SAS programs for constructing of optimal, efficient, and robust crossover designs for comparing test treatments to a control treatment
*Supported by NSF Grant DMS-0304661
The two programs are for searching efficient/robust crossover design when comparing several test treatments with a control. For given parameter t, p, and n, the constructed design, in which rows correspond to periods and columns correspond to subjects, will be given. The given design is not only efficient under carryover model, but also under self and mixed carryover effects model, two-way elimination model (without carryover effects), one-way elimination model (without carryover effects and period effects), zero-way elimination model (treatment effects only), etc.
The corresponding efficiency, A-efficiency, is given. Notice that the true efficiencies are bigger or equal to the given efficiencies. The two programs are corresponding Algorithm I and II. When n is big, say more than 2t, the program for Algorithm I is recommended since it is fast and efficiency is high enough. When n is small (no more than 2t), the program for Algorithm II is recommended since the efficiency is high compared with the first program. But the second program usually takes much longer time although we can use for larger n.
There are two efficiencies under traditional first order carryover effects model. The first one is based on the lower bound of the subclass, in which no treatment is allowed immediately followed by itself and the control treatment appears equivalent often in each period. The second one is based on the nearly entire class, in which the control treatment appears equivalent often in each period. (Treatment can be immediately followed by itself). The result design may not belong to the subclass since the control treatment may not appear equivalent often in each period.
There are also two efficiencies under self and mixed carryover effects model and the model without period effects, which are similar as these of the traditional first order carryover effects model.
The efficiencies under all other models are for entire class.
Notice this program is designed for practical parameters. There is some restriction on t and p. p ranges from 3 to 5. t ranges from p-1 to 9. n can be any positive number. But it should be at least t or t+1 depends on the situation. If you need to construct a design which is not covered by the program, you are welcome to contact the authors for assistance.
%efficient_design(p= , t= , n= , method= ) or
%efficient_design_detail(p= , t= , n= , method= )
p is the number of periods. t is the number of test treatments. n is the number of experimental subjects. The results of the two commands are the same. In addition to the design and efficiencies which the first command can give you, the second command can give you detail information about the design, such as the inverse matrix of information matrix, which is proportional to the variance-covariance matrix.
You can choose 5 different Methods: 0, 1, 2, 3, 4. The default is method 0; Method 0 is the fastest one, it based on the sequences which have been already arranged by author. Method 1 and 2 are similar. The difference is that: In method 1, selection without replacement; in method 2, selection with replacement. Method 3 and 4 are similar. The difference is that: In method 3, selection without replacement; in method 4, selection with replacement. Method 3 and 4 can make sure the control treatment appear approximately same frequency in each period. But method 1 and 2 cannot.
The result design is contained in the dataset DESIGN_Pp_Tt_Nn_method. For example dataset DESIGN_P3_T5_N89_2 means the constructed design for 3 period, 5 test treatments, and 89 subjects by Method 2. In the constructed design, 0 refers to the control treatment; 1 to t refers to the t test treatments.
%efficient_design_small_n(p=, t=, n=, w_trad=, w_mix=, w_two=, precision=) or
%efficient_design_small_n_detail(p=,t=, n=, w_trad=, w_mix=, w_two=, precision=);
Compared with first program, we do not have the parameter method. But we have w_trad, w_mix, w_two, and precision. They are the corresponding weight for traditional model, mixed carryover effects model, two-way elimination model, and the pre-set cut value for the efficiency improvement.
Although the second program can be run for any n, it is recommended for small n. Since for a larger n, it may take a long time to run the program and the improvement of the efficiency is insignificant compared with that of first program.