In response surface experiments, the prediction variance over the range of the factors is more important than the variance of the coefficients. The prediction variance over the design space is addressed by Ioptimality. An Ioptimal design tends to place fewer runs at the extremes of the design space than does a Doptimal design. For details about D and Ioptimality, see Optimality Criteria in Custom Designs.
By default, Custom Design uses the Recommended option for the Optimality Criterion. Custom Design uses the Ioptimality criterion as the Recommended criterion whenever you add quadratic effects using the RSM button in the Model outline. Otherwise, Custom Design uses the Doptimality criterion as the Recommended criterion. See Optimality Criteria in Custom Designs.
Construct a response surface design for three continuous factors that you have identified as active. You want to find process settings to maintain your response(Y) within specifications. The lower and upper specification limits for Y are 54 and 56, respectively, with a target of 55.
1.

Select DOE > Custom Design.

2.

3.

4.

Leave Importance blank.

5.

Type 3 next to Add N Factors.

6.

Click Add Factor > Continuous.

7.

Click Continue.

8.

In the Model outline, click the RSM button.

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.
9.

(Optional) From the Custom Design red triangle menu, select Set Random Seed, type 929281409, and click OK.

10.

(Optional) From the Custom Design red triangle menu, select Number of Starts, type 40, and click OK.

11.

Click Make Design.

In order to estimate quadratic effects, a response surface design uses three levels for each factor. Note that the design in RSM Design is a facecentered Central Composite Design with two center points.
12.

Open the Design Evaluation > Design Diagnostics outline.

13.

Open the Design Evaluation > Prediction Variance Profile outline.

The vertical axis shows the relative prediction variance of the expected value of the response. The relative prediction variance is the prediction variance divided by the error variance. When the relative prediction variance is one, its absolute variance equals the error variance of the regression model.
14.

Select Maximize Desirability from the red triangle menu next to Prediction Variance Profile outline.

15.

Open the Design Evaluation > Fraction of Design Space Plot outline.

The Custom RSM.jmp sample data table contains the results of the experiment. The Model script opens a Fit Model window showing all of the effects specified in the DOE window’s Model outline. This script was saved to the data table by the Custom Design platform.
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2.

3.

Click Run.

The report shows that X1, X2, X1*X1, and X2*X2 are significant at the 0.01 level. None of the other effects are significant at even the 0.10 level. Reduce the model by removing these insignificant effects.
4.

5.

Click Remove.

The Fit Least Squares report is updated to show a model containing only the significant effects: X1, X2, X1*X1, and X2*X2.
Use the Prediction Profiler (at the bottom of the Fit Least Squares window) to explore how the predicted response (Y) changes as you vary the factors X1 and X2. Note the quadratic behavior of Y across the values of X1 and X2.
Remember that you entered response limits for Y in the Responses outline of the Custom Design window. As a result, the Response Limit column property is attached to the Y column in the design table. The Desirability function for Y (in the top plot at right) is based on the information contained in the Response Limit column property. JMP uses this function to calculate Desirability as a function of the settings of X1 and X2. The traces of the Desirability function appear in the bottom row of plots.
6.

In the Prediction Profiler report, select Maximize Desirability from the red triangle options.

The predicted response achieves the target value of 55 at the process settings shown in red above X1 and X2. Prediction Profiler with Desirability Maximized shows that a value of X1 near 0.65 also achieves a predicted value of 55 when X2 = 0.75062. In fact, your Prediction Profiler might show different settings as those that maximize desirability. This is because the predicted response is 55 for many settings of X1 and X2.
7.

Select Factor Profiling > Contour Profiler from the red triangle next to Response Y.

The settings of X1 and X2 that correspond to the red contour have predicted response values of 55. You might want to select from among these process settings based on cost efficiency.
1.

Select DOE > Custom Design.

2.

Type 2 next to Add N Factors.

3.

Click Add Factor > Continuous.

4.

5.

Click Continue.

The Default number of runs is 12. The Factors outline updates to show three levels for the Blocking factor, X3. Because you required X3 to have four runs per block, the 12 runs allow three blocks.
6.

Click RSM.

7.

Click OK to dismiss the message.

Quadratic and interactions terms for X1 and X2 are added to the model. Because you added RSM terms, the Recommended optimality criterion changes from DOptimal to IOptimal. You can see this later in the Design Diagnostics outline.
Note: Setting the Random Seed in step 8 and Number of Starts in step 9 reproduces the exact results shown in this example. In constructing a design on your own, these steps are not necessary.
8.

(Optional) From the Custom Design red triangle menu, select Set Random Seed, type 1415408414, and click OK.

9.

(Optional) From the Custom Design red triangle menu, select Number of Starts, type 21, and click OK.

10.

Click Make Design.

11.

Open the Design Evaluation > Design Diagnostics outline.

12.

Click Make Table.

Because the default Run Order was Randomize within Blocks, the levels of the blocking factor (X3) are sorted.
13.

In the Tables panel of the design table, click the red triangle next to Model and select Run Script.

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The blocking factor (X3) is entered as an effect.

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No interactions involving X3 are included.

1.

Select DOE > Custom Design.

2.

Type 2 next to Add N Factors.

3.

Click Add Factor > Continuous.

4.

Click Continue.

5.

Click RSM.

Quadratic and interactions terms for X1 and X2 are added to the model. Because you added RSM terms, the Recommended optimality criterion changes from DOptimal to IOptimal. You can see this later in the Design Diagnostics outline.
Note: Setting the Random Seed in step 6 and Number of Starts in step 7 reproduces the exact results shown in this example. In constructing a design on your own, these steps are not necessary.
6.

(Optional) From the Custom Design red triangle menu, select Set Random Seed, type 383570403, and click OK.

7.

8.

Click Make Design.

In this Ioptimal design, runs 1, 4, 7, and 10 are at the center point (X1 = 0 and X2 = 0). Ioptimal designs tend to place more runs in the center (and consequently fewer runs at the extremes) of the design space than do Doptimal designs. You can compare this design to the Doptimal design shown in DOptimal Design.
9.

10.

Open the Fraction of Design Space Plot outline.

The Fraction of Design Space Plot appears on the left in Fraction of Design Space Plots (IOptimal on left, DOptimal on right). When the Fraction of Space is 0.95, the vertical coordinate of the blue curve is about 0.5. This means that for about 95% of the design space, the relative prediction variance is below 50% of the error variance.
1.

In the Custom Design window containing your Ioptimal design, from the Custom Design red triangle menu, select Save Script to Script Window.

2.

In this new script window, select Edit > Run Script.

4.

From the red triangle next to Custom Design, select Optimality Criterion > Make DOptimal Design.

5.

Click Make Design.

In this Doptimal design, run 7 is the only run at the center point. Doptimal designs tend to place more runs at the extremes of the design space than do Ioptimal designs. Recall that the Ioptimal design had four center runs (IOptimal Design).
6.

At the center of the design region, the relative prediction variance is 0.53562, as compared to 0.208333 for the Ioptimal design (Prediction Variance Profile for IOptimal Model). This means that the relative standard error is 0.732 for the Doptimal design and 0.456 for the Ioptimal design. All else being equal, at the center of the design region, confidence intervals for the expected response based on the Doptimal design are about 60% wider than those based on the Ioptimal design.
The Design outline shows that the Doptimal design has nine design points, one for every combination of X1 and X2 set to 1, 0, 1. The Doptimality criterion attempts to keep the relative prediction variance low at each of these design points. Explore the variance at the extremes of the design region by moving the sliders for X1 and X2 to 1 and 1. Note that the variance at these extreme points is usually smaller than the variance for the Ioptimal design at these points.
7.

The Fraction of Design Space Plot appears on the right in Fraction of Design Space Plots (IOptimal on left, DOptimal on right).
8.

Rightclick in the Fraction of Design Space Plot for your Ioptimal design. Select Edit > Copy Frame Contents.

9.

Rightclick in the Fraction of Design Space Plot for your Doptimal design. Select Edit > Paste Frame Contents.
