Some bootstrap methods in nonlinear mixed-effect 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.

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.

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 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.

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.

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.

Using ultrashort echo time (UTE) magnetic resonance imaging (MRI), T2∗ can be measured and used to evaluate cartilage degradation. In general, high-quality MRI is required for accurate T2∗ measurement. However, such high-quality MRI requires long scan times which make the clinical utility of T2∗ measurement infeasible. Thus, a method that can produce satisfactory T2∗ maps within reasonable scan times (i.e. limited data) is required. Non-linear mixed-effects model (NLME) is a powerful tool for analyzing incomplete longitudinal data. Since NLME contains both fixed and random effects, it is preferred over more traditional methods. The aim of this study is to investigate whether T2∗ map can be generated using NLME. The simulation results show that NLME is a feasible method for T2∗ measurement. Moreover, NLME can improve the accuracy and precision of parameter estimates compared to the traditional non-linear least square fit.

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.

Model for estimation of total nitrogen content in sandalwood leaves based on nonlinear mixed effects and dummy variables using multispectral images

Cancer cell lines (CCLs) have been widely used to study of cancer. Recent studies have called into question the reliability of data collected on CCLs. Hence, we set out to determine CCLs that tend to be overly sensitive or resistant to a majority of drugs utilizing a nonlinear mixed-effects (NLME) modeling framework. Using drug response data collected in the Cancer Cell Line Encyclopedia (CCLE) and the Genomics of Drug Sensitivity in Cancer (GDSC), we determined the optimal functional form for each drug. Then, a NLME model was fit to the drug response data, with the estimated random effects used to determine sensitive or resistant CCLs. Out of the roughly 500 CCLs studies from the CCLE, we found 17 cell lines to be overly sensitive or resistant to the studied drugs. In the GDSC, we found 15 out of the 990 CCLs to be excessively sensitive or resistant. These results can inform researchers in the selection of CCLs to include in drug studies. Additionally, this study illustrates the need for assessing the dose-response functional form and the use of NLME models to achieve more stable estimates of drug response parameters.

Comparison of nonparametric methods in nonlinear mixed effects models

Variability is an intrinsic property of biological systems and is often at the heart of their complex behaviour. Examples range from cell-to-cell variability in cell signalling pathways to variability in the response to treatment across patients. A popular approach to model and understand this variability is nonlinear mixed effects (NLME) modelling. However, estimating the parameters of NLME models from measurements quickly becomes computationally expensive as the number of measured individuals grows, making NLME inference intractable for datasets with thousands of measured individuals. This shortcoming is particularly limiting for snapshot datasets, common e.g. in cell biology, where high-throughput measurement techniques provide large numbers of single cell measurements. We introduce a novel approach for the estimation of NLME model parameters from snapshot measurements, which we call filter inference. Filter inference uses measurements of simulated individuals to define an approximate likelihood for the model parameters, avoiding the computational limitations of traditional NLME inference approaches and making efficient inferences from snapshot measurements possible. Filter inference also scales well with the number of model parameters, using state-of-the-art gradient-based MCMC algorithms such as the No-U-Turn Sampler (NUTS). We demonstrate the properties of filter inference using examples from early cancer growth modelling and from epidermal growth factor signalling pathway modelling.

The characterization of the dynamics of canopy coverage represents a relevant matter of study in the field of crop physiology. The objective of this work was to calibrate a model able to simulate potato canopy coverage as a function of thermal time, but including the error structure in such model. This was accomplished by using a mixed effects modelling approach where random effects were added to the average response model. By applying this modelling approach, the structure of the data was taken into account. Calibration data for the model were obtained from canopy coverage estimates derived from image processing analysis. Digital images were taken periodically within 11 potato fields located in the Mantaro Valley (Peru) during the 2005–2006 growing season. This model gave a better fit in comparison with the traditional fixed parameters model. An additional uncertainty analysis with the objective of estimating the confidence region for the predictions of the mixed effects model was carried out. By exploring the data structure, a more comprehensive overview of the potato canopy coverage was achieved with the mixed effects model.