Estimation and variable selection in high dimension in nonlinear mixed-effects models

Pharmacokinetic-Pharmacodynamic Modeling and Simulation

Different mixed-effects models were compared to evaluate the population dose-response and relative potency of two albuterol inhalers. Bronchodilator response was measured after ascending doses of each inhaler in 37 asthmatic patients. A linear mixed-effects model was developed based on the approach proposed by Finney for the evaluation of bioassay data. A nonlinear mixed-effects (Emax) model with interindividual and interoccasion variability (IOV) in the different pharmacodynamic parameters was also fit to the data. Both methods produced a similar estimate of relative potency. However, the estimate of relative potency was 22% lower with the nonlinear mixed-effects model if IOV was not taken into account. Monte Carlo simulations based on a similar study design demonstrated that more biased and variable estimates of ED50 and relative potency were obtained when the nonlinear mixed-effects model ignored the presence of IOV in the data. Furthermore, the linear mixed-effects model that did not account for IOV produced confidence intervals for relative potency that were too narrow and thus could lead to erroneous conclusions. These problems were avoided when the estimation model could account for IOV. Results of the simulations were consistent with those of the experimental data. Although the linear or the nonlinear mixed-effects model may be used to evaluate population dose-response and relative potency, there are important differences in the assumptions made by each method.

In the previous chapters, we discussed autoregressive linear mixed effects models. In this section, we discuss the relationships between the autoregressive linear mixed effects models and nonlinear mixed effects models, growth curves, and differential equations. The autoregressive model shows a profile approaching an asymptote, where the change is proportional to the distance remaining to the asymptote. Autoregressive models in discrete time correspond to monomolecular curves in continuous time. Autoregressive linear mixed effects models correspond to monomolecular curves with random effects in the baseline and asymptote, and special error terms. The autoregressive coefficient is a nonlinear parameter, but all random effects parameters in the model are linear. Therefore, autoregressive linear mixed effects models are nonlinear mixed effects models without nonlinear random effects and have a closed form of likelihood. When there are time-dependent covariates, autoregressive linear mixed effects models are represented by a differential equation and random effects. The monomolecular curve is one of the popular growth curves. We introduce other growth curves, such as the logistic curves and von Bertalanffy curves, and generalizations of growth curves. Re-parameterization is often performed in nonlinear models, and various representations of re-parameterization in monomolecular and other curves are provided herein.

The aim of this study was to investigate the use of nonlinear mixed effects (NLME) models in a real bioequivalence study and compare it to noncompartmental analysis (NCA) which is proposed by regulatory agencies. NCA requires few hypotheses but a large number of samples per subject. On the other hand, NLME approach is more complex than NCA but it has some advantages such as it requires few samples per subject. A real data application was provided for the study, which was get from Ege University Drug Development and Pharmacokinetics Research Center. For NLME, we used the stochastic approach to expectation-maximization (SAEM) algorithm, whereas linear trapezoidal rule was used for NCA. We estimate pharmacokinetic parameters, area under the curve (AUC0-∞) and maximum concentration (Cmax), and perform a bioequivalence tests using NCA and NLME. According to real data analysis, NLME approach has smaller within subject error, narrower confidence intervals than non-compartmental analysis. However, NLME models have some limitations because of increasing type I error. Therefore, caution is needed for small sample size and data with high variability.

In this software review, we provide a brief overview of four R functions to estimate nonlinear mixed-effects programs: nlme (linear and nonlinear mixed-effects model), nlmer (from the lme4 package, linear mixed-effects models using Eigen and S4), saemix (stochastic approximation expectation maximization), and brms (Bayesian regression models using Stan). We briefly describe the approaches used, provide a sample code, and highlight strengths and weaknesses of each.

Some bootstrap methods in nonlinear mixed-effect models

An individual-tree diameter growth model was developed for Cunninghamia lanceolata in Fujian province, southeast China. Data were obtained from 72 plantation-grown China-fir trees in 24 single-species plots. Ordinary non-linear least squares regression was used to choose the best base model from among 5 theoretical growth equations; selection criteria were the smallest absolute mean residual and root mean square error and the largest adjusted coefficient of determination. To account for autocorrelation in the repeated-measures data, we developed one-level and nested two-level nonlinear mixed-effects (NLME) models, constructed on the selected base model; the NLME models incorporated random effects of the tree and plot. The best random-effects combinations for the NLME models were identified by Akaike's information criterion, Bayesian information criterion and −2 logarithm likelihood. Heteroscedasticity was reduced with two residual variance functions, a power function and an exponential function. The autocorrelation was addressed with three residual autocorrelation structures: a first-order autoregressive structure [AR(1)], a combination of first-order autoregressive and moving average structures [ARMA(1,1)] and a compound symmetry structure (CS). The one-level (tree) NLME model performed best. Independent validation data were used to test the performance of the models and to demonstrate the advantage of calibrating the NLME models.

IntroductionAccurate estimation of individual tree aboveground biomass (AGB) is essential for tree species selection, carbon accounting, and precision forestry. Unmanned aerial vehicle (UAV) LiDAR provides rapid access to detailed tree structural information, offering a promising tool for high-frequency biomass assessment.MethodsIn this study, a nonlinear mixed-effects (NLME) model integrating UAV LiDAR and field measurements was developed to quantify the influence of genetic heterogeneity and environmental factors on AGB estimation of Catalpa bungei. Data from 2,941 trees across 79 genotypes were collected in Henan Province, including LiDAR-derived tree height (LH), LiDAR-derived crown diameter (LCD), and AGB. By incorporating genotype as a random effect and planting density as a dummy variable, the NLME model significantly outperformed traditional dummy-variable models.ResultsGenotype effects explained significant AGB variation, achieving high accuracy (R²=0.7916, RMSE = 3.7095) and reducing TRE by 23.29% compared to the basic power function model. Leave-one-genotype-out cross-validation confirmed robustness. Calibration with the four largest trees yielded the best performance (TRE = 13.09%), while a simplified scheme using only two trees per genotype maintained high accuracy (TRE = 13.24%), markedly reducing field effort.DiscussionThese results highlight the superiority of NLME AGB models over linear approaches and demonstrate that accounting for genotype effects is critical for reliable biomass estimation. The proposed framework provides an efficient and cost-effective solution for biomass monitoring, tree breeding, carbon sink assessment, and precision forestry.

In the current study, a novel approach combining quantile regression with nonlinear mixed-effects (QR-NLME) modeling was applied to predict the maximum crown width (cwmax) of three economically important forest species—the European beech (Fagus sylvatica L.), the black pine (Pinus nigra Arn.), and the Calabrian pine (Pinus brutia Ten.) at tree level. A power QR-NLME model was fitted first to a dataset including 1414 European beech trees obtained from 29 randomly distributed sample plots, 770 black pine trees from 25 sample plots, and 1880 Calabrian pine trees from 41 sample plots in Greece, to predict the cwmax at tree level. Additionally, a nonlinear mixed-effects model (NLME) was fitted to the same dataset to predict the average crown width at tree level for all species. In the second stage, the crown competition factor (CCF) was estimated based on the population average response of the cwmax predictions. The proposed approach presented sound results when compared with the outcomes of relevant models from other regions fitted to open-grown tree data, and therefore, it can be well implemented on clustered data structures, in cases of absence of open-grown tree data.

IntroductionDiameter at breast height (DBH) is a key parameter for assessing tree growth, carbon storage, and ecological functions. Traditional ground surveys are inefficient, labor-intensive, and terrain-limited, making them unsuitable for large-scale monitoring. Airborne LiDAR, as an advanced remote sensing tool, provides an efficient and non-destructive method for DBH estimation. However, most existing LiDAR-based models overlook the influence of genotype differences, limiting prediction accuracy.MethodsIn this study, we used data from 2,899 Catalpa bungei trees of different genotypes to develop a nonlinear mixed-effects (NLME) model that incorporates genotype as a random effect. This approach improved model generalizability by using LiDAR-derived tree height (LH) and LiDAR-derived crown diameter (LCD) as core predictors. Multiple sampling strategies were also evaluated to assess their impact on model performance.ResultsThe results showed that, considering genotype effects, the proposed NLME model outperformed both traditional regression models and dummy-variable models (R2 = 0.8624, RMSE = 1.1330, TRE = 3.9555), demonstrating the important role of genotype differences in improving model accuracy. Random sampling further improved prediction accuracy while effectively reducing measurement costs.DiscussionThis research introduces a new framework for integrating genotype variability into DBH prediction models and offers valuable insights for future LiDAR-based studies in genetically heterogeneous plantations. The findings provide technical support for forest management and ecosystem monitoring, as well as a methodological foundation for predicting tree growth under varying site and genetic conditions.

Longitudinal data are data collected repeatedly from each subject for a particular response variable over a certain time period. Specifically, in longitudinal data analysis, the researchers are interested in changes in the response levels over time and the differences in these changes among factor levels or covariates. Because of within-subject correlations, analysis methods considering the correlations or variance-covariance structures have been developed. One of the approaches is the use of mixed effects models that take into account between-subject heterogeneity by random effects. In population pharmacokinetics, the response variable corresponds to drug concentration and is analysed typically using nonlinear mixed effects models. In this article, longitudinal data analysis with a continuous response variable is introduced focusing on population pharmacokinetics. Longitudinal data analysis, linear mixed effects models, nonlinear mixed effects models, and population pharmacokinetics are discussed from a biostatistical point of view. This article is expected to be of interest to biostatisticians, pharmacologists, pharmacokineticists, and those in related fields.

Long-Term Growth of the Neoaortic Root After Arterial Switch Operation

BackgroundLongitudinal data analysis can improve our understanding of the influences on health trajectories across the life-course. There are a variety of statistical models which can be used, and their fitting and interpretation can be complex, particularly where there is a nonlinear trajectory. Our aim was to provide an accessible guide along with applied examples to using four sophisticated modelling procedures for describing nonlinear growth trajectories.MethodsThis expository paper provides an illustrative guide to summarising nonlinear growth trajectories for repeatedly measured continuous outcomes using (i) linear spline and (ii) natural cubic spline linear mixed-effects (LME) models, (iii) Super Imposition by Translation and Rotation (SITAR) nonlinear mixed effects models, and (iv) latent trajectory models. The underlying model for each approach, their similarities and differences, and their advantages and disadvantages are described. Their application and correct interpretation of their results is illustrated by analysing repeated bone mass measures to characterise bone growth patterns and their sex differences in three cohort studies from the UK, USA, and Canada comprising 8500 individuals and 37,000 measurements from ages 5–40 years. Recommendations for choosing a modelling approach are provided along with a discussion and signposting on further modelling extensions for analysing trajectory exposures and outcomes, and multiple cohorts.ResultsLinear and natural cubic spline LME models and SITAR provided similar summary of the mean bone growth trajectory and growth velocity, and the sex differences in growth patterns. Growth velocity (in grams/year) peaked during adolescence, and peaked earlier in females than males e.g., mean age at peak bone mineral content accrual from multicohort SITAR models was 12.2 years in females and 13.9 years in males. Latent trajectory models (with trajectory shapes estimated using a natural cubic spline) identified up to four subgroups of individuals with distinct trajectories throughout adolescence.ConclusionsLME models with linear and natural cubic splines, SITAR, and latent trajectory models are useful for describing nonlinear growth trajectories, and these methods can be adapted for other complex traits. Choice of method depends on the research aims, complexity of the trajectory, and available data. Scripts and synthetic datasets are provided for readers to replicate trajectory modelling and visualisation using the R statistical computing software.

The population pharmacokinetics of gentamicin in neonates was determined using a nonlinear, mixed-effects model (NONMEM). The final regression equations derived to estimate clearance (Cl) and volume of distribution (Vd) were Cl = 0.120 * (WT/2.4)1.36 L/hr and Vd = 0.429 * (WT) L. The interindividual variability (% CV) for clearance was 26.2% and for volume of distribution 15.9%. Intraindividual variability was 11.0%. In a separate group of 30 neonates, the predictive ability of the NONMEM-generated population variables was compared to the predictions from a standard two-stage population analysis. The trough concentrations predicted using NONMEM-generated parameters were significantly less biased and more precise; there were no significant differences between the methods in predicting peaks. NONMEM is a useful tool for determining population pharmacokinetics and appears to be consistent across populations using routine clinical data and limited observation.

Accurate knowledge of static parameters forms the basis for nearly all applications in the energy management system. This paper proposes an efficient method for the simultaneous identification and correction of multiple network parameter errors based on a linear mixed-effects (LME) model and the generalized least squares (GLS) method. An LME model for parameter error identification is formulated using the residual equations of multiple-snapshot state estimation with equality constraints. The parameter errors are considered as the fixed effects and the measurement errors are considered as the random effects. Then, using the measurement error variances estimated by the LME model, the GLS method is used to estimate the parameter errors along with a hypothesis testing to infer whether each parameter error is zero. The semi-supervised adversarial autoencoder is used for bad data detection in the presence of erroneous parameters and limited labels such that only measurement snapshots that are free of any bad data are used. The proposed methodology is efficient in that the LME model is only used to estimate the variances of the measurement errors using a small number of measurement snapshots, therefore the huge computation burden needed to solve a large-scale LME model is avoided. In addition, the GLS only involves inversion of low-dimension matrices, which is very efficient even a large number of measurement snapshots are used. Thorough tests of the proposed methodology on a large number of scenarios are provided to show the effectiveness of the proposed methodology with promising results.