Addition or deletion?

Abstract

Suppose it is desired to have an optimal' resolution III fraction of a 2p factorial in N runs where N ≡ 2 (mod 4). A design for this purpose can be obtained by adding two runs optimally to the n × p matrix derived by a suitable choice of p columns of Hn, a Hadamard matrix of order n. Alternatively, one can think of deleting two runs in an optimal manner from the (n + 4) × p matrix derived from Hn+4. A natural question then arises: do these two strategies give designs that are equally efficient in terms of a well defined optimality criterion? We show that for p = 2 or 3, the design obtained by deletion is as good as the addition design under the A- or the D-optimality criterion. However, for p ⩾ 4, the performance of the deletion design compared to the optimal addition design is rather poor as per the D-criterion, especially for large values of p. Under the A-criterion, the addition design is always better than the deletion design for p ⩾ 4, but the loss of efficiency using the deletion design is not too large for moderate values of p.