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

Abstract

Candidate locally D-optimal designs for the binary two-variable logistic model with no interaction, which comprise 3 and 4 support points lying in the first quadrant of the two-dimensional Euclidean space, were introduced by Haines et al. (D-optimal designs for logistic regression in two variables. In: Lopez-Fidalgo J, Rodrigez-Diaz JM, Torsney B, editors. MODA8 – advances in model-oriented designs and analysis. Heidelberg: Physica-Verlag; 2007. p. 91–98). The authors proved algebraically the global D-optimality of the 3-point design for the special case in which the intercept parameter is equal to−1.5434. However for other selected values of the intercept parameter, the global D-optimality of the proposed 3- and 4-point designs was only demonstrated numerically. In this paper, we provide analytical proofs of the D-optimality of these 3- and 4-point designs for all negative and zero intercept parameters of the binary two-variable logistic model with no interaction. The results are extended to the construction of D-optimal designs on a rectangular design space and illustrated by means of two examples of which one is a real example taken from the literature.

Keywords:

• trapezium design
• D-optimality
• binary two-variable logistic model
• no interaction
• stand-ardized variance function

Acknowledgements

The authors are very grateful to the editor and two anonymous referees for their valuable suggestions. Any opinion, finding and conclusion or recommendation expressed in this material is that of the authors and the NRF does not accept liability in this regard.

Additional information

Funding

The authors would like to thank the Medical Research Council of South Africa, the University of KwaZulu-Natal, the University of Cape Town, the University of South Africa, the National Research Foundation (NRF) of South Africa [Grant (UID) 85456], and the University of Johannesburg for financial support.