zbMATH — the first resource for mathematics

Integral approximations for computing optimum designs in random effects logistic regression models. (English) Zbl 06975458
Summary: In the context of nonlinear models, the analytical expression of the Fisher information matrix is essential to compute optimum designs. The Fisher information matrix of the random effects logistic regression model is proved to be equivalent to the information matrix of the linearized model, which depends on some integrals. Some algebraic approximations for these integrals are proposed, which are consistent with numerical integral approximations but much faster to be evaluated. Therefore, these algebraic integral approximations are very useful from a computational point of view. Locally \(D\)-, \(A\)-, \(c\)-optimum designs and the optimum design to estimate a percentile are computed for the univariate logistic regression model with Gaussian random effects. Since locally optimum designs depend on a chosen nominal value for the parameter vector, a Bayesian \(D\)-optimum design is also computed. In order to find Bayesian optimum designs it is essential to apply the proposed integral approximations, because the use of numerical approximations makes the computation of these optimum designs very slow.

62 Statistics
Full Text: DOI
[1] Abdelbasit, K. M.; Plackett, R. L., Experimental design for binary data, Journal of the American Statistical Association, 78, 90-98, (1983) · Zbl 0501.62071
[2] Biedermann, S.; Dette, H.; Pepelyshev, A., Some robust design strategies for percentile estimation in binary response models, The Canadian Journal of Statistics, 34, 603-622, (2006) · Zbl 1115.62069
[3] Debusho, L. K.; Haines, L. M., \(D\)- and \(V\)-optimal population designs for the quadratic regression model with a random intercept term, Journal of Statistical Planning and Inference, 41, 889-898, (2011) · Zbl 1353.62082
[4] Demidenko, E., Mixed models: theory and applications, (2004), John Wiley & Sons Hoboken, New Jersey · Zbl 1055.62086
[5] Elfving, G., Optimum allocation in linear regression theory, Annals of Mathematical Statistics, 23, 255-262, (1952) · Zbl 0047.13403
[6] Fedorov, V. V.; Hackl, P., Model-oriented design of experiments, (1997), Springer-Verlag New York · Zbl 0878.62052
[7] Ford, I.; Torsney, B.; Wu, C. F.J., The use of a canonical form in the construction of locally optimal designs for nonlinear problems, Journal of the Royal Statistical Society. Series B, 54, 569-583, (1992) · Zbl 0774.62080
[8] Graßhoff, U.; Holling, H.; Schwabe, R., On optimal design for a heteroscedastic model arising from random coefficients, (Ermakov, S. M.; Melas, V. B.; Pepelyshev, A. N., Proceedings of the 6th St. Petersburg Workshop on Simulation, (2009), VVM Com. Ltd. St. Petersburg), 387-392
[9] Holland-Letz, T.; Dette, H.; Pepelyshev, A., A geometric characterization of optimal designs for regression models with correlated observations, Journal of the Royal Statistical Society. Series B, 73, 2, 239-252, (2011)
[10] López-Fidalgo, J.; Rodríguez-Díaz, J. M., Elfving method for computing \(c\)-optimal designs in more than two dimensions, Metrika, 59, 235-244, (2004) · Zbl 1147.62358
[11] Mentré, F.; Mallet, A.; Baccar, D., Optimal design in random-effects regression models, Biometrika, 84, 429-442, (1997) · Zbl 0882.62069
[12] Minkin, S., Optimal designs for binary data, Journal of the American Statistical Association, 82, 1098-1103, (1987)
[13] Ouwens, M. J.N. M.; Tan, F. E.S.; Berger, M. P.F., A maximin criterion for the logistic random intercept model with covariates, Journal of Statistical Planning and Inference, 136, 962-981, (2006) · Zbl 1077.62061
[14] Patan, M.; Bogacka, B., Efficient sampling windows for parameter estimation in mixed effects models, (Lopez-Fidalgo, L.; Rodriguez-Diaz, J. M.; Torsney, B., MODa 8-Advances in Model-Oriented Design and Analysis, (2007), Physica Heidelberg), 147-155
[15] Schmelter, T.; Benda, N.; Schwabe, R., Some curiosities in optimal designs for random slopes, (Lopez-Fidalgo, J.; Rodriguez-Diaz, J. M.; Torsney, B., MODa 8-Advances in Model-Oriented Design and Analysis, (2007), Physica Heidelberg), 189-195
[16] Silvey, S. D., Optimal design, (2004), Chapman and Hall London, New York · Zbl 0391.62054
[17] Sitter, R. R.; Fainaru, I., Optimal designs for the logit and probit models for binary data, Canadian Journal of Statistics, 25, 175-190, (1997) · Zbl 0891.62053
[18] Sitter, R. R.; Wu, C. F.J., Optimal designs for binary response experiments: fieller-, \(D\)- and \(A\)-criteria, Scandinavian Journal of Statistics, 20, 329-341, (1993) · Zbl 0790.62080
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.