single center run to a single block.
• If there are categorical factors, the unblocked definitive screening design requires the addition of two center runs to the foldover pairs defined by the conference matrix. See “Conference Matrices and the Number of Runs”. To construct the blocked design without extra center runs, these two center runs are added to a single block.
Because the only requirement on block size is that a block contains a foldover pair, the number of blocks can range from 2 to k, if k is even and from 2 to k+1, if k is odd. See “Conference Matrices and the Number of Runs”. JMP attempts to construct blocks of equal size.
Design
The Design outline shows the runs for the definitive screening design. The runs are given in a standard order. To change the run order for your design table, you can select Run Order options in the Output Options panel before generating the table.
Note: Definitive screening designs for four or fewer factors are based on a five-factor design. See “Definitive Screening Designs for Four or Fewer Factors”.
Design Evaluation
The Design Evaluation outline provides a number of ways to evaluate the properties of the generated design. Open the Design Evaluation outline to see the following options:
Power Analysis
Enables you to explore your ability to detect effects of given sizes.
Prediction Variance Profile
Shows the prediction variance over the range of factor settings.
Fraction of Design Space Plot
Shows how much of the model prediction variance lies below (or above) a given value.
Prediction Variance Surface
Shows a surface plot of the prediction variance for any two continuous factors.
Estimation Efficiency
For each parameter, gives the fractional increase in the length of a confidence interval compared to that of an ideal (orthogonal) design, which might not exist. Also gives the relative standard error of the parameters.
Alias Matrix
Gives coefficients that indicate the degree by which the model parameters are biased by effects that are potentially active, but not in the model.
Color Map on Correlations
Shows the absolute correlation between effects on a plot using an intensity scale.
Design Diagnostics
Indicates the optimality criterion used to construct the design. Also gives efficiency measures for your design.
For more details about the Design Evaluation panel, see “Design Evaluation” in the “Evaluate Designs” chapter.
Output Options
Use the Output Options panel to perform the following tasks:
• specify the order for the runs in the design data table
• construct the design table
• return to a previous point in the Definitive Screening Design window
Figure 7.19 Output Options Panel
The Output Options panel contains these options:
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 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.
Make Table
Click Make Table to construct the Definitive Screening Design data table.
In the Definitive Screening Design table, the Table panel (in the upper left) contains the following scripts. To run a script, click the green triangle next to the script name.
Fit Definitive Screening
Runs the DOE > Definitive Screening > Fit Definitive Screening platform. See Chapter 8, “The Fit Definitive Screening Platform”.
Evaluate Design
Runs the DOE > Design Diagnostics > Evaluate Design platform. See Chapter 15, “Evaluate Designs”.
DOE Dialog
Re-creates the Definitive Screening Design window that you used to generate the design table.
Figure 7.20 Definitive Screening Design Table Showing Scripts
Back
The Back button takes you back to where you were before clicking Make Design. You can make changes to the previous outlines and regenerate the design.
Definitive Screening Design Options
The red triangle menu in the Definitive Screening 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.
‒ A Model window opens 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.
For additional details, see “Simulate Responses”.
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
Specify 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 Definitive Screening Design window and saves it in an open script window.
Simulate Responses
When you click Make Table to create your design table, the Simulate Responses option does the following for each response:
• It adds random response values to the response column in your design table.
• It adds a new a column containing a simulation model formula to the design table. The formula and values are based on a main effects model.
A Model window opens where you can add and remove effects to define a model, specify parameter values, and select a response distribution for simulation. When you click Apply in the Model window, each column containing a simulation model formula is updated.
Control Window
Figure 7.21 shows the Model window for a design with two continuous factors (X1 and X2) and one two-level categorical factor (X3).
Figure 7.21 Simulate Responses Control Window
The window has three outlines:
• Factors
• Simulate Responses
• Distribution
The initial Simulate Responses outline shows terms for a main effects model with values of 1 for all coefficients. The Distribution outline shows a Normal distribution with error standard deviation equal to 1. If you have set Anticipated Coefficients as part of Power Analysis under Design Evaluation in the DOE window, then the initial values in the Simulate Responses outline are the values that you specified as Anticipated Coefficients and Anticipated RMSE (Error Std) in the Power Analysis outline.
Factors
Add terms to the simulation model using the Factors outline.
Interactions
Select factors in the list. Then select the interaction order from the Interactions menu. Those interactions are added to the list of Effects in the Simulate Responses outline.
RSM
Adds all possible response surface terms to the list of Effects in the Simulate Responses outline.
Powers
Select factors in the list. Then select the order from the Powers menu. Those powers are added to the list of Effects in the Simulate Responses outline.
Simulate Responses
To specify a model for simulated values, do the following:
1. For each term in the list of Effects, enter coefficients for the linear model used to simulate the response values. These define a linear function, L(x, β) = x‘β. See the Simulate Responses outline in Figure 7.21:
‒ The vector x consists of the terms that define the effects listed under Effects.
‒ The vector β is the vector of model coefficients that you specify under Y.
2. Under Distribution, select a response distribution.
3. Click Apply. A <Y> Simulated column containing simulated values and their formula is added to the design table, where Y is the name of the response column.
Reset coefficients
Sets all coefficients to 0.
Remove Term
Remove terms from the list of Effects. Select the effects to remove and click Remove Term. Note that you cannot remove main effects.
Distribution
Choose from one of the following distributions in the Simulate Responses window:
Normal
Simulates values from a normal distribution. Enter a value for Error σ, the standard deviation of the normal error distribution. If you have designated factors to have Changes of Hard in the Factors outline, you can enter a value for Whole Plots σ, the whole plot error. If you have designated factors to have Changes of Hard and Very Hard, you can enter values for both the subplot and whole plot errors. When you click Apply, random values and a formula containing a random response vector based on the model are entered in the column <Y> Simulated.
Binomial
Simulates values from a binomial distribution. Enter a value for N, the number of trials. Random integer values are generated according to a binomial distribution based on N trials with probability of success 1/(1 + exp(-L(x, β)). When you click Apply, random values and their formula are entered in the column <Y> Simulated. A column called N Trials that contains the value N is also added to the data table.
Poisson
Simulates random integer values according to a Poisson distribution with parameter exp((L(x, β)). When you click Apply, random values and their formula are entered in the column <Y> Simulated.
Note: You can set a preference to simulate responses every time you click Make Table. To do so, select File > Preferences > Platforms > DOE. Select Simulate Responses.
Technical Details
This section contains technical details for the following topics:
Structure of Definitive Screening Designs
Figure 7.22 shows an example of a definitive design with eight continuous factors. and four Extra Runs that correspond to fake factors. Notice the following:
• Each pair of rows is a foldover pair; each even-numbered row is -1 times the previous row. The foldover aspect of the design removes the confounding of two-factor interactions and main effects.
• Each factor is set at its center value for three runs; this, together with the design’s construction, makes all quadratic effects estimable.
• Rows 17 through 20 are the Extra Runs that correspond to the fake factors.
• Adding the center run in the last row enables you to fit a model that includes an intercept and all main and quadratic effects.
This structure is typical of definitive screening designs for continuous factors.
Figure 7.22 Definitive Screening Design for Eight Continuous Factors
Conference Matrices and the Number of Runs
Definitive screening designs in JMP are constructed using conference matrices (Xiao et al., 2012). A conference matrix is an m x m matrix C where m is even. The matrix C has 0s on the diagonal, off-diagonal entries equal to 1 or –1, and satisfies 
.
.Note: For certain even values of m, it is not known if a conference matrix exists.
Suppose that the number of factors, k, is five or larger. For the case of k ≤ 4 factors, see “Definitive Screening Designs for Four or Fewer Factors”.
Consider the case of k continuous factors and suppose that a conference matrix is available.
• When k is even, the k x k conference matrix is used to define k runs of the design. Its negative, –C, defines the foldover runs. A center point is added to the design to ensure that a model containing an intercept, main effects, and quadratic effects is estimable. So, for k even, the minimum number of runs in the definitive screening design is 2k + 1.
• When k is odd, a (k+1) x (k+1) conference matrix is used, with its last column deleted. A center point is added. Thus, for k odd, the minimum number of runs in the screening design is 2k + 3.
A similar procedure is used when some factors are categorical and a conference matrix is available. See Jones and Nachtsheim (2013).
• Instead of a single center point, two additional runs are required. These two runs are center runs where all continuous factors are set at their middle values.
• When there are k factors and k is even, the number of runs in the design is 2k + 2.
• When k is odd, the number of runs is 2k + 4.
For those values of m for which a conference matrix is not available, a definitive screening design can be constructed using the next largest conference matrix. As a result, the required number of runs might exceed 2k + 3, in the continuous case, and 2k + 4, in the categorical case.
Extra Runs
Extra runs are constructed using fictitious, or fake, factors. Adding k1 fake factors to a design results in 2k1 additional runs.
Denote the number of factors in your experimental study by k. Extra runs are constructed by creating a design for k + k1 factors, as described in “Conference Matrices and the Number of Runs”, and then dropping the last k1 columns. If k1 = 2, four extra runs are added. If k1 = 3, eight extra runs are added. As few as four extra runs can be highly beneficial in model selection.
For information about how extra runs are used, see “Effective Model Selection for DSDs” in the “The Fit Definitive Screening Platform” chapter.
Definitive Screening Designs for Four or Fewer Factors
Definitive screening designs for four or fewer factors are constructed using the five-factor definitive screening design as a base. This is because designs for k ≤ 4 factors constructed strictly according to the conference matrix approach have undesirable properties. In particular, it is difficult to separate second-order effects.
If you specify k ≤ 4 factors, a definitive screening design for five factors is constructed and unnecessary columns are dropped. For this reason, the number of runs for an unblocked design with k ≤ 4 factors is 13 if all factors are continuous or 14 if some factors are categorical.
Analysis of Experimental Data
In general, you want to fit a model that permits the possibility that two-way interactions are active. You also might want to include pure quadratic terms in your model. You might want to postulate a full second-order model, or you might want to specify an a priori model containing only certain second-order terms.
Two-Way Interactions
In fitting such a model, you need to be mindful of two facts:
• two-way interaction effects and quadratic effects are often correlated
• two-way interaction and quadratic effects cannot all be estimated simultaneously
Figure 7.23 shows a Color Map on Correlations for the design with eight continuous factors shown in Figure 7.22. The color map is for a full quadratic design. The eight pure quadratic effects are listed to the far right. You can construct this plot by using DOE > Design Diagnostics > Evaluate Design and entering the appropriate terms into the Alias Terms list. See “Alias Terms” in the “Evaluate Designs” chapter for details.
Figure 7.23 Color Map on Correlations for Full Quadratic Model
Use your cursor to hold your mouse pointer over the cells of the color map in order to see the absolute correlations between effects. You see that main effects are uncorrelated with all two-way interaction and pure quadratic effects. You also see that none of the effects are completely confounded with other effects because the only red cells are on the main diagonal. But note that some of the absolute correlations between two-factor interactions are substantial, with some at 0.75. Note also that absolute correlations between two-factor interactions and pure quadratic effects are either 0 or 0.3118.
If only main and pure quadratic effects are active, you can fit a saturated model that contains main effects and quadratic effects. This model will result in effect estimates that are unbiased, assuming no active three-way or higher order effects.
Because of the correlations involving second-order effects, you must be careful in fitting a model with two-way interactions. Analysis methodologies include the following, where the first is preferred:
• The method of Efficient Model Selection performs well, especially if many effects are likely to be active. See “Effective Model Selection for DSDs” in the “The Fit Definitive Screening Platform” chapter.
• Forward stepwise or all possible subsets regression performs adequately if the following conditions hold:
‒ The number of active effects is no more than half the number of runs.
‒ There are at most two active two-way interactions or at most one active quadratic effect.
Forward Stepwise Regression or All Possible Subsets Regression
This method consists of first specifying a full response surface model. Then do one of the following:
• Use forward stepwise regression with the Stopping Rule set to Minimum AICc and the Rules set to Combine to ensure model heredity.
• Use All Possible Models regression, where you select the option that imposes the heredity restriction and use the AICc criterion for model selection.
You cannot fit the full response surface model because the number of runs is less than the number of parameters. So your analysis depends on the assumption of effect sparsity, where you assume that the number of active effects is less than the number of runs. This approach has some limitations:
• If the number of active effects exceeds half the number of runs, both stepwise and all possible models regression have difficulty finding the correct model.
• The power of tests to detect moderate quadratic effects is low. A quadratic effect must exceed three times the error standard deviation for the power to exceed 0.9.
• Because of effect confounding, several models might be equivalent. Additional runs will be necessary to resolve the confounding.
..................Content has been hidden....................
You can't read the all page of ebook, please click here login for view all page.