A note on the D-optimality and D-efficiency of nonorthogonal blocked main effects plans

This paper considers main effects plans used to study m two-level factors using n runs which are partitioned into b blocks of equal size k = n/b. The assumptions are adopted that n ≡ 2 (mod 8) and k > 2 is even. Certain designs not having all main effects orthogonal to blocks were shown by Jacroux (2011a) to be D-optimal when (m − 2)(k − 2) + 2 ⩽ n ⩽ (m − 1)(k − 2) + 2. Here, we extend that result. For (m − 3)(k − 2) + 2 ⩽ n < (m − 2)(k − 2) + 2, the D-optimality of those designs is proved. Moreover, their D-efficiency is shown to be close to one for 2(m + 1) ⩽ n < (m − 3)(k − 2) + 2, indicating their good performance under the criterion of D-optimality.