Contents 
Screening Design Examples
This chapter is divided into two sections. The first section consists of two examples using screening designs. The second section outlines the procedures to follow to create a screening design to match your needs.
Using Two Continuous Factors and One Categorical Factor
Suppose an engineer wants to investigate a process that uses an electron beam welding machine to join two parts. The engineer fits the two parts into a welding fixture that holds them snugly together. A voltage applied to a beam generator creates a stream of electrons that heats the two parts, causing them to fuse. The ideal depth of the fused region is 0.17 inches. The engineer wants to study the welding process to determine the best settings for the beam generator to produce the desired depth in the fused region.
For this study, the engineer wants to explore the following three inputs, which are the factors for the study:
• Operator, who is the technician operating the welding machine
• Rotation Speed, which is the speed at which the part rotates under the beam
• Beam Current, which is a current that affects the intensity of the beam
After each processing run, the engineer cuts the part in half. This reveals an area where the two parts have fused. The Length of this fused area is the depth of penetration of the weld. This depth of penetration is the response for the study.
The goals of the study are to:
• find which factors affect the depth of the weld
• quantify those effects
• find specific factor settings that predict a weld depth of 0.17 inches
To begin this example, select DOE > Screening Design from the main menu. Note that in the Responses panel, there is a single default response called Y. Change the default response as follows:
1. Double-click the response name and change it to Depth (In.).
2. The default goal for the single default response is Maximize, but the goal of this process is to get a target value of 0.17 inches with a lower bound of 0.12 and an upper bound of 0.22. Click the Goal box and choose Match Target from the drop-down menu, as shown in Figure 5.1.
Figure 5.1 Screening Design Response With Match Target Goal
3. Click the Lower Limit text edit area and enter 0.12 as the lower limit (minimum acceptable value). Then click the Upper Limit text edit area and enter 0.22 as the upper limit (maximum acceptable value).
This example has one categorical factor (Operator) and two continuous factors (Speed and Current).
4. Add the categorical factor by clicking the Add button beside 2-Level Categorical.
5. Add two continuous factors by typing 2 in the Continuous box and clicking the associated Add button.
6. Double-click the factor names and rename them Operator, Speed, and Current.
7. Set high and low values for Speed to 3 and 5 rpm. Set high and low values for Current to 150 and 165 amps, and assign John and Mary as values for the categorical factor called Operator, as shown in Figure 5.2.
Figure 5.2 Screening Design with Two Continuous and One Categorical Factor
8. Click Continue.
9. Select Full Factorial in the list of designs, as shown in Figure 5.3, and then click Continue.
Figure 5.3 List of Screening Designs for Two Continuous and One Categorical Factors
In the Output Options report, click on the Run Order menu and select Sort Left to Right. This arranges the runs in the JMP design data table (see Figure 5.7). Then click Make Table to create the JMP table that contains the specified design.
The table in Figure 5.4 appears. The table uses the names for responses, factors, and levels you specified. The Pattern variable shows the coded design runs. You can also see the table produced in this example by selecting Help > Sample Data > Design of Experiments > DOE Example 1. (You can also open DOE Example 1.jmp from the sample data directory.)
Figure 5.4 The Design Data Table
Using Five Continuous Factors
As illustrated in the previous section, experiments for screening the effects of many factors usually consider only two levels of each factor. This allows the examination of many factors with a minimum number of runs.
The following example, adapted from Meyer, et al. (1996), demonstrates how to use JMP’s screening designer when you have many factors. In this study, a chemical engineer investigates the effects of five factors on the percent reaction of a chemical process. The factors are:
• feed rate, the amount of raw material added to the reaction chamber in liters per minute
• percentage of catalyst
• stir rate, the RPMs of a propeller in the chamber
• reaction temperature in degrees Celsius
• concentration of reactant
To start the example:
1. Select DOE > Screening Design.
2. You see one default response called Y. Change the default response name (Y) to Percent Reacted.
3. The Goal is to maximize the response, but change the minimum acceptable reaction percentage to 90 (Lower Limit), and upper limit to 99 (Upper Limit), as shown in Figure 5.5.
4. Add five continuous factors.
5. Change the default factor names (X1-X5) to Feed Rate, Catalyst, Stir Rate, Temperature, and Concentration.
6. Enter the high and low values, as shown in Figure 5.5.
Figure 5.5 Screening for Many Factors
7. Click Continue. Now, JMP lists the designs for the number of factors you specified, as shown to the left in Figure 5.6.
8. Select the first item in the list, which is an 8-run fractional factorial design with no blocks.
9. Click Continue to see the Output Options outline on the right in Figure 5.6.
Figure 5.6 Two-level Screening Design (left) and design output options (right)
The design dialog has options shown in Figure 5.7 that can modify the final design table.
10. Select Sort Left to Right from the Run Order Menu.
Figure 5.7 Output Options for Design Table
11. Click Make Table to create the data table shown in Figure 5.8 that lists the runs for the design you selected. Note that it also has a column called Percent Reacted for recording experimental results, showing as the rightmost column of the data table.
Figure 5.8 JMP Table of Runs for Screening Example
Of the five factors in the reaction percentage experiment, you expect a few to stand out in comparison to the others. Let’s take an approach to the analysis that looks for active effects.
12. To run the model generated by the data shown in Figure 5.8, open Reactor 8 Runs.jmp from the Design Experiment folder found in the sample data that was installed with JMP. This table has the design runs and the results of the experiment.
13. In the design data table, click the Screening script that shows on the upper left of the data table, and select Run Script. Or, you can choose Analyze > Modeling > Screening to analyze the data. Select Percent Reacted as Y and all other continuous variables as X. Click OK.The report is shown in Figure 5.9.
Figure 5.9 Report for Screening Example
Note: Analysis of the screening data is covered in the section Create a Plackett-Burman design at the end of this chapter.
Creating a Screening Design
To begin, select DOE > Screening Design, or click the Screening Design button on the JMP Starter DOE page. Then, see the following sections for each step to create a screening design:
Enter Responses
To enter responses, follow the steps in Figure 5.10.
1. To enter one response at a time, click and then select a goal type. Possible goal types are Maximize, Match Target, Minimize, or None.
2. (Optional) Double-click to edit the response name.
3. (Optional) Click to change the response goal.
4. Click to enter lower and upper limits and importance weights.
Figure 5.10 Entering Responses
Tip: To quickly enter multiple responses, click the Number of Responses button and enter the number of responses you want.
Specifying Goal Types and Lower and Upper Limits
When entering responses, you can tell JMP that your goal is to obtain the maximum or minimum value possible, to match a specific value, or that there is no goal.
The following description explains the relationship between the goal type (step 3 in Figure 5.10) and the lower and upper limits (step 4 in Figure 5.10):
• For responses such as strength or yield, the best value is usually the largest possible. A goal of Maximize supports this objective.
• The Minimize goal supports an objective of having the smallest value be the most desirable, such as when the response is impurity or defects.
• The Match Target goal supports the objective when the best value for a response is a specific target value, such as dimensions of a manufactured part. The default target value is assumed to be midway between the lower and upper limits.
Note: If your target range is not symmetric around the target value, you can alter the default target after you make a table from the design. In the data table, open the response’s Column Info dialog by double-clicking the column name, and enter an asymmetric target value.
Understanding Importance Weights
When computing overall desirability, JMP uses the value you enter as the importance weight (step 4 in Figure 5.10) as the weight of each response. If there is only one response, then specifying importance is unnecessary. With two responses you can give greater weight to one response by assigning it a higher importance value.
Enter Factors
Next, you enter factors. The Factors panel’s appearance depends on the design you select. Entering factors is the same in Screening Design, Space Filling Design, Mixture Design, and Response Surface Design. This process is described below, in Figure 5.11.
1. To enter factors, type the number of factors and click Add.
2. Highlight the factor and click the Remove Selected button to remove a factor in the list.
3. Double-click to edit the factor name.
4. Click to enter factor values. To remove a level, click it, press the delete key on your keyboard, then press the Return or Enter key on your keyboard.
Figure 5.11 Entering Factors
Types of Factors
In general, when designing experiments, you can enter different types of factors in the model. Below is a description of factor types from which you can choose when creating screening designs:
Continuous
Continuous factors have numeric data types only. In theory, you can set a continuous factor to any value between the lower and upper limits you supply.
Categorical
Categorical factors (either numeric or categorical data types) have no implied order. If the values are numbers, the order is the numeric magnitude. If the values are character, the order is the sorting sequence. The settings of a categorical factor are discrete and have no intrinsic order. Examples of categorical factors are machine, operator, and gender.
After you enter responses and factors, click Continue.
Choose a Design
The list of screening designs you can use includes designs that group the experimental runs into blocks of equal sizes where the size is a power of two. Highlight the type of screening design you want to use and click Continue.
Figure 5.12 Choosing a Type of Screening Design
The screening designer provides the following types of designs:
Two-Level Full Factorial
A full factorial design has runs for all combinations of the levels of the factors. The samples size is the product of the levels of the factors. For two-level designs, this is 2k where k is the number of factors. This can be expensive if the number of factors is greater than 3 or 4.
These designs are orthogonal. This means that the estimates of the effects are uncorrelated. If you remove an effect in the analysis, the values of the other estimates remain the same. Their p-values change slightly, because the estimate of the error variance and the degrees of freedom are different.
Full factorial designs allow the estimation of interactions of all orders up to the number of factors. Most empirical modeling involves first- or second-order approximations to the true functional relationship between the factors and the responses. The figure to the left in Figure 5.13 is a geometric representation of a two-level factorial.
Two-Level Fractional Factorial
A fractional factorial design also has a sample size that is a power of two. If k is the number of factors, the number of runs is 2k – p where p < k. The fraction of the full factorial is 2-p. Like the full factorial, fractional factorial designs are orthogonal.
The trade-off in screening designs is between the number of runs and the resolution of the design. If price is no object, you can run several replicates of all possible combinations of m factor levels. This provides a good estimate of everything, including interaction effects to the mth degree. But because running experiments costs time and money, you typically only run a fraction of all possible levels. This causes some of the higher-order effects in a model to become nonestimable. An effect is nonestimable when it is confounded with another effect. In fact, fractional factorials are designed by deciding in advance which interaction effects are confounded with the other interaction effects.
Resolution Number: The Degree of Confounding
In practice, few experimenters worry about interactions higher than two-way interactions. These higher-order interactions are assumed to be zero.
Experiments can therefore be classified by resolution number into three groups:
• Resolution = 3 means that main effects are confounded with one or more two-way interactions, which must be assumed to be zero for the main effects to be meaningful.
• Resolution = 4 means that main effects are not confounded with other main effects or two-factor interactions. However, two-factor interactions are confounded with other two-factor interactions.
• Resolution ≥ 5 means there is no confounding between main effects, between two-factor interactions, or between main effects and two-factor interactions.
A minimum aberration design is one in which there are a minimum number of confoundings for a given resolution. For DOE experts, the minimum aberration design of a given resolution minimizes the number of words in the defining relation that are of minimum length.
The figure on the right in Figure 5.13 is geometric representation of a two-level fractional factorial design.
Figure 5.13 Representation of Full Factorial (Left) and Two-Level Fractional Factorial (Right) Designs
Plackett-Burman Designs
Plackett-Burman designs are an alternative to fractional factorials for screening. One useful characteristic is that the sample size is a multiple of four rather than a power of two. There are no two-level fractional factorial designs with sample sizes between 16 and 32 runs. However, there are 20-run, 24-run, and 28-run Plackett-Burman designs.
The main effects are orthogonal and two-factor interactions are only partially confounded with main effects. This is different from resolution-three fractional factorial where two-factor interactions are indistinguishable from main effects.
In cases of effect sparsity, a stepwise regression approach can allow for removing some insignificant main effects while adding highly significant and only somewhat correlated two-factor interactions. The new Screening platform in JMP, Analyze > Modeling > Screening, is a streamlined approach for looking at sparse data. This platform can accept multiple responses and multiple factors, then automatically fits a two-level design and shows significant effects with plots and statistics. See the chapter in the Modeling and Multivariate Methods book on the Screening platform for more information.
Mixed-Level Designs
If you have qualitative factors with three values, then none of the classical designs discussed previously are appropriate. For pure three-level factorials, JMP offers fractional factorials. For mixed two-level and three-level designs, JMP offers complete factorials and specialized orthogonal-array designs, listed in Table 4.1
If you have fewer than or equal to the number of factors for a design listed in the table, you can use that design by selecting an appropriate subset of columns from the original design. Some of these designs are not balanced, even though they are all orthogonal.
 
Table 5.1 Table of Mixed-Level Designs 
Design
Two–Level Factors
Three–Level Factors
L18 John
1
7
L18 Chakravarty
3
6
L18 Hunter
8
4
L36
11
12
Cotter Designs
Cotter designs are used when you have very few resources and many factors, and you believe there may be interactions. Suppose you believe in effect sparsity— that very few effects are truly nonzero. You believe in this so strongly that you are willing to bet that if you add up a number of effects, the sum will show an effect if it contains an active effect. The danger is that several active effects with mixed signs will cancel and still sum to near zero and give a false negative.
Cotter designs are easy to set up. For k factors, there are 2k + 2 runs. The design is similar to the “vary one factor at a time” approach many books call inefficient and naive.
A Cotter design begins with a run having all factors at their high level. Then follow k runs each with one factor in turn at its low level, and the others high. The next run sets all factors at their low level and sequences through k more runs with one factor high and the rest low. This completes the Cotter design, subject to randomizing the runs.
When you use JMP to generate a Cotter design, the design also includes a set of extra columns to use as regressors. These are of the form factorOdd and factorEven where factor is a factor name. They are constructed by adding up all the odd and even interaction terms for each factor. For example, if you have three factors, A, B, and C:
 
Table 5.2 Cotter Design Table
AOdd = A + ABC
AEven = AB + AC
BOdd = B + ABC
BEven = AB + BC
COdd = C + ABC
CEven = BC + AC
Because these columns in a Cotter design make an orthogonal transformation, testing the parameters on these combinations is equivalent to testing the combinations on the original effects. In the example of factors listed above, AOdd estimates the sum of odd terms involving A. AEven estimates the sum of the even terms involving A, and so forth.
Because Cotter designs have a false-negative risk, many statisticians discourage their use.
How to Run a Cotter Design
By default, JMP does not include a Cotter design in the list of available screening designs (Figure 5.12). However, if you want to make a Cotter design:
1. Immediately after entering responses and factors (and before clicking Continue), click the red triangle icon in the Screening Design title bar.
2. Select Supress Cotter Designs (to uncheck it).
Changing the setting via the red triangle menu applies only to the current design. To alter the setting for all screening designs:
1. Select File > Preferences.
2. Click the Platforms icon.
3. Click DOE to highlight it.
4. Uncheck the box beside Suppress Cotter Designs.
Display and Modify a Design
After you select a design type, open the Display and Modify Design outline.
Figure 5.14 Display and Modification Options
Change Generating Rules
Controls the choice of different fractional factorial designs for a given number of factors.
Aliasing of Effects
Shows the confounding pattern for fractional factorial designs.
Coded Design
Shows the pattern of high and low values for the factors in each run.
Aliasing of Effects
To see which effects are confounded with which other effects, open the Aliasing of Effects outline. It shows effects and confounding up to two-factor interactions.
Figure 5.15 Generating Rules and Aliasing of Effects Panel
For example, a full factorial with five factors requires 25 = 32 runs. Eight runs can only accommodate a full factorial with three two-level factors. It is necessary to construct the two additional factors in terms of the first three factors.
The price of reducing the number of runs from 32 to eight is effect aliasing (confounding). Confounding is the direct result of the assignment of new factor values to products of the coded design columns.
In the example above, the values for Temperature are the product of the values for Feed Rate and Concentration. This means that you can’t tell the difference of the effect of Temperature and the synergistic (interactive) effect of Feed Rate and Concentration.
In the example shown in Figure 5.15, all the main effects are confounded with two-factor interactions. This is characteristic of resolution-three designs.
Look at the Confounding Pattern
JMP can create a data table that shows the aliasing pattern for a specified level. To create this table:
1. Click the red triangle at the bottom of the Aliasing of Effects area.
2. Select Show Confounding Pattern (Figure 5.16).
Figure 5.16 Show Confounding Patterns
3. Enter the order of confounding you want to see (Figure 5.17).
Figure 5.17 Enter Order of Confounding in Text Edit Box
4. Click OK.
Figure 5.18 shows the third order aliasing for the five-factor reactor example. The effect names begin with C (Constant) and are shown by their order number in the design. Thus, Temperature appears as “4”, with second order aliasing as “1 5” (Feed Rate and Concentration), and third order confounding as “1 2 3” (Feed Rate, Catalyst, and Stir Rate).
Figure 5.18 The Third Level Alias for the Five-Factor Reactor Example
Understanding Design  Codes
In the Coded Design panel, each row represents a run. Plus signs designate high levels and minus signs represent low levels. As shown in Figure 5.19, rows for the first three columns of the coded design, which represent Feed Rate, Catalyst, and Stir Rate are all combinations of high and low values (a full factorial design). The fourth column (Temperature) of the coded design is the element-by-element product of the first three columns. Similarly, the last column (Concentration) is the product of the second and third columns.
Figure 5.19 Default Coded Designs
Changing the Coded Design
In the Change Generating Rules panel, changing the check marks and clicking Apply changes the coded design; it changes the choice of different fractional factorial designs for a given number of factors. The Coded Design table in Figure 5.19 shows how the last two columns are constructed in terms of the first three columns. The check marks in the Change Generating Rules table shown in Figure 5.20 for Temperature now show it is a function of Feed Rate, and Catalyst. The check marks for Concentration show it is a function of Feed Rate and Stir Rate.
If you check the options as shown in Figure 5.20 and click Apply, the Coded Design panel changes. The first three columns of the coded design remain a full factorial for the first three factors (Feed Rate, Catalyst, and Stir Rate). Temperature is now the product of Feed Rate and Catalyst, so the fourth column of the coded design is the element by element product of the first two columns. Concentration is a function of Feed Rate and Stir Rate.
Figure 5.20 Modified Coded Designs and Generating Rules
Specify Output Options
Use the Output Options panel (Figure 5.21) to specify how you want the output data table to appear. When the options are the way you want them, click Make Table.
Figure 5.21 Select the Output Options
Run Order
Lets you designate the order you want the runs to appear in the data table when it is created. Choices are:
Keep the Same the rows (runs) in the output table appear as they do in the Coded Design panel.
Sort Left to Right the rows (runs) in the output table appear sorted from left to right.
Randomize the rows (runs) in the output table appear in a random order.
Sort Right to Left the rows (runs) in the output table appear sorted from right to left.
Randomize within Blocks the rows (runs) in the output table will appear in random order within the blocks you set up.
Number of Center Points
Specifies additional runs placed at the center points.
Number of Replicates
Specify the number of times to replicate the entire design, including center points. Type the number of times you want to replicate the design in the associated text box. One replicate doubles the number of runs.
View the Design Table
Click Make Table to create a data table that contains the runs for your experiment. In the table, the high and low values you specified are displayed for each run.
Figure 5.22 The Design Data Table
The name of the table is the design type that generated it. Run the Screening script to screen for active effects. The column called Pattern shows the pattern of low values denoted “–” and high values denoted “+”. Pattern is especially useful as a label variable in plots.
Create a Plackett-Burman design
The previous example shows an 8-run fractional factorial design for five continuous factors. But suppose you can afford 4 additional runs. First, repeat the steps shown in the previous sections. This time, use the Load Responses and Load Factors commands to define the design, as follows:
1. Select DOE > Screening Design.
2. Select Load Responses from the red triangle menu on the Screening Design title bar. Navigate to the Design Experiment folder in the Sample Data installed with JMP and open the file called Reactor Response.jmp.
3. Next, select Load Factors from red triangle menu on the Screening Design title bar. Navigate to the Design Experiment folder in the Sample Data installed with JMP and open the file called Reactor Factors.jmp.
These two commands complete the DOE screening dialog for you, with the correct response and factor names, goal and limits for the response, and the values for the factors.
4. Click Continue on the completed Screening design dialog to see the list of designs in Figure 5.23, and chose the Plackett-Burman, as shown.
Figure 5.23 Design List for 5-factor Plackett-Burman Screening Design
5. Click Continue.
After you select the model from the Design list, the outlines for modifying and evaluating the model appear. In the Custom designer, you have the ability to form any model effects you want. The Screening designer creates the design effects based on the design you choose. In particular, the full factorial with all two-factor interactions has no aliasing of the included interactions, as shown in Figure 5.24.
Figure 5.24 No aliasing of the included interactions
A complete discussion of the Design Evaluation options is in the Building Custom Designs.
To continue with this example, do the following:
6. Choose Sort Left to Right in the Output Options panel.
7. Click Make Table to see the design runs shown in Figure 5.25.
Examine the data table and note the Pattern variable to see the arrangement of plus and minus signs that define the runs. This table is used in the analysis sections that follow.
Figure 5.25 Listing of a 5-factor Placket-Burman Design Table
Analysis of Screening Data
After creating and viewing the data table, you can now run analyses on the data. As an example, open the data table called Plackett-Burman.jmp, found in Design Experiment folder in the Sample Data installed with JMP. This table contains the design runs and the Percent Reacted experimental results for the 12-run Plackett-burman design created in the previous section.
Using the Screening Analysis Platform
The data table has two scripts called Screening and Model, showing in the upper-left corner of the table, that were created by the DOE Screening designer. You can use these scripts to analyze the data, however it is simple to run the analyses yourself.
1. Select Analyze > Modeling > Screening to see the completed launch dialog shown in Figure 5.26. When you create a DOE design table, the variable roles are saved with the data table and used by the launch platform to complete the dialog.
Figure 5.26 Launch Dialog for the Screening Platform
2. Click OK to see the Screening platform result shown in Figure 5.27.
The Contrasts section of the Screening platform results lists all possible model effects, a contrast value for each effect, Lenth t-ratios (calculated as the contrast value divided by the Lenth PSE (pseudo-standard error), individual and simultaneous p-values, and aliases if there are any. Significant and marginally significant effects are highlighted. See the chapter on analyzing Screening designs in the Modeling and Multivariate Methods book for complete documentation of the Screening analysis platform.
Figure 5.27 Results of the Screening Analysis
3. Examine the Half Normal plot in Figure 5.27.
Using the Fit Model Platform
The Make Model button beneath the Half Normal Plot creates a Fit Model dialog that includes all the highlighted effects. However, note that the Catalyst*Stir Rate interaction is highlighted, but the Stir Rate main effect is not. Therefore, you should add the Stir Rate main effect to the model.
4. Click the Make Model Button beneath the Half Normal Plot.
5. Select Stir Rate and click Add on the Fit Model dialog.
6. The Emphasis might change to Effect Screening when you add Stir Rate. Change it back to Effect Leverage. The dialog is shown in Figure 5.28.
7. Then click Run to see the analysis results.
Figure 5.28 Create Fit Model Dialog and Remove Unwanted Effect
The Actual-by-Predicted Plot
The Whole Model actual-by-predicted plot, shown in Figure 5.29, appears at the top of the Fit Model report. You see at a glance that this model fits well. The blue line falls outside the bounds of the 95% confidence curves (red-dotted lines), which tells you the model is significant. The model p-value (p = 0.0208), R2, and RMSE appear below the plot. The RMSE is an estimate of the standard deviation of the process noise, assuming that the unestimated effects are negligible.
Figure 5.29 An Actual-by-Predicted Plot
The Scaled Estimates Report
To see a scaled estimates report, use Effect Screening > Scaled Estimates found in the red triangle menu on the Response Percent Reacted title bar. When there are quadratic or polynomial effects, the coefficients and the tests for them are more meaningful if effects are scaled and coded. The Scaled Estimates report includes a bar chart of the individual effects embedded in a table of parameter estimates. The last column of the table has the p-values for each effect.
Figure 5.30 Example of a Scaled Estimates Report
A Power Analysis
The Fit Model report has outline nodes for the Catalyst and Temperature effects. To run a power analysis for an effect, click the red triangle icon on its title bar and select Power Analysis.
This example shows a power analysis for the Catalyst variable, using default value for α (0.05), the root mean square error and parameter estimate for Catalyst, for a sample size of 12. The resulting power is 0.8926, which means that in similar experiments, you can expect an 89% chance of detecting a significant effect for Catalyst.
Figure 5.31 Example of a Power Analysis
Refer to Modeling and Multivariate Methods for details.
..................Content has been hidden....................

You can't read the all page of ebook, please click here login for view all page.
Reset