sk−p Fractional factorial designs in sb blocks

Draper and Guttman (1997) shows that for basic 2^k−p designs, p≥0, k − p replicates of blocks designs of size two are needed to estimate all the usual (estimable) effects. In this work, we provide an algebraic formal proof for the two-level blocks designs results and present results applicable to the general case; that is, for the case of s ^k factorial (p=0) or s ^k−p fractional factorial (p >0) designs in s ^b blocks, where 0<b<k− p, at least replicates are needed to clear up all possible effects. Through the theoretical development presented in this work, it can provide a clearer view on why those results would hold. We will also discuss the estimation equations given in Draper and Guttman (1997).  Research supported in part by the National Science Council of Taiwan, R.O.C., Grant No. NSC 89-2118-M110-010. Acknowledgement. The authors would like to thank the referee for very helpful comments.