Overview of Custom Design
Use the Custom Design platform to construct an optimal design custom built for your specific experimental needs.
You can include a wide range of factor types, including the following:
• Continuous
• Discrete numeric (with any number of levels)
• Categorical (with any number of levels)
• Blocking (with an arbitrary number of runs per block)
• Covariate
• Mixture
• Constant
• Uncontrolled
Specify the Region of Operability
You can restrict your experimental region to reflect your operating conditions using linear factor constraints or disallowed combinations. In particular, restrictions can be specified for categorical, continuous, and discrete numeric factors. See “Define Factor Constraints”.
Specify Factors with Hard-to-Change Levels
For continuous, discrete numeric, categorical, and mixture factors, you can indicate two levels of difficult-to-change factors. These difficulty levels are represented by whole plots or whole plots and split plots. You can also specify hard-to-change covariates.
Specify the Effects of Primary Interest
You can explicitly specify your assumed model. Your assumed model is an initial model that ideally contains all the effects that you want to estimate. Your model can contain any combination of main effects, interactions, response surface effects, and polynomial effects (up to the fifth power). You can specify the effects for which estimability is necessary and those for which estimability is desired. Custom Design uses a Bayesian optimality approach to estimate effects whose estimability is desired, subject to the number of runs. See “Model”.
Specify the Number of Runs
The Custom Design platform enables you to specify the number of runs that matches the budget for your experimental situation. The platform indicates the minimum number of runs that can be used to estimate the required effects and provides a default number of runs. These values can serve as a guide for determining a feasible number of runs. See “Design Generation”.
Construct the Appropriate Design Type
Custom Design can construct a wide variety of design types. These include classical designs and random block designs. For examples of different design types, see the “Examples of Custom Designs” chapter.
Construct an Optimal Design
Given your specific requirements, the Custom Design platform constructs a design that is optimal. The algorithm supports several optimality criteria:
• D optimality
• I optimality
• Bayesian D and I optimality (using If Possible effects)
• Alias optimality
Designs are constructed using the coordinate-exchange algorithm (Meyer and Nachtsheim, 1995). See “Coordinate-Exchange Algorithm”.
Example of a Custom Design
The following example describes a wine tasting experiment. Your employer grows two varieties of Pinot Noir grapes that can be processed in different ways. Your goal is to determine which factors affect the taste of Pinot Noir wine. Before the grapes are processed, you set up your experimental design. Once processed, the wine samples are aged for 12 months, then filtered and bottled. At this point, the wine samples are rated for quality by expert wine tasters.
Response
Most of your vineyard’s product is sold to five large wine distributors. You arrange for a wine-tasting expert from each distributor to evaluate the wine samples for quality. To maximize the number of factors that you can study, you decide that each expert must rate eight different samples. This means that your design needs to have 40 wine samples, or runs.
The ratings follow a 0 – 20 scale, where 0 is the worst and 20 is the best. Rating, the variable consisting of the experts’ ratings, is the response of interest. You want to identify the wine-related factors that maximize the response.
Blocking Factor
A blocking factor is used to account for variation that is not necessarily of direct interest. A blocking factor is particularly effective when observations taken at one factor level are expected to be more similar than observations at different levels. In your experiment, ratings by one expert are likely to have similar characteristics and to differ from ratings by a different expert. Yet, you are interested in which properties of the wine lead to high ratings by all experts.
Because each rater tastes eight wines, Rater is a blocking factor with eight runs per block. For this experiment, only these five raters are of concern. You are not interested in generalizing to a larger population of raters.
Process Factors
You have identified nine process factors for the study. These include the grape variety, the field on which the grapes were grown, and seven other factors related to processing. You can experiment with any combination of these factors. Also, the factors can be varied at will as part of the experiment. Relative to the experiment, these factors are all “Easy” to change. For information about specifying factor changes, see “Changes and Random Blocks”.
The factors and their levels appear in Table 4.1. Note that all of these factors are categorical. The factors and their levels are also given in the factor table Wine Factors.jmp in the Design Experiment folder of Sample Data.
To experiment with all possible combinations of these factors would require a staggering 
runs. However, in this example, you are able to construct a compelling design in only 40 runs.
runs. However, in this example, you are able to construct a compelling design in only 40 runs.|
Factor
|
Levels
|
|
Variety
|
Bernard, Dijon
|
|
Field
|
1, 2, 3, 4
|
|
De-Stem
|
No, Yes
|
|
Yeast
|
Cultured, Wild
|
|
Temperature
|
High, Low
|
|
Press
|
Hard, Soft
|
|
Barrel Age
|
New, 2 Years
|
|
Barrel Seasoning
|
Air, Kiln
|
|
Filtering
|
No, Yes
|
Create the Design
Note: In order to introduce and describe the Custom Design outlines, this example works through the outlines in succession.
To create the custom design, follow the steps in these sections:
• “Model”
• “Design”
For information about the complete DOE workflow, see “The DOE Workflow: Describe, Specify, Design” in the “Starting Out with DOE” chapter.
Responses
Add your response, the response Goal, and, if appropriate, the Lower Limit, Upper Limit, and Importance. Here, the response is Rating.
1. Select DOE > Custom Design.
2. Double-click Y under Response Name and type Rating.
Note that the default Goal is Maximize. Because you want to maximize the taste rating, do not change the goal.
3. Click under Lower Limit and type 0.
The least desirable rating is 0.
4. Click under Upper Limit and type 20.
The most desirable rating is 20.
5. Leave the area under Importance blank.
Because there is only one response, that response is given Importance 1 by default.
Figure 4.2 shows the completed Responses outline.
Factors
Enter factors either manually or automatically using a pre-existing table that contains the factors and settings.
• If you are designing a new experiment, you must first enter the factors manually. See “Entering Factors Manually”.
• Once you have saved the factors using the Save Factors option, you can load them automatically using the saved table. See “Entering Factors Using Load Factors”.
Both methods add these four outlines to the Custom Design window: Define Factor Constraints, Model, Alias Terms, and Design Generation.
Entering Factors Manually
1. First, add the blocking factor, Rater. Click Add Factor > Blocking > 8 runs per block.
2. Type Rater over the default Name of X1.
Note that Role is set to Blocking. Note also that only one setting for Values appears. This is because the number of blocks cannot be determined until the desired number of runs is specified. Once you specify the Number of Runs in the Design Generation outline, the number of levels for Rater updates to what is required.
3. Click Add Factor > Categorical > 2 Level.
4. Type Variety over the default Name of X2.
Note that Role is set to Categorical, as requested, and that Changes is set to Easy by default.
5. Click L1 and L2 and change them to Bernard and Dijon.
6. Click Add Factor > Categorical > 4 Level.
7. Type Field over the default Name of X3.
8. Click L1, L2, L3, and L4, and change them to 1, 2, 3, and 4.
9. Click Add Factor > Categorical > 2 Level.
10. Type De-Stem over the default Name of X4.
11. Click L1 and L2 and change them to No and Yes.
Add the rest of the factors as follows:
12. Type 6 next to Add N Factors, and then click Add Factor > Categorical > 2 Level.
13. Type the following names and values over the default ones:
‒ Yeast (Cultured and Wild)
‒ Temperature (High and Low)
‒ Press (Hard and Soft)
‒ Barrel Age (New and Two Years)
‒ Barrel Seasoning (Air and Kiln)
‒ Filtering (No and Yes)
The completed Factors outline appears in Figure 4.2.
Figure 4.2 Completed Responses and Factors Outlines
14. Click Continue.
The following outlines are added to the Custom Design window:
• Define Factor Constraints (not used in this example)
• Model
• Alias Terms
• Design Generation
Entering Factors Using Load Factors
To enter factors using a table containing factor information, proceed as follows:
1. Select Help > Sample Data Library and open Design Experiment/Wine Factors.jmp.
2. From the Custom Design red triangle menu, select Load Factors.
After loading the factors, the Custom Design window automatically updates. The following outlines are added to the Custom Design window:
• Define Factor Constraints (not used in this example)
• Model
• Alias Terms
• Design Generation
Model
The Model outline shows all main effects as Necessary, indicating that the design needs to be capable of estimating all main effects. For this example, your assumed model reflects your interest in main effects only. However, if you wanted to estimate other effects, you could add them to the Model outline. See “Model”.
Figure 4.3 Model Outline
Alias Terms
The Alias Terms outline specifies the effects to be shown in the Alias Matrix, which appears later. See “Alias Matrix”. The Alias Matrix shows the aliasing relationships between the Model terms and the effects listed in the Alias Terms outline. Open the Alias Terms outline node to verify that all two-factor interactions are listed.
Figure 4.4 Partial View of the Alias Terms Outline
Duplicate Results (Optional)
In the next step, you generate your design. Because the Custom Design algorithm begins with a random starting design, your design might differ from the one shown in Figure 4.5. If you want to obtain a design with exactly the same runs and run order, perform the following steps:
1. From the Custom Design red triangle menu, select Set Random Seed.
2. Type 100526291 (the random seed).
3. Click OK.
4. From the Custom Design red triangle menu, select Number of Starts.
5. Type 2.
6. Click OK.
Note: Setting the Random Seed and Number of Starts reproduces the exact results shown in this example. In constructing a design on your own, these steps are not necessary.
Proceed to the Design Generation section.
Design Generation
In the Design Generation outline, you can enter additional details about the structure and size of your design. In this example, the Default design shows 16 runs. But you have five raters, each of whom can sample eight wines. This means that you want a design with 40 runs. Change the number of runs as follows:
1. Under Number of Runs, type 40 in the User Specified box.
Because you do not want to replicate runs, leave the Number of Replicate Runs set to 0.
2. Click Make Design.
The Design and Design Evaluation outlines are added to the Custom Design window. The Output Options panel also appears.
Design
The Design outline shows the runs in the design that you have constructed. Later, you are able to randomize the order under Output Options. For now, verify that this design is appropriate for your experiment. For example, check that each of five Raters evaluates eight wines, that all necessary factors are shown, and that none of the settings represent infeasible combinations.
Figure 4.5 Design for Wine Experiment
Design Evaluation
The Design Evaluation outline provides different ways to evaluate your design.
Note: For details about the Design Evaluation outline, see the “Evaluate Designs” chapter.
For this example, open the Design Evaluation outline, and examine the Color Map on Correlations, the Alias Matrix, and Design Diagnostics.
Color Map on Correlations
The Color Map on Correlations shows the absolute value of the correlation between any two effects that appear in either the Model or the Alias Terms outline. (The colors shown in Figure 4.6 are the JMP default colors.)
Figure 4.6 Color Map on Correlations
The main effects are represented by the 15 terms in the upper left corner of the map. The deep blue color corresponding to the correlations of main effects with other main effects indicate correlations of 0. This means that all main effects are orthogonal and can be estimated independently.
The only red in Figure 4.6 is on the main diagonal. The color indicates absolute correlations of one, reflecting that each term is perfectly correlated with itself. It follows that no main effect is completely confounded with any two-way interaction. In fact, the absolute values of the correlations of main effects with two-way interactions are fairly low. This means that estimates of main effects might be only slightly biased by the presence of active two-way interactions.
Tip: Position your mouse pointer over cells in the color map to see the absolute correlations between effects.
Alias Matrix
In the Alias Matrix, model effects are listed in the column on the left. For a given model effect, a column entry indicates the degree to which the column effect (if active) biases the estimate of the model effect.
Figure 4.7 Partial View of Alias Matrix
For example, consider the model effect Barrel Seasoning. If Variety*Press is active, then the expected value of the estimate for the Barrel Seasoning effect differs from an unbiased estimate of that effect. The amount by which it differs is equal to 0.4 times the effect of Variety*Press. Therefore, what appears to be a significant Barrel Seasoning estimated effect could in reality be a significant Variety*Press effect.
Design Diagnostics
The Design Diagnostics outline provides information about the efficiency of the design. Efficiency measures compare your design to a theoretically optimal design, which might not exist. The efficiency values are ratios, expressed as percents, of the efficiency of your design to the efficiency of this optimal design. For details about the efficiency measures, see “Estimation Efficiency” in the “Evaluate Designs” chapter.
Figure 4.8 Design Diagnostics Outline
Notice that the D-, G-, and A-efficiency values are all 100%. Because your design is orthogonal for main effects, the design is optimal for the main effects model relative to all three efficiency criteria.
The first line in the Design Diagnostics outline indicates that your design was constructed to optimize the D-efficiency criterion. For more details, see the Optimality Criterion description in “Custom Design Options”. In this case, your design has D Efficiency of 100%.
Output Options
Specify the order of runs in your data table using the Output Options panel. The default selection, Randomize within Blocks, is appropriate for this example. Simply click Make Table.
A Custom Design table is created and opens, similar to the one in Figure 4.9.
Note: Your table might look different because the algorithm that creates it uses a random starting design. To obtain the precise table shown in Figure 4.9, follow the steps in “Duplicate Results (Optional)”.
Figure 4.9 Custom Design Table
Note the following:
• In the Table panel, the Model, Evaluate Design, and DOE Dialog scripts are added during the design creation process. The Model script opens a Fit Model window containing the effects that you specified as Necessary in the Custom Design dialog. The DOE Dialog script re-creates the window used to generate the design table.
• In the Columns panel, the asterisks to the right of the factors and response indicate column properties that have been saved to the columns in the data table. These column properties are used in the analysis of the data. For details about column properties, see “Factors” and “Factor Column Properties”.
Analyze the Data
Now you are ready to run your experiment, gather the Rating data, and insert the results in the Rating column of your Custom Design table.
1. Select Help > Sample Data Library and open Design Experiment/Wine Data.jmp.
The Wine Data.jmp table is exactly the same as the Custom Design table shown in Figure 4.9, except that it contains your recorded experimental results.
2. In the Table panel, click the green triangle next to the Model script.
Figure 4.10 Fit Model Dialog for Wine Experiment
Notice that Rater, the blocking factor, is added as a fixed effect, rather than as a random block effect. This is appropriate because the five raters were specifically chosen and are not a random sample from a larger population.
3. Click Run.
Interpret the Full Model Results
The results are shown below.
Figure 4.11 Partial Model Fit Results
Note the following:
• The Actual by Predicted Plot shows no obvious evidence of lack of fit.
• The model is significant, as indicated by the Actual by Predicted Plot and by the P value beneath it.
• The Effect Tests report indicates that seven of the model terms are significant at the 0.05 level. Field, Temperature, and Barrel Age are not significant.
• The Effect Summary report lists these effects in decreasing order of significance. Larger LogWorth values correspond to smaller PValues and greater significance.
Reduce the Model
Reduce the model by removing the effects that you identified as inactive:
1. In the Effect Summary report, press the Control key and hold it as you select Temperature, Field, and Barrel Age.
2. Click Remove.
The report updates to show the model fit with these three effects removed.
Interpret the Reduced Model Results
The Actual by Predicted Plot for the reduced model shows no lack of fit issues. The Effect Summary and the Effect Test report show that the remaining seven terms are significant at the 0.05 level.
Figure 4.12 shows the Prediction Profiler. Recall that you specified a response goal of Maximize, with lower and upper limits of 0 and 20. Setting these limits caused a Response Limits column property to be saved to the Rating column in the Custom Design table. The Prediction Profiler uses the Response Limits information to construct a Desirability function, which appears in the right-most plot in the top row in Figure 4.12. The bottom row displays Desirability traces.
The first six plots in the top row show traces of the predicted model. For each factor, the line in the plot shows how Rating varies when all other factors are set at the values defined by the red dashed vertical lines. By default, the profiler appears with categorical factors set at their low settings. By varying the settings for the factors, you can see how the predicted Rating for wines changes. Notice that a confidence interval is given for the mean predicted Rating.
Observe that Rater is not included among the factors shown in the profiler. This is because Rater is a block variable. You included Rater to explain variation, but Rater is not of direct interest in terms of optimizing process factor settings. The predicted Rating for a wine with the given settings is the average of the predicted ratings for that wine by all raters.
Figure 4.12 Profiler for Reduced Model
Optimize Factor Settings
You would like to identify settings that maximize Rating across raters.
1. From the Prediction Profiler red triangle menu, select Optimization and Desirability > Maximize Desirability.
The red dashed vertical lines in the Prediction Profiler update to show optimal settings for each factor. The optimal settings result in a predicted rating of 19.925. In general, there can be different sets of factor settings that result in the same optimal value.
Figure 4.13 Prediction Profiler with Factor Settings Optimized
2. To see predicted ratings for all runs, save the Prediction Formula. From the Response Rating red triangle menu, select Save Columns > Prediction Formula.
A column called Pred Formula Rating is added to the data table. Note that one of the runs, row 33, was given the maximum rating of 20 by Rater 5. The predicted rating for that run by Rater 5 is 19.550. But the row 33 trial was run at the optimal settings. The predicted value of 19.925 given for these settings in the Prediction Profiler is obtained by averaging the predicted ratings for that run over all five raters.
Lock a Factor Level
When you maximized desirability, you learned that the optimal rating is achieved with the Dijon variety of grapes. See Figure 4.13. Your manager points out that it would be cost-prohibitive to replant the fields that are growing Bernard grapes with young Dijon vines. Therefore, you need to find optimal process settings and the predicted rating for Bernard grapes.
1. In the Variety plot of the Prediction Profiler, drag the red dashed vertical line to Bernard.
2. Press Control and click in one of the Variety plots.
The Factor Settings window appears.
3. Select Lock Factor Setting and click OK.
4. From the Prediction Profiler red triangle menu, select Optimization and Desirability > Maximize Desirability.
Figure 4.14 Prediction Profiler with Optimal Settings for Bernard Variety
The optimal settings are unchanged because the model contains no interaction terms. The predicted rating at these settings is 17.975.
Profiler with Rater
If you want to see the Profiler traces for the levels of Rater, perform the following steps:
1. From the Prediction Profiler red triangle menu, select Reset Factor Grid.
A Factor Settings window appears with columns for all of the factors, including Rater. The box under Rater and next to Show is not checked. This indicates that Rater is not shown in the Prediction Profiler.
2. Check the box under Rater in the row corresponding to Show.
3. Deselect the box under Rater in the row corresponding to Lock Factor Setting.
4. Click OK.
The Profiler updates to show a plot for Rater.
5. Click in either plot above Rater.
Figure 4.15 Profiler for Reduced Model Showing Rater
A dashed vertical red line appears. Drag this line to see the traces for each of the raters. Keep in mind that Variety is still locked at Bernard.To unlock Variety, press Control and click in one of the Variety plots. In the Factor Settings window that appears, deselect Lock Factor Setting.
Summary
In your wine tasting experiment, using only 40 runs, you have identified six (out of nine) factors that have an effect on ratings for Pinot Noir grapes. You found that you could achieve a predicted rating of 19.925 (out of a possible 20) at the optimal settings for those factors. You also identified optimal settings for both varieties of grapes.
In this section, you constructed a design using the outlines in the Custom Design window. The next section explains each outline and the design steps in more detail.
Custom Design Window
The Custom Design window updates as you work through the design steps. The outlines that appear, separated by buttons that update the window, follow the flow in Figure 4.16.
Figure 4.16 Custom Design Flow
This section describes the outlines in the Custom Design window.
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 4.17 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.
Tip: When you have completed the Factors outline, consider selecting Save Factors from the red triangle menu. This saves the factor names, roles, changes, and values in a data table that you can later reload.
Figure 4.18 Factors Outline
Add Factor
Select the factor type. For details, see “Factor Types”.
Remove
Removes the selected factors.
Note: If you attempt to remove all factors after clicking the Continue or Back button, one continuous factor remains. You can delete it after you add new factors.
Add N Factors
Adds multiple factors. Enter the number of factors to add, click Add Factor, and then select the factor type. Repeat Add N Factors to add multiple factors of different types.
Factors Outline
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.
Changes
Indicates whether the factor levels are Easy, Hard, or Very Hard to change. Click on the default value of Easy to change it. When you specify factors as Hard or Very Hard to change, your design reflects these restrictions on randomization. A factor cannot be designated as Very Hard unless the Factors list contains a factor designated as Hard. The Factor Changes column property is saved to the data table. For more details, see “Changes and Random Blocks”.
Values
The experimental settings for the factors. To insert Values, click on the default values and enter the desired values.
Editing the Factors Outline
In the Factors outline, note the following:
• To edit a factor name, double-click the factor name.
• Categorical factors have a down arrow to the left of the factor name. Click the arrow to add a level.
• To remove a factor level, click the value, click Delete, and click outside the text box.
• To modify the entry under Changes, click the value in the Changes column and select the appropriate entry.
• To edit a value, click the value in the Values column.
Factor Types
To choose a factor type, click Add Factor in Custom Design.
Note: A Design Role column property containing each factor’s role is added to that factor’s column in the design table that is generated. The Design Role column property ensures that the factor is modeled correctly.
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 process and measurement system.
Discrete Numeric
Numeric data types only. A discrete numeric factor can assume only a discrete number of values. These values have an implied order.
The default values for a discrete numeric factor with k levels, where 
, are the integers 
. The default values for a discrete numeric factor with 
levels are -1 and 1. Replace the default values with the settings that you plan to use in your experiment.
, are the integers . The default values for a discrete numeric factor with levels are -1 and 1. Replace the default values with the settings that you plan to use in your experiment.Note: Not all levels of a discrete numeric factor appear in the design. The levels that appear are determined by your specifications in the Model outline. If you need all levels to appear in your design, consider using the Screening Design platform.
In the assumed model, the effects for a discrete numeric factor with k levels include polynomial terms in that effect through order k-1. For k greater than 6, powers up to the 5th level are included. The Estimability for polynomial effects (powers of two or higher) is set to If Possible. This allows the algorithm to use the multiple levels as permitted by the run size. If the polynomial terms are not included, then a main effects only design is created. For more details about how discrete numeric factors are treated in the assumed model, see “Model”.
Fit Model treats a discrete numeric factor as a continuous predictor. The Model script that is saved to the design table does not contain any polynomial terms of order greater than two.
Categorical
Either numeric or character data types. The data type in the resulting data table is categorical. The value ordering of the levels 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.
Blocking
Either numeric or character data types. A blocking factor is a special type of categorical factor that can enter the model only as a main effect. When you define a blocking factor, you specify the number of runs per block. The RunsPerBlock column property is saved to the design table. The Default run size always assumes that there are at least two blocks. If you specify a run size that is not an integer multiple of the number of runs per block, JMP tries to balance the design to the extent possible. In balancing the design, JMP ensures that there are at least two runs per block.
Covariate
Either numeric or character data types. The values of a covariate factor are measurements on experimental units that are known in advance of an experiment. Covariate values are selected to ensure the optimality of the resulting design relative to the optimality criterion. See “Changes and Random Blocks” and “Covariates with Hard-to-Change Levels”.
JMP obtains the covariate factors and their values from a data table that contains the measured covariates for the available experimental units. Make this data table your current data table. When you select Covariate, a list of columns in the current data table opens, and you select the columns containing covariates from this list.
In some situations, you may want to select a small set of design points from a larger set of candidate settings. For example, you may have multiple measurement columns (factors) for a large batch of units. You want to treat the measurements for each unit as a candidate run. From these candidate runs, you want to select a small but optimal collection for which you will measure a response. In this case, make the data table of all candidate runs the active table, select Add Factor > Covariate, and enter all of your measurement columns as covariates. Specify your desired run size. The Custom Design platform will identify an optimal collection of design settings.
Note: You cannot specify a Number of Runs or Number of Whole Plots that exceeds the number of rows in the covariate’s data table.
Mixture
Continuous factors that represent ingredients in a mixture. The values for a mixture factor must sum to a constant. By default, the values for all mixture factors sum to one. To set the sum of the mixture components to some other positive value, select Advanced Options > Mixture Sum from the red triangle menu. The Mixture column property is saved to the data table.
Constant
Either numeric or character data types. A constant factor is a factor whose values are fixed during an experiment. Constant factors are not included in the Model outline or in the Model script that is saved to the data table.
Uncontrolled
Either numeric or character data types. An uncontrolled factor is one whose values cannot be controlled during production, but it is a factor that you want to include in the model. It is assumed that you can record the factor's value for each experimental run.
An empty column with a Continuous Modeling Type is created in the design table. You can change the column’s Data Type and Modeling Type in the Column Info window if required. Enter your data in this column. Uncontrolled factors are included in the Model outline and the Model script that is saved to the data table.
Changes and Random Blocks
Specifying the relative difficulty of changing a factor from run to run is useful in industrial experimentation. It is often convenient to make several runs while keeping factors that are hard-to-change fixed at some setting. A Changes value of Hard results in a split-plot design. A Changes value of Very Hard results in a split-split-plot design or a two-way split-plot design.
You can set Changes for Continuous, Discrete Numeric, Categorical, and Mixture factors to Hard and Very Hard. To set a factor to Very Hard, the list must contain another factor that is set to Hard.
You can set Changes for a Covariate factor to Hard. In this case, all other covariates are also set to Hard and the remaining factors are set to Easy. The algorithm requires a combination of row exchange and coordinate exchange. For this reason, even moderately sized designs might take some time to generate.
For designs with Hard or Very Hard to change factors, Custom Design strives to find a design that is optimal, given your specified optimality criterion. See “Optimality Criteria”. For details about the methodology used to generate split-plot designs, see Jones and Goos (2007). For details relating to designs with hard-to-change covariates, see Jones and Goos (2015).
Figure 4.19 shows a split-split-plot scenario, using the factors from the Cheese Factors.jmp sample data table (located in the Design Experiment folder).
Figure 4.19 Factors and Design Generation Outline for a Split-Split-Plot Design
If you assign Changes as Hard for one or more factors, but no factors have Changes assigned as Very Hard, a categorical factor called Whole Plots is added to the design. This situation results in a split-plot design:
• Each level of Whole Plots corresponds to a block of constant settings of the hard-to-change factors.
• The Model script in the design table applies the Random Effect attribute to the factor Whole Plots.
• The factor Whole Plots is assigned the Design Role column property with a value of Random Block.
When you designate Changes as both Hard and Very Hard, categorical factors called Subplots and Whole Plots are added to the design. This situation results in a split-split-plot design:
• Each level of Subplots corresponds to a block of constant settings of the hard-to-change factors.
• Each level of Whole Plots corresponds to a block of constant settings of the very-hard-to-change factors.
• The Model script in the design table applies the Random Effect attribute to the Whole Plots and Subplots effects.
• The levels of the hard-to-change factor are assumed to be nested within the levels of the very-hard-to-change factor by default.
• In the design table, both of the factors Whole Plots and Subplots are assigned the Design Role column property with a value of Random Block.
To construct a two-way split-plot design, select the Hard to change factors can vary independently of Very Hard to change factors option under Design Generation. The option crosses the levels of the hard-to-change factor with the levels of the very-hard-to-change factor. See “Two-Way Split-Plot Designs”.
Use the Number of Whole Plots and Number of Subplots text boxes to specify values for the numbers of whole plots or subplots. These boxes are initialized to suggested numbers of whole plots and subplots. For information about how these values are obtained, see “Numbers of Whole Plots and Subplots”.
For more details and scenarios that illustrate random block split-plot, split-split-plot, and two-way split-plot designs, see “Designs with Randomization Restrictions”. For details about designs with hard-to-change covariates, see “Covariates with Hard-to-Change Levels”.
Factor Column Properties
For each factor, various column properties are saved to the data table. You can find details about these column properties and related examples in Appendix A, “Column Properties”.
Design Role
Each factor is given 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 random block under Design Generation, that factor is assigned the Random Block 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. For details, see “Design Role” in the “Column Properties” appendix.
Factor Changes
Each factor is assigned the Factor Changes column property. The value that you specify under Changes determines the value of its Factor Changes column property. The Factor Changes property reflects how the factor is used in modeling the experimental data. Factor Changes values are used in the Augment Design and Evaluate Design platforms. For details, see “Factor Changes” in the “Column Properties” appendix.
Coding
If the Role is Continuous, Discrete Numeric, a continuous Covariate, or Uncontrolled, 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. For details, see “Coding” in the “Column Properties” appendix.
Value Ordering
If the 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. For details, see “Value Ordering” in the “Column Properties” appendix.
Mixture
If the Role is Mixture, the Mixture column property for the factor is saved. This property indicates the limits for the factor and the mixture sum. It also enables you to choose the coding for the mixture factors. For details, see “Mixture” in the “Column Properties” appendix.
RunsPerBlock
For a blocking factor, indicates the maximum allowable number of runs in each block. When a Blocking factor is specified in the Factors outline, the RunsPerBlock column property is saved for that factor. For details, see “RunsPerBlock” in the “Column Properties” appendix.
Define Factor Constraints
Note: If you are working in Covering Arrays, see the “Covering Arrays” chapter for more information.
Use Define Factor Constraints to restrict the design space. Unless you have loaded a constraint or included one as part of a script, the None option is selected. To specify constraints, select one of the other options:
Specify Linear Constraints
Specifies inequality constraints on linear combinations of factors. Only available for factors with a Role of Continuous or Mixture. See “Specify Linear Constraints”.
Note: When you save a script for a design that involves a linear constraint, the script expresses the linear constraint as a less than or equal to inequality (
).
).Use Disallowed Combinations Filter
Defines sets of constraints based on restricting values of individual factors. You can define both AND and OR constraints. See “Use Disallowed Combinations Filter”.
Use Disallowed Combinations Script
Defines disallowed combinations and other constraints as Boolean JSL expressions in a script editor box. See “Use Disallowed Combinations Script”.
Specify Linear Constraints
In cases where it is impossible to vary continuous factors independently over the design space, you can specify linear inequality constraints. Linear inequalities describe factor level settings that are allowed.
Click Add to enter one or more linear inequality constraints.
Add
Adds a template for a linear expression involving all the continuous factors in your design. Enter coefficient values for the factors and select the direction of the inequality to reflect your linear constraint. Specify the constraining value in the box to the right of the inequality. To add more constraints, click Add again.
Note: The Add option is disabled if you have already constrained the design region by specifying a Sphere Radius.
Remove Last Constraint
Removes the last constraint.
Check Constraints
Checks the constraints for consistency. This option removes redundant constraints and conducts feasibility checks. A JMP alert appears if there is a problem. If constraints are equivalent to bounds on the factors, a JMP alert indicates that the bounds in the Factors outline have been updated.
Use Disallowed Combinations Filter
This option uses an adaptation of the Data Filter to facilitate specifying disallowed combinations. For detailed information about using the Data Filter, see the JMP Reports chapter in the Using JMP book.
Select factors from the Add Filter Factors list and click Add. Then specify the disallowed combinations by using the slider (for continuous factors) or by selecting levels (for categorical factors).
The red triangle options for the Add Filter Factors menu are those found in the Select Columns panel of many platform launch windows. See the Get Started chapter in the Using JMP book for additional details about the column selection menu.
When you click Add, the Disallowed Combinations control panel shows the selected factors and provides options for further control. Factors are represented as follows, based on their modeling types:
Continuous Factors
For a continuous factor, a double-arrow slider that spans the range of factor settings appears. An expression that describes the range using an inequality appears above the slider. You can specify disallowed settings by dragging the slider arrows or by clicking on the inequality bounds in the expression and entering your desired constraints. In the slider, a solid blue highlight represents the disallowed values.
Categorical Factor
For a categorical factor, the possible levels are displayed either as labeled blocks or, when the number of levels is large, as list entries. Select a level to disallow it. To select multiple levels, hold the Control key. The block or list entries are highlighted to indicate the levels that have been disallowed. When you add a categorical factor to the Disallowed Combinations panel, the number of levels of the categorical factor is given in parentheses following the factor name.
Disallowed Combinations Options
The control panel has the following controls:
Clear
Clears all disallowed factor level settings that you have specified. This does not clear the selected factors.
Start Over
Removes all selected factors and returns you to the initial list of factors.
AND
Opens the Add Filter Factors list. Selected factors become an AND group. Any combination of factor levels specified within an AND group is disallowed.
To add a factor to an AND group later on, click the group’s outline to see a highlighted rectangle. Select AND and add the factor.
To remove a single factor, select Delete from its red triangle menu.
OR
Opens the Add Filter Factors list. Selected factors become a separate AND group. For AND groups separated by OR, a combination is disallowed if it is specified in at least one AND group.
Red Triangle Options for Factors
A factor can appear in several OR groups. An occurrence of the factor in a specific OR group is referred to as an instance of the factor.
Delete
Removes the selected instance of the factor from the Disallowed Combinations panel.
Clear Selection
Clears any selection for that instance of the factor.
Invert Selection
Deselects the selected values and selects the values not previously selected for that instance of the factor.
Display Options
Available only for categorical factors. Changes the appearance of the display. Options include:
‒ Blocks Display shows each level as a block.
‒ List Display shows each level as a member of a list.
‒ Single Category Display shows each level.
‒ Check Box Display adds a check box next to each value.
Find
Available only for categorical factors. Provides a text box beneath the factor name where you can enter a search string for levels of the factor. Press the Enter key or click outside the text box to perform the search. Once Find is selected, the following Find options appear in the red triangle menu:
‒ Clear Find clears the results of the Find operation and returns the panel to its original state.
‒ Match Case uses the case of the search string to return the correct results.
‒ Contains searches for values that include the search string.
‒ Does not contain searches for values that do not include the search string.
‒ Starts with searches for values that start with the search string.
‒ Ends with searches for values that end with the search string.
Use Disallowed Combinations Script
Use this option to disallow particular combinations of factor levels using a JSL script. This option can be used with continuous factors or mixed continuous and categorical factors.
This option opens a script window where you insert a script that identifies the combinations that you want to disallow. The script must evaluate as a Boolean expression. When the expression evaluates as true, the specified combination is disallowed.
When forming the expression for a categorical factor, use the ordinal value of the level instead of the name of the level. If a factor’s levels are high, medium, and low, specified in that order in the Factors outline, their associated ordinal values are 1, 2, and 3. For example, suppose that you have two continuous factors, X1 and X2, and a categorical factor X3 with three levels: L1, L2, and L3, in order. You want to disallow levels where the following holds:
Enter the expression (Exp(X1) + 2*X2 < 0) & (X3 == 2) into the script window.
Figure 4.20 Expression in Script Editor
(In the figure, unnecessary parentheses were removed by parsing.) Notice that functions can be entered as part of the Boolean expression.
Model
Specify your assumed model (which contains all the effects that you want to estimate) in the Model outline. For each effect that you specify, you can designate that effect’s Estimability. The Estimability value indicates whether it is Necessary to estimate that effect, or if you are content to estimate that effect If Possible.
When the Model outline opens, for most factors only the main effects appear. If you have entered a discrete numeric factor, polynomial terms also appear. The Estimability of second-and higher-order terms is set to If Possible. If you want to ensure that these terms are estimable, change their Estimability to Necessary.
Note: You can ensure that the estimability of discrete numeric polynomial terms is always set to Necessary. Select File > Preferences > Platforms > DOE. Check Discrete Numeric Powers Set to Necessary.
Figure 4.21 Model Outline
When you construct your design table, JMP saves a Model script to the data table. Except for discrete numeric factors, the Model script contains the effects shown in the Model outline. For a discrete numeric factor, the Model script contains only its main effect and quadratic term.
The Model outline contains the following buttons and fields:
Main Effects
Adds main effects for all factors in the model, and polynomial terms for discrete numeric factors.
Interactions
Adds interaction effects. If no factors are selected in the Factors outline, select 2nd, 3rd, 4th, or 5th to add all appropriate interactions up to that order. Add interactions up to a given order for specific factors by selecting the factor names in the Factors outline, selecting Interactions, and then specifying the appropriate order. Interactions between non-mixture and mixture factors, and interactions with blocking and constant factors, are not added.
RSM
Adds interaction and quadratic terms up to the second order (response surface model terms) for continuous factors. Categorical factors are not included in RSM terms. Main effects for non-mixture factors that interact with all the mixture factors are removed.
Cross
Adds specific interaction terms. Select factor names in the Factors outline and effect names in the Model outline. Click Cross to add the crossed terms to the Model outline.
Powers
Adds polynomial terms. If no factor names are selected in the Factors outline, adds polynomial terms for all continuous factors. If factor names are selected in the Factors outline, adds polynomial terms for only those factors. Select 2nd, 3rd, 4th, or 5th to add polynomial terms of that order.
Scheffé Cubic
Adds Scheffé cubic terms for all mixture factors. These terms are used to specify a mixture model with third-degree polynomial terms.
Remove Term
Removes selected effects.
Name
Name of the effect.
Estimability
A designation of your need to estimate the effect.
‒ If Estimability is set to Necessary, the algorithm ensures that the effect is estimable.
‒ If Estimability is set to If Possible, the algorithm attempts to make that effect estimable, as permitted by the number of runs that you specify.
Except for polynomial terms for discrete numeric factors, all effects are specified as Necessary by default. Click an effect’s Estimability value to change it.
Bayesian D-Optimality and Estimation of If Possible Effects
The Bayesian D-Optimal design approach obtains precise estimation of all Necessary terms while providing omnibus detectability (and some estimability) for If Possible terms. For more detail, see “Response Surface Experiments” in the “Examples of Custom Designs” chapter and “Bayesian D-Optimality”.
Alias Terms
It is possible that effects not included in your assumed model are active. In the Alias Terms outline, add potentially active effects that are not in your assumed model but might bias the estimates of model terms. Once you generate your design, the Alias Matrix outline appears under Design Evaluation. The Alias Matrix entries represent the degree of bias imparted to model parameters by the effects that you specified in the Alias Terms outline. For details, see the “The Alias Matrix” in the “Technical Details” appendix.
By default, the Alias Terms outline includes all two-way interaction effects that are not in your Model outline (with the exception of terms involving blocking factors). Add terms using the buttons. For a description of how to use these buttons to add effects to the Alias Terms table, see “Model”.
For example, suppose that you specify a design with three continuous factors. Your assumed model, specified in the Model outline, contains only those three main effects. You can afford only six runs. You want to see how estimates of the main effects might be biased by active two-way interactions and the three-way interaction.
The Alias Terms table includes all two-way interactions by default. You can add the three-way interaction by selecting Interactions > 3rd.
Figure 4.22 Alias Terms Outline
Once you specify six runs in the Design Generation outline and click Make Design, the Design Evaluation outline appears. Open the Design Evaluation outline and the Alias Matrix outline. See Figure 4.23.
Figure 4.23 Alias Matrix
The Alias Matrix indicates that each main effect is partially aliased with two of the interactions. See “Alias Matrix” in the “Evaluate Designs” chapter and “The Alias Matrix” in the “Technical Details” appendix.
Design Generation
The Design Generation outline gives you choices relating to the size and structure of the design. Typically, the input area has two parts:
• Design structure options
• Number of runs options
Figure 4.24 Design Generation Outline
Design Structure Options
Group runs into random blocks of size
(Not available if a blocking factor is specified) To construct a random block design, enter the number of runs that you want in each block. When you specify the sample size, a factor called Random Block is created. Its levels define blocks of a size that is consistent with the block size that you entered, given the specified number of runs. If the number of runs is an integer multiple of the block size, the block sizes equal your specified value.
Number of Whole Plots
Appears when you specify a hard or very-hard-to-change factor. The factor Whole Plots corresponds to the very-hard-to-change factors (split-split-plot design), if there are any, otherwise to the hard-to-change factors (split-plot design). JMP suggests a value for the number of whole plots that maximizes the information about the coefficients in the model. Or, you can enter a value for the number of whole plots. For details, see “Numbers of Whole Plots and Subplots”.
Number of Subplots
Appears when you specify a very-hard-to-change factor. The factor Subplots corresponds to the hard-to-change factors in the split-split-plot design. JMP suggests values for the number of whole plots and subplots that maximize the information about the coefficients in the model. Or, you can enter a value for the number of subplots. For details, see “Numbers of Whole Plots and Subplots”.
Hard to change factors can vary independently of Very Hard to change factors
Select this option to create a strip-plot (also known as two-way split-plot or split block) design. This option creates a design where the hard-to-change factors are randomized within the levels of the very-hard-to-change factors. They are not nested within the very-hard-to-change factors.
Number of Center Points
Appears only if the design contains factors with a Continuous or Mixture factor type. Specify how many additional runs you want to add as center points to the design. A center point is a run whose setting for each continuous factor is midway between the high and low settings. See “Center Points, Replicate Runs, and Testing” in the “Starting Out with DOE” chapter.
If a design contains both continuous and other types of factors, center points might not be balanced relative to the levels of the other factors. Custom Design chooses the center points to maximize the D-, I-, or alias efficiency of the design.
Number of Replicate Runs
Specify the number of replicate trials that you want to add to the design. This does not replicate the entire design, but chooses the optimal design points to replicate. See “Center Points, Replicate Runs, and Testing” in the “Starting Out with DOE” chapter.
Number of Runs Options
Minimum
A lower bound on the number of runs necessary to avoid failures in design generation. When you select Minimum, the resulting design is saturated. There are no degrees of freedom for error.
Note: If you select the Minimum number of runs, there will be no error term for testing. You will not be able to test parameter estimates. This choice is appropriate only when the cost of additional runs is prohibitive.
Default
Suggests the number of runs. This value is based on heuristics for creating a balanced design with at least four runs more than the Minimum number of runs.
User Specified
Specify the number of runs that you want. Enter that value into the Number of Runs text box. This option enables you to balance the cost of additional runs against the potential gain in information.
Number of Runs
This is the only option that appears when a covariate factor with Changes set to Easy is specified. The number of runs shown is the number of rows in the data table associated with your covariate or covariates. You can specify a smaller number of runs. In that case, the covariate runs that are selected optimize the design criterion.
Make Design
Once you have completed the Design Generation outline, click Make Design. Custom 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, allowing you to create the design table.
Note: Sometimes several designs can optimize the optimality criterion. When this is the case, the design algorithm might generate different designs when you click the Back and Make Design buttons repeatedly.
Design
The Design outline shows the runs for a design that is optimal, given the conditions that you have specified. The runs might not appear to be properly randomized. You can select Run Order options in the Output Options panel before generating your design table.
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 idealized (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. You specify the terms representing potentially active effects in the Alias Terms table. See “The Alias Matrix” in the “Technical Details” appendix.
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. See Optimality Criterion in “Custom Design Options” and “Optimality Criteria”.
Note: The Design Diagnostics outline does not provide the following statistics when the model includes factors with Changes set to Hard or Very Hard or with Estimability set to If Possible: D Efficiency, G Efficiency, A Efficiency.
For more details about the Design Evaluation outline, see “Design Evaluation” in the “Evaluate Designs” chapter.
Output Options
Use the Output Options panel to perform the following tasks:
• specify how you want the custom design data table to appear
• construct the design table
• return to a previous point in the Custom Design window
In most cases, the Output Options panel appears as shown in Figure 4.25.
Figure 4.25 Output Options Panel
The Output Options panel contains these options:
• “Back”
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 custom design data table. In the Custom Design table, the Table panel (in the upper left) can contain scripts, as appropriate given your design. The Model, Evaluate Design, and DOE Dialog scripts are always provided. To run a script, click the green triangle next to the script name.
Figure 4.26 Custom Design Table Showing Scripts
Possible scripts include the following:
Model
Runs the Analyze > Fit Model platform. The model described by the script is determined by your choices in the Model outline and by the type of design.
Evaluate Design
Runs the DOE > Design Diagnostics > Evaluate Design platform. The model described by the script is determined by your choices in the Model outline and by the type of design.
Constraint
Shows model constraints that you entered in the Define Factor Constraints outline using the Specify Linear Constraints option.
Disallowed Combinations
Shows model constraints that you entered in the Define Factor Constraints outline using the Use Disallowed Combinations Filter or the Use Disallowed Combinations Script options.
DOE Dialog
Re-creates the Custom Design window that you used to generate the design table. The script also contains the random seed used to generate your design.
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.
Note: If you attempt to remove all factors after clicking the Back button, one continuous factor remains. You can delete the continuous factor after new factors are added.
Custom Design Options
This section describes the options available under the Custom Design red triangle menu.
Description of Options
The Custom Design red triangle menu contains the following 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 have 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 have saved using the Save Factors option.
Save Constraints
(Unavailable for some platforms) Saves factor constraints that you have 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 have 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:
‒ 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.Save X Matrix
Saves scripts called Moments Matrix and Model Matrix to the design data table. These scripts contain the moments and design matrices. See “Save X Matrix”.
Caution: For a design with nominal factors, the matrix in the Model Matrix script saved by the Save X Matrix option is not the coding matrix used in fitting the linear model. You can obtain the coding matrix used for fitting the model by selecting the option Save Columns > Save Coding Table in the Fit Model report that you obtain when you run the Model script.
Optimality Criterion
Changes the design optimality criterion. The default criterion, Recommended, specifies D-optimality for all design types, unless you added quadratic effects using the RSM button in the Model outline. For more information about the D-, I-, and alias-optimal designs, see “Optimality Criteria”.
Note: You can set a preference to always use a given optimality criterion. Select File > Preferences > Platforms > DOE. Check Optimality Criterion and select your preferred criterion.
Number of Starts
Enables you to specify the number of random starts used in constructing the design. See “Number of Starts”.
Design Search Time
Maximum number of seconds spent searching for a design. The default search time is based on the complexity of the design. See “Design Search Time” and “Number of Starts”.
If the iterations of the algorithm require more than a few seconds, a Computing Design progress window appears. If you click Cancel in the progress window, the calculation stops and gives the best design found at that point. The progress window also displays D-efficiency for D-optimal designs that do not include factors with Changes set to Hard or Very Hard or with Estimability set to If Possible.
Note: You can set a preference for Design Search Time. Select File > Preferences > Platforms > DOE. Check Design Search Time and enter the maximum number of seconds. In certain situations where more time is required, JMP extends the search time.
Sphere Radius
Constrains the continuous factors in a design to a hypersphere. Specify the radius and click OK. Design points are chosen so that their distance from 0 equals the Sphere Radius. Select this option before you click Make Design.
Note: Sphere Radius constraints cannot be combined with constraints added using the Specify Linear Constraints option. Also, the option is not available when hard-to-change factors are included (split-plot designs).
Advanced Options > Mixture Sum
Set the sum of the mixture factors to any positive value. Use this option to keep a component of a mixture constant throughout an experiment.
Advanced Options > Split Plot Variance Ratio
Specify the ratio of the variance of the random whole plot and the subplot variance (if present) to the error variance. Before setting this value, you must define a hard-to-change factor for your split-plot design, or hard and very-hard-to-change factors for your split-split-plot design. Then you can enter one or two positive numbers for the variance ratios, depending on whether you have specified a split-plot or a split-split-plot design.
Advanced Options > Prior Parameter Variance
(Available only when the Model outline is available) Specify the weights that are used for factors whose Estimability is set to If Possible. The option updates to show the default weights when you click Make Design. Enter a positive number for each of the terms for which you want to specify a weight. The value that you enter is the square root of the reciprocal of the prior variance. A larger value represents a smaller variance and therefore more prior information that the effect is not active.
Bayesian D- or I-optimality is used in constructing designs with If Possible factors. The default values used in the algorithm are 0 for Necessary terms, 4 for interactions involving If Possible terms, and 1 for If Possible terms. For more details, see “The Alias Matrix” in the “Technical Details” appendix and “Optimality Criteria”.
Advanced Options > D Efficiency Weight
Specify the relative importance of D-efficiency to alias optimality in constructing the design. Select this option to balance reducing the variance of the coefficients with obtaining a desirable alias structure. Values should be between 0 and 1. Larger values give more weight to D-Efficiency. The default value is 0.5. This option has an effect only when you select Make Alias Optimal Design as your Optimality Criterion.
For the definition of D-efficiency, see “Optimality Criteria”. For details about alias optimality, see “Alias Optimality”.
Advanced Options > Set Delta for Power
Specify the difference in the mean response that you want to detect for model effects. See “Set Delta for Power”.
Save Script to Script Window
Creates the script for the design that you specified in the Custom Design window and places 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 the model that is specified in the design window.
A Model window opens where you can 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 4.27 shows the Model window for a design with one continuous factor (X1) and one three-level categorical factor (X2). Notice that X2 is represented by two model terms.
Figure 4.27 Simulate Responses Control Window
The initial window shows values for the coefficients of either 1 or -1, and 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 default 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. If it is not possible to fit the model specified in the data table’s Model script, the intercept and coefficients have default values of 0.
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 4.27:
‒ 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.
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.
Save X Matrix
This option saves scripts called Moments Matrix and Model Matrix that contain the moments matrix and the model matrix. The moments matrix and the model matrix are used to calculate the Average Variance of Prediction, which appears in the Design Diagnostics section of the Design Evaluation outline. For details, see Goos and Jones (2011). If the design is a split-plot design, a V Inverse script is also saved. The V Inverse script contains the inverse of the covariance matrix of the responses.
Caution: For a design with nominal factors, the matrix in the Model Matrix script saved by the Save X Matrix option is not the coding matrix used in fitting the linear model. You can obtain the coding matrix used for fitting the model by selecting the option Save Columns > Save Coding Table in the Fit Model report that you obtain when you run the Model script.
Note: You can set a preference to always save the matrix script. Select File > Preferences > Platforms > DOE. Check Save X Matrix.
Model Matrix
The model matrix describes the design for the experiment. The model matrix has a row for each run and a column for each term of the model specified in the Model outline. For each run, the corresponding row of the model matrix contains the coded values of the model terms:
• Continuous terms are coded to range from -1 to 1.
• Nominal terms are coded by applying the Gram-Schmidt orthogonalization procedure to the coding for nominal effects that is used in fitting linear models.
Because of how nominal terms are coded for constructing optimal designs, when a design contains nominal factors, the model matrix coding differs from the coding used in fitting linear models. For information about the coding used for nominal terms in fitting linear models, see the Standard Least Squares chapter in the Fitting Linear Models book.
Moments Matrix
The moments matrix is dependent upon the model effects but is independent of the design. It is defined as follows:
where 
denotes the model effects corresponding to factor combinations of the vector of factors, 
, and R denotes the design space. For additional details concerning moments and design matrices, see Goos and Jones (2011, pp 88-90) and Myers et al. (2009). Note that the moments matrix is called a matrix of region moments in Myers et al. (2009, p. 376).
denotes the model effects corresponding to factor combinations of the vector of factors, , and R denotes the design space. For additional details concerning moments and design matrices, see Goos and Jones (2011, pp 88-90) and Myers et al. (2009). Note that the moments matrix is called a matrix of region moments in Myers et al. (2009, p. 376).Scripts
From the Custom Design red triangle menu, select Save X Matrix. After the design and the table are created, in the Custom Design table, the Moments Matrix and Model Matrix scripts, and if the design is a split plot, the V Inverse script, are saved as table properties.
• Select Edit from the red triangle next to either the Moments Matrix, Model Matrix, or V Inverse script. The script shows the corresponding matrix. You can copy this matrix into scripts that you write.
• When you run the Moments Matrix script, the log shows the number of rows in the moments matrix, called Moments.
• When you run the script Model Matrix, the log displays the number of rows in the model matrix, called X.
• When you run the script V Inverse, the log displays the number of rows in the inverse covariance matrix, called V Inverse.
Example
Follow these steps to illustrate these features:
Tip: To see the log, select View > Log (Window > Log on the Macintosh).
1. Select DOE > Custom Design.
2. Add 3 continuous factors and click Continue.
3. Click Interactions > 2nd.
4. From the Custom Design red triangle menu, select Save X Matrix.
5. Using the Default Number of Runs (12), click Make Design and then Make Table.
6. In the Table panel, right click the Moments Matrix script and select Edit.
The script appears in a script window. The script shows the moments matrix, which is called Moments.
Figure 4.28 Moments Matrix Script
7. If it is not already open, select View > Log (Window > Log on the Macintosh).
8. In the Table panel, click the green triangle next to the Moments Matrix script.
The number of rows appear in the log as N Row(Moments)=7.
9. In the Table panel, right click the Model Matrix script and select Edit.
The script appears in a script window. The script shows the model matrix, which is called X.
10. Click Run.
The number of rows appears in the log as N Row(X)=12.
11. To view the Model Matrix as a data table, add these lines to the script:
dt = New Table( "Model Matrix" );
dt << Set Matrix( X );
12. Click Run.
Number of Starts
The number of starts is the number of times that the coordinate-exchange algorithm initiates with a new design. See “Coordinate-Exchange Algorithm”. You can specify your own value using the Number of Starts option. Increasing the number of random starts tends to improve the optimality of the resulting design.
Unless you specify a value for Number of Starts and click OK, the number of starts is controlled by Design Search Time. To see how many starts were used to construct a design, click Make Design. Then select Number of Starts. The value in the text box is the number of starts used to construct the specific design.
In certain special cases, the globally optimal design is known from theory. If the coordinate-exchange algorithm detects that it has found an optimal design, it stops searching and returns that design.
Tip: To reproduce a specific design, you need to specify the Number of Starts and the Random Seed originally used to produce the design. Obtain these values from the red triangle options after you click Make Design.
In examples of custom designs in the documentation, the random seed and number of starts are often provided so that you can reconstruct the exact design being discussed.
Design Search Time
Design Search Time is the amount of time allocated to finding an optimal design. Custom Design’s coordinate-exchange algorithm consists of finding near-optimal designs based on random starting designs. See “Coordinate-Exchange Algorithm”. The Design Search Time determines how many designs are constructed based on random starting designs.
You can specify your own value using the Design Search Time option. Increasing the search time tends to improve the optimality of the resulting design.
Keep in mind that designs produced by rerunning the algorithm can differ. Even with the same random seed, the numbers of starting designs used to construct the final design might differ because of variations in computing speed and other factors.
Note: The number of starting designs is given by the value in the Number of Starts text box. However, this value is not updated until after you construct your design by clicking Make Design.
In certain special cases, the globally D-optimal design is known from theory. These cases include:
• Two-level fractional factorial designs or nonregular orthogonal arrays. These are globally D-optimal for all main effect and two-factor interaction models.
• Latin-square designs. These are D-optimal for main effect models assuming the right sample size and numbers of levels of the factors.
• Plackett-Burman designs. These are D-optimal for main effect models.
If the coordinate-exchange algorithm detects that it has found an optimal design, it stops searching and returns that design.
Set Delta for Power
This option specifies the difference in the mean response that you want to be able to detect for model effects. Power calculations appear in the Power Analysis outline within the Design Evaluation outline. Power is calculated for each model parameter based on detecting the specified difference of delta. For categorical effects, the power calculation is based on detecting a maximum change of delta between any two levels.
For example, suppose that you want to detect a change of 3 units in the mean response. All of your factors are continuous. Because your factors are expressed in coded units (the coded levels are -1 and 1), a change of 3 units in the response corresponds to parameter coefficient values of 1.5. When you specify 3 in the Set Delta for Power text box, the Anticipated Coefficients in the Power Analysis outline are set to 1.5. For each parameter, the probability of detecting the change of 3 units in the response appears in the Power column to the right of the parameter.
By default, delta is set to 2. The default coefficient for each continuous effect is set to 1. An n-level categorical factor is represented by n–1 indicator variables. The default coefficients for the n–1 terms (which represent the categorical factor) are alternating values of 1 and -1. The default coefficients for an interaction effect with more than one degree of freedom are also alternating values of 1 and -1.
Note: The order in which parameters appear in the Power Analysis report might not be identical to their order in the Parameter Estimates report obtained using Standard Least Squares. This difference can occur only when the model contains an interaction with more than one degree of freedom.
Given a specified value of delta, each coefficient in the Anticipated Coefficients list is set at delta/2 multiplied by the default coefficient. For a continuous factor, this assignment ensures that a difference of delta is detected with the calculated power. For a categorical factor, this assignment of coefficients ensures that a maximum difference of delta between any two levels is detected with the calculated power.
Technical Details
This section contains technical details for the following topics:
Designs with Randomization Restrictions
This section describes how the Custom Design platform handles various types of designs where random assignment of experimental units to factor level settings is restricted. Random block designs and various types of split-plot designs are included.
Random Block Designs
A random block design groups the runs of an experiment into blocks that are considered to be randomly chosen from a larger population. Runs within a block of runs are usually more homogeneous than runs in different blocks. In these instances, you are often better able to discern other effects if you account for the variation explained by the blocking variables.
Scenario for a Random Block Design
Goos (2002) presents an example involving a pastry dough mixing experiment. The purpose of the experiment is to understand how certain properties of the dough depend on three factors: feed flow rate, initial moisture content, and rotational screw speed. Since it was possible to only conduct four runs a day, the experiment required several days to run. It is likely that random day-to-day differences in environmental variables have some effect on all of the runs that are performed on a given day. To account for the day-to-day variation, the runs were grouped into blocks of size four so that this variation would not compromise the information about the three factors.
The blocking factor, Day, consists of each day's runs. The days on which the trials were conducted are representative of a large population of days with different environmental conditions. It follows that Day is a random blocking factor.
Setup for a Random Block Design
To create a random block design, use the Custom Design platform to enter responses and factors and define your model as usual. In the Design Generation outline, select the Group runs into random blocks of size option and enter the number of runs you want in each block. See “Design Structure Options”.
Note: To define a fixed blocking factor, enter a blocking factor in the Factors outline. To define a random blocking factor, do not enter a blocking factor in the Factors outline. Instead, select the Group runs into random blocks of size option under Design Generation.
Split-Plot Designs
Split-plot designs are used in situations where the settings of certain factors are held constant for groups of runs. In industry, these are usually factors that are difficult or expensive to change from run to run. Factors whose settings need to be held constant for groups of runs are classified as hard-to-change in JMP.
Because certain factors are hard-to-change, it is not practical to randomly allocate them to experimental units. Instead, they are allocated to groups of units. This imposes a restriction on randomization that must be considered in generating a design and in analyzing the results.
Scenario for a Split-Plot Design
Box et al. (2005) presents an experiment to study the corrosion resistance of steel bars. The bars are placed in a furnace for curing. Afterward, a coating is applied to increase resistance to corrosion. The two factors of interest are:
• Furnace Temp in degrees centigrade, with levels 360, 370, and 380
• Coating, with levels C1, C2, C3, and C4 depicting four different types of coating
Furnace Temp is a hard-to-change factor, due to the time it takes to reset the temperature in the furnace. For this reason, four bars are processed for each setting of furnace temperature. At a later stage, the four coatings are randomly assigned to the four bars.
The experimental units are the bars. Furnace Temp is a hard-to-change factor whose levels define whole plots. Within each whole plot, the Coating factor is randomly assigned to the experimental units to which the whole plot factor was applied.
Figure 4.29 Factors and Design Outlines for Split-Plot Design
The Factors outline for the corrosion experiment has Changes set to Hard for Furnace Temp and Easy for Coating. The 15-run design consists of five whole plots, within which the settings of Temperature are held constant.
Setup for a Split-Plot Design
In general, several factors can be applied to a processing step where settings are hard-to-change. In the furnace example, you might consider a furnace location factor, as well as temperature. In the Factors outline, under the Changes column, you would specify a Changes value of Hard for such factors.
When a custom design involves only easy-to-change and hard-to-change factors, the runs of the hard-to-change factors are grouped using a new factor called Whole Plots. The values of Whole Plots designate blocks of runs with identical settings for the hard-to-change factors. The Model script that is saved to the design table treats Whole Plots as a random effect. For details, see “Changes” and “Design Structure Options”.
For an example of creating a split-plot design and analyzing the experimental data, see “Split-Plot Experiment” in the “Examples of Custom Designs” chapter.
Split-Split-Plot Designs
A split-split-plot design is used when there are two levels of factors that are hard-to-change. In industry, such designs often occur when batches of material or experimental units from one processing stage pass to a second processing stage. Factors are applied to batches of material at the first stage. Then those batches are divided for second-stage processing, where additional factors are studied. The first stage factors are considered very-hard-to-change, and the second-stage factors are considered hard-to-change. Additional factors can be applied to experimental units after the second processing stage. These factors are considered easy-to-change.
In a split-split-plot design, the batches are considered to be random blocks. Since the batches are divided for second-stage processing, the second-stage factors are nested within the first-stage factors.
Scenario for a Split-Split-Plot Design
Schoen (1999) presents an example of a split-split-plot design that relates to cheese quality. The factors are given in the Cheese Factors.jmp data table found in the Design Experiment folder. The experiment consists of three stages of processing:
• Milk is received from farmers and stored in a large tank.
• Milk from this tank is distributed to smaller tanks used for curd processing.
• The curds from each tank are transported to presses for processing individual cheeses.
The experiment consists of testing:
• Two factors that are applied when the milk is in the large storage tank.
• Five factors that are applied to the smaller curd processing tanks.
• Three factors that are applied to the individual cheeses from a curds processing tank.
Notice that the levels of factors applied to the curd processing tanks (subplots) are nested within the levels of factors applied to the milk storage tank (whole plots).
The Factors outline for the cheese experiment have Changes set as follows:
• Very Hard for the two storage tank factors
• Hard for the five curd processing tank factors
• Easy for the three factors that can be randomly assigned to cheeses
Figure 4.30 Factors and Design Generation Outline for Split-Split-Plot Design
The default number of whole plots is 5 and the default number of subplots is 11. Click Make Design to see a 22-run design.
Figure 4.31 Split-Split-Plot Design for Cheese Scenario
The five whole plots correspond to the storage factors, storage 1 and storage 2. The settings of the storage factors are constant within a whole plot. If consecutive whole plots have the same setting for a whole plot factor, the factor should be reset between the plots. For example, you should reset the level for storage 1 between runs 10 and 11 and between runs 14 and 15, and your should reset the level for storage 2 between runs 18 and 19. Resetting the factor between whole plots, even when the specified settings are the same, is required in order to capture whole plot variation.
The 11 subplots correspond to the curds factors. Within a subplot, the settings of the curds factors are constant. Each level of Subplots only appears within one level of Whole Plots, indicating that the levels of Subplots are nested within the levels of Whole Plots.
Levels of the cheese factors vary randomly from run to run.
Setup for a Split-Split-Plot Design
In a split-split-plot design, the Factors outline contains factors with Changes set to Very Hard and Hard. The design can also contain factors with Changes set to Easy. Two factors are created:
• A factor called Whole Plots represents the blocks of constant levels of the factors with Changes set to Very Hard.
• A factor called Subplots represents the blocks of constant levels of the factors with Changes set to Hard.
• The factor Subplots reflects the nesting of the levels of the factors with Changes set to Hard within the levels of the factors with Changes set to Very Hard.
• The levels of factors with Changes set to Easy are randomly assigned to units within subplots.
• The factors Whole Plots and Subplots are treated as random effects in the Model script that is saved to the design table.
For details, see the Changes description under “Factors Outline” and “Design Structure Options”.
Two-Way Split-Plot Designs
A two-way split-plot (also known as strip plot or split block) design consists of two split-plot components. In industry, these designs arise when batches of material or experimental units from one processing stage pass to a second processing stage. But, after the first processing stage, it is possible to divide the batches into sub-batches. The second-stage processing factors are applied randomly to these sub-batches. For a specific second-stage experimental setting, all of the sub-batches assigned to that setting can be processed simultaneously. Additional factors can be applied to experimental units after the second processing stage.
In contrast to a split-split-plot design, the second-stage factors are not nested within the first-stage factors. After the first stage, the batches are subdivided and formed into new batches. Therefore, both the first- and second-stage factors are applied to whole batches.
Although factors at both stages might be equally hard-to-change, to distinguish these factors, JMP denotes the first stage factors as very-hard-to-change, and the second-stage factors as hard-to-change. Additional factors applied to experimental units after the second processing stage are considered easy-to-change.
Scenario for a Two-Way Split-Plot Design
Vivacqua and Bisgaard (2004) describe an experiment to improve the open circuit voltage in battery cells. Two stages of processing are of interest:
• First stage: A continuous assembly process
• Second stage: A curing process with a 5-day cycle time
The engineers want to study six two-level factors:
• Four factors, X1, X2, X3, and X4, that are applied to the assembly process
• Two factors, X5 and X6, that are applied to the curing process
A full factorial design with all factors at two levels would require 26 = 64 runs, and would require a prohibitive 64*5 = 320 days. Also, it is not practical to vary assembly conditions for individual batteries. However, assembly conditions can be changed for large batches, such as batches of 2000 batteries.
Both the first- and second-stage factors are hard-to-change. In a sense, there are two split-plot designs. However, the batches of 2,000 batteries from the first-stage experiment can be divided into four sub-batches of 500 batteries each. These sub-batches can be randomly assigned to the four settings of the two second-stage factors. All of the batches assigned to a given set of curing conditions can be processed simultaneously. In other words, the first- and second-stage factors are crossed.
To distinguish between the first- and second-stage factors, you designate the Changes for the first-stage factors as Very Hard, and the Changes for the second-stage factors as Hard. See Figure 4.32. Also, under Design Generation, note the following option: Hard to change factors can vary independently of Very Hard to change factors. If this is not checked, the design is treated as a split-split-plot design, with nesting of factors at the two levels. Check this option to create a two-way split-plot design.
Figure 4.32 Factors and Design Generation Outline for Two-Way Split Plot Design
The default number of whole plots is 7; the default number of subplots is 14. Click Make Design to see the 28-run design.
Figure 4.33 Two-Way Split-Plot Design for Battery Cells
The seven whole plots correspond to the first-stage factors, X1, X2, X3, and X4. The settings of these factors are constant within a whole plot. The 14 subplots correspond to the second-stage factors, X5 and X6. For example, the sub-batches for runs 1 and 15 (from different whole plots) are subject to the same subplot treatment, where X5 is set at 1 and X6 at -1.
Setup for a Two-Way Split-Plot Design
A two-way split-plot design requires factors with Changes set to Very Hard and to Hard. As described in “Setup for a Split-Split-Plot Design”, factors called Whole Plots and Subplots are created. However, in a two-way split-plot design, Subplots does not nest the levels of factors with Changes set to Hard within the levels of factors with Changes set to Very Hard. Both Whole Plots and Subplots are treated as random effects in the Model script that is saved to the design table.
You need to ensure that the factor Subplots is not nested within the factor Whole Plots. Select the option Hard to change factors can vary independently of Very Hard to change factor in the Design Generation outline (Figure 4.32). For more details, see “Changes” and “Design Structure Options”.
For an example of creating a split-plot design and analyzing the experimental data, see “Two-Way Split-Plot Experiment” in the “Examples of Custom Designs” chapter.
Covariates with Hard-to-Change Levels
Suppose that you have measurements on batches of material that are available for use in testing experimental factors. Or suppose that you have measurements on individuals who might be selected to participate in testing experimental factors. The measurements on batches or individuals are known in advance of the experiment and are considered to be covariates.
The batches or individuals correspond to whole plots. You might want to use only some of these whole plots in your experiment. Because information about the whole plots in the form of covariates is available, the design should choose the whole plots in an optimal fashion.
In the Factors outline, the Custom Design platform enables you to designate covariates as hard-to-change. The model, as given by the terms that you include in the Model outline, can include interactions and powers constructed using covariates and experimental factors.
Note: When you set Changes for a Covariate factor to Hard, all other covariates are also set to Hard The remaining factors must be set to Easy. Because the algorithm requires a combination of row exchange and coordinate exchange, even moderately sized designs might take some time to generate.
Scenario for an Experiment with a Hard-to-Change Covariate
An experiment involving batches of polypropylene plates is discussed in Goos and Jones (2011, Chapter 9) and Jones and Goos (2015). Large batches of polypropylene plates are produced according to various formulations determined by several variables. Some plates are used immediately, and the remainder are stored for future experimental purposes. The compositions of these stored batches are known.
A customer has certain requirements regarding the plate formulation. Future experiments involve customizing the gas plasma treatment to the types of formulations required by the customer. The composition variables are considered hard-to-change covariates. Gas plasma treatment factors can be applied to sub-batches of plates with a given formulation.
The optimal design identifies the batches (defined by the covariates) to use, determines the number of plates from each batch to use, and provides settings for the gas plasma levels. Note that the optimal number of batches and plates from a given batch depend on the covariates.
An example is provided in “Examples of Custom Designs” chapter.
Numbers of Whole Plots and Subplots
JMP suggests default values for the Number of Whole Plots and Number of Subplots. These values are based on heuristics for creating a balanced design that allows for estimation of the effects specified in the Model outline.
If you enter missing values for Number of Whole Plots or Number of Subplots, JMP chooses values that maximize the D-efficiency of the design. The algorithm uses the values specified in the Split Plot Variance Ratio option. See “Advanced Options > Split Plot Variance Ratio”. The D-efficiency is given by the determinant of
, where V -1 is the inverse of the variance matrix of the responses. For further details, see Goos, 2002.
, where V -1 is the inverse of the variance matrix of the responses. For further details, see Goos, 2002.If you enter values for the Number of Whole Plots and Number of Subplots, Custom Design attempts to maximize the optimality of the resulting design. For details about split-plot designs, see Jones and Goos (2007). For details about designs with hard-to-change covariates, see Jones and Goos (2015).
Optimality Criteria
This section provides information about the following designs:
D-Optimality
By default, the Custom Design platform optimizes the D-optimality criterion except when a full quadratic model is created using the RSM button. In that case, an I-optimal design is constructed.
The D-optimality criterion minimizes the determinant of the covariance matrix of the model coefficient estimates. It follows that D-optimality focuses on precise estimates of the effects. This criterion is desirable in the following cases:
• screening designs
• experiments that focus on estimating effects or testing for significance
• designs where identifying the active factors is the experimental goal
The D-optimality criterion is dependent on the assumed model. This is a limitation because often the form of the true model is not known in advance. The runs of a D-optimal design optimize the precision of the coefficients of the assumed model. In the extreme, a D-optimal design might be saturated, with the same number of runs as parameters and no degrees of freedom for lack of fit.
Specifically, a D-optimal design maximizes D, where D is defined as follows:
and where X is the model matrix as defined in “Simulate Responses”.
D-optimal split-plot designs maximize D, where D is defined as follows:
and 
is the block diagonal covariance matrix of the responses (Goos 2002).
is the block diagonal covariance matrix of the responses (Goos 2002).Since a D-optimal design focuses on minimizing the standard errors of coefficients, it might not allow for checking that the model is correct. For example, a D-optimal design does not include center points for a first-order model. When there are potentially active terms that are not included in the assumed model, a better approach is to specify If Possible terms and to use a Bayesian D-optimal design.
Bayesian D-Optimality
Bayesian D-optimality is a modification of the D-optimality criterion. The Bayesian D-optimality criterion is useful when there are potentially active interactions or non-linear effects. See DuMouchel and Jones (1994) and Jones et al (2008).
Bayesian D-optimality estimates a specified set of model parameters precisely. These are the effects whose Estimability you designate as Necessary in the Model outline. But at the same time, Bayesian D-optimality has the ability to estimate other, typically higher-order effects, as allowed by the run size. These are the effects whose Estimability you designate as If Possible in the Model outline. To the extent possible given the run size restriction, a Bayesian D-optimal design allows for detecting inadequacy in a model that contains only the Necessary effects.
The Bayesian D-optimality criterion is most effective when the number of runs is larger than the number of Necessary terms, but smaller than the sum of the Necessary and If Possible terms. When this is the case, the number of runs is smaller than the number of parameters that you would like to estimate. Using prior information in the Bayesian setting allows for precise estimation of all of the Necessary terms while providing the ability to detect and estimate some If Possible terms.
To allow for a meaningful prior distribution to apply to the parameters of the model, responses and factors are scaled to have certain properties (DuMouchel and Jones, 1994, Section 2.2).
Consider the following notation:
• K is a diagonal matrix with values as follows:
‒ k = 0 for Necessary terms
‒ k = 1 for If Possible main effects, powers, and interactions involving a categorical factor with more than two levels
‒ k = 4 for all other If Possible terms
The prior distribution imposed on the vector of If Possible parameters is multivariate normal, with mean vector 0 and diagonal covariance matrix with diagonal entries 
. Therefore, a value 
is the reciprocal of the prior variance of the corresponding parameter.
. Therefore, a value is the reciprocal of the prior variance of the corresponding parameter.The values for k are empirically determined. If Possible main effects, powers, and interactions with more than one degree of freedom have a prior variance of 1. Other If Possible terms have a prior variance of 1/16. In the notation of DuMouchel and Jones, 1994, 
.
.To control the weights for If Possible terms, select Advanced Options > Prior Parameter Variance from the red triangle menu. See “Advanced Options > Prior Parameter Variance”.
The posterior distribution for the parameters has the covariance matrix 
. The Bayesian D-optimal design is obtained by maximizing the determinant of the inverse of the posterior covariance matrix:
. The Bayesian D-optimal design is obtained by maximizing the determinant of the inverse of the posterior covariance matrix:
I-Optimality
I-optimal designs minimize the average variance of prediction over the design space. The I-optimality criterion is more appropriate than D-optimality if your primary experimental goal is not to estimate coefficients, but rather to do the following:
• predict a response
• determine optimum operating conditions
• determine regions in the design space where the response falls within an acceptable range
In these cases, precise prediction of the response takes precedence over precise estimation of the parameters.
The prediction variance relative to the unknown error variance at a point 
in the design space can be calculated as follows:
in the design space can be calculated as follows:
where X is the model matrix as defined in “Simulate Responses”.
I-optimal designs minimize the integral I of the prediction variance over the entire design space, where I is given as follows:
Here M is the moments matrix:
See “Simulate Responses”. For further details, see Goos and Jones (2011).
The moments matrix does not depend on the design and can be computed in advance. The row vector f (x)’ consists of a 1 followed by the effects corresponding to the assumed model. For example, for a full quadratic model in two continuous factors, f (x)’ is defined as follows:
Bayesian I-Optimality
The Bayesian I-optimal design minimizes the average prediction variance over the design region for Necessary and If Possible terms.
Consider the following notation:
• K is a diagonal matrix with values as follows:
‒ k = 0 for Necessary terms
‒ k = 1 for If Possible main effects, powers, and interactions involving a categorical factor with more than two levels
‒ k = 4 for all other If Possible terms
The prior distribution imposed on the vector of If Possible parameters is multivariate normal, with mean vector 0 and diagonal covariance matrix with diagonal entries 
. (See “Bayesian D-Optimality” for more details about the values k.)
. (See “Bayesian D-Optimality” for more details about the values k.)The posterior variance of the predicted value at a point x0 is as follows:
The Bayesian I-optimal design minimizes the average prediction variance over the design region, as follows:
where M is the moments matrix. See “Simulate Responses”.
Alias Optimality
Alias optimality seeks to minimize the aliasing between effects that are in the assumed model and effects that are not in the model but are potentially active. Effects that are not in the model but that are of potential interest are called alias effects. For details about alias-optimal designs, see Jones and Nachtsheim (2011).
Specifically, let X1 be the model matrix corresponding to the terms in the assumed model, as defined in “Simulate Responses”. The design defines the model that corresponds to the alias effects. Denote the matrix of model terms for the alias effects by X2.
The alias matrix is the matrix A, defined as follows:
The entries in the alias matrix represent the degree of bias associated with the estimates of model terms. See “The Alias Matrix” in the “Technical Details” appendix for the derivation of the alias matrix.
The sum of squares of the entries in A provides a summary measure of bias. This sum of squares can be represented in terms of a trace as follows:
Designs that reduce the trace criterion generally have lower D-efficiency than the D-optimal design. Consequently, alias optimality seeks to minimize the trace of 
subject to a lower bound on D-efficiency. For the definition of D-efficiency, see “Optimality Criteria”. The lower bound on D-efficiency is given by the D-efficiency weight, which you can specify under Advanced Options. See “Advanced Options > D Efficiency Weight”.
subject to a lower bound on D-efficiency. For the definition of D-efficiency, see “Optimality Criteria”. The lower bound on D-efficiency is given by the D-efficiency weight, which you can specify under Advanced Options. See “Advanced Options > D Efficiency Weight”.D-Efficiency
Let X denote the design, or model, matrix for a given assumed model with p parameters. For the definition of the model matrix, see “Simulate Responses”. Let 
denote the model matrix for a D-optimal design for the assumed model. Then the D-efficiency of the design given by X is as follows:
denote the model matrix for a D-optimal design for the assumed model. Then the D-efficiency of the design given by X is as follows:
Coordinate-Exchange Algorithm
Custom Design constructs a design that seeks to optimize one of several optimality criteria. (See “Optimality Criteria”.) To optimize the criterion, Custom Design uses the coordinate-exchange algorithm (Meyer and Nachtsheim, 1995). The algorithm begins by randomly selecting values within the specified design region for each factor and each run to construct a starting design.
Suppose your study requires continuous factors, no factor constraints, and a main-effects model. An iteration consists of testing each value of the model matrix, as follows:
• The current value of each factor is replaced by its two most extreme values.
• The optimality criterion is computed for both of these replacements.
• If one of the values increases the optimality criterion, this value replaces the old value.
The process continues until no replacement occurs for an entire iteration.
Appropriate adjustments are made to the algorithm to account for polynomial terms, nominal factors, and factor constraints.
The design obtained using this process is optimal in a large class of neighboring designs. But it is only locally optimal. To improve the likelihood of finding a globally optimal design, the coordinate-exchange algorithm is repeated a large number of times. Goos and Jones (2011, p. 36) recommend using at least 1,000 random starts for all but the most trivial design situations. The number of starting designs is controlled by the Number of Starts option. See “Number of Starts”. Custom Design provides the design that maximizes the optimality criterion among all the constructed designs.
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