Statistica Sinica 3(1993), 197-207

ESTIMABILITY AND EFFICIENCY IN NEARLY

ORTHOGONAL DELETION DESIGNS

Joan M. Mahoney and John E. Angus

University of California and Claremont Graduate School

Abstract: This article considers single replicate factorial experimental designs in incomplete blocks. A single replicate deletion design in 3 incomplete blocks is obtained from a single replicate 3^m(m=m₁+m₂) preliminary design by deleting all runs (or treatment combinations) with the first m₁ factors at level two. A systematic method for determining the unbiasedly estimable (u.e.) and not-unbiasedly estimable (n.u.e.) factorial effects is provided. Specifically, it is shown that, for m₂>0 all factorial effects of the form , where α=0,1 for i=1,...m₁, α_i=0,1,2, for i=m₁+1,...,m, with (α₁...α_m)≠(0...0), and for α=1,2, are u.e. and the remaining effects are n.u.e. The result that identifies u.e. factorial effects is derived as a special case of a theorem developed herein for general deletion designs obtained from a single replicate p_m preliminary design. It is noted that factorial effects of factorial experiments, and factorial effects of factorial experiments, which are embedded in experiments, are u.e. The 2 ×3^m-1 deletion designs were considered in the work by Voss (1986). By defining the single-degree-of-freedom components F(α₁...α_m) of the factorial effects of a factorial experiment in a form different from that of Voss (1986), our simple representation of u.e. and n.u.e. effects identifies more u.e. effects than is done in the representation by Voss (1986). The relative efficiency expressions, and their bounds, in the estimation of factorial effects of deletion designs are also given, along with methods for adjusting n.u.e. effects to be u.e. when certain higher order effects are assumed negligible.

Key words and phrases: Confounding, factorial experiment, single replicate, unbiasedly estimable.