A general method to determine sampling windows for nonlinear mixed effects models with an application to population pharmacokinetic studies

Optimal design methods have been proposed to determine the best sampling times when sparse blood sampling is required in clinical pharmacokinetic studies. However, the optimal blood sampling time points may not be feasible in clinical practice. Sampling windows, a time interval for blood sample collection, have been proposed to provide flexibility in blood sampling times while preserving efficient parameter estimation. Because of the complexity of the population pharmacokinetic models, which are generally nonlinear mixed effects models, there is 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 attains a stationary distribution rapidly and provides time-sensitive windows around the optimal design points. The proposed method is applicable to determine sampling windows for any nonlinear mixed effects model although our work focuses on an application to population pharmacokinetic models.     

Nonlinear mixed effects modelling for estimation of steady state attainment

Assessment of the time needed to attain steady state is a key pharmacokinetic objective during drug development. Traditional approaches for assessing steady state include ANOVA‐based methods for comparing mean plasma concentration values from each sampling day, with either a difference or equivalence test. However, hypothesis‐testing approaches are ill suited for assessment of steady state. This paper presents a nonlinear mixed effects modelling approach for estimation of steady state attainment, based on fitting a simple nonlinear mixed model to observed trough plasma concentrations. The simple nonlinear mixed model is developed and proposed for use under certain pharmacokinetic assumptions. The nonlinear mixed modelling estimation approach is described and illustrated by application to trough data from a multiple dose trial in healthy subjects. The performance of the nonlinear mixed modelling approach is compared to ANOVA‐based approaches by means of simulation techniques. Copyright © 2005 John Wiley & Sons, Ltd.     

Sample‐size calculations for multi‐group comparison in population pharmacokinetic experiments

This paper describes an approach for calculating sample size for population pharmacokinetic experiments that involve hypothesis testing based on multi‐group comparison detecting the difference in parameters between groups under mixed‐effects modelling. This approach extends what has been described for generalized linear models and nonlinear population pharmacokinetic models that involve only binary covariates to more complex nonlinear population pharmacokinetic models. The structural nonlinear model is linearized around the random effects to obtain the marginal model and the hypothesis testing involving model parameters is based on Wald's test. This approach provides an efficient and fast method for calculating sample size for hypothesis testing in population pharmacokinetic models. The approach can also handle different design problems such as unequal allocation of subjects to groups and unbalanced sampling times between and within groups. The results obtained following application to a one compartment intravenous bolus dose model that involved three different hypotheses under different scenarios showed good agreement between the power obtained from NONMEM simulations and nominal power. Copyright © 2009 John Wiley & Sons, Ltd.     

A generalisation of T‐optimality for discriminating between competing models with an application to pharmacokinetic studies

The T‐optimality criterion is used in optimal design to derive designs for model selection. To set up the method, it is required that one of the models is considered to be true. We term this local T‐optimality. In this work, we propose a generalisation of T‐optimality (termed robust T‐optimality) that relaxes the requirement that one of the candidate models is set as true. We then show an application to a nonlinear mixed effects model with two candidate non‐nested models and combine robust T‐optimality with robust D‐optimality. Optimal design under local T‐optimality was found to provide adequate power when the a priori assumed true model was the true model but poor power if the a priori assumed true model was not the true model. The robust T‐optimality method provided adequate power irrespective of which model was true. The robust T‐optimality method appears to have useful properties for nonlinear models, where both the parameter values and model structure are required to be known a priori, and the most likely model that would be applied to any new experiment is not known with certainty. Copyright © 2012 John Wiley & Sons, Ltd.     

Design issues for population growth models

We briefly review and discuss design issues for population growth and decline models. We then use a flexible growth and decline model as an illustrative example and apply optimal design theory to find optimal sampling times for estimating model parameters, specific parameters and interesting functions of the model parameters for the model with two real applications. Robustness properties of the optimal designs are investigated when nominal values or the model is mis-specified, and also under a different optimality criterion. To facilitate use of optimal design ideas in practice, we also introduce a website for generating a variety of optimal designs for popular models from different disciplines.     

Penalized calibration in survey sampling: Design-based estimation assisted by mixed models

Calibration techniques in survey sampling, such as generalized regression estimation (GREG), were formalized in the 1990s to produce efficient estimators of linear combinations of study variables, such as totals or means. They implicitly lie on the assumption of a linear regression model between the variable of interest and some auxiliary variables in order to yield estimates with lower variance if the model is true and remaining approximately design-unbiased even if the model does not hold. We propose a new class of model-assisted estimators obtained by releasing a few calibration constraints and replacing them with a penalty term. This penalization is added to the distance criterion to minimize. By introducing the concept of penalized calibration, combining usual calibration and this ‘relaxed’ calibration, we are able to adjust the weight given to the available auxiliary information. We obtain a more flexible estimation procedure giving better estimates particularly when the auxiliary information is overly abundant or not fully appropriate to be completely used. Such an approach can also be seen as a design-based alternative to the estimation procedures based on the more general class of mixed models, presenting new prospects in some scopes of application such as inference on small domains.     

Robust Bayesian nonlinear mixed‐effects modeling of time to positivity in tuberculosis trials

Early phase 2 tuberculosis (TB) trials are conducted to characterize the early bactericidal activity (EBA) of anti‐TB drugs. The EBA of anti‐TB drugs has conventionally been calculated as the rate of decline in colony forming unit (CFU) count during the first 14 days of treatment. The measurement of CFU count, however, is expensive and prone to contamination. Alternatively to CFU count, time to positivity (TTP), which is a potential biomarker for long‐term efficacy of anti‐TB drugs, can be used to characterize EBA. The current Bayesian nonlinear mixed‐effects (NLME) regression model for TTP data, however, lacks robustness to gross outliers that often are present in the data. The conventional way of handling such outliers involves their identification by visual inspection and subsequent exclusion from the analysis. However, this process can be questioned because of its subjective nature. For this reason, we fitted robust versions of the Bayesian nonlinear mixed‐effects regression model to a wide range of TTP datasets. The performance of the explored models was assessed through model comparison statistics and a simulation study. We conclude that fitting a robust model to TTP data obviates the need for explicit identification and subsequent “deletion” of outliers but ensures that gross outliers exert no undue influence on model fits. We recommend that the current practice of fitting conventional normal theory models be abandoned in favor of fitting robust models to TTP data.     

CVX‐based algorithms for constructing various optimal regression designs

CVX‐based numerical algorithms are widely and freely available for solving convex optimization problems but their applications to solve optimal design problems are limited. Using the CVX programs in MATLAB, we demonstrate their utility and flexibility over traditional algorithms in statistics for finding different types of optimal approximate designs under a convex criterion for nonlinear models. They are generally fast and easy to implement for any model and any convex optimality criterion. We derive theoretical properties of the algorithms and use them to generate new A‐, c‐, D‐ and E‐optimal designs for various nonlinear models, including multi‐stage and multi‐objective optimal designs. We report properties of the optimal designs and provide sample CVX program codes for some of our examples that users can amend to find tailored optimal designs for their problems. The Canadian Journal of Statistics 47: 374–391; 2019 © 2019 Statistical Society of Canada     

LASSO-type estimators for semiparametric nonlinear mixed-effects models estimation

Parametric nonlinear mixed effects models (NLMEs) are now widely used in biometrical studies, especially in pharmacokinetics research and HIV dynamics models, due to, among other aspects, the computational advances achieved during the last years. However, this kind of models may not be flexible enough for complex longitudinal data analysis. Semiparametric NLMEs (SNMMs) have been proposed as an extension of NLMEs. These models are a good compromise and retain nice features of both parametric and nonparametric models resulting in more flexible models than standard parametric NLMEs. However, SNMMs are complex models for which estimation still remains a challenge. Previous estimation procedures are based on a combination of log-likelihood approximation methods for parametric estimation and smoothing splines techniques for nonparametric estimation. In this work, we propose new estimation strategies in SNMMs. On the one hand, we use the Stochastic Approximation version of EM algorithm (SAEM) to obtain exact ML and REML estimates of the fixed effects and variance components. On the other hand, we propose a LASSO-type method to estimate the unknown nonlinear function. We derive oracle inequalities for this nonparametric estimator. We combine the two approaches in a general estimation procedure that we illustrate with simulations and through the analysis of a real data set of price evolution in on-line auctions.     

Functional approach of flexibly modelling generalized longitudinal data and survival time

We propose a flexible functional approach for modelling generalized longitudinal data and survival time using principal components. In the proposed model the longitudinal observations can be continuous or categorical data, such as Gaussian, binomial or Poisson outcomes. We generalize the traditional joint models that treat categorical data as continuous data by using some transformations, such as CD4 counts. The proposed model is data-adaptive, which does not require pre-specified functional forms for longitudinal trajectories and automatically detects characteristic patterns. The longitudinal trajectories observed with measurement error or random error are represented by flexible basis functions through a possibly nonlinear link function, combining dimension reduction techniques resulting from functional principal component (FPC) analysis. The relationship between the longitudinal process and event history is assessed using a Cox regression model. Although the proposed model inherits the flexibility of non-parametric methods, the estimation procedure based on the EM algorithm is still parametric in computation, and thus simple and easy to implement. The computation is simplified by dimension reduction for random coefficients or FPC scores. An iterative selection procedure based on Akaike information criterion (AIC) is proposed to choose the tuning parameters, such as the knots of spline basis and the number of FPCs, so that appropriate degree of smoothness and fluctuation can be addressed. The effectiveness of the proposed approach is illustrated through a simulation study, followed by an application to longitudinal CD4 counts and survival data which were collected in a recent clinical trial to compare the efficiency and safety of two antiretroviral drugs.     

A note on Läuter-type robust design for polynomial regression models

In this paper, we consider the problem of model robust design for simultaneous parameter estimation among a class of polynomial regression models with degree up to k. A generalized D-optimality criterion, the Ψ_α‐optimality criterion, first introduced by Läuter (1974) is considered for this problem. By applying the theory of canonical moments and the technique of maximin principle, we derive a model robust optimal design in the sense of having highest minimum Ψ_α‐efficiency. Numerical comparison indicates that the proposed design has remarkable performance for parameter estimation in all of the considered rival models.     

Sample reuse simulation in optimal design for Tmax in pharmacokinetic experiments

T _max and C _max are important pharmacokinetic parameters in drug development processes. Often a nonparametric procedure is needed to estimate them when model independence is required. This paper proposes a simulation-based optimal design procedure for finding optimal sampling times for nonparametric estimates of T _max and C _max for each subject, assuming that the drug concentration follows a non-linear mixed model. The main difficulty of using standard optimal design procedures is that the property of the nonparametric estimate is very complicated. This procedure uses a sample reuse simulation to calculate the design criterion, which is an integral of multiple dimension, so that effective optimization procedures such as Newton-type procedures can be used directly to find optimal designs. This procedure is used to construct optimal designs for an open one-compartment model. An approximation based on the Taylor expansion is also derived and showed results that were consistent with those based on the sample reuse simulation.     

A nonsequential design procedure for parameter estimation and model discrimination in nonlinear regression models

This paper presents and illustrates a new nonsequential design procedure for simultaneous parameter estimation and model discrimination for a collection of nonlinear regression models. This design criterion is extended to make it robust to initial parameter choices by using a Bayesian design approach, and is also extended to yield efficient estimation-discrimination designs which take account of curvature.     

Conditional Akaike information under covariate shift with application to small area estimation

In this study, we consider the problem of selecting explanatory variables of fixed effects in linear mixed models under covariate shift, which is when the values of covariates in the model for prediction differ from those in the model for observed data. We construct a variable selection criterion based on the conditional Akaike information introduced by Vaida & Blanchard (2005). We focus especially on covariate shift in small area estimation and demonstrate the usefulness of the proposed criterion. In addition, numerical performance is investigated through simulations, one of which is a design‐based simulation using a real dataset of land prices. The Canadian Journal of Statistics 46: 316–335; 2018 © 2018 Statistical Society of Canada     

Modified D-optimal design for logistic model

Asymptotic theory of using the Fisher information matrix may provide poor approximation to the exact variance matrix of maximum likelihood estimation in nonlinear models. This may be due to not obtaining an efficient D-optimal design. In this article, we propose a modified D-optimality criterion, using a more accurate information matrix, based on the Bhattacharyya matrix. The proposed information matrix and its properties are given for two parameters simple logistic model. It is shown that the resulted modified locally D-optimal design is more efficient than the previous one; particularly, for small sample size experiments.     

Bias‐reduced marginal Akaike information criteria based on a Monte Carlo method for linear mixed‐effects models

In linear mixed‐effects (LME) models, if a fitted model has more random‐effect terms than the true model, a regularity condition required in the asymptotic theory may not hold. In such cases, the marginal Akaike information criterion (AIC) is positively biased for (?2) times the expected log‐likelihood. The asymptotic bias of the maximum log‐likelihood as an estimator of the expected log‐likelihood is evaluated for LME models with balanced design in the context of parameter‐constrained models. Moreover, bias‐reduced marginal AICs for LME models based on a Monte Carlo method are proposed. The performance of the proposed criteria is compared with existing criteria by using example data and by a simulation study. It was found that the bias of the proposed criteria was smaller than that of the existing marginal AIC when a larger model was fitted and that the probability of choosing a smaller model incorrectly was decreased.     

Small area estimation using unmatched sampling and linking models

The authors use a hierarchical Bayes approach to area level unmatched sampling and Unking models for small area estimation. Empirically they compare inferences under unmatched models with those obtained under the customary matched sampling and linking models. They apply the proposed method to Canadian census undercoverage estimation, developing a full hierarchical Bayes approach using Markov Chain Monte Carlo sampling methods. They show that the method can provide efficient model‐based estimates. They use posterior predictive distributions to assess model fit.     

Bayesian density regression for discrete outcomes

We develop Bayesian models for density regression with emphasis on discrete outcomes. The problem of density regression is approached by considering methods for multivariate density estimation of mixed scale variables, and obtaining conditional densities from the multivariate ones. The approach to multivariate mixed scale outcome density estimation that we describe represents discrete variables, either responses or covariates, as discretised versions of continuous latent variables. We present and compare several models for obtaining these thresholds in the challenging context of count data analysis where the response may be over‐ and/or under‐dispersed in some of the regions of the covariate space. We utilise a nonparametric mixture of multivariate Gaussians to model the directly observed and the latent continuous variables. The paper presents a Markov chain Monte Carlo algorithm for posterior sampling, sufficient conditions for weak consistency, and illustrations on density, mean and quantile regression utilising simulated and real datasets.     

A generalized Bayesian nonlinear mixed‐effects regression model for zero‐inflated longitudinal count data in tuberculosis trials

In this paper, we investigate Bayesian generalized nonlinear mixed‐effects (NLME) regression models for zero‐inflated longitudinal count data. The methodology is motivated by and applied to colony forming unit (CFU) counts in extended bactericidal activity tuberculosis (TB) trials. Furthermore, for model comparisons, we present a generalized method for calculating the marginal likelihoods required to determine Bayes factors. A simulation study shows that the proposed zero‐inflated negative binomial regression model has good accuracy, precision, and credibility interval coverage. In contrast, conventional normal NLME regression models applied to log‐transformed count data, which handle zero counts as left censored values, may yield credibility intervals that undercover the true bactericidal activity of anti‐TB drugs. We therefore recommend that zero‐inflated NLME regression models should be fitted to CFU count on the original scale, as an alternative to conventional normal NLME regression models on the logarithmic scale.     

$$D_s$$-optimality in copula models

Optimum experimental design theory has recently been extended for parameter estimation in copula models. The use of these models allows one to gain in flexibility by considering the model parameter set split into marginal and dependence parameters. However, this separation also leads to the natural issue of estimating only a subset of all model parameters. In this work, we treat this problem with the application of the \(D_s\)-optimality to copula models. First, we provide an extension of the corresponding equivalence theory. Then, we analyze a wide range of flexible copula models to highlight the usefulness of \(D_s\)-optimality in many possible scenarios. Finally, we discuss how the usage of the introduced design criterion also relates to the more general issue of copula selection and optimal design for model discrimination.