Factorial clinical trials test the effect of two or more treatments simultaneously using various combinations of the treatments. The simplest factorial design is known as a 2x2 factorial design, whereby participants are randomly allocated to one of four combinations of two interventions (A and B, say). These combinations are A alone, B alone, both A and B; neither A nor B (control). This design allows the investigators to compare the experimental interventions with the control, compare the experimental interventions with each other, and investigate possible interactions between them (that is, comparison of the sum of the effects of A and B given separately with the effects of the combination).
If there is no interaction between the interventions, the factorial design will have greater statistical power than a traditional multiple arm trial. This implies that two interventions can be studied in the same trial without increasing the required number of participants. The downside is that it is often very difficult to guarantee that no interaction took place and so results can be difficult to interpret. See further reading for more details.
Illustrative example  From the Lancet

Patients were randomised to one of 4 groups – intraveneous streptokinase, oral aspirin, both or neither – and mortality was measured. (Lancet 1988;ii:34960 ) 