The OPTEX Procedure

## Example 10.1 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 in habitat during month :

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} c1-c4;
array s{3} s1-s3;
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 four cosine variables (c1, c2, c3, and c4) and three sine variables (s1, s2, and s3). The following statements produce Output 10.1.1:

```proc optex seed=193030034 data=a;
class    Habitat;
model    Habitat Month c1-c4 s1-s3 / noint;
generate n=12;
run;
```

Output 10.1.1 Sampling Wildlife Habitats over Time
The OPTEX Procedure

Design Number D-Efficiency A-Efficiency G-Efficiency 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 19.7379 57.7350 1.3229
4 31.6103 19.3793 57.7350 1.3229
5 31.6103 19.2916 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 (D-efficiency) was found in 6 out of the 10 tries. Thus, you can be confident that this is the best achievable determinant. Only the A-efficiency distinguishes among the designs listed in Output 10.1.1. The best design has an A-efficiency of 19.74%, whereas another design has the same D-efficiency but a slightly smaller A-efficiency of 19.03%, or about 96% relative A-efficiency. 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 10.1.2 and Output 10.1.3:

```   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 10.1.2 The Best Design
The OPTEX Procedure

Class Level Information
Class Levels Values
Habitat 4 1 2 3 4

Factor Ranges
Factor Low Value High Value
Month 1.000000 12.000000
c1 -1.000000 1.000000
c2 -1.000000 1.000000
c3 -1.000000 1.000000
c4 -1.000000 1.000000
s1 -1.000000 1.000000
s2 -1.000000 1.000000
s3 -1.000000 1.000000

The OPTEX Procedure

Design Number D-Efficiency A-Efficiency G-Efficiency 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 19.7379 57.7350 1.3229
4 31.6103 19.3793 57.7350 1.3229
5 31.6103 19.2916 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

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 10.1.3 Design with Lower A-Efficiency
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 10.1.2. 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 10.1.3 is as D-efficient 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 A-optimal design directly. This takes more time but (with only 48 candidate points) is not too large a problem. The following statements produce Output 10.1.4:

```proc optex seed=193030034 data=a;
class    Habitat;
model    Habitat Month c1-c4 s1-s3 / noint;
generate n=12 criterion=A;
run;
```

Output 10.1.4 Searching Directly for an A-Efficient Design
The OPTEX Procedure

Design Number D-Efficiency A-Efficiency G-Efficiency 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 A-efficient than the one found previously.

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