Sequences of Two-Level Fractional Factorial Plans

when investigating the effects of a number of factors in an industrial experiment is to partition the treatment combinations into blocks, confounding the higher order interaction effects with the block effects, and to evaluate each block in succession as soon as it is completed. This same suggestion was reiterated in the text edited by Davies (1954). Daniel (1957, 1962) and John (1965) have also discussed the use of sequences of some two-level fractional factorial plans. The main reason for running a sequence of blocks and evaluating the sequence after each block of treatment combinations has been run is to discover large effects quickly and hence terminate the experiment as soon as conclusive results are obtained. The additional blocks are added to the sequence in order to obtain estimates of additional parameters as well as new estimates of previously estimable parameters and to decrease the variance of these estimates. When a full factorial plan is partitioned into blocks, each block is a fractional replicate of the full factorial plan. Thus, a sequence of the blocks that make up a full factorial plan can be considered to be a sequence of fractional replicate plans, the treatment combinations of each being distinct. A complete sequence of fractional replicate plans that contain distinct treatment combinations can be identified by a single family of defining contrasts. The particular family to which a defining contrast belongs determines the alias or confounding pattern in the first block of the sequence. The order in which the individual members of the family of defining contrasts appear in the sequence determines the manner in which the aliased effects become individually estimable. To illustrate the notion of a single family of defining contrasts, consider the