Fractional Randomized Block Design

A fractional randomized block design (also called a randomized block fractional factorial design) reduces the number of treatment combinations that must be included in a multitreatment experiment to some fraction (1_2, ⅓, ¼, ⅛, 1/9, and so on) of the total number of treatment combinations. Consider an experiment with five treatments, denoted by the letters A, B, C, D, and E. If each treatment has two levels, the number of treatment combinations in the experiment is 2 × 2 × 2 × 2 × 2 = 32. By using a ½ or a ¼ fractional randomized block design, the number of treatment combinations can be reduced to 16 or 8, respectively. However, the reduction in the size of the experiment comes at a price: Considerable ambiguity may exist in interpreting the results of the experiment. Ambiguity occurs because in the case of a ½ fractional design, two names, called aliases, can be given to each source of variation. For example, a sum of squares could be attributed to the effects of treatment A and the BCDE interaction. In a one-fourth fractional randomized block design, each source of variation has four aliases. Treatments are customarily aliased with higher-order interactions that are assumed to equal zero. This helps minimize but does not eliminate ambiguity in interpreting the outcome of an experiment. One can never be sure that the higher-order interaction is really equal to zero. Because the interpretation of fractional randomized block designs always involves some ambiguity, the designs are most useful for pilot experiments and for exploratory research situations that permit follow-up experiments to be performed. Thus, a large[Page 376] number of treatments, typically seven or more, can be investigated efficiently in an initial experiment, with subsequent smaller experiments designed to clarify the results or follow up on the most promising independent variables.

A ½ fractional randomized block design in which each treatment has two levels is denoted by 2k–1, where 2k indicates that each of the k treatments has 2 levels. The –1 in 2k–1 indicates that the design is a one-half fraction of a complete 2k factorial design. This follows because the designation for a one-half fraction of 2k can be written as ½2k = 2–12k = 2k–1. A one-fourth fractional randomized block design is denoted by 2k–2 because ¼2k = ½22k = 2–22k = 2k–2.