Optimal Design And Robust Design Of Linear Mixed Effect Model

Mixed models attract an increasing popularity in many fields of applications like, e.g., pop-ulation pharmacokinetics, agriculture, psychology, market research, medical diagnostics where repeated measurements are available for each observed individual and both individual curves and reference curves describing the whole population are of interest (Mentre et al.(1997), Schmelter (2007)). The quality of the analysis of these models highly depends on the experimental de-signs, i.e., on the experimental settings (like time points of measurements, the dosage in biologi-cal studies), which are in many of these applications under the control of the investigator. Unlike fixed-effects models, many problems will be encountered in designing for mixed-effects models. First, optimal designs for mixed-effects models may depend on unknown parameters: the variance components; Second, the observations for the same individual are correlated; Third, designs for mixed-effects models have two layouts: the within-subject layout and the between-subject lay-out. Possibly having these difficulties, many design problems for mixed-models remain unsolved. Among the existing work on experimental designs for mixed models most of them had put atten-tion to model fitting, their designs purposes were effectively estimating the fixed effects and the random effects were looked to capturing the variabilities of the individuals. In practice, however, prediction may be the ultimate aims (Das, Jiang and Rao (2004), Taylor et al.(1994)). But few researchers have taken this as the design aims. In this paper we will consider optimal designs for prediction besides estimating the population parameters.In optimal experimental design studies we often need to take into account some situations which the experimenter should watch out for in constructing design criteria. These situations include:(1) misspecified parameter values which the constructed design depends upon;(2) mis-specification in a regression function;(3) uncertainty of the fitted models etc. Designs constructed to safeguard against these situations are called robust designs in the literature. In this paper we also consider design techniques for constructing robust designs for mixed-effects models. We arrange our paper as follows:In Chapter2, an introduction to the well-known topic of optimal experimental designs for the ordinary linear model is given. In Chapter3, the linear mixed models that are considered within this thesis are introduced.In Chapter4, we exploit the de la Garaz phenomenon (DLG phenomenon) and Loewner order domination (LOD) in reducing the dimension of the problem of determining optimal re-gression designs for estimation the fixed effect, the random effects and prediction the future ob- servation of an individual, respectively, in balanced linear mixed-effects models. Assuming the variance parameters to be known, we tabulate optimal designs for various combinations of the variance parameters.In Chapter5, we derive the general equivalence theorems in the construction of optimum population designs and the balanced optimal designs for estimation population effects, respec-tively. We also obtain the sufficient and necessary condition for prediction a future observation. Robust design strategies are proposed to handle the dependence of design criteria on the unknown variance parameters. Deriving optimal designs for simultaneously estimating fixed effects and variance parameters in a special case of our considered model is also considered.In Chapter6, we consider deriving the optimal designs for predicting the linear combinations of the random effects and the fixed effects in random coefficients models. Under the assumption of the variance parameters known(unknown), we give the design criteria. An equivalence theorem is provided for validating the optimality of a given design under the variance known. Specifically, we consider the design problems for predicting individual curve and the future observation of an individual. An interesting result is obtained for the random intercept model.In Chapter7, we consider optimal design problems for predicting the future observations at the points outside of the design space in longitudinal models encountered in modeling of CD4counts. Two design problems are investigated: one is to predict the population mean and the other is to predict the future individual observation. We give two design strategies, Maximin and Bayesian, to obtain robust designs when the parameters in the design criteria are uncertain for a given model, the design methods to derive robust designs for the situation that the design models and the parameters in criteria are both unknown and for the situation that the predicted points are uncertain, respectively. A simulated annealing algorithm is given to compute the optimal designs.In Chapter8, we investigate the problem of finding robust designs for linear random intercept regression model with possible misspecification in the response and possible correlated errors on discrete design spaces. Two robust design strategies are proposed to handle the possible bias in the response and correlated errors. One is the Minimax method. The other is the Bayesian method. The maximum value or average value of the mean squared error is obtained analytically, and the final robust population designs can be computed through an annealing algorithm. Several examples are given to show robust designs for polynomial regression. A real-life example is provided to demonstrate the usefulness of our robust design strategies.In Chapter9, we investigate the problem of designing for a mixed effects regression model with a random intercept, when the assumed model form is only an approximation to an unknown true model. Model-robust design criterion, Bayesian criterion and T-optimal design criterion are derived to overcome the drawback of the model uncertainty. In the spirit of Goos et al.[2005. Model-robust and model-sensitive designs. Computational Statistics&Data Analysis49,201-216.], new compound criteria are proposed in order to achieve model estimation, discrimination and the ability of protecting possible bias in the model. Simulated annealing algorithms are pro-posed and used to compute the optimal designs. The performance of the algorithms and the prop-erties of the optimal designs obtained from our criteria are investigated through several examples.