Abstract

The main purpose of this paper is to investigate D-optimal population designs in multi-response linear mixed models for longitudinal data. Observations of each response variable within subjects are assumed to have a first-order autoregressive structure, possibly with observation error. The equivalence theorems are provided to characterise the D-optimal population designs for the estimation of fixed effects in the model. The semi-Bayesian D-optimal design which is robust against the serial correlation coefficient is also considered. Simulation studies show that the correlation between multi-response variables has tiny effects on the optimal design, while the experimental costs are important factors in the optimal designs.

References

1. Atkinson, A. C., Donev, A. N., & Tobias, R. D. (2007). Optimum experimental designs, with SAS. Oxford University Press. [Google Scholar]
2. Azurduy, S. A. O. (2009). Robust designs for longitudinal studies. University of Maastricht. [Google Scholar]
3. Berger, M. P. F., Ouwens, M. J. N. M., & Tan, F. E. S. (2003). Robust designs for longitudinal mixed effects models. In H. Yanai, A. Okada, K. Shigemasu, Y. Kano, & J. J. Meulman (Eds.), New developments in psychometrics. Springer. [Crossref], [Google Scholar]
4. Castañeda, L. M. E., & López-Rios, V. I. (2016). Optimal population designs for discrimination between two nested nonlinear mixed effects models. Revista Ciencia En Desarrollo, 7(1), 71–81. https://doi.org/10.19053/01217488.4233 [Crossref], [Google Scholar]
5. Chi, E. M., & Reinsel, G. C. (1989). Models for longitudinal data with random effects and AR(1) errors. Journal of the American Statistical Association, 84(406), 452–459. https://doi.org/10.1080/01621459.1989.10478790 [Taylor & Francis Online], [Web of Science ®], [Google Scholar]
6. Diggle, P. J. (1988). An approach to the analysis of repeated measurements. Biometrics, 44(4), 959–971. https://doi.org/10.2307/2531727 [Crossref], [Web of Science ®], [Google Scholar]
7. Fedorov, V. V., Gagnon, R. C., & Leonov, S. L. (2002). Design of experiments with unknown parameters in variance. Applied Stochastic Models in Business and Industry, 18(3), 207–218. https://doi.org/10.1002/(ISSN)1526-4025 [Crossref], [Web of Science ®], [Google Scholar]
8. Gagnon, R., & Leonov, S. (2005). Optimal population designs for PK models with serial sampling. Journal of Biopharmaceutical Statistics, 15(1), 143–163. https://doi.org/10.1081/BIP-200040853 [Taylor & Francis Online], [Web of Science ®], [Google Scholar]
9. Liang, K. Y., & Zeger, S. L. (1986). Longitudinal data analysis using generalized linear models. Biometrika, 78(1), 13–22. https://doi.org/10.1093/biomet/73.1.13 [Crossref], [Google Scholar]
10. Liu, X., Yue, R. X., & Wong, W. K. (2019). D-optimal designs for multi-response linear mixed models. Metrika, 82, 87–98. https://doi.org/10.1007/s00184-018-0679-7 [Crossref], [Web of Science ®], [Google Scholar]
11. Molenberghs, G., & Verbeke, G. (2005). Models for discrete longitudinal data. Springer Science+Business Media, Inc. [Google Scholar]
12. Nyberg, J., Höglund, R., Bergstrand, M., Karlsson, M. O., & Hooker, A. C. (2012). Serial correlation in optimal design for nonlinear mixed effects models. Journal of Pharmacokinetics and Pharmacodynamics, 39(3), 239–249. https://doi.org/10.1007/s10928-012-9245-5 [Crossref], [Web of Science ®], [Google Scholar]
13. Ogungbenro, K., Gueorguieva, I., Majid, O., Graham, G., & Aarons, L. (2007). Optimal design for multiresponse pharmacokinetic-pharmacodynamic models – dealing with unbalanced designs. Journal of Pharmacokinetics and Pharmacodynamic, 34(3), 313–331. https://doi.org/10.1007/s10928-006-9048-7 [Crossref], [Web of Science ®], [Google Scholar]
14. Roy, A. (2006). Estimating correlation coefficient between two variables with repeated observations using mixed effects model. Biometrical Journal, 48(2), 286–301. https://doi.org/10.1002/(ISSN)1521-4036 [Crossref], [Web of Science ®], [Google Scholar]
15. Schmelter, T. (2007a). The optimality of single-group designs for certain mixed models. Metrika, 65, 183–193. https://doi.org/10.1007/s00184-006-0068-5 [Crossref], [Web of Science ®], [Google Scholar]
16. Schmelter, T. (2007b). Considerations on group-wise identical designs for linear mixed models. Journal of Statistical Planning and Inference, 137(12), 4003–4010. https://doi.org/10.1016/j.jspi.2007.04.017 [Crossref], [Web of Science ®], [Google Scholar]
17. Tan, F. E. S., & Berger, M. P. F. (1999). Optimal allocation of time points for the random effects model. Communications in Statistics-Simulation and Computation, 28(2), 517–540. https://doi.org/10.1080/03610919908813563 [Taylor & Francis Online], [Web of Science ®], [Google Scholar]
18. Tekle, F. B., Tan, F. E. S., & Berger, M. P. F. (2008). D-optimal cohort designs for linear mixed-effects model. Statistics in Medicine, 27(14), 2586–2600. https://doi.org/10.1002/sim.v27:14 [Crossref], [Web of Science ®], [Google Scholar]
19. Verbeke, G., Fieuws, S., Molenberghs, G., & Davidian, M. (2014). The analysis of multivariate longitudinal data: A review. Statistical Methods in Medical Research, 23(1), 42–59. https://doi.org/10.1177/0962280212445834 [Crossref], [Web of Science ®], [Google Scholar]
20. Verbeke, G., & Molenberghs, G. (2000). Linear mixed models for longitudinal data, Springer Series in Statistics. Springer. [Google Scholar]
21. Wang, W. L., & Fan, T. H. (2010). ECM-based maximum likelihood inference for multivariate linear mixed models with autoregressive errors. Computational Statistics and Data Analysis, 54(5), 1328–1341. https://doi.org/10.1016/j.csda.2009.11.021 [Crossref], [Web of Science ®], [Google Scholar]