Small screening design when the overall variance is unknown

Consider the problem of a design for estimating all main effects and the overall error variance. The conventional choice is a saturated orthogonal array supplemented by several center points (as popularized in Response Surface Methodology). We propose an alternative design — the projection of a larger Hadamard matrix. In this article, we prove that the proposed design is optimal for estimating error variance, and thus is preferable over the conventional choice. Under various common error distributions, theoretical values of Var(σˆ²) are evaluated to illustrate our theory. Simulation results are provided to demonstrate that the proposed design achieves a more reliable estimate of error variance, as well as a reliable significant test. Furthermore, we study the optimal follow-up plan. Relevant optimality theories are established and a convenient construction method for optimal follow-up design is proposed.