Design and statistical analysis of some confounded factorial experiments

The factorial design is used for the study of the effects of two or more factors simultaneously. It has distinct advantages over a series of simple experiments, each designed to test a single factor. As an example, let us consider the case in which we wish to study experimentally the effect of three different kinds of fertilizers, A, Band C, on the yield of a certain crop. There are altogether eight fertilizer combinations: A0B0C0, A0B0C1, A0B1C0, A0B1C1, A1B0C0, A1B0C1, A1B1C0 and A1B1C1, where A0 means the absence and A1 means the presence of the fertilizer A, and so forth. Suppose that 3 trials are made with each fertilizer combination; then for any particular fertilizer, we have for comparison 12 observations in which the fertilizer is applied and 12 in which it is not applied. So a factorial experiment can be regarded as a number of simple experiments superimposed on each other. Every observation supplies information on each of the factors studied. One factorial experiment serves the purpose of a number of simple experiments of the same size. But a series of simple experiments will throw no light whatever on the interaction of the different factors, and this will be shown in a factorial experiment. If, for example, the presence of A were advantageous in the presence of B, but were ineffective in its absence, we could hope to learn this fact only by carrying out our test of A both in the presence and in the absence of B