I-optimal Designs
This paper was presented at the IASI conference in Orlando, FL, February 1998.

Using I-Optimal Designs for Narrower Confidence Limits
William D. Kappele
President, Math Options Inc.
336 36th Street # PMB 179
Bellingham WA 98225
1-888-764-3958

Introduction
Industrial experiments demand high quality predictions of product performance from a small amount of data. Scientists and engineers need experiment designs which provide narrow confidence limits on predictions to meet this demand. The effect of a design on the quality of predictions is seldom considered when selecting a design. This paper shows the advantages of I-optimal designs over conventional designs used in industry. Confidence intervals for predictions will show that I-optimal designs are better.

What are I-Optimal Designs?
Trials in designed experiments are "well spread out" from each other. Classic designs spread the points out in space. I-optimal designs spread the points out to provide the minimum average variance of predictions over the region of interest. The variance is the square of the standard deviation of the prediction from the fitted model.

I-optimal designs can be difficult to make. Dave Doehlert, Neil Sloane and Ron Hardin have developed a means of making I-optimal designs relatively quickly. Their designs are called "spherical code designs" or "Hardin-Sloane designs." Doehlert, Hardin and Sloane have made it feasible for industrial experimenters to use I-optimal designs in their work.

Design Comparison Criteria
In this paper I will compare I-optimal designs with three designs provided to us by experimenters in industry. Industrial experiments demand high quality predictions of product performance, so the criteria for comparison should be the quality of the predictions.

Predictions with narrower confidence limits are better quality. The experiment design affects the width of the confidence limits on predictions. A superior design will have a smaller average confidence limit width on its predictions.

I will compare a conventional design with a corresponding I-optimal design for three different cases: case 1 is a simple process factor design; case 2 is a mixture design; and case 3 is a process factor design with a constraint. For each case I have selected a random point. You will see the confidence limits for the prediction at this random point for each design. You will also see the average confidence limits for predictions at 1000 different points in the region of interest. To focus the comparisons only on the contribution of the design to the confidence limits, I have standardized t to 2.57 and s to 1.00. These confidence limits will allow you to compare the quality of the designs for industrial applications.

Another measure of the quality of a design is the "integrated variance" (IV). This is the average variance for the entire region of interest. Smaller IV indicates a better experiment design.

Case 1: A Simple Process Design with 4 Factors
The conventional design for case 1 is listed in table 1 below. It is a 4 factor design for the quadratic model with 2-factor interactions included. It has 25 runs: 20 trials, 5 of which have 2 replicates each.

Conventional Process Design
Table 1

The corresponding I-optimal design is listed in table 2 on the next page. Table 3 summarizes the comparison of the two designs.

I-optimal Process Design
Table 2

In table 3, notice that the confidence limits on the random point for the conventional design are 60% wider than for the I-optimal design. Also notice that the average confidence limit width for the conventional design is 54% wider than for the I-optimal design. The integrated variance for the I-optimal design is also better. The I-optimal design provides better precision of prediction in the same number of runs.

Summary
Table 3

Case 2: A Mixture Design in 4 Factors
Table 4 has a listing of the conventional mixture design for 4 factors. It has 15 runs: 13 trials, 2 of which have 2 replicates each. The limits on the factors are,

X1 = 0.740 - 0.890

X2 = 0.030 - 0.130

X3 = 0.055 - 0.160

X4 = 0.025 - 0.035.

The implied constraint for a mixture is X1 + X2 + X3 + X4 = 1.000. The model is a three factor (X2, X3, X4) quadratic model with all 2 factor interactions. X1 is implied by the constraint. The corresponding I-optimal design is listed in table 5. Table 6 shows a comparison of the two designs.

Conventional Mixture Design
Table 4

I-optimal Mixture Design
Table 5

Please notice, in table 6, that the confidence limits on the random point are 41% wider for the conventional design than for the I-optimal design. Also notice that the average confidence limits for the conventional design are 16% wider than for the I-optimal design. The I-optimal design also has a lower IV.

Summary
Table 6

Case 3: A Process Design with A Constraint and Restricted Levels
Case three has 4 continuous process factors. X4 is restricted to 2 levels. The factor limits are

• X1 = 2 - 4
• X2 = 0.0 - 3.0
• X3 = 1.0 - 3.4
• X4 = 4 or 6 (only 4 and 6 may be in the design, but the model must predict from 4 to 6).

The experimenter required the constraint, 10 < (2X1 + X4) <12. The experimenter also required these two trials to be in the design:

X1 = 2, X2 = 0, X3 = 3.4, X4 = 6, and X1 = 4, X2 = 0, X3 = 3.4, X4 = 4.

The model is,

Y = b0 + b1X1 + b2X2 + b3X3 +b4X4 + b12X1X2 + b13X1X3 + b23X2X3 + b11X12 + b22X22 + b33X32.

The conventional design is listed in table 7 and the corresponding I-optimal design is listed in table 8. Table 9 summarizes the comparison of the two designs.

Conventional Design with Constraint
Table 7

I-optimal Design with Constraint
Table 8

Notice in table 9 that the confidence limits on the random point prediction for the conventional design are 81% wider than for the I-optimal design. Also notice that the average confidence limits on the predictions for the region of interest are 13% wider than for the I-optimal design. The IV for the I-optimal design is also smaller.

It is interesting to note that the conventional design here is D-optimal. D-optimal designs provide narrow confidence limits on the b coefficients, rather than on predictions. Narrow confidence limits on predictions are more important in industrial experiments.

Summary
Table 9

Conclusion
This paper has shown I-optimal designs to be superior to conventional designs for industrial experiments using three real cases. I-optimal designs are superior because they give narrower confidence limits on predictions, on average, producing higher quality predictions of product performance. We supply I-optimal designs to all of our clients.