The Construction of Locally D-Optimal Designs by Canonical Forms to an Extension for the Logistic Model

Irene García Camacha Gutiérrez, Raúl Martín Martín

Mathematics Department, Castilla-La Mancha University, Toledo, Spain.

DOI: 10.4236/am.2014.55078                

Abstract

Logistic regression models for binary response problems are present in a wide variety of industrial, biological, social and medical experiments; therefore, optimum designs are a valuable tool for experimenters, leading to estimators of parameters with minimum variance. Our interest in this contribution is to provide explicit formulae for the D-optimal designs as a function of the unknown parameters for the logistic model where q is an indicator variable. We have considered an experiment based on the dose-response to a fly insecticide in which males and females respond in different ways, proposed in Atkinson et al. (1995) [1]. To find the D-optimal designs, this problem has been reduced to a canonical form.

Keywords

Binomial Response; D-Optimal Design; Canonical Form

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Atkinson, A.C., Demetrio, C.G.B. and Zochhi, S.S. (1995) Optimun Dose Levels When Males and Females Differ en Response. Applied Statistics, 44, 213-226.
[2]	McCullagh, P. and Nelder, J.A. (1992) Generalized Linear Models. Chapman and Hall, London.
[3]	Silvey, S.D. (1980) Optimal Design. Chapman and Hall, London. http://dx.doi.org/10.1007/978-94-009-5912-5
[4]	Ford, I., Tosney, B. and Wu, C.F.J. (1992) The Use of a Canonical form in the Construction of Locally Optimal Designs for Non-Linear Problems. Journal of the Royal Statistical Society: Series B, 54, 569-583.
[5]	Fedorov, V.V. (1972) Theory of Optimal Experiments. Academic Press, New York.
[6]	Ardanuy, R., Lopez-Fidalgo, J., Laycock, P.J. and Wong, W.K. (1999) When Is an Equally-Weihted Design D-Optimal? Annals of the Institute of Statistical Mathematics, 51, 531-540. http://dx.doi.org/10.1023/A:1003954207112
[7]	Chernoff, H. (1953) Locally Optimal Designs for Estimating Parameters. The Annals of Mathematical Statistics, 24, 586-602. http://dx.doi.org/10.1214/aoms/1177728915
[8]	Kiefer, J. and Wolfowitz, J. (1960) The Equivalence of Two Extremun Problems. Canadian Journal of Mathematics, 12, 363-366.
[9]	Chaloner, K. and Larntz, K. (1989) Optimal Bayesian Design Applied to Logistic Regression Experiments. Journal of Statistical Planning and Inference, 21, 191-208. http://dx.doi.org/10.1016/0378-3758(89)90004-9
[10]	Sibson, R. (1972) Discussion on Results in the Theory and Construction of D-Optimun Experimental Designs (by H. P. Wynn). Journal of the Royal Statistical Society, 34, 174-175.
[11]	Silvey, S.D. and Titterington, D.M. (1973) A Geometrical Approach to Optimal Design Theory. Biometrika, 60, 21-32. http://dx.doi.org/10.1093/biomet/60.1.21
[12]	Torsney, B. and Musrati, A.K. (1993) On the Construction of Optimal Designs with Applications to Binary Response and to Weighted Regression Models. Model Oriented Data Analysis, Physica-Verlag, Heidelberg.