Nonlinear models for repeated measurement data: An overview and update

A nonlinear mixed-effect mixture model for functional mapping of dynamic traits

Functional mapping has emerged as a next-generation statistical tool for mapping quantitative trait loci (QTL) that affect complex dynamic traits. In this article, we incorporated the idea of nonlinear mixed-effect (NLME) models into the mixture-based framework of functional mapping, aimed to generalize the spectrum of applications for functional mapping. NLME-based functional mapping, implemented with the linearization algorithm based on the first-order Taylor expansion, can provide reasonable estimates of QTL genotypic-specific curve parameters (fixed effect) and the between-individual variation of these parameters (random effect). Results from simulation studies suggest that the NLME-based model is more general than traditional functional mapping. The new model can be useful for the identification of the ontogenetic patterns of QTL genetic effects during time course.

Nonlinear Change Models in Populations with Unobserved Heterogeneity

When unobserved heterogeneity exists in populations where the phenomenon of interest is governed by a functional form of change linear in its parameters, the growth mixture model (GMM) is useful for modeling change conditional on latent class. However, when the functional form of interest is nonlinear in its parameters, the GMM is not very useful because it is based on a system of equations linear in its parameters. The nonlinear change mixture model (NCMM) is proposed, which explicitly addresses unobserved heterogeneity in situations where change follows a nonlinear functional form. Due to the integration of nonlinear multilevel models and finite mixture models, neither of which generally have closed form solutions, analytic solutions do not generally exist for the NCMM. Five methods of parameter estimation are developed and evaluated with a comprehensive Monte Carlo simulation study. The simulation showed that the parameters of the NCMM can be accurately estimated with several of the proposed methods, and that the method of choice depends on the precise question of interest.

International variation in the use of blood transfusion in patients with non-ST-segment elevation acute coronary syndromes

The purpose of this study was to determine international patterns of blood transfusion in patients with acute coronary syndrome (ACS). Previous studies showed geographic heterogeneity in some aspects of ACS care. Data for variability in the use of blood transfusion in ACS management are limited. Pooled data from 3 international randomized trials of patients with non-ST-segment elevation ACS (n = 23,906) were analyzed to determine the association between non-United States (US) location and blood transfusion after stratifying by the use of invasive procedures. The analysis adjusted for differences in patient characteristics and was repeated using a 2-stage mixed-model approach and in patients who underwent in-hospital coronary artery bypass grafting. Compared with US patients, both unadjusted and adjusted hazards for blood transfusion were significantly lower in non-US patients who did not undergo invasive procedures (unadjusted hazard ratio [HR] 0.23, 95% confidence interval [CI] 0.17 to 0.33; adjusted HR 0.20, 95% CI 0.14 to 0.28). This was also true in non-US patients who underwent invasive procedures (unadjusted HR 0.34, 95% CI 0.27 to 0.44; adjusted HR 0.31, 95% CI 0.23 to 0.42). Results were similar in both validation analyses. In conclusion, there was substantial international variation in blood transfusion use in patients with ACS. These results, along with the controversy regarding the appropriate use of transfusion in patients with coronary heart disease, emphasize the need for understanding the role of blood transfusion in the management of patients with ACS and factors that influence transfusion decisions.

Exterior exposure estimation using a one-compartment toxicokinetic model with blood sample measurements

Exposure assessment of individuals exposed to certain chemicals plays an important role in the analysis of occupational-as well as environmental-health problems. Biological monitoring, as an alternative to direct environmental measurements, may be applied to relate the exterior exposure with the amount of individual intake. In this paper, we estimate individuals' (inhalation) exposure retrospectively from their blood concentrations via a simplified one-compartment toxicokinetic model. Considering stochastic variations to the toxicokinetic model, the solution to the resultant stochastic differential equation (SDE), together with measurement error, is transformed into a dynamic linear state-space model. The unknown model parameters and the mean inhalation concentration are then estimated via Markov Chain Monte Carlo (MCMC) simulations. The proposed method is used in the analysis of the styrene data (Wang et al. in Occup Environ Med 53:601-605, 1996) to backward estimate the inhalation concentration, assuming it is unknown. The data analysis showed that the internal stochastic variations, often ignored in toxicokinetic model analysis, outweighed in standard deviation almost twice that of the measurement error. Also, the simulation results showed that the method performed relatively well to the approach considering measurement error only.

Overview of model-building strategies in population PK/PD analyses: 2002-2004 literature survey

AIMS

A descriptive survey of published population pharmacokinetic and/or pharmacodynamic (PK/PD) analyses from 2002 to 2004 was conducted and an evaluation made of how model building was performed and reported.

METHODS

We selected 324 articles in Pubmed using defined keywords. A data abstraction form (DAF) was then built comprising two parts: general characteristics including article identification, context of the analysis, description of clinical studies from which the data arose, and model building, including description of the processes of modelling. The papers were examined by two readers, who extracted the relevant information and transmitted it directly to a MySQL database, from which descriptive statistical analysis was performed.

RESULTS

Most published papers concerned patients with severe pathology and therapeutic classes suffering from narrow therapeutic index and/or high PK/PD variability. Most of the time, modelling was performed for descriptive purposes, with rich rather than sparse data and using NONMEM software. PK and PD models were rarely complex (one or two compartments for PK; E(max) for PD models). Covariate testing was frequently performed and essentially based on the likelihood ratio test. Based on a minimal list of items that should systematically be found in a population PK-PD analysis, it was found that only 39% and 8.5% of the PK and PD analyses, respectively, published from 2002 to 2004 provided sufficient detail to support the model-building methodology.

CONCLUSIONS

This survey allowed an efficient description of recent published population analyses, but also revealed deficiencies in reporting information on model building.

REML Estimation of Variance Parameters in Nonlinear Mixed Effects Models Using the SAEM Algorithm

Nonlinear mixed effects models are now widely used in biometrical studies, especially in pharmacokinetic research or for the analysis of growth traits for agricultural and laboratory species. Most of these studies, however, are often based on ML estimation procedures, which are known to be biased downwards. A few REML extensions have been proposed, but only for approximated methods. The aim of this paper is to present a REML implementation for nonlinear mixed effects models within an exact estimation scheme, based on an integration of the fixed effects and a stochastic estimation procedure. This method was implemented via a stochastic EM, namely the SAEM algorithm. The simulation study showed that the proposed REML estimation procedure considerably reduced the bias observed with the ML estimation, as well as the residual mean squared error of the variance parameter estimations, especially in the unbalanced cases. ML and REML based estimators of fixed effects were also compared via simulation. Although the two kinds of estimates were very close in terms of bias and mean square error, predictions of individual profiles were clearly improved when using REML vs. ML. An application of this estimation procedure is presented for the modelling of growth in lines of chicken.

Using mathematical modeling and control to develop structured treatment interruption strategies for HIV infection

The goal of this article is to suggest that mathematical models describing biological processes taking place within a patient over time can be used to design adaptive treatment strategies. We demonstrate using the key example of treatment strategies for human immunodeficiency virus type-1 (HIV) infection. Although there has been considerable progress in management of HIV infection using highly active antiretroviral therapies, continuous treatment with these agents involves significant cost and burden, toxicities, development of drug resistance, and problems with adherence; these latter complications are of particular concern in substance-abusing individuals. This has inspired interest in structured or supervised treatment interruption (STI) strategies, which involve cycles of treatment withdrawal and re-initiation. We argue that the most promising STI strategies are adaptive treatment strategies. We then describe how biological mechanisms governing the interaction over time between HIV and a patient's immune system may be represented by mathematical models and how control methods applied to these models can be used to design adaptive STI strategies seeking to maintain long-term suppression of the virus. We advocate that, when such mathematical representations of processes underlying a disease or disorder are available, they can be an important tool for suggesting adaptive treatment strategies for clinical study.

Analysis of nonlinear patterns of change with random coefficient models

Nonlinear patterns of change arise frequently in the analysis of repeated measures from longitudinal studies in psychology. The main feature of nonlinear development is that change is more rapid in some periods than in others. There generally also are strong individual differences, so although there is a general similarity of patterns for different persons over time, individuals exhibit substantial heterogeneity in their particular response. To describe data of this kind, researchers have extended the random coefficient model to accommodate nonlinear trajectories of change. It can often produce a statistically satisfying account of subject-specific development. In this review we describe and illustrate the main ideas of the nonlinear random coefficient model with concrete examples.

Stenseth N. Statistical analysis of the dynamics of antibody loss to a disease-causing agent: plague in natural populations of great gerbils as an example

We propose a new stochastic framework for analysing the dynamics of the immunity response of wildlife hosts against a disease-causing agent. Our study is motivated by the need to analyse the monitoring time-series data covering the period from 1975 to 1995 on bacteriological and serological tests-samples from great gerbils being the main host of Yersinia pestis in Kazakhstan. Based on a four-state continuous-time Markov chain, we derive a generalized nonlinear mixed-effect model for analysing the serological test data. The immune response of a host involves the production of antibodies in response to an antigen. Our analysis shows that great gerbils recovered from a plague infection are more likely to keep their antibodies to plague and survive throughout the summer-to-winter season than throughout the winter-to-summer season. Provided the seasonal mortality rates are similar (which seems to be the case based on a mortality analysis with abundance data), our finding indicates that the immune function of the sampled great gerbils is seasonal.

Evaluation of uncertainty parameters estimated by different population PK software and methods

The uncertainty associated with parameter estimations is essential for population model building, evaluation, and simulation. Summarized by the standard error (SE), its estimation is sometimes questionable. Herein, we evaluate SEs provided by different non linear mixed-effect estimation methods associated with their estimation performances. Methods based on maximum likelihood (FO and FOCE in NONMEM, nlme in Splus, and SAEM in MONOLIX) and Bayesian theory (WinBUGS) were evaluated on datasets obtained by simulations of a one-compartment PK model using 9 different designs. Bootstrap techniques were applied to FO, FOCE, and nlme. We compared SE estimations, parameter estimations, convergence, and computation time. Regarding SE estimations, methods provided concordant results for fixed effects. On random effects, SAEM and WinBUGS, tended respectively to under or over-estimate them. With sparse data, FO provided biased estimations of SE and discordant results between bootstrapped and original datasets. Regarding parameter estimations, FO showed a systematic bias on fixed and random effects. WinBUGS provided biased estimations, but only with sparse data. SAEM and WinBUGS converged systematically while FOCE failed in half of the cases. Applying bootstrap with FOCE yielded CPU times too large for routine application and bootstrap with nlme resulted in frequent crashes. In conclusion, FO provided bias on parameter estimations and on SE estimations of random effects. Methods like FOCE provided unbiased results but convergence was the biggest issue. Bootstrap did not improve SEs for FOCE methods, except when confidence interval of random effects is needed. WinBUGS gave consistent results but required long computation times. SAEM was in-between, showing few under-estimated SE but unbiased parameter estimations.

Estimation of population pharmacokinetic parameters of saquinavir in HIV patients with the MONOLIX software

In nonlinear mixed-effects models, estimation methods based on a linearization of the likelihood are widely used although they have several methodological drawbacks. Kuhn and Lavielle (Comput. Statist. Data Anal. 49:1020-1038 (2005)) developed an estimation method which combines the SAEM (Stochastic Approximation EM) algorithm, with a MCMC (Markov Chain Monte Carlo) procedure for maximum likelihood estimation in nonlinear mixed-effects models without linearization. This method is implemented in the Matlab software MONOLIX which is available at http://www.math.u-psud.fr/~lavielle/monolix/logiciels. In this paper we apply MONOLIX to the analysis of the pharmacokinetics of saquinavir, a protease inhibitor, from concentrations measured after single dose administration in 100 HIV patients, some with advance disease. We also illustrate how to use MONOLIX to build the covariate model using the Bayesian Information Criterion. Saquinavir oral clearance (CL/F) was estimated to be 1.26 L/h and to increase with body mass index, the inter-patient variability for CL/F being 120%. Several methodological developments are ongoing to extend SAEM which is a very promising estimation method for population pharmacockinetic/pharmacodynamic analyses.

Comment on "Computational Improvements Reveal Great Bacterial Diversity and High Metal Toxicity in Soil"

Gans et al. (Reports, 26 August 2005, p. 1387) provided an estimate of soil bacterial species richness two orders of magnitude greater than previously reported values. Using a re-derived mathematical model, we reanalyzed the data and found that the statistical error exceeds the estimate by a factor of 26. We also note two potential sources of error in the experimental data collection and measurement procedures.

QTL methodology for response curves on the basis of non-linear mixed models, with an illustration to senescence in potato

The improvement of quantitative traits in plant breeding will in general benefit from a better understanding of the genetic basis underlying their development. In this paper, a QTL mapping strategy is presented for modelling the development of phenotypic traits over time. Traditionally, crop growth models are used to study development. We propose an integration of crop growth models and QTL models within the framework of non-linear mixed models. We illustrate our approach with a QTL model for leaf senescence in a diploid potato cross. Assuming a logistic progression of senescence in time, two curve parameters are modelled, slope and inflection point, as a function of QTLs. The final QTL model for our example data contained four QTLs, of which two affected the position of the inflection point, one the senescence progression-rate, and a final one both inflection point and rate.

Non-linear mixed effects modeling - from methodology and software development to driving implementation in drug development science

Few scientific contributions have made significant impact unless there was a champion who had the vision to see the potential for its use in seemingly disparate areas-and who then drove active implementation. In this paper, we present a historical summary of the development of non-linear mixed effects (NLME) modeling up to the more recent extensions of this statistical methodology. The paper places strong emphasis on the pivotal role played by Lewis B. Sheiner (1940-2004), who used this statistical methodology to elucidate solutions to real problems identified in clinical practice and in medical research and on how he drove implementation of the proposed solutions. A succinct overview of the evolution of the NLME modeling methodology is presented as well as ideas on how its expansion helped to provide guidance for a more scientific view of (model-based) drug development that reduces empiricism in favor of critical quantitative thinking and decision making.