In a saturated design, the number of runs equals the number of model terms. In a supersaturated design, the number of model terms exceeds the number of runs (Lin, 1993). A supersaturated design can examine dozens of factors using fewer than half as many runs as factors. This makes it an attractive choice for factor screening when there are many factors and experimental runs are expensive.
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If the number of active factors is more than half the number of runs in the experiment, then it is likely that these factors will be impossible to identify. A general rule is that the number of runs should be at least four times larger than the number of active factors. In other words, if you expect that there might be as many as five active factors, you should plan on at least 20 runs.
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Analysis of supersaturated designs cannot yet be reduced to an automatic procedure. However, using forward stepwise regression is reasonable. In addition, the Screening platform (Analyze > Specialized Modeling > Specialized DOE Models > Fit Two Level Screening) offers a streamlined analysis.
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Note: This example is for illustration only. You should have at least 14 runs in any supersaturated design. If there are as many as four active factors, it is very difficult to interpret the results of an 8-run design. See Limitations of Supersaturated Designs.
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1.
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Select DOE > Custom Design.
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2.
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Type 12 next to Add N Factors.
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3.
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Click Add Factor > Continuous.
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4.
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Click Continue.
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6.
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Setting the effects to If Possible ensures that JMP uses the Bayesian D-optimality criterion to obtain the design.
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7.
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8.
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Select Simulate Responses from the red triangle menu.
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Note: Setting the Random Seed in step 9 and Number of Starts in step 10 reproduces the exact results shown in this example. In constructing a design on your own, these steps are not necessary.
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9.
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(Optional) From the Custom Design red triangle menu, select Set Random Seed, type 1008705125, and click OK.
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10.
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(Optional) From the Custom Design red triangle menu, select Number of Starts, type 100, and click OK.
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11.
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Click Make Design.
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12.
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Click Make Table.
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The design table (Design Table with Simulated Responses) and the Simulate Responses window (Simulate Responses Window) appear.
The response column, Y, contains simulated values. These are randomly generated using the model defined by the parameter values in the Simulate Responses window.
The Simulate Responses window shows coefficients of 0 for all terms, with an Error Std of 1. The values in the Y column currently reflect only random variation. Notice that the model coefficients are set to 0 because not all coefficients are estimable.
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13.
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14.
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Click Apply.
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Note: If you did not set the random seed and the number of starts, or if you click Apply more than once, your response values will not match those in Response Column with X1 and X11 Active.
In your simulation, you specified X1 and X11 as active factors with large effects relative to the error variation. For this reason, your analysis of the data should identify these two factors as active.
The Screening platform provides a way to identify active factors. The design table in Response Column with X1 and X11 Active contains three scripts. The Screening script analyzes your data using the Screening platform (located under the Analyze > Specialized Modeling > Specialized DOE Models > Fit Two Level Screening menu).
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The factors X1 and X11 have large contrast and Lenth t-Ratio values. Also, their Simultaneous p-Values are small. In the Half Normal Plot, both X1 and X11 fall far from the line. The Contrasts and the Half Normal Plot reports indicate that X1 and X11 are active. Although X12 has an Individual p-Value less than 0.05, its effect is much smaller than that of X1 and X11.
Because the design is supersaturated, p-values might be smaller than they would be in a model where all effects are estimable. This is because effect estimates are biased by other potentially active main effects. In Screening Report for Supersaturated Design, a note directly above the Make Model button warns you of this possibility.
You might also want to check whether the effects that appear active could be highly correlated with other effects. When this occurs, one effect can mask the true significance of another effect. The Color Map in Color Map on Correlations Outline displays absolute correlations between effects.
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2.
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Click Make Model.
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3.
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Click Run in the Model Specification window.
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Note that the parameter estimates for X11 and X1 are close to the theoretical values that you used to simulate the model. See Parameter Values for Simulated Responses, where you specified a model with X1 = 10 and X11 = 10. The significance of the factor X12 is an example of a false positive.
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4.
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In your Custom Design window, open the Design Evaluation > Color Map on Correlations outline.
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With your cursor, place your mouse pointer over cells to see the absolute correlations. Notice that X1 has correlations as high as 0.5 with other main effects (X4, X5, X7). (Color Map on Correlations Outline uses JMP default colors.)
Stepwise regression is another way to identify active factors. The design table in Response Column with X1 and X11 Active contains three scripts. The Model script analyzes your data using stepwise regression in the Fit Model platform.
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3.
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Click Run.
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4.
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5.
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Click Go.
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Stepwise Regression for Supersaturated Design shows that the selected model consists of the two active factors, X1 and X11. The step history appears in the bottom part of the report. Keep in mind that correlations between X1 and X11 and other factors could mask the effects of other active factors. See Color Map on Correlations Outline.