Skip to content

Workshop Programme

for period 9 - 12 August 2011

Optimum Design for Mixed Effects Non-linear and Generalised Linear Models

9 - 12 August 2011

Timetable

Tuesday 9 August
09:00-09:25 Registration
09:25-09:30 Opening remarks and welcome from John Toland (INI Director Designate)
Chair: B. Bogacka
09:30-10:30 Bates, D (Wisconsin-Madison)
  Lecture on modelling: Mixed effects non-linear and generalized linear models Sem 1
 

Mixed-effects models are defined by the distributions of two vector-valued random variables, an n-dimensional response vector, Y and an unobserved q-dimensional random-effects vector, B. The mean of the conditional distribution, Y|B=b, depends on a linear predictor expression of the form Xß+Zb where ß is a p-dimensional fixed-effects parameter vector and the fixed and known model matrices, X and Z, are of the appropriate dimension. For linear mixed-effects models the conditional mean is the linear predictor; for generalized linear mixed-effects models the conditional mean is the value of an inverse link function applied to the linear predictor and for a nonlinear mixed-effects model the conditional mean is the result of applying a nonlinear model function for which the parameter vector is derived from the linear predictor. We describe the formulation of these mixed-effects models and provide computationally effective expressions for the profiled deviance function through which the maximum likelihood parameter estimates can be determined. In the case of the linear mixed-effects model the profiled deviance expression is exact. For generalized linear or nonlinear mixed-effects models the profiled deviance is approximated, either through a Laplace approximation or, at the expense of somewhat greater computational effort, through adaptive Gauss-Hermite quadrature.

 
10:30-11:00 Morning coffee
Chair: E. Demidenko
11:00-11:45 Schwabe, R (Otto Von Guericke)
  "To estimate or to predict" - implications on the design for linear mixed models Sem 1
 

During the last years mixed models have attracted an increasing popularity in many fields of applications due to advanced computer facilities. Although the main theme of the present workshop is devoted to optimal design of experiments for non-linear mixed models, it may be illustrative to elaborate the specific features of mixed models already in the linear case: Besides the estimation of population (location) parameters for the mean behaviour of the individuals a prediction of the response for the specific individuals under investigation may be of prior interest, for example in oncology studies to determine the further treatment of the patients investigated. While there have been some recent developments in optimal design for estimating the population parameters, the problem of optimal design for prediction has been considered as completely solved since the seminal paper by Gladitz and Pilz (1982). However, the optimal designs obtained there require the population parameters to be known or may be considered as an approximation, if the number of individuals is large. The latter may be inadequate, when the resulting "optimal design" fails to allow for estimation of the population parameters. Therefore we will develop the theory and solutions for finite numbers of individuals. Finally we will illustrate the trade-off in optimal designs caused by the two competing aims of estimation and prediction by a simple example. Gladitz, J. and J. Pilz (1982): Construction of optimal designs in random coefficient regression models. Statistics 13, 371-385.

 
11:45-12:30 Donev, A; Loeza-Serrano, S (Manchester)
  Experimental designs for estimating variance components Sem 1
 

Many experiments are designed to estimate as precise as possible the fixed parameters in the required models. For example, D-optimum designs ensure that the volume of the confidence ellipsoid for these parameters is minimized. In some cases, only some of the fixed parameters are of interest. DS-optimality is then required. However, little attention has been given to the accuracy of the estimation of the variance components in the models, while they are very important for the interpretation of the results and in some cases it is their estimation that is the reason for the studies to be carried out. We give examples of such studies and focus on the design of experiments where only the variance components are important. The resulting DV-optimum designs are useful to use in crossed or split-plot validation experiments where fixed effects can be regarded as nuisance parameters. We conclude with some considerations about the implications of our results on the design of experiments where both the fixed parameters and the variance components are important.

 
12:30-13:30 Lunch at Wolfson Court
Chair: A. Donev
14:00-14:45 Gilmour, S (Southampton)
  GLMs and GLMMs in the analysis of randomized experiments Sem 1
 

The Normal linear model analysis is usually used as an approximation to the exact randomization analysis and extended to structures, such as nonorthogonal split-plot designs, as a natural approximation. If the responses are counts or proportions a generalized linear model (GLM) is often used instead. It will be shown that GLMs do not in general arise as natural approximations to the randomization model. Instead the natural approximation is a generalized linear mixed model (GLMM).

 
14:45-15:30 Woods, D (Southampton)
  Block designs for non-normal data via conditional and marginal models Sem 1
 

Many experiments in all areas of science, technology and industry measure a response that cannot be adequately described by a linear model with normally distributed errors. In addition, the further complication often arises of needing to arrange the experiment into blocks of homogeneous units. Examples include industrial manufacturing experiments with binary responses, clinical trials where subjects receive multiple treatments and crystallography experiments in early-stage drug discovery. This talk will present some new approaches to the design of such experiments, assuming both conditional (subject-specific) and marginal (population-averaged) models. The different methods will be compared, and some advantages and disadvantages highlighted. Common issues, including the impact of correlations and the dependence of the design on the values of model parameters, will also be discussed.

 
15:30-16:00 Afternoon tea
Chair: S. Gilmour
16:00-16:45 Waite, T (Southampton)
  Designs for mixed models with binary response Sem 1
 

For an experiment measuring a binary response, a generalized linear model such as the logistic or probit is typically used to model the data. However these models assume that the responses are independent. In blocked experiments, where responses in the same block are potentially correlated, it may be appropriate to include random effects in the predictor, thus producing a generalized linear mixed model (GLMM). Obtaining designs for such models is complicated by the fact that the information matrix, on which most optimality criteria are based, is computationally expensive to evaluate (indeed if one computes naively, the search for an optimal design is likely to take several months). When analyzing GLMMs, it is common to use analytical approximations such as marginal quasi-likelihood (MQL) and penalized quasi-likelihood (PQL) in place of full maximum likelihood. In this talk, we consider the use of such computationally cheap approximations as surrogates for the true information matrix when producing designs. This reduces the computational burden substantially, and enables us to obtain designs within a much shorter time frame. However, other issues also need to be considered such as the accuracy of the approximations and the dependence of the optimal design on the unknown values of the parameters. In particular, we evaluate the effectiveness of designs found using these analytic approximations through comparison to designs that are found using a more computationally expensive numerical approximation to the likelihood.

 
16:45-17:30 Welcome wine reception
Wednesday 10 August
Chair: S. Leonov
09:30-10:30 Demidenko, E (Dartmouth College)
  Lecture on estimation and inference: Non-maximum likelihood estimation and statistical inference for linear and nonlinear mixed models Sem 1
 

Traditionally linear and nonlinear mixed effects models are estimated by maximum likelihood assuming normal distribution. The goal of this lecture is to discuss non-iterative methods for estimation of linear mixed models and simplified methods for estimation of generalized linear and nonlinear mixed models. In particular, we will talk about testing the presence of random effects, often overlooked fundamental test in the framework of mixed effects model. Simplified methods for generalized linear mixed models (GLMM), such as conditional logistic regression models with random intercepts and Poisson model for count data will be discussed. Limitations of popular generalized estimating equation (GEE) approach are uncovered. On the other hand, it is shown that this approach is valid for Poisson mixed model. Fixed sample maximum likelihood approach is introduced and its statistical properties are investigated via statistical simulations. Open problems and future work on statistical inference for mixed models are outlined.

 
10:30-11:00 Morning coffee
Chair: D. Woods
11:00-11:45 Ueckert, S (Uppsala)
  Optimal design of clinical trials in Alzheimer's disease Sem 1
 

Clinical trials in Alzheimer's disease are long, costly and have a low success rate. In this setting, optimal design theory constitutes a valuable tool to help planning of clinical studies efficiently. Our work presents how the theory of optimal design can be applied to a complex situation like an Alzheimer's trial. We illustrate how to handle challenges like dropout, covariate relationships and clinically relevant constraints. Furthermore, we illustrate how clinical trials can be directly optimized for drug effect detection power.

 
11:45-12:30 Mentré, F (Paris)
  Optimal designs for pharmacokinetic and viral dynamic nonlinear mixed effect models in HIV treatment Sem 1
 

Background: Nonlinear mixed effects models (NLMEM) are increasingly used for analysis of dose-exposure-response models. Methods for “population” designs evaluation/ optimisation are needed for complex models to limit the number of samples in each patient. Approaches for population designs optimisation based on the Fisher information matrix for NLMEM are developed, using mostly first order approximation of the model. Antiretroviral treatment in patients with HIV infection is complex and show large inter -individual variability. Pharmacokinetic and viral dynamic models are available to describe evolution of concentrations, viral loads and CD4 counts. Parameters of these models are estimated through NLMEM.

Objectives: 1) to evaluate and optimise designs in patients for the pharmacokinetic study of an antiretroviral drug (zidovudine) and its active metabolite using cost functions, 2) to evaluate and optimise designs for viral dynamic response and study power to compare treatments efficacy.

Methods: We used the models and estimated parameters from data of patients of the COPHAR 2 - ANRS 111 trial. Measuring active metabolite concentration of zidovudine is costly, as they are intracellular, and we explored D-optimal population designs using various cost functions. The viral dynamic model is a complex model written in ordinary differential equations. We proposed sparse designs with limited number of visits per patient during the one year follow up. We studied the predicted power to compare two treatments. These analyses were performed using PFIM3.2, an R function that we developed for population designs.

Results: We found a design with only three samples for zidovudine and two samples for its active metabolite and showed that optimal designs varied with cost functions. For the viral dynamic model, we showed that a design with 6 visits, if optimally located, can provide good information on response. We evaluated the power to compare two treatments and computed the number of subject needed to get adequate power.

Conclusion: We showed that population design optimisation provides efficient designs respecting clinical constraints in multi responses nonlinear mixed effects models.

 
12:30-13:30 Lunch at Wolfson Court
Chair: D. Bates
14:00-14:45 Rosner, G (Johns Hopkins)
  Bayesian enrichment strategies for randomized discontinuation trials Sem 1
 

We propose optimal choice of the design parameters for random discontinuation designs (RDD) using a Bayesian decision-theoretic approach. We consider applications of RDDs to oncology phase II studies evaluating activity of cytostatic agents. The design consists of two stages. The preliminary open-label stage treats all patients with the new agent and identi?es a possibly sensitive subpopulation. The subsequent second stage randomizes, treats, follows, and compares outcomes among patients in the identi?ed subgroup, with randomization to either the new or a control treatment. Several tuning parameters characterize the design: the number of patients in the trial, the duration of the preliminary stage, and the duration of follow-up after randomization. We de?ne a probability model for tumor growth, specify a suitable utility function, and develop a computational procedure for selecting the optimal tuning parameters.

 
14:45-15:30 Mueller, P (Texas at Austin)
  Sequential stopping for high-throughput experiments Sem 1
 

In high-throughput experiments sample size is typically chosen informally. Although formal sample size calculations have been proposed, they depend critically on prior knowledge. We propose a sequential strategy which, by updating knowledge when new data is available, depends less critically on prior assumptions. Compared to fixed sample size approaches, our sequential strategy stops experiments early when enough evidence has been accumulated, and recommends continuation when additional data is likely to provide valuable information. The approach is based on a decision-theoretic framework, guaranteeing that the chosen design proceeds in a coherent fashion. We propose a utility function based on the number of true positives which is straightforward to specify and intuitively appealing. As for most sequential design problems, an exact solution is computationally prohibitive. To address the computational challenge and also to limit the dependence on an arbitrarily chosen utility function we propose instead a simulation-based approximation with decision boundaries. The approach allows us to determine good designs within reasonable computing time and is characterized by intuitively appealing decision boundaries. We apply the method to next-generation sequencing, microarray and reverse phase protein array studies. We show that it can lead to substantial increases in posterior expected utility. An implementation of the proposed approach is available in the Bioconductor package gaga.

 
15:30-16:00 Afternoon tea
16:00-16:30 Bates, D (Wisconsin)
  Profiling the deviance to assess variability of parameter estimates in mixed models Sem 1
Chair: B. Bogacka
16:00-17:30 Poster storm and Posters
Thursday 11 August
Chair: P. Mueller
09:30-10:30 Fedorov, V (GlaxoSmithKline)
  Mixed models: design of experiments Sem 1
 

After a short discussion of commonalities between the mixed effect models and the Bayesian setting I define two design problems. The first one is related to the estimation of the population parameters and is often used in comparison of different treatments or in dose response studies. The necessity to estimate individual parameters (for a specific experimental unit like a clinical center or even a patient) leads to another optimization problem. I compare various criteria of optimality for both settings and derive elemental information matrices for various special cases. The latter allows to apply the standard machinery of optimal design of experiments.

 
10:30-11:00 Morning coffee
Chair: G. Rosner
11:00-11:45 Patan, M (Zielona Gora, Poland)
  Group sensor scheduling for parameter estimation of random-effects distributed systems Sem 1
 

The problem of sensor location for monitoring network with stationary nodes used for estimating unknown parameters of distributed-parameter system is addressed. In particular, the situation is considered, when the system parameters at the experimentation stage may randomly change according to the slight fluctuations of experimental conditions or differences in individual properties of observed distributed systems. A proper theoretical formulation of the sensor scheduling problem is provided together with a characterization of the optimal solutions. The theory is applicable to those practical situations in which a distributed system is sensitive to sampling or gives a different response at each run of the experiment. In the presented approach, some results from experimental design theory for dynamic systems are extended for the purpose of configuring a sensor grid in order to obtain practical and numerically tractable representation of optimum designs for estimation of the mean values of the parameters. A suitable computational scheme is illustrated by numerical example on a sensor scheduling problem for a two-dimensional example of dynamical distributed process representing the performance of magnetic brake.

 
11:45-12:30 Latif, M (Dhaka, Bangladesh)
  Optimum designs for transform-both-sides nonlinear mixed effects model in the presence of covariates Sem 1
 

In the early stage of drug developing process, pharmaceutical companies are interested in whether the candidate compounds show interactions with other drugs. Since most of the drugs are metabolized in human liver, these early stage pharmacokinetic experiments are conducted at different levels of concentrations of the compound under study with randomly selected liver tissues. As enzymes play a vital role in metabolizing drugs, examination of association between compound and different enzymes could be useful in understanding the compound's potentiality of adverse drug reactions. Michaelis-Menten model is often used to examine the association between enzymes and compound. In many cases transform-both-sides Michales-Menten model fits pharmacokinetic data well compared to the regular Michaelis-Menten model. In this talk, we will discuss optimum designs for such transform-both-sides Michaelis-Menten model when information on covariates associated with randomly selected liver tissues are available.

 
12:30-13:30 Lunch at Wolfson Court
14:00-17:30 Cambridge Colleges Tour
19:00-22:00 Conference dinner at Peterhouse
Friday 12 August
Session: Population Optimum Design of Experiments
09:30-09:45 Intro by B Bogacka and S Leonov
09:45-10:30 Leonov, S (Vertex)
  Approximation of the individual Fisher information matrix and its use in design of population PK/PD studies Sem 1
 

We continue a discussion started at PODE 2010 meeting about different types of approximation of the individual Fisher information matrix and their use in design of population PK/PD studies which are described by nonlinear mixed effects models. We focus on several Monte Carlo-based options and provide examples of their performance.

 
11:00-11:45 Mielke, T (Otto-von-Guericke)
  Approximation of the Fisher information and design in nonlinear mixed effects models Sem 1
 

The missing closed form representation of the probability density of the observations is one main problem in the analysis of Nonlinear Mixed Effects Models. Often local approximations based on linearizations of the model are used to approximately describe the properties of estimators. The Fisher Information is of special interest for designing experiments, as its inverse yields a lower bound of the variance of any unbiased estimator. Different linearization approaches for the model yield different approximations of the true underlying stochastical model and the Fisher Information (Mielke and Schwabe (2010)). Target of the presentation are alternative motivations of Fisher-Information approximations, based on conditional moments. For an individual design, known inter-individual variance and intra-individual variance, the Fisher Information for estimating the population location parameter vector results in an expression depending on conditional moments, such that approximations of the expectation of the conditional variance and the variance of the conditional expectation yield approximations of the Fisher Information, which are less based on distribution assumptions. Tierney et. al. (1986) described fully exponential Laplace approximations as an accurate method for approximating posterior moments and densities in Bayesian models. We present approximations of the Fisher Information, obtained by approximations of conditional moments with a similar heuristic and compare the impact of different Fisher Information approximations on the optimal design for estimating the population location parameters in pharmacokinetic studies.

 
11:45-12:30 Nyberg, J (Uppsala)
  Improved conditional approximations of the population Fisher information matrix Sem 1
 

We present an extended approximation of the Fisher Information Matrix (FIM) for nonlinear mixed effects models based on a first order conditional (FOCE) approximation of the population likelihood. Unlike previous FOCE based FIM, we use the empirical Bayes estimates to derive the FIM. In several examples, compared to the old FOCE based FIM, the improved FIM predicts parameter uncertainty much closer to simulation based empirical parameter uncertainty. Furthermore, this approach seems more robust against other approximations of the FIM, i.e. (Full/Reduced FIM). Finally, the new FOCE derived FIM is slightly closer to the simulated empirical precision than the FO based FIM.

 
14:00-14:45 Waterhouse, T (Eli Lilly & Company)
  Experiences in optimal design for population PK/PD models Sem 1
 

In all stages of contemporary drug development, the use of mixed effects ("population") models has become crucial to the understanding of pharmacokinetic (PK) and pharmacodynamic (PD) data. Population PK/PD models allow for the use of sparse sampling (i.e., fewer samples per subject), and they can be used to explain different sources of variability, ultimately leading to the possibility of dose optimisation for special populations or even individuals.

The design of trials involving population PK/PD models is often assessed via simulation, although the use of optimal design is gaining prominence. In recent years there have been a number of methodological advances in this area, but this talk will focus on more practical considerations of optimal design in the setting of a pharmaceutical company, from time and cost constraints to awareness and acceptance of optimal design methods. Several examples will be presented for illustration.

 
14:45-15:30 Duffull, S (Otago)
  A general method to determine sampling windows for nonlinear mixed effects models Sem 1
 

Many clinical pharmacology studies require repeated measurements to be taken on each patient and analysis of the data are conducted within the framework of nonlinear mixed effects models. It is increasingly common to design these studies using information theoretic principles due to the need for parsimony because of the presence of many logistical and ethical constraints. D-optimal design methods are often used to identify the best possible study conditions, such as the dose and number and timing of blood sample collection. However, the optimal times for collecting blood samples may not be feasible in clinical practice. Sampling windows, a time interval for blood sample collection, have been proposed to provide flexibility while preserving efficient parameter estimation. Due to the complexity of nonlinear mixed effects models there is generally no analytical solution available to determine sampling windows. We propose a method for determination of sampling windows based on MCMC sampling techniques. The proposed method reaches the stationary distribution rapidly and provides time-sensitive windows around the optimal design points. The proposed method is applicable to determine windows around any continuous design variable for which repeated measures per run are required. This has particular importance for clinical pharmacology studies.

 
10:30-11:00 Morning coffee
12:30-13:30 Lunch at Wolfson Court
15:30-16:00 Afternoon tea
Chair: S. Duffull
16:00-16:30 Mentré, F (Paris)
  Software for the population optimum design of experiments Sem 1
Chair: S. Duffell
16:30-17:00 Discussion

Back to top ∧