Factorial designs are used efficiently and effectively to carry out experiments involving multiple independent variables needed to explain complex human behavior. These designs are not feasible, however, when a study requires a large and, in some cases, prohibitive number of measurements on the same subjects (within-subjects designs) or on groups of different subjects (between-subjects designs). For example, a 2 factorial design, not unusual in some behavioral research, produces 1,024 treatment combinations. If the experimenter can assume, on theoretical and empirical grounds, that higher order interactions are negligible, only a fraction of the complete factorial design is needed to estimate main effects and lower order interactions. For example, in a 2 factorial experiment, if the highest order interaction is considered to be negligible, the number of observations or groups can be reduced in half, from 1,024 to 512. If another interaction can be sacrificed, the number of observations can be halved again, to 256 (1,4 fraction), and so on. Moreover, "in cases where ambiguity is present because of confounding, the initial experiment may be supplemented by follow-up experiments specificallydesigned to clarify such ambiguities" (Winer, 1971). A factorial design with n factors, each with two levels, contains 2 treatment combinations in its full replication. However, in a V2 P fraction of a 2 design (where p= I in a V2 fraction, p=2 in a 1,4 fraction, and so on), a total of 2 p treatment combinations are actually tested with 2 p-1 degrees of freedom among them. In such fractional designs, 2 -lout of the total of 2 -I effects and interactions are confounded with the grand mean. Thus, effects will be mutually confounded in groups of 2 (Kempthome, 1952, p. 405). That is, in a half replicate, each effect is confounded with one other effect: the confounding is in sets of two; in a quarter replicate, each effect is confounded with three others: the confounding is in sets of four; and so on. The effects that are confounded with each other are called aliases, because the same source of variation has more than one name due to confounding of sources of variation. "The problem of deciding to which alias an effect is to be attributed is one that the experimenter must face