Overview of Full Factorial Design
In a full factorial design, you perform an experimental run at every combination of the factor levels. The sample size is the product of the numbers of levels of the factors. For example, a factorial experiment with a two-level factor, a three-level factor, and a four-level factor has 2 x 3 x 4 = 24 runs.
The Full Factorial Design platform supports both continuous factors and categorical factors with arbitrary numbers of levels. It is assumed that you can run the trials in a completely random fashion.
Full factorial designs are the most conservative of all design types. Unfortunately, because the sample size grows exponentially with the number of factors, full factorial designs are often too expensive to run. Custom designs, definitive screening designs, and screening designs are less conservative but more efficient and cost-effective.
Example of a Full Factorial Design
In this example, you construct a full factorial design to study the effects of five two-level factors (Feed Rate, Catalyst, Stir Rate, Temperature, and Concentration) on the yield of a reactor. Because there are five factors, each at two levels, the full factorial design includes at least 25 = 32 runs. For smaller screening designs involving this experimental situation, see “Additional Examples of Screening Designs” in the “Screening Designs” chapter.
In this example you load the responses and factors from existing tables. When designing a new experiment on your own, enter the responses and factors manually. See “Responses” and “Factors”.
Construct the Design
1. Select DOE > Classical > Full Factorial Design.
2. Select Help > Sample Data Library and open Design Experiment/Reactor Response.jmp.
3. Click the Full Factorial Design red triangle and select Load Responses.
4. Select Help > Sample Data Library and open Design Experiment/Reactor Factors.jmp.
5. Click the Full Factorial Design red triangle and select Load Factors.
6. Click Continue.
Figure 12.2 Full Factorial Example Response and Factors Panels
Note: Setting the Random Seed in step 7 ensures that the runs in your design table appear in the same order as in this example. In constructing a design on your own, this step is not necessary.
7. (Optional) Click the Full Factorial Design red triangle menu and select Set Random Seed. Type 12345 and click OK.
The Run Order in the Output Options panel is set to Randomize. The order of runs in the design table will be random, as determined by the random seed.
The Number of Runs is set to 32. This is the number of all possible factor level combinations.
8. Click Make Table.
The first column in the design data table shows the factor level combination for each run in terms of + and - signs, indicating high and low factor settings. The table also has an empty Y column named Percent Reacted for entering response values as you conduct the experiment.
Figure 12.3 Full Factorial Design for Reactor Experiment
Analyze the Experimental Data
Next, proceed to analyze the data from the completed experiment. You will use two methods to analyze the results: Screening and Stepwise Regression. Then you will find optimal settings using the Prediction Profiler.
Analysis Using Screening Platform
1. Select Help > Sample Data Library and open Design Experiment/Reactor 32 Runs.jmp.
2. Run the Screening script.
The Screening report shows a Contrasts report and a Half Normal Plot. The Contrasts report shows estimates for all 31 potential effects, up to the five-way interaction.
Figure 12.4 Contrasts Report for Reactor 32 Runs.jmp
Note: Because the p-values in the Contrasts report are obtained using a Monte Carlo simulation, you will not obtain the same values as shown in Figure 12.4. For more information, see “Lenth’s Pseudo-Standard Error” in the “The Fit Two Level Screening Platform” chapter.
The six highlighted effects in the Contrasts outline are labeled in the Half Normal Plot.
Figure 12.5 Half Normal Plot for Reactor 32 Runs.jmp
The Half Normal Plot provides strong evidence that at least five of the labeled effects are larger than would be expected if they were the result of random variation. This suggests that these effects are active. The plot does not provide a clear indication that the three-way Concentration*Feed Rate*Stir Rate interaction is active.
In the Contrasts outline in Figure 12.4, the Individual p-Value for the three-way Concentration*Feed Rate*Stir Rate interaction is 0.0705 and its Simultaneous p-Value is 0.7592. Because the effect does not stand out on the Half Normal Plot and because its p-values are large, you decide not to include this effect in your model.
3. In the Half Normal Plot, drag a rectangle to select all labeled effects except for Concentration*Feed Rate*Stir Rate.
4. Click Make Model to open a Fit Model window containing the five effects.
5. Click Run.
The Actual by Predicted plot shows no evidence of lack of fit and the Effect Summary outline shows that all five effects are significant.
Analysis Using Stepwise Regression
1. Return to the Reactor 32 Runs.jmp data table, or reopen it by selecting Help > Sample Data Library and opening Design Experiment/Reactor 32 Runs.jmp.
2. Run the Model script.
The Construct Model Effects list contains only up to two-way interactions. You want to consider all interactions.
3. From the Select Columns list, select Feed Rate through Concentration.
4. Click Macros > Full Factorial.
All possible effects are added in the Construct Model Effects list.
5. Change the Personality to Stepwise.
6. Click Run.
7. Change the Stopping Rule to Minimum AICc.
For designed experiments, AICc is preferred to BIC. This is because BIC is typically a more lenient stopping rule than AICc as it tends to allow inactive effects into the model.
8. Click Go.
The Stepwise procedure selects seven effects as potentially active.
9. Click Run Model.
This fits a model using the seven effects. The Effect Summary outline indicates that Catalyst*Temperature*Concentration is not significant, because its p-value is 0.83042.
10. Select Catalyst*Temperature*Concentration in the Effect Summary outline and click Remove.
The effect Catalyst*Concentration has a p-value of 0.08960. You decide to remove it too.
11. Select Catalyst*Concentration in the Effect Summary outline and click Remove.
The five remaining effects are all highly significant. These are the same five effects that you identified using the Screening platform (“Analysis Using Screening Platform”).
Optimal Settings Using the Prediction Profiler
Now, find optimal settings for the three active factors involved in the five significant effects that you retained in your model.
1. In the Reactor 32 Runs.jmp data table, run the Reduced Model script.
The Reduced Model script opens a Fit Model window for the five-effect model that you identified in “Analysis Using Screening Platform” and “Analysis Using Stepwise Regression”.
2. Click Run.
The Prediction Profiler report displays Desirability because in the Full Factorial window, you specified a Goal of Maximize when you defined your response. The desirability function shown in the rightmost cell in the top row of the profiler shows that a value of 100 is most desirable and a value of 90 or below is least desirable.
3. Click the Prediction Profiler red triangle and select Optimization and Desirability > Maximize Desirability.
Figure 12.6 Prediction Profiler Showing Settings That Optimize Desirability
The predicted mean Percent Reacted at the settings that are shown is 95.875, with a confidence interval of 92.91 to 98.84. Note that, for all three factors, the settings that are identified are at the extremes of the ranges used in the experiment. In a future experiment, you should explore the process behavior beyond these settings.
Full Factorial Design Window
The Full Factorial design window is updated as you work through the design steps. For more information, see “The DOE Workflow: Describe, Specify, Design”. The outlines, separated by buttons that update the outlines, follow the flow in Figure 12.7.
Figure 12.7 Full Factorial Design Flow
The following sections describe the steps in creating a full factorial design:
Responses
Use the Responses outline to specify one or more responses.
Tip: When you have completed the Responses outline, consider selecting Save Responses from the red triangle menu. This option saves the response names, goals, limits, and importance values in a data table that you can later reload in DOE platforms.
Figure 12.8 Responses Outline
Add Response
Enters a single response with a goal type of Maximize, Match Target, Minimize, or None. If you select Match Target, enter limits for your target value. If you select Maximize or Minimize, entering limits is not required but can be useful if you intend to use desirability functions.
Remove
Removes the selected responses.
Number of Responses
Enters additional responses so that the number that you enter is the total number of responses. If you have entered a response other than the default Y, the Goal for each of the additional responses is the Goal associated with the last response entered. Otherwise, the Goal defaults to Match Target. Click the Goal type in the table to change it.
The Responses outline contains the following columns:
Response Name
The name of the response. When added, a response is given a default name of Y, Y2, and so on. To change this name, double-click it and enter the desired name.
Goal, Lower Limit, Upper Limit
The Goal tells JMP whether you want to maximize your response, minimize your response, match a target, or that you have no response goal. JMP assigns a Response Limits column property, based on these specifications, to each response column in the design table. It uses this information to define a desirability function for each response. The Profiler and Contour Profiler use these desirability functions to find optimal factor settings. For further details, see the Profiler chapter in the Profilers book and “Response Limits” in the “Column Properties” appendix.
‒ A Goal of Maximize indicates that the best value is the largest possible. If there are natural lower or upper bounds, you can specify these as the Lower Limit or Upper Limit.
‒ A Goal of Minimize indicates that the best value is the smallest possible. If there are natural lower or upper bounds, you can specify these as the Lower Limit or Upper Limit.
‒ A Goal of Match Target indicates that the best value is a specific target value. The default target value is assumed to be midway between the Lower Limit and Upper Limit.
‒ A Goal of None indicates that there is no goal in terms of optimization. No desirability function is constructed.
Note: If your target response is not midway between the Lower Limit and the Upper Limit, you can change the target after you generate your design table. In the data table, open the Column Info window for the response column (Cols > Column Info) and enter the desired target value.
Importance
When you have several responses, the Importance values that you specify are used to compute an overall desirability function. These values are treated as weights for the responses. If there is only one response, then specifying the Importance is unnecessary because it is set to 1 by default.
Editing the Responses Outline
In the Responses outline, note the following:
• Double-click a response to edit the response name.
• Click the goal to change it.
• Click on a limit or importance weight to change it.
• For multiple responses, you might want to enter values for the importance weights.
Response Limits Column Property
The Goal, Lower Limit, Upper Limit, and Importance that you specify when you enter a response are used in finding optimal factor settings. For each response, the information is saved in the generated design data table as a Response Limits column property. JMP uses this information to define the desirability function. The desirability function is used in the Prediction Profiler to find optimal factor settings. For further details about the Response Limits column property and examples of its use, see “Response Limits” in the “Column Properties” appendix.
If you do not specify a Lower Limit and Upper Limit, JMP uses the range of the observed data for the response to define the limits for the desirability function. Specifying the Lower Limit and Upper Limit gives you control over the specification of the desirability function. For more details about the construction of the desirability function, see the Profiler chapter in the Profilers book.
Factors
Factors in a full factorial design can be continuous or categorical.
Tip: When you have completed the Factors outline, consider selecting Save Factors from the red triangle menu. This option saves the factor names, roles, changes, and values in a data table that you can later reload in DOE platforms.
Figure 12.9 Factors Outline
Continuous
Adds a Continuous factor. The data type in the resulting data table is Numeric. 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.
Categorical
Adds a Categorical factor. Click to select or specify the number of levels. The data type in the resulting data table is Character. 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.
The default values for a categorical factor are L1, L2, ..., Lk, where k is the number of levels that you specify. Replace the default values with level names that are relevant for your experiment.
Remove
Removes the selected 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
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. The Role of the factor determines other factor properties that are saved to the data table. For details, see “Factor Column Properties”.
Values
The experimental settings for the factors.
Editing the Factors Outline
In the Factors outline, note the following:
• To edit a factor name, double-click the factor name.
• To edit a value, click the value in the Values column.
Factor Column Properties
For each factor, various column properties are saved to the design table after you create the design by selecting Make Table in the Screening Design window. These properties are also saved automatically to the data table that is created when you select the Save Factors option. You can find details about these column properties and related examples in Appendix A, “Column Properties”.
Coding
If the Role is Continuous, the Coding column property for the factor is saved. This property transforms the factor values so that the low and high values correspond to –1 and +1, respectively. For details, see “Coding” in the “Column Properties” appendix.
Value Ordering
If the Role is Categorical or if a Block variable is constructed, 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.
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. 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 with the value of Easy. 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.
Select Output Options
After you enter your responses and factors and click Continue, you can make selections for your design table in the Output Options outline. The structure of the full factorial design appears at the top of the outline.
Figure 12.10 Output Options Panel
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 sorted from left to right.
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.
Center Points and Replicates
Number of Runs
Shows the number of runs in the design before you add center points or replicates.
Number of Center Points
Specifies how many additional runs to add as center points to the design. A center point is a run where every continuous factor is set at the center of the factor’s range.
Suppose that your design includes both continuous and categorical factors. If you request center points in the Output Options panel, the center points are distributed as follows:
1. The settings for the categorical factors are ordered using the value ordering specified in the Factors outline.
2. One center point is assigned to each combination of the settings of the categorical factors in order, and this is repeated until all center points are assigned.
Number of Replicates
The number of times to replicate the entire design, including center points. One replicate doubles the number of runs.
Make Table
Clicking Make Table creates a data table that contains the runs for your experiment. The example in Figure 12.11 shows a full factorial design with five center points for three factors: X1 (a two-level continuous factor), X2 (a three-level continuous factor), and X3 (a two-level categorical factor). The design uses the default values for the factor levels. The center points are in rows 1, 5, 8, 10, and 15. See “Pattern Column”.
Figure 12.11 Design Data Table
The name of the table, shown in the upper left corner, is the design type that generated it.
Design Table Scripts
The design table includes the following scripts:
Model
Runs the Analyze > Fit Model platform.
Evaluate Design
Runs the DOE > Design Diagnostics > Evaluate Design platform.
DOE Dialog
Re-creates the Full Factorial Design window that you used to generate the design table. The script also contains the random seed used to generate your design.
Run the Screening or Model scripts to analyze the data.
Pattern Column
The Pattern column contains entries that summarize the run in the given row. You can use Pattern as a label variable in plots.
• For a two-level continuous factor, the low setting is denoted by “–”, the high setting by “+”, and a center point by “0”.
• For a continuous factor with more than two levels:
‒ For a non-center point, the factor setting is denoted by an integer that corresponds to the value level for the run.
‒ For a center point, the factor setting is denoted by a “0”.
• For a categorical factor, the factor setting is denoted by an integer that corresponds to the value level for the run.
Full Factorial Design Options
The red triangle menu in the Full Factorial Design platform contains these options:
Save Responses
Saves the information in the Responses panel to a new data table. You can then quickly load the responses and their associated information into most DOE windows. This option is helpful if you anticipate re-using the responses.
Load Responses
Loads responses that you saved using the Save Responses option.
Save Factors
Saves the information in the Factors panel to a new data table. Each factor’s column contains its levels. Other information is stored as column properties. You can then quickly load the factors and their associated information into most DOE windows.
Note: It is possible to create a factors table by entering data into an empty table, but remember to assign each column an appropriate Design Role. Do this by right-clicking on the column name in the data grid and selecting Column Properties > Design Role. In the Design Role area, select the appropriate role.
Load Factors
Loads factors that you saved using the Save Factors option.
Save Constraints
(Unavailable for some platforms) Saves factor constraints that you defined in the Define Factor Constraints or Linear Constraints outline into a data table, with a column for each constraint. You can then quickly load the constraints into most DOE windows.
In the constraint table, the first rows contain the coefficients for each factor. The last row contains the inequality bound. Each constraint’s column contains a column property called ConstraintState that identifies the constraint as a “less than” or a “greater than” constraint. See “ConstraintState” in the “Column Properties” appendix.
Load Constraints
(Unavailable for some platforms) Loads factor constraints that you saved using the Save Constraints option.
Set Random Seed
Sets the random seed that JMP uses to control certain actions that have a random component. These actions include the following:
‒ simulating responses using the Simulate Responses option
‒ randomizing Run Order for design construction
‒ selecting a starting design for designs based on random starts
To reproduce a design or simulated responses, enter the random seed that generated them. For designs using random starts, set the seed before clicking Make Design. To control simulated responses or run order, set the seed before clicking Make Table.
Note: The random seed associated with a design is included in the DOE Dialog script that is saved to the design data table.
Simulate Responses
Adds response values and a column containing a simulation formula to the design table. Select this option before you click Make Table.
When you click Make Table, the following occur:
‒ A set of simulated response values is added to each response column.
‒ For each response, a new a column that contains a simulation model formula is added to the design table. The formula and values are based on the model that is specified in the design window.
‒ A Model window appears where you can set the values of coefficients for model effects and specify one of three distributions: Normal, Binomial, or Poisson.
‒ A script called DOE Simulate is saved to the design table. This script re-opens the Model window, enabling you to re-simulate values or to make changes to the simulated response distribution.
Make selections in the Model window to control the distribution of simulated response values. When you click Apply, a formula for the simulated response values is saved in a new column called <Y> Simulated, where Y is the name of the response. Clicking Apply again updates the formula and values in <Y> Simulated.
For additional details, see “Simulate Responses” in the “Custom Designs” chapter.
Note: 
You can use Simulate Responses to conduct simulation analyses using the JMP Pro Simulate feature. For information about Simulate and some DOE examples, see the Simulate chapter in the Basic Analysis book.
You can use Simulate Responses to conduct simulation analyses using the JMP Pro Simulate feature. For information about Simulate and some DOE examples, see the Simulate chapter in the Basic Analysis book.Advanced Options
None available.
Save Script to Script Window
Creates the script for the design that you specified in the Full Factorial Design window and saves it in an open script window.
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