Suppose that d experimental treatments, each at a number of levels, are applied to a group of subjects in such a way that each treatment combination is assigned to one, and only one, subject. This is the makeup of a factorial experiment. If the assignment of treatment combinations to subjects is random, standard techniques have long been available for examining the effects of differing levels within a treatment. In this thesis we seek a technique for examining the effects of differing combinations of levels between treatments;For the simplest case, that of two treatments (or factors), we describe a number of techniques; some based on mathematical models, and some based on the rank of a matrix. We extend these techniques to the case where not all levels of one of the treatments are used, but only a random sample of levels. We also supply a procedure for choosing the appropriate technique;When more than one response variable is measured on each subject, the problem becomes a bit more complicated, since these variables may well be correlated. However, we are still able to provide a partial solution to the above problem;The case of three treatments, which, when solved, should suggest a general technique, is also examined here. Although by no means a complete solution, the techniques described present a solid basis for further study.