Abstract
While generalized linear mixed models are useful, optimal design questions for such models are challenging due to complexity of the information matrices. For longitudinal data, after comparing three approximations for the information matrices, we propose an approximation based on the penalized quasi-likelihood method. We evaluate this approximation for logistic mixed models with time as the single predictor variable. Assuming that the experimenter controls at which time observations are to be made, the approximation is used to identify locally optimal designs based on the commonly used A- and D-optimality criteria. The method can also be used for models with random block effects. Locally optimal designs found by a Particle Swarm Optimization algorithm are presented and discussed. As an illustration, optimal designs are derived for a study on self-reported disability in older women. Finally, we also study the robustness of the locally optimal designs to mis-specification of the covariance matrix for the random effects.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abebe, H.T., Tan, F.E.S., Breukelen, G.J.P.V., et al.: Robustness of Bayesian D-optimal design for the logistic mixed model against misspecification of autocorrelation. Comput. Stat. 29(6), 1667–1690 (2014). https://doi.org/10.1007/s00180-014-0512-3
Atkinson, A.C., Woods, D.C.: Designs for generalized linear models. In: Dean, A., Morris, M., Stufken, J., et al. (eds.) Handbook of Design and Analysis of Experiments. Chapman and Hall/CRC, New York (2015)
Breslow, N.E., Clayton, D.G.: Approximate inference in generalized linear mixed models. J. Am. Stat. Assoc. 88(421), 9–25 (1993). https://doi.org/10.2307/2290687
Carrière, I., Bouyer, J.: Choosing marginal or random-effects models for longitudinal binary responses: application to self-reported disability among older persons. BMC Med. Res. Methodol. 2, 15 (2002). https://doi.org/10.1186/1471-2288-2-15
Green, P.J.: Penalized likelihood for general semi-parametric regression models. Int. Stat. Rev. 55, 245–259 (1987). https://doi.org/10.2307/1403404
Liang, K.Y., Zeger, S.L.: Longitudinal data analysis using generalized linear models. Biometrika 73(1), 13–22 (1986). https://doi.org/10.1093/biomet/73.1.13
Mak, S., Joseph, V.R.: Support points. Ann. Stat. 46, 2562–2592 (2018)
Mielke, T.: Approximations of the Fisher information for the construction of efficient experimental designs in nonlinear mixed effects models. PhD thesis, Faculty of Mathematics, Otto-von-Guericke University, Magdeburg (2012)
Miller, K.S.: On the inverse of the sum of matrices. Math. Mag. 54(2), 67–72 (1981). https://doi.org/10.2307/2690437
Molenberghs, G., Verbeke, G.: Models for Discrete Longitudinal Data. Springer, New York (2005)
Niaparast, M.: On optimal design for a Poisson regression model with random intercept. Stat. Probab. Lett. 79(6), 741–747 (2009). https://doi.org/10.1016/j.spl.2008.10.035
Niaparast, M., Schwabe, R.: Optimal design for quasi-likelihood estimation in Poisson regression with random coefficients. J. Stat. Plann. Inference 143(2), 296–306 (2013). https://doi.org/10.1016/j.jspi.2012.07.009
Niaparast, M., MehrMansour, S., Schwabe, R.: V-optimality of designs in random effects Poisson regression models. Metrika (2023). https://doi.org/10.1007/s00184-023-00896-3
Provost, F., Fawcett, T.: Analysis and visualization of classifier performance: comparison under imprecise class and cost distributions. Proceedings of the Third International Conference on Knowledge Discovery and Data Mining 43-48 (1999)
Schmelter, T.: The optimality of single-group designs for certain mixed models. Metrika 65(2), 183–193 (2007). https://doi.org/10.1007/s00184-006-0068-5
Sinha, S.K., Xu, X.: Sequential D-optimal designs for generalized linear mixed models. J. Stat. Plann. Inference 141(4), 1394–1402 (2011). https://doi.org/10.1016/j.jspi.2010.10.008
Tekle, F.B., Tan, F.E., Berger, M.P.: Maximin d-optimal designs for binary longitudinal responses. Comput. Stat. Data Anal. 52(12), 5253–5262 (2008). https://doi.org/10.1016/j.csda.2008.04.037
Ueckert, S., Mentré, F.: A new method for evaluation of the fisher information matrix for discrete mixed effect models using Monte Carlo sampling and adaptive Gaussian quadrature. Comput. Stat. Data Anal. 111, 203–219 (2017). https://doi.org/10.1016/j.csda.2016.10.011
Waite, T.W., Woods, D.C.: Designs for generalized linear models with random block effects via information matrix approximations. Biometrika 102(3), 677–693 (2015). https://doi.org/10.1093/biomet/asv005
Yang, M., Stufken, J.: Support points of locally optimal designs for nonlinear models with two parameters. Ann. Stat. 37(1), 518–541 (2009). https://doi.org/10.1214/07-AOS560
Zeger, S.L., Liang, K.Y., Albert, P.S.: Models for longitudinal data: a generalized estimating equation approach. Biometrics 44(4), 1049–1060 (1988). https://doi.org/10.2307/2531734
Zhou, X., Wang, Y., Yue, R.: Optimal designs for discrete-time survival models with random effects. Lifetime Data Anal. 27, 300–332 (2021). https://doi.org/10.1007/s10985-020-09512-2
Funding
The work by JS was partially funded by NSF grants DMS-1935729 and DMS-2304767.
Author information
Authors and Affiliations
Contributions
All authors contributed equally to the study that results in this manuscript, and have read and approved this final version.
Corresponding author
Ethics declarations
Conflict of interest
The authors have no competing interests to declare that are relevant to the content of this article.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary Information
Below is the link to the electronic supplementary material.
Online Resource
The file Supplementary Material contains optimal designs for (n=2) and (n=5) when the slope parameter is (\beta_1=-2) and (beta_1=1) rather than (beta_1=-1) as used in Section~4 of the paper. (13,820 KB)
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Shi, Y., Yu, W. & Stufken, J. Optimal designs for generalized linear mixed models based on the penalized quasi-likelihood method. Stat Comput 33, 114 (2023). https://doi.org/10.1007/s11222-023-10279-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11222-023-10279-3