Using I-Optimal Designs for Narrower Confidence Limits
A Common Data Interchange Format for Experiment Designers
and Analyzers
Short Courses
Conference Notes
Jobs
-----------------------------
First, we would like to apologize to Dr. Selden B. Crary for leaving the byline off the
excellent article that he wrote last month comparing the efficiencies of classical designs
and I-optimal designs. Those that enjoyed this article may also enjoy a letter that
we got from Dr. Greg Piepel in response to Selden’s article.
http://www.processbuilder.com/doe/Newsletters/BackIssues/October_1999.htm
http://www.processbuilder.com/doe/Newsletters/Letters/piepel2.htm
Next, we would like to thank Bill Kappele for his excellent article below. Note that
this has a lot of tables and special formatting. If they do not show up in the
version you receive, read the article on-line at our web site. Note the announcement
at the end of the article on the free
teleclass on I-optimal designs .
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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.
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.
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.
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.
|
Trial |
X1 |
X2 |
X3 |
X4 |
|
20 |
-1 |
0 |
-1 |
1 |
|
3 |
1 |
-1 |
1 |
-1 |
|
4 |
-1 |
1 |
1 |
-1 |
|
8 |
-1 |
1 |
1 |
1 |
|
9 |
1 |
1 |
1 |
0 |
|
1 |
-1 |
-1 |
-1 |
-1 |
|
14 |
1 |
0 |
0 |
1 |
|
1 |
-1 |
-1 |
-1 |
-1 |
|
2 |
1 |
1 |
-1 |
-1 |
|
11 |
-1 |
1 |
-1 |
0 |
|
2 |
1 |
1 |
-1 |
-1 |
|
15 |
0 |
1 |
0 |
1 |
|
4 |
-1 |
1 |
1 |
-1 |
|
10 |
1 |
-1 |
-1 |
0 |
|
12 |
-1 |
-1 |
1 |
0 |
|
19 |
-1 |
0 |
1 |
-1 |
|
3 |
1 |
-1 |
1 |
-1 |
|
5 |
-1 |
-1 |
-1 |
1 |
|
7 |
1 |
-1 |
1 |
1 |
|
13 |
0 |
0 |
1 |
1 |
|
17 |
0 |
1 |
-1 |
-1 |
|
18 |
1 |
-1 |
0 |
-1 |
|
6 |
1 |
1 |
-1 |
1 |
|
16 |
0 |
-1 |
-1 |
1 |
|
5 |
-1 |
-1 |
-1 |
1 |
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.
|
Trial |
X1 |
X2 |
X3 |
X4 |
|
1 |
-1.00 |
-0.39 |
1.00 |
-0.45 |
|
2 |
-1.00 |
-0.39 |
1.00 |
-0.45 |
|
3 |
-0.01 |
-1.00 |
0.01 |
-0.10 |
|
4 |
-0.01 |
-1.00 |
0.01 |
-0.10 |
|
5 |
1.00 |
0.30 |
0.04 |
0.88 |
|
6 |
1.00 |
0.30 |
0.04 |
0.88 |
|
7 |
1.00 |
-1.00 |
-1.00 |
0.37 |
|
8 |
1.00 |
-1.00 |
-1.00 |
0.37 |
|
9 |
-0.06 |
0.12 |
-0.08 |
0.00 |
|
10 |
-0.06 |
0.12 |
-0.08 |
0.00 |
|
11 |
0.04 |
-0.01 |
1.00 |
0.43 |
|
12 |
0.00 |
-0.01 |
-1.00 |
-1.00 |
|
13 |
1.00 |
1.00 |
1.00 |
-0.03 |
|
14 |
-1.00 |
-1.00 |
0.19 |
1.00 |
|
15 |
1.00 |
-0.15 |
-0.05 |
-1.00 |
|
16 |
-1.00 |
1.00 |
-1.00 |
0.00 |
|
17 |
-1.00 |
-1.00 |
-1.00 |
-1.00 |
|
18 |
1.00 |
-1.00 |
1.00 |
-1.00 |
|
19 |
1.00 |
-1.00 |
1.00 |
1.00 |
|
20 |
-1.00 |
-0.25 |
-1.00 |
1.00 |
|
21 |
-1.00 |
1.00 |
0.04 |
-1.00 |
|
22 |
-0.03 |
1.00 |
1.00 |
-1.00 |
|
23 |
-1.00 |
1.00 |
1.00 |
1.00 |
|
24 |
1.00 |
1.00 |
1.00 |
-0.92 |
|
25 |
0.34 |
1.00 |
-1.00 |
1.00 |
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.
|
|
Conventional |
I-Optimal |
|
95% CL for random point |
+/- 2.96 |
+/- 1.86 |
|
Average 95% CL |
+/- 2.23 |
+/- 1.45 |
|
Integrated Variance (IV) |
0.77 |
0.32 |
Summary
Table 3
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.
|
Trial |
X1 |
X2 |
X3 |
X4 |
|
9 |
0.835 |
0.030 |
0.100 |
0.035 |
|
1 |
0.740 |
0.075 |
0.160 |
0.025 |
|
6 |
0.740 |
0.065 |
0.160 |
0.035 |
|
11 |
0.780 |
0.130 |
0.055 |
0.035 |
|
12 |
0.830 |
0.080 |
0.055 |
0.035 |
|
10 |
0.780 |
0.030 |
0.160 |
0.030 |
|
5 |
0.740 |
0.130 |
0.095 |
0.035 |
|
8 |
0.790 |
0.080 |
0.100 |
0.030 |
|
2 |
0.790 |
0.130 |
0.055 |
0.025 |
|
4 |
0.785 |
0.030 |
0.160 |
0.025 |
|
2 |
0.790 |
0.130 |