A geometric characterization of optimal designs for regression models with correlated observations (original) (raw)
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The Annals of Applied Statistics, 2010
We consider the problem of constructing optimal designs for population pharmacokinetics which use random effect models. It is common practice in the design of experiments in such studies to assume uncorrelated errors for each subject. In the present paper a new approach is introduced to determine efficient designs for nonlinear least squares estimation which addresses the problem of correlation between observations corresponding to the same subject. We use asymptotic arguments to derive optimal design densities, and the designs for finite sample sizes are constructed from the quantiles of the corresponding optimal distribution function. It is demonstrated that compared to the optimal exact designs, whose determination is a hard numerical problem, these designs are very efficient. Alternatively, the designs derived from asymptotic theory could be used as starting designs for the numerical computation of exact optimal designs. Several examples of linear and nonlinear models are presented in order to illustrate the methodology. In particular, it is demonstrated that naively chosen equally spaced designs may lead to less accurate estimation.
A New Approach to Optimal Design for Linear Models With Correlated Observations
Journal of the American Statistical Association, 2010
In the common linear regression model the problem of determining optimal designs for least squares estimation is considered in the case where the observations are correlated. A necessary condition for the optimality of a given design is provided, which extends the classical equivalence theory for optimal designs in models with uncorrelated errors to the case of dependent data. For one parameter models this condition is also shown to be sufficient in many cases and for several models optimal designs can be identified explicitly. For the multi-parameter regression models a simple relation which allows verifying the necessary optimality condition is established. Moreover, it is proved that the arcsine distribution is universally optimal for the polynomial regression model with a correlation structure defined by the logarithmic potential. It is also shown that for models in which the regression functions are eigenfunctions of an integral operator induced by the correlation kernel of the error process, designs satisfying the necessary conditions of optimality can be found explicitly. To the best knowledge of the authors these findings provide the first explicit results on optimal designs for regression models with correlated observations, which are not restricted to the location scale model.
A new approach to optimal designs for models with correlated observations
We consider the problem of designing experiments for the estimation of the mean in the location model in the presence of correlated observations. For a fixed correlation structure approximate optimal designs are determined, and it is demonstrated that under the model assumptions made by for the determination of asymptotic optimal design, the designs derived in this paper converge weakly the measures obtained by these authors.
Optimal Designs for Two Nested Pharmacokinetic Models with Correlated Observations
Communications in Statistics - Simulation and Computation, 2012
Two nested pharmacokinetic models are considered in this work. Several observations are taken on the same subject so they are correlated. The covariance function assumed is an exponential covariance function. Optimal exact designs are computed with different criteria, both for discriminating between models and estimating parameters. Compound criteria to estimate the parameters and nonlinear functions of the parameters are used. An iterative algorithm based on T-optimality and an algorithm from Brimkulov, Krug and Savanov [?] are combined in order to compute T-optimal designs with correlated observations. Finally, compound designs to discriminate between the models and estimate the nonlinear functions are considered. A test power study is performed to adjust the compound parameter.
Optimal design for linear models with correlated observations
The Annals of Statistics, 2013
In the common linear regression model the problem of determining optimal designs for least squares estimation is considered in the case where the observations are correlated. A necessary condition for the optimality of a given design is provided, which extends the classical equivalence theory for optimal designs in models with uncorrelated errors to the case of dependent data. For one parameter models this condition is also shown to be sufficient in many cases and for several models optimal designs can be identified explicitly. For the multi-parameter regression models a simple relation which allows verifying the necessary optimality condition is established. Moreover, it is proved that the arcsine distribution is universally optimal for the polynomial regression model with a correlation structure defined by the logarithmic potential. It is also shown that for models in which the regression functions are eigenfunctions of an integral operator induced by the correlation kernel of the error process, designs satisfying the necessary conditions of optimality can be found explicitly. To the best knowledge of the authors these findings provide the first explicit results on optimal designs for regression models with correlated observations, which are not restricted to the location scale model.
Journal of Statistical Planning and Inference, 2008
In this paper VV- and DD-optimal population designs for the simple linear regression model with a random intercept term are considered. This is done with special reference to longitudinal data, that is data measured repeatedly at specified time points. Individual designs comprising up to k+1k+1 distinct and equally spaced values of the explanatory variable are assumed to be available. The problem of constructing a population design which allocates weights to these individual designs in such a way that the variances associated with the mean marginal responses at a given vector of time points are in some sense minimized is addressed. VV-optimal designs are obtained and a geometric approach to confirm the global optimality or otherwise of these designs is introduced. The study is extended to the DD-optimal case. It is noted that the VV- and DD-optimal population designs are robust to the value of the intraclass correlation coefficient.
An Evaluation of Population D-Optimal Designs Via Pharmacokinetic Simulations
Annals of biomedical …, 2003
One goal of large scale clinical trials is to determine how a drug is processed by, and cleared from, the human body [i.e., its pharmacokinetic (PK) properties] and how these PK properties differ between individuals in a population (i.e., its population PK properties). Due to the high cost of these studies and the limited amount of data (e.g., blood samples) available from each study subject, it would be useful to know how many measurements are needed and when those measurements should be taken to accurately quantify population PK model parameters means and variances. Previous studies have looked at optimal design strategies of population PK experiments by developing an optimal design for an individual study (i.e., no interindividual variability was considered in the design), and then applying that design to each individual in a population study (where interindividual variability is present). A more algorithmically and informationally intensive approach is to develop a population optimal design, which inherently includes the assessment of interindividual variability. We present a simulation-based evaluation of these two design methods based on nonlinear Gaussian population PK models. Specifically, we compute standard individual and population D-optimal designs and compare population PK model parameter estimates based on simulated optimal design measurements. Our results show that population and standard D-optimal designs are not significantly different when both designs have the same number of samples per individual. However, population optimal designs allow for sampling schedules where the number of samples per individual is less than the number of model parameters, the theoretical limit allowed in standard optimal design. These designs with a low number of samples per individual are shown to be nearly as robust in parameter estimation as standard D-optimal designs. In the limit of just one sample per individual, however, population D-optimal designs are shown to be inadequate. © 2003 Biomedical Engineering Society.
Optimal designs in multivariate linear models
Statistics & Probability Letters, 2007
The purpose of this paper is to study optimality of an experimental design under the multivariate models with a known or unknown dispersion matrix. In the case of unknown dispersion matrix optimality is considered with respect to the precision in maximum likelihood estimation. We show relations between optimality of designs in univariate models and in their multivariate extensions. r
Efficient and optimal designs for correlated observations
2000
This thesis considers some aspects of the problem of finding efficient and optimal designs when observations are correlated. The two main areas that are examined are nested row-column (NRC) designs and early generation variety trials (EGVTs). In NRC designs, the experimental area is divided into b blocks, and each block is divided into P1 rows and P2 columns (blocks of size P1 x P2). Here, optimal NRC designs, which can be constructed from semi-balanced arrays, are obtained under the assumption that within-block observations are correlated. For a stationary reflection symmetric dependence structure, optimal NRC designs with blocks of size 2 x 2 are obtained for models with fixed block effects, which may also include row and/or column effects. It is shown that the efficiency of binary designs can be very low for some correlation values. Also, optimal NRC designs for blocks of size 3 x 3 and P1 x 2 (P1 ≥ 3 ) are determined. The optimality region for blocks of size P1 x P2 (P1 P2 ≥ 2) ...
Marginally restricted D-optimal designs for correlated observations
Journal of Applied Statistics, 2008
Two practical degrees of complexity may arise when designing an experiment for a model of a real life case. First, some explanatory variables may be not under the control of the practitioner. Second, the responses may be correlated. In this paper three real life cases in this situation are considered. Different covariance structures are studied and some designs are computed adapting the theory of marginally restricted designs for correlated observations. An exchange algorithm given by Brimkulov's algorithm is also adapted to Marginally restricted D-optimality and it is applied to a complex situation.