Overview of the Fit Two Level Screening Platform
The analysis of screening designs depends on effect sparsity, where most effects are assumed to be inactive. Using this assumption, effects with small estimates can help estimate the error in the model and determine whether the larger effects are active. Basically, if all the effects are inactive, they should vary randomly, with no effect deviating substantially from the other effects.
Data from a screening experiment can be analyzed using Fit Model (Analyze > Fit Model) or Fit Two Level Screening (Analyze > Specialized Modeling > Specialized DOE Models > Fit Two Level Screening). Use the Fit Two Level Screening platform to analyze data from screening experiments In accordance with the following guidelines:
• If your factors are all two-level and orthogonal, then all of the statistics in the Fit Two Level Screening platform are appropriate.
• For highly supersaturated main effect designs, the Fit Two Level Screening platform is effective in selecting factors, but is not as effective at estimating the error or the significance. The Monte Carlo simulation to produce p-values uses assumptions that are not valid for this case.
• If you have a categorical or a discrete numeric factor with more than two levels, then the Fit Two Level Screening platform is not an appropriate way to analyze the data. JMP treats the associated model terms as continuous. The variation for the factor is scattered across main and polynomial effects for that term.
• If your data are not orthogonal, then the constructed estimates are different from standard regression estimates. JMP can pick out big effects, but it does not effectively test each effect. This is because later effects are artificially orthogonalized, making earlier effects look more significant.
• The Fit Two Level Screening platform is not appropriate for mixture designs.
An Example Comparing Fit Two Level Screening and Fit Model
Consider the Reactor Half Fraction.jmp sample data table. The data are derived from a design discussed in Box, Hunter, and Hunter (1978). You are interested in a model with main effects and two-way interactions. This example uses a model with fifteen parameters for a design with sixteen runs.
For this example, select all continuous factors, except the response, Percent Reacted, as the screening effects, X. Select Percent Reacted as the response Y. The screening platform constructs interactions automatically. This is in contrast to Fit Model, where you manually specify the interactions that you want to include in your model.
Figure 10.2 shows the result of using the Fit Model platform, where a factorial to degree 2 model is specified. Since there are not enough observations to estimate an error term, it is not possible to conduct standard tests.
Figure 10.2 Traditional Saturated Half Reactor.jmp Design Output
JMP can calculate parameter estimates, but because there are no degrees of freedom for error, standard errors, t-ratios, and p-values are all missing. Rather than use Fit Model, you can use the Fit Two Level Screening platform, which specializes in getting the most information out of these situations, leading to a better model. The report from the Fit Two Level Screening platform for the same data is shown in Figure 10.3.
Figure 10.3 Half Reactor.jmp Fit Two Level Screening Design Report
Note the following features of the Screening report:
• Estimates labeled Contrast. Effects whose individual p-value is less than 0.10 are highlighted.
• A t-ratio is calculated using Lenth’s PSE (pseudo-standard error). The Lenth PSE is shown below the Half Normal Plot.
• Both individual and simultaneous p-values are shown. Those that are less than 0.05 are shown with an asterisk.
• The Half Normal Plot enables you to quickly examine the effects. Effects initially highlighted in the effects list are also labeled in this plot.
• You can highlight effects by clicking on them in the Contrasts outline.
• The Make Model button opens the Fit Model window and populates it with the selected effects. The Run Model button runs the model based on the selected effects.
For this example, Catalyst, Temperature, and Concentration, along with two of their two-factor interactions, are selected.
Launch the Fit Two Level Screening Platform
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.
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 Help > Sample Data Library and open Design Experiment/Plackett-Burman.jmp.
2. Select Analyze > Specialized Modeling > Specialized DOE Models > Fit Two Level Screening.
The populated launch window appears. When you use the Screening Design platform to create a design, a Screening script is saved to the design table. This allows JMP to populate the screening launch window.
Figure 10.4 Launch Window for the Fit Two Level Screening Platform
3. Click OK.
The Screening report, shown in Figure 10.5, appears.
Figure 10.5 Screening Report
If all effects are inactive, their estimates are random normal noise. Their estimates (contrasts) should fall close to the line shown in the Half Normal plot in Figure 10.5. Effects that fall far from the line are likely not noise, and so may represent active effects. Note that effects with Individual p-Values that fall below 0.10 are highlighted in the Contrasts outline. These effects are labeled in the Half Normal Plot and then tend to fall far from the line.
The Screening Report
The Screening report has two parts: The Contrasts outline and the Half Normal Plot outline.
Contrasts
The Contrasts outline lists 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. Effects are entered into the analysis following a hierarchical ordering. See “Order of Effect Entry” for details. Effects with Individual p-Value less than 0.10 are highlighted.
Term
Name of the factor.
Contrast
Estimate for the factor. For orthogonal designs, this number is the same as the regression parameter estimate. This is not the case for non-orthogonal designs. An asterisk might appear next to the contrast, indicating a lack of orthogonality.
Bar Chart
Shows the Lenth t-ratios with blue vertical lines indicating a value that is significant at the 0.10 level.
Lenth t-Ratio
Lenth’s t-ratio, calculated as 
, where PSE is Lenth’s Pseudo-Standard Error. See “Lenth’s Pseudo-Standard Error” for details.
, where PSE is Lenth’s Pseudo-Standard Error. See “Lenth’s Pseudo-Standard Error” for details.Individual p-Value
Analogous to the standard p-values for a linear model. Small values of this value indicate a significant effect. Refer to “Lenth’s Pseudo-Standard Error” for details.
Do not expect the p-values to be exactly the same if the analysis is re-run. The Monte Carlo method should give similar, but not identical, values if the same analysis is repeated.
Simultaneous p-Value
Similar to the individual p-value, but multiple-comparison adjusted.
Aliases
Appears only if there are exact aliases of later effects to earlier effects.
Half Normal Plot
The Half Normal Plot shows the absolute value of the contrasts plotted against the absolute value of quantiles for the half-normal distribution. Significant effects appear separated from the line towards the upper right of the graph.
The Half Normal Plot is interactive. Select different model effects by dragging a rectangle around the effects of interest, or hold down CTRL and click on an effect name in the report.
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.
1. Open the Plackett-Burman.jmp sample data table, found in Design Experiment folder.
2. Select Analyze > Specialized Modeling > Specialized DOE Models > Fit Two Level Screening.
3. Click OK.
4. Click the Make Model Button beneath the Half Normal Plot.
Note that the Catalyst*Stir Rate interaction is highlighted, but the Stir Rate main effect is not. In accordance with the principle of Effect Heredity, add the Stir Rate main effect to the model. See “Effect Heredity” in the “Starting Out with DOE” chapter.
5. Select Stir Rate and click Add in the Fit Model window.
6. Click Run.
The Actual-by-Predicted Plot
The Whole Model actual-by-predicted plot, shown in Figure 10.6, 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, assuming that the unestimated effects are negligible.
Figure 10.6 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 10.7 Example of a Scaled Estimates Report
A Power Analysis
Open the Effect Details disclosure icon to see 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 10.8 Example of a Power Analysis
Refer to the Standard Least Squares chapter in the Fitting Linear Models book for details.
Additional Fit Two Level Screening Analysis Examples
This section provides examples of using the Fit Two Level Screening platform.
Analyzing a Plackett-Burman Design
Plackett-Burman designs are an alternative to fractional-factorial screening designs. Two-level fractional factorial designs must, by their nature, have a number of runs that are a power of two. Plackett-Burman designs exist for 12-, 24-, and 28-run designs.
The Weld-Repaired Castings.jmp sample data table uses a Plackett-Burman design, and is found in textbooks such as Giesbrecht and Gumpertz (2004) and Box, Hunter, and Hunter (1978). Seven factors are thought to be influential on weld quality. The seven factors include Initial Structure, Bead Size, Pressure Treatment, Heat Treatment, Cooling Rate, Polish, and Final Treatment. A Plackett-Burman design with 12 runs is used to investigate the importance of the seven factors. The response is 
. (There are also four terms that were used to model error that are not used in this analysis.)
. (There are also four terms that were used to model error that are not used in this analysis.)Using the Fit Two Level Screening platform, select the seven effects as X and Log Life as Y. (If terms are automatically populated in the Fit Two Level Screening platform launch window, remove the four error terms listed as effects.) Click OK. Figure 10.9 appears, showing only a single significant effect.
Figure 10.9 Screening Report for Weld-Repaired Castings.jmp
Note asterisks mark four terms, indicating that they are not orthogonal to effects preceding them, and the obtained contrast value was after orthogonalization. So, they would not match corresponding regression estimates.
Analyzing a Supersaturated Design
Supersaturated designs have more factors than runs. The objective is to determine which effects are active. They rely heavily on effect sparsity for their analysis, so the Fit Two Level Screening platform is ideal for their analysis.
As an example, look at Supersaturated.jmp, from the sample data folder, a simulated data set with 18 factors but only 12 runs. Y is generated by
where ε ~ N(0,1). So, Y has been constructed with three active factors.
To detect the active factors, run the Fit Two Level Screening platform with X1–X18 as X and Y as Y. The report shown in Figure 10.10 appears.
Figure 10.10 Screening Report for Supersaturated.jmp
Note that the three active factors have been highlighted. One other factor, X18, has also been highlighted. It shows in the Half Normal plot close to the blue line, indicating that it is close to the 0.1 cutoff significance value. The 0.1 critical value is generous in its selection of factors so you don’t miss those that are possibly active.
The contrasts of 5.1, –3, and 1.8 are close to their simulated values (5, –3, 2). However, the similarity of these values can be increased by using a regression model, without the effect of orthogonalization.
The p-values, while useful, are not entirely valid statistically, since they are based on a simulation that assumes orthogonal designs, which is not the case for supersaturated designs.
Technical Details
Order of Effect Entry
The Fit Two Level Screening platform has a carefully defined order of operations.
• First, the main effect terms enter according to the absolute size of their contrast. All effects are orthogonalized to the effects preceding them in the model. The method assures that their order is the same as it would be in a forward stepwise regression. Ordering by main effects also helps in selecting preferred aliased terms later in the process.
• After main effects, all second-order interactions are brought in, followed by third-order interactions, and so on. The second-order interactions cross with all earlier terms before bringing in a new term. For example, with size-ordered main effects A, B, C, and D, B*C enters before A*D. If a factor has more than two levels, square and higher-order polynomial terms are also considered.
• An effect that is an exact alias for an effect already in the model shows in the alias column. Effects that are a linear combination of several previous effects are not displayed. If there is partial aliasing (a lack of orthogonality) the effects involved are marked with an asterisk.
• The process continues until n effects are obtained, where n is the number of rows in the data table, thus fully saturating the model. If complete saturation is not possible with the factors, JMP generates random orthogonalized effects to absorb the rest of the variation. They are labeled Null n where n is a number. For example, this situation occurs if there are exact replicate rows in the design.
Fit Two Level Screening as an Orthogonal Rotation
Mathematically, the Fit Two Level Screening platform takes the n values in the response vector and rotates them into n new values. The rotated values are then mapped by the space of the factors and their interactions.
Contrasts = T’ × Responses
where T is an orthonormalized set of values starting with the intercept, main effects of factors, two-way interactions, three-way interactions, and so on, until n values have been obtained. Since the first column of T is an intercept, and all the other columns are orthogonal to it, these other columns are all contrasts, that is, they sum to zero. Since T is orthogonal, it can serve as X in a linear model. It does not need inversion, since T’ is also T-1 and (T’T)T’. The contrasts are the parameters estimated in a linear model.
• If no effect in the model is active other than the intercept, the contrasts are just an orthogonal rotation of random independent variates into different random independent variates. These newly orthogonally rotated variates have the same variance as the original random independent variates. To the extent that some effects are active, the inactive effects still represent the same variation as the error in the model. The hope is that the effects and the design are strong enough to separate the active effects from the random error effects.
Lenth’s Pseudo-Standard Error
Lenth’s method (Lenth, 1989), known as the Lenth Pseudo Standard Error (PSE), constructs an estimate of the residual standard error using effects that appear to be inactive. Lenth’s PSE can be used to estimate the standard error for experiments where contrasts are independent and have a common variance.
If there are n rows, the platform constructs n - 1 contrasts. Denote these contrasts by 
, where i = 1, ... , n - 1.
, where i = 1, ... , n - 1.To obtain Lenth’s PSE, first calculate the following:
Lenth’s PSE is based on the effects that are likely to be inactive and is given by:
The value for Lenth’s PSE is shown at the bottom of the Screening report.
Lenth t-Ratios
For each contrast, a t-ratio is computed by dividing the contrast by the PSE. The reference distribution of these t-ratios under the null hypothesis is not computationally tractable, and so it is obtained by simulation. The method, described below, is based on a discussion in Ye and Hamada (2000).
Denote the t-ratio for the ith contrast by ti:
Of primary importance in screening experiments is the individual error rate, namely the probability of declaring a given effect as active when it is not. For the ith effect, this occurs when |ti| is large, falling in the upper tail of it’s reference distribution.
Because the platform constructs a relatively large number of effects, the experimentwise error rate is also of importance. The experimentwise error rate is the probability of declaring any effect as active when no effects are active. An experimentwise error occurs when no effects are active and the maximum of the absolute t-ratios, max|ti|, is large and falls in the upper tail of its reference distribution.
The Fit Two Level Screening platform obtains reference distributions for both types of error rates using Monte Carlo simulation. Consider a set of n - 1 values that is simulated from a normal distribution with mean 0 and standard deviation equal to PSE. This set of values represents potential contrast values for the experiment under the null hypothesis of no active effects. In all, 10,000 sets of contrast values are generated.
Individual p-Values
Because the contrast distributions are identical, all of the 10,000*(n - 1) values obtained in the simulation are generated from the distribution of values for any specific contrast under the null hypothesis that the contrast is not active.
Consider the ith contrast. Lenth t-ratios are constructed using each simulated value. The reference distribution for the individual error rate is approximated by the absolute values of these t-ratios. The p-value given in the Individual p-Value column of the report is the interpolated fractional position of the observed absolute Lenth t-Ratio among the 10,000*(n - 1) simulated absolute t-ratios arranged in descending order. This approximates the area to the right of the absolute value of the observed absolute Lenth t-Ratio with respect to the reference distribution.
Simultaneous p-Values
An experimentwise error occurs if any t-ratio leads to rejecting the null hypothesis when all effects are inactive. Equivalently, an experimentwise error occurs if the maximum of the absolute t-ratios, max|ti|, leads to rejecting the null hypothesis.
To obtain a reference distribution in this case, in each of the 10,000 simulations, the maximum of the absolute t-ratios is computed. These 10,000 maximum values form the reference distribution. The p-value given in the Simultaneous p-Value column of the report is the interpolated fractional position of the observed absolute Lenth t-Ratio among the 10,000 simulated maximum absolute t-ratios arranged in descending order. This approximates the area to the right of the absolute value of the absolute Lenth t-Ratio with respect to the reference distribution based on the simulated maximum absolute t-ratios.
Controlling the Monte Carlo Simulation
To change the number of default sets of simulations from 10,000, you must assign a value to a global JSL variable named LenthSimN. As an example, do the following:
1. Select Help > Sample Data Library and open Reactor Half Fraction.jmp.
2. Select Analyze > Specialized Modeling > Specialized DOE Models > Fit Two Level Screening.
3. Select Percent Reacted as the response variable, Y.
4. Select all the other continuous variables as effects, X.
5. Click OK.
6. Click the Screening for Percent Reacted red triangle and select Save Script > To Script Window.
7. At the top of the Script Window (above the code), type: LenthSimN=50000;
8. Highlight LenthSimN=50000; and the remaining code.
9. Right-click in the script window and select Run Script.
Note: If LenthSimN=0, the standard t-distribution is used and simultaneous p-values are not provided (not recommended).
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