D-optimal Designs for Logistic Regression in Two Variables

The analytic construction ofD-optimal designs for the two-variable binary logistic regression model without interaction

“…The aim of the present study has been to prove, analytically, that the candidate D-optimal designs for the two-variable binary logistic model, u = logit(p) = β 0 + β 1 d 1 + β 2 d 2 , on the design region (d 1 , d 2 ) ∈ [0, ∞) × [0, ∞) with parameters chosen such that β 0 ≤ 0, β 1 > 0 and β 2 > 0, proposed by Haines et al, [15] are indeed globally optimal. Specifically, it has been demonstrated that, for β 0 < −1.5434, the D-optimal designs comprise 4 support points, with pairs of equally weighted points at the intersection of logit lines with the d 1 -and d 2 -axes and that, for −1.5434 ≤ β 0 ≤ 0, the designs are based on 3 support points, necessarily equally weighted, one at the origin and two at the intersection of the same logit line with the d 1 -and d 2 -axes.…”

“…Haines et al [15] conjectured that such a design comprises 3 support points, equally weighted, as shown in Figure 1(b). Specifically, the point A lies at the origin and the points C and D lie at the intersection of the same logit line with the z 2 and z 1 axes, respectively.…”

“…Haines et al [15] introduced the candidate 4-point D-optimal design for the model specified by (1) with design space D = {(z 1 , z 2 ) : 0 ≤ z 1 < ∞, 0 ≤ z 1 < ∞} as the design taking the form shown in Figure 1(a), with equal weights placed on the support points A and B and, separately, on C and D.…”

“…The aim of the present study is to complete the work of Haines et al [15] by providing algebraic proofs of the global optimality of the 3-and 4-points candidate D-optimal designs for the two-variable binary logistic regression model with the design space the positive quadrant and then to apply the results to the case of a bounded rectangular design space. The paper is organized as follows.…”

“…Atkinson et al [14] presented further numerical results which reinforce those of Atkinson and Haines. [11] As a counterpoint to the numerical studies, Haines et al [15] provided an analytic construction of candidate D-optimal designs with 3 and 4 support points for the two-variable binary logistic model without interaction and with the design space restricted to the positive quadrant. However, they only proved global optimality in the special case of a 3-point design with the intercept parameter given by the unique negative root of the function 1 + β 0 + e β 0 − β 0 e β 0 , that is −1.5434.…”