Optimal Designs for Binary Data

Abstract

Abstract This article is concerned with the problem of selecting the values of the explanatory variable in logistic regression to obtain likelihood-based confidence regions of minimum area. One way of finding analytic solutions is by replacing the log-likelihood by a quadratic surface around the maximum, approximating in this way the likelihood regions by ellipses. The optimal allocation for this local approximation is derived without unnecessary restrictions. Since the implementation of the optimal allocation requires accurate initial estimates, a two-stage procedure that uses the second stage to complement the first is recommended. An alternative approach is to deal directly with the likelihood regions. In this case, the identification of the globally optimal allocation calls for numerical integration and optimization. The results presented here attempt to strengthen those introduced by Abdelbasit and Plackett (1983), who derived optimal allocation for the local criterion under the restriction of symmetry and suggested a two-stage procedure. Here it is shown that although for even sample size the optimal allocation is in fact symmetric, for odd sample size the restriction of symmetry leads to suboptimal designs (one important case is only 93% efficient). For the two-stage procedure, their proposal does not attempt to compensate in the second stage. This may lead to a serious loss of efficiency (below 80% in extreme cases). The numerical results corresponding to the global criterion (i.e., using the likelihood regions) indicate that the globally best allocation corresponds to less extreme probabilities than those prescribed by the local criterion.