# Factorial Designs

Completely Randomized Factorial Design

Factorial designs differ from those described previously in that two or more treatments can be evaluated simultaneously in an experiment. The simplest factorial design from the standpoint of randomization, data analysis, and model assumptions is based on a completely randomized design and, hence, is called a completely randomized factorial design. A two-treatment completely randomized factorial design is denoted by the letters CRF-pq, where p and q denote the number of levels, respectively, of treatments A and B.

In the weight-loss experiment, a researcher might be interested in knowing also whether walking on a treadmill for 20 minutes a day would contribute to losing weight, as well as whether the difference between the effects of walking or not walking on the treadmill would be the same for each of the three diets. To answer these additional questions, a researcher can use a two-treatment completely randomized factorial design. Let treatment A consist of the three diets (a:, a2, and a3) and treatment B consist of no exercise on the treadmill (b1) and exercise for 20 minutes a day on the treadmill (b2). This design is a CRF-32 design, where 3 is the number of levels of treatment A and 2 is the number of levels of treatment B. The layout for the design is obtained by combining the treatment levels of a CR-3 design with those of a CR-2 design so that each treatment level of the CR-3 design appears once with each level of the CR-2 design and vice versa. The resulting design has 3 x 2 = 6 treatment combinations as follows: a1b1, axb2, a2bv a2b2, a3bv a3b2. When treatment levels are combined in this way, the treatments are said to be crossed. The use of crossed treatments is a characteristic of all factorial designs. The layout of the design with 30 girls randomly assigned to the six treatment combinations is shown in Figure 1.8.

The classical model equation for the weight-loss experiment is

Yijk = ^ + aj + Pk + (ap)jk + e; ( jk) (i = 1,...,n; j = 1,..., p; k = 1, where ijk

(ap)jk is the weight loss for participant i in treatment combination ajbk.

is the grand mean of the six weight-loss population means.

is the treatment effect for population aj and is equal to Vj. — It reflects the effect of diet a;-. is the treatment effect for population bk and is equal to vk — It reflects the effects of exercise condition bk.

is the interaction effect for populations aj and bk and is equal to Vjk — Vj. — — Interaction effects are discussed later.

Groupj Group2 Group3 Group4 Group 5 Group6 