2 X 2 Factorial experiments in incomplete groups for use in biological assays.

Among the simpler experimental designs one of the most useful is the 2 x 2 factorial for testing two levels of two factors in all combinations. In developing an insecticidal treatment, for example, two concentrations of an oil emulsion may be applied with two different stickers. In a biological assay one drug may be compared with another at each of two comparable dosage levels. The two-dose design has been used widely in assays based upon a graded response and provides adequate estimates of relative potency when the form of the dosage-response curve is reasonably well known. The two-dose assay owes some of its popularity to the greater ease of assembling homogeneous groups of four than of six or eight responses, which would be required for three or four dose assays. Even groups of four may be inconveniently large. When they consist of four successive reactions of the same animal separated by rest periods, the experiment is time-consuming and some individuals may die before it can be completed. When the groups are litters of the same sex the number with four males or four females may be limited. For these and other reasons groups of three or two may be preferred. Some experimenters [5, 6, 7] have used pairs and confounded one of the treatment effects with differences between pairs. This increases the error of the confounded effect. Frequently one would rather keep the same precision in all treatment comparisons and yet reduce the size of the group. The experimental design known as balanced incomplete blocks or groups meets this need. As originally described by Yates [9], only the information within groups was utilized. The method was adapted to two-dose assays arranged in pairs by Bliss and Rose [2]. However, groups with different treatment combinations also provide estimates of the effect of treatment. More recently Yates [10] has described statistical methods for recovering inter-block information and has applied them to determining the adjusted mean response for each treatment and its error. In the 2 x 2 factorial design interest centers in three specific comparisons. Given this more limited objective, the calculation can be reduced to a convenient form. Two restrictions in design. Frequently two restrictions are re-