Details of the OPTEX Procedure 
Example 24.1: Nonstandard Linear Model
See OPTEX3 in the SAS/QC Sample Library

The following example is based on an example in
Mitchell (1974a). An animal scientist wants to compare wildlife
densities in four different
habitats over a year. However, due to the cost of experimentation,
only 12 observations can be made. The following model is postulated for
the density y_{j}(t) in habitat j during month t:
This model includes the habitat as a classification variable, the
effect of time with an overall linear drift term , and cyclic
behavior in the form of a Fourier series. There is no intercept term
in the model.
The OPTEX procedure is used since there are no standard designs
that cover this situation. The candidate set is the full
factorial arrangement of four habitats by 12 months, which can be
generated with a DATA step, as follows:
data a;
drop theta pi;
array c{4} c1c4;
array s{3} s1s3;
pi = arcos(1);
do habitat=1 to 4;
do month=1 to 12;
theta = pi * month / 4;
do i=1 to 4; c{i} = cos(i*theta); end;
do i=1 to 3; s{i} = sin(i*theta); end;
output;
end;
end;
run;
Data set A contains the 48 candidate points and includes the cosine
variables (C1, C2, C3, and C4) and sine variables (S1, S2, S3, S4).
The following statements produce Output 24.1.1:
proc optex seed=193030034 data=a;
class habitat;
model habitat month c1c4 s1s3 / noint;
generate n=12;
run;
Output 24.1.1: Sampling Wildlife Habitats Over Time
Design Number 
DEfficiency 
AEfficiency 
GEfficiency 
Average Prediction Standard Error 
1 
31.6103 
19.7379 
57.7350 
1.3229 
2 
31.6103 
19.7379 
57.7350 
1.3229 
3 
31.6103 
19.3793 
57.7350 
1.3229 
4 
31.6103 
19.2916 
57.7350 
1.3229 
5 
31.6103 
19.2626 
57.7350 
1.3229 
6 
31.6103 
19.0335 
57.7350 
1.3229 
7 
30.1304 
14.8837 
44.7214 
1.4907 
8 
30.1304 
14.2433 
44.7214 
1.5092 
9 
30.1304 
13.1687 
44.7214 
1.5456 
10 
28.1616 
9.8842 
40.8248 
1.7559 

The best determinant (Defficiency)
was found in 6 out of the 10 tries. Thus, you can be confident that this
is the best achievable determinant. Only the Aefficiency distinguishes
among the designs
listed in Output 24.1.1. The best design has an Aefficiency of 19.74%,
whereas another design has the same Defficiency but a slightly smaller
Aefficiency of 19.03%, or about 96% relative Aefficiency. To explore
the differences, you can save the designs in data sets and print them.
Since the OPTEX procedure is interactive, you
need to submit only the following statements (immediately after the
preceding statements) to produce Output 24.1.2 and Output 24.1.3:
output out=d1 number=1;
run;
output out=d6 number=6;
run;
proc sort data=d1;
by month habitat;
proc print data=d1;
var month habitat;
run;
proc sort data=d6;
by month habitat;
proc print data=d6;
var month habitat;
run;
Output 24.1.2: The Best Design
Obs 
month 
habitat 
1 
1 
3 
2 
2 
2 
3 
3 
4 
4 
4 
1 
5 
5 
4 
6 
6 
1 
7 
7 
2 
8 
8 
3 
9 
9 
4 
10 
10 
1 
11 
11 
2 
12 
12 
3 

Output 24.1.3: Design with Lower AEfficiency
Obs 
month 
habitat 
1 
1 
4 
2 
2 
2 
3 
3 
3 
4 
4 
1 
5 
5 
1 
6 
6 
4 
7 
7 
4 
8 
8 
1 
9 
9 
2 
10 
10 
1 
11 
11 
4 
12 
12 
3 

Note the structure of the best design in Output 24.1.2. One habitat
is sampled in each month, each habitat is sampled three times, and the
habitats are sampled in consecutive complete blocks. Even though the
design in Output 24.1.3 is as
Defficient as the best, it has almost
none of this structure; one habitat is sampled each month, but habitats
are not sampled an equal number of times. This demonstrates the
importance of choosing a final design on the basis of more than one
criterion.
You can try searching for the Aoptimal design directly.
This takes more time but (with only 48 candidate points) is not too large a
problem. The following statements produce Output 24.1.4:
proc optex seed=193030034 data=a;
class habitat;
model habitat month c1c4 s1s3 / noint;
generate n=12 criterion=A;
run;
Output 24.1.4: Searching Directly for an Aefficient Design
Design Number 
DEfficiency 
AEfficiency 
GEfficiency 
Average Prediction Standard Error 
1 
31.6103 
19.7379 
57.7350 
1.3229 
2 
30.1304 
17.8273 
52.2233 
1.3894 
3 
30.1304 
17.7943 
52.2233 
1.3944 
4 
30.1304 
17.6471 
52.2233 
1.4093 
5 
28.1616 
15.7055 
44.7214 
1.4860 
6 
28.1616 
14.5289 
44.7214 
1.5343 
7 
28.1616 
13.8603 
39.2232 
1.5811 
8 
25.0891 
11.6152 
37.7964 
1.8143 
9 
25.0891 
10.7563 
37.7964 
1.8143 
10 
25.0891 
10.5437 
33.3333 
1.8930 

The best design found is no more Aefficient than the one found
previously.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.