Nonlinear Mixed-Effects Models for Repairable Systems Reliability

Mixed-effects models,also called random-effects models,are a regression type of analysis which enables the analyst to not only describe the trend over time within each subject,but also to describe the variation among different subjects.Nonlinear mixed-effects models provide a powerful and flexible tool for handling the unbalanced count data.In this paper,nonlinear mixed-effects models are used to analyze the failure data from a repairable system with multiple copies.By using this type of models,statistical inferences about the population and all copies can be made when accounting for copy-to-copy variance.Results of fitting nonlinear mixed-effects models to nine failure-data sets show that the nonlinear mixed-effects models provide a useful tool for analyzing the failure data from multi-copy repairable systems.

Mixed-Effects Parametric Proportional Hazard Model with Generalized Log-Logistic Baseline Distribution

Clustered survival data are widely observed in a variety of setting. Most survival models incorporate clustering and grouping of data accounting for between-cluster variability that creates correlation in order to prevent underestimate of the standard errors of the parameter estimators but do not include random effects. In this study, we developed a mixed-effect parametric proportional hazard (MEPPH) model with a generalized log-logistic distribution baseline. The parameters of the model were estimated by the application of the maximum likelihood estimation technique with an iterative optimization procedure (quasi-Newton Raphson). The developed MEPPH model’s performance was evaluated using Monte Carlo simulation. The Leukemia dataset with right-censored data was used to demonstrate the model’s applicability. The results revealed that all covariates, except age in PH models, were significant in all considered distributions. Age and Townsend score were significant when the GLL distribution was used in MEPPH, while sex, age and Townsend score were significant in MEPPH model when other distributions were used. Based on information criteria values, the Generalized Log-Logistic Mixed-Effects Parametric Proportional Hazard model (GLL-MEPPH) outperformed other models.

Empirical Likelihood for Mixed-effects Error-in-variables Model

This paper mainly introduces the method of empirical likelihood and its applications on two different models. We discuss the empirical likelihood inference on fixed-effect parameter in mixed-effects model with error-in-variables. We first consider a linear mixed-effects model with measurement errors in both fixed and random effects. We construct the empirical likelihood confidence regions for the fixed-effects parameters and the mean parameters of random-effects. The limiting distribution of the empirical log likelihood ratio at the true parameter is X2p＋q, where p, q are dimension of fixed and random effects respectively. Then we discuss empirical likelihood inference in a semi-linear error-in-variable mixed-effects model. Under certain conditions, it is shown that the empirical log likelihood ratio at the true parameter also converges to X2p＋q. Simulations illustrate that the proposed confidence region has a coverage probability more closer to the nominal level than normal approximation based confidence region.

Bayesian quantile semiparametric mixed-effects double regression models

Semiparametric mixed-effects double regression models have been used for analysis of longitu-dinal data in a variety of applications,as they allow researchers to jointly model the mean and variance of the mixed-effects as a function of predictors.However,these models are commonly estimated based on the normality assumption for the errors and the results may thus be sensitive to outliers and/or heavy-tailed data.Quantile regression is an ideal alternative to deal with these problems,as it is insensitive to heteroscedasticity and outliers and can make statistical analysis more robust.In this paper,we consider Bayesian quantile regression analysis for semiparamet-ric mixed-effects double regression models based on the asymmetric Laplace distribution for the errors.We construct a Bayesian hierarchical model and then develop an efficient Markov chain Monte Carlo sampling algorithm to generate posterior samples from the full posterior dis-tributions to conduct the posterior inference.The performance of the proposed procedure is evaluated through simulation studies and a real data application.

Linear mixed-effects model for longitudinal complex data with diversified characteristics

The increasing richness of data encourages a comprehensive understanding of economic and financial activities,where variables of interest may include not only scalar(point-like)indicators,but also functional(curve-like)and compositional(pie-like)ones.In many research topics,the variables are also chronologically collected across individuals,which falls into the paradigm of longitudinal analysis.The complicated nature of data,however,increases the difficulty of modeling these variables under the classic longitudinal frame-work.In this study,we investigate the linear mixed-effects model(LMM)for such complex data.Different types of variables arefirst consistently represented using the corresponding basis expansions so that the classic LMM can then be conducted on them,which gener-alizes the theoretical framework of LMM to complex data analysis.A number of simulation studies indicate the feasibility and effectiveness of the proposed model.We further illustrate its practical utility in a real data study on Chinese stock market and show that the proposed method can enhance the performance and interpretability of the regression for complex data with diversified characteristics.

Maximum likelihood estimation of nonlinear mixed-effects models with crossed random effects by combining first-order conditional linearization and sequential quadratic programming

Nonlinear mixed-eirects (NLME) modek have become popular in various disciplines over the past several decades.However,the existing methods for parameter estimation imple-mented in standard statistical packages such as SAS and R/S-Plus are generally limited k) single-or multi-level NLME models that only allow nested random effects and are unable to cope with crossed random effects within the framework of NLME modeling.In t his study,wc propose a general formulation of NLME models that can accommodate both nested and crassed random effects,and then develop a computational algorit hm for parameter estimation based on normal assumptions.The maximum likelihood estimation is carried out using the first-order conditional expansion (FOCE) for NLME model linearization and sequential quadratic programming (SCJP) for computational optimization while ensuring positive-definiteness of the estimated variance-covariance matrices of both random effects and error terms.The FOCE-SQP algorithm is evaluated using the height and diameter data measured on trees from Korean larch (L.olgeiisis var,Chang-paienA.b) experimental plots aa well as simulation studies.We show that the FOCE-SQP method converges fast with high accuracy.Applications of the general formulation of NLME models are illustrated with an analysis of the Korean larch data.

Monte Carlo Likelihood Estimation of Mixed-Effects State Space Models with Application to HIV Dynamics

The statistical inference for generalized mixed-effects state space models （MESSM） are investigated when the random effects are unknown. Two filtering algorithms are designed both of which are based on mixture Kalman filter. These algorithms are particularly useful when the longitudinal ts are sparse. The authors also propose a globally convergent algorithm for parameter estimation of MESSM which can be used to locate the initial value of parameters for local while more efficient algorithms. Simulation examples are carried out which validate the efficacy of the proposed approaches. A data set from the clinical trial is investigated and a smaller mean square error is achieved compared to the existing results in literatures.

BLUP Estimation of Linear Mixed-effects Models with Measurement Errors and Its Applications to the Estimation of Small Areas

The linear mixed-effects model （LMM） is a very useful tool for analyzing cluster data. In practice, however, the exact values of the variables are often difficult to observe. In this paper, we consider the LMM with measurement errors in the covariates. The empirical BLUP estimator of the linear combination of the fixed and random effects and its approximate conditional MSE are derived. The application to the estimation of small area is provided. Simulation study shows good performance of the proposed estimators.

Generalized or general mixed-effect modelling of tree morality of Larix gmelinii subsp.principis-rupprechtii in Northern China

Tree mortality models play an important role in predicting tree growth and yield,but existing mortality models for Larix gmelinii subsp.principis-rupprechtii,an important species used for regeneration and afforestation in northern China,have overlooked potential regional influences on tree mortality.This study used data acquired from 102 temporary sample plots(TSPs)in natural stands of Prince Rupprecht larch in the state-owned Guandi Mountain Forest(n=67)and state-owned Boqiang Forest(n=35)in northern China.To model stand-level tree mortality,we compared seven model forms of county data.Three continuous(dominant height,plot mean diameter,and basal area per hectare)and one dummy variable with two levels(region)were used as fixed effects variables.Tree morality variations caused by forest blocks were accounted for using forest blocks as a random effect in selected models.Results showed that tree mortality significantly positively correlated with stand basal area and dominant height,but negatively correlated with stand mean diameter.Incorporating both the dummy variables and random effects into the tree mortality models significantly increased the fitting improvements,and Hurdle Poisson mixed-effects model showed the most attractive fit statistics(largest R^(2)and smallest RMSE)when employing leave-one-out cross-validation.These mixed-effects dummy variable models will be useful for accurately predicting Larix tree mortality in different regions.

Analysis of the inflection points of height-diameter models

The inflection point is an important feature of sigmoidal height-diameter(H-D)models.It is often cited as one of the properties favoring sigmoidal model forms.However,there are very few studies analyzing the inflection points of H-D models.The goals of this study were to theoretically and empirically examine the behaviors of inflection points of six common H-D models with a regional dataset.The six models were the Wykoff(WYK),Schumacher(SCH),Curtis(CUR),HossfeldⅣ(HOS),von Bertalanffy-Richards(VBR),and Gompertz(GPZ)models.The models were first fitted in their base forms with tree species as random effects and were then expanded to include functional traits and spatial distribution.The distributions of the estimated inflection points were similar between the two-parameter models WYK,SCH,and CUR,but were different between the threeparameter models HOS,VBR,and GPZ.GPZ produced some of the largest inflection points.HOS and VBR produced concave H-D curves without inflection points for 12.7%and 39.7%of the tree species.Evergreen species or decreasing shade tolerance resulted in larger inflection points.The trends in the estimated inflection points of HOS and VBR were entirely opposite across the landscape.Furthermore,HOS could produce concave H-D curves for portions of the landscape.Based on the studied behaviors,the choice between two-parameter models may not matter.We recommend comparing seve ral three-parameter model forms for consistency in estimated inflection points before deciding on one.Believing sigmoidal models to have inflection points does not necessarily mean that they will produce fitted curves with one.Our study highlights the need to integrate analysis of inflection points into modeling H-D relationships.

Ecoregional height-diameter models for Scots pine in Turkiye

Ecoregion-based height-diameter models were developed in the present study for Scots pine(Pinus sylves-tris L.)stands in Turkiye and included several ecological factors derived from a pre-existing ecoregional classification system.The data were obtained from 2831 sample trees in 292 sample plots.Ten generalized height–diameter models were developed,and the best model(HD10)was selected according to statistical criteria.Then,nonlinear mixed-effects modeling was applied to the best model.The R2 for the generalized height‒diameter model(Richards function)modified by Sharma and Parton is 0.951,and the final model included number of trees,dominant height,and diameter at breast height,with a random parameter associated with each ecoregion attached to the inverse of the mean basal area.The full model predictions using the nonlinear mixed-effects model and the reduced model(HD10)predictions were compared using the nonlinear sum of extra squares test,which revealed significant differences between ecore-gions;ecoregion-based height–diameter models were thus found to be suitable to use.In addition,using these models in appropriate ecoregions was very important for achieving reliable predictions with low prediction errors.

National-scale data-driven rainfall induced landslide susceptibility mapping for China by accounting for incomplete landslide data

China is one of the countries where landslides caused the most fatalities in the last decades. The threat that landslide disasters pose to people might even be greater in the future, due to climate change and the increasing urbanization of mountainous areas. A reliable national-scale rainfall induced landslide susceptibility model is therefore of great relevance in order to identify regions more and less prone to landsliding as well as to develop suitable risk mitigating strategies. However, relying on imperfect landslide data is inevitable when modelling landslide susceptibility for such a large research area. The purpose of this study is to investigate the influence of incomplete landslide data on national scale statistical landslide susceptibility modeling for China. In this context, it is aimed to explore the benefit of mixed effects modelling to counterbalance associated bias propagations. Six influencing factors including lithology, slope,soil moisture index, mean annual precipitation, land use and geological environment regions were selected based on an initial exploratory data analysis. Three sets of influencing variables were designed to represent different solutions to deal with spatially incomplete landslide information: Set 1(disregards the presence of incomplete landslide information), Set 2(excludes factors related to the incompleteness of landslide data), Set 3(accounts for factors related to the incompleteness via random effects). The variable sets were then introduced in a generalized additive model(GAM: Set 1 and Set 2) and a generalized additive mixed effect model(GAMM: Set 3) to establish three national-scale statistical landslide susceptibility models: models 1, 2 and 3. The models were evaluated using the area under the receiver operating characteristics curve(AUROC) given by spatially explicit and non-spatial cross-validation. The spatial prediction pattern produced by the models were also investigated. The results show that the landslide inventory incompleteness had a substantial impact on the outcomes of the statistical landslide susceptibility models. The cross-validation results provided evidence that the three established models performed well to predict model-independent landslide information with median AUROCs ranging from 0.8 to 0.9.However, although Model 1 reached the highest AUROCs within non-spatial cross-validation(median of 0.9), it was not associated with the most plausible representation of landslide susceptibility. The Model 1 modelling results were inconsistent with geomorphological process knowledge and reflected a large extent the underlying data bias. The Model 2 susceptibility maps provided a less biased picture of landslide susceptibility. However, a lower predicted likelihood of landslide occurrence still existed in areas known to be underrepresented in terms of landslide data(e.g., the Kuenlun Mountains in the northern Tibetan Plateau). The non-linear mixed-effects model(Model 3) reduced the impact of these biases best by introducing bias-describing variables as random effects. Among the three models, Model 3 was selected as the best national-scale susceptibility model for China as it produced the most plausible portray of rainfall induced landslide susceptibility and the highest spatially explicit predictive performance(median AUROC of spatial cross validation 0.84) compared to the other two models(median AUROCs of 0.81 and 0.79, respectively). We conclude that ignoring landslide inventory-based incompleteness can entail misleading modelling results and that the application of non-linear mixed-effect models can reduce the propagation of such biases into the final results for very large areas.

Modelling tree mortality across diameter classes using mixedeffects zero-inflated models

The mortality of trees across diameter class model is a useful tool for predicting changes in stand structure.Mortality data commonly contain a large fraction of zeros and general discrete models thus show more errors.Based on the traditional Poisson model and the negative binomial model,different forms of zero-inflated and hurdle models were applied to spruce-fir mixed forests data to simulate the number of dead trees.By comparing the residuals and Vuong test statistics,the zero-inflated negative binomial model performed best.A random effect was added to improve the model accuracy;however,the mixed-effects zero-inflated model did not show increased advantages.According to the model principle,the zeroinflated negative binomial model was the most suitable,indicating that the"0"events in this study,mainly from the sample"0",i.e.,the zero mortality data,are largely due to the limitations of the experimental design and sample selection.These results also show that the number of dead trees in the diameter class is positively correlated with the number of trees in that class and the mean stand diameter,and inversely related to class size,and slope and aspect of the site.

Large scale mapping of forest attributes using heterogeneous sets of airborne laser scanning and National Forest Inventory data

Background:The Norwegian forest resource map(SR16)maps forest attributes by combining national forest inventory(NFI),airborne laser scanning(ALS)and other remotely sensed data.While the ALS data were acquired over a time interval of 10 years using various sensors and settings,the NFI data are continuously collected.Aims of this study were to analyze the effects of stratification on models linking remotely sensed and field data,and assess the accuracy overall and at the ALS project level.Materials and methods:The model dataset consisted of 9203 NFI field plots and data from 367 ALS projects,covering 17 Mha and 2/3 of the productive forest in Norway.Mixed-effects regression models were used to account for differences among ALS projects.Two types of stratification were used to fit models:1)stratification by the three main tree species groups spruce,pine and deciduous resulted in species-specific models that can utilize a satellite-based species map for improving predictions,and 2)stratification by species and maturity class resulted in stratum-specific models that can be used in forest management inventories where each stand regularly is visually stratified accordingly.Stratified models were compared to general models that were fit without stratifying the data.Results:The species-specific models had relative root-mean-squared errors(RMSEs)of 35%,34%,31%,and 12% for volume,aboveground biomass,basal area,and Lorey’s height,respectively.These RMSEs were 2-7 percentage points(pp)smaller than those of general models.When validating using predicted species,RMSEs were 0-4 pp.smaller than those of general models.Models stratified by main species and maturity class further improved RMSEs compared to species-specific models by up to 1.8 pp.Using mixed-effects models over ordinary least squares models resulted in a decrease of RMSE for timber volume of 1.0-3.9 pp.,depending on the main tree species.RMSEs for timber volume ranged between 19%-59% among individual ALS projects.Conclusions:The stratification by tree species considerably improved models of forest structural variables.A further stratification by maturity class improved these models only moderately.The accuracy of the models utilized in SR16 were within the range reported from other ALS-based forest inventories,but local variations are apparent.

Modelling height-diameter relationships in complex tropical rain forest ecosystems using deep learning algorithm

Modelling tree height-diameter relationships in complex tropical rain forest ecosystems remains a challenge because of characteristics of multi-species, multi-layers, and indeterminate age composition. Effective modelling of such complex systems required innovative techniques to improve prediction of tree heights for use for aboveground biomass estimations. Therefore, in this study, deep learning algorithm (DLA) models based on artificial intelligence were trained for predicting tree heights in a tropical rain forest of Nigeria. The data consisted of 1736 individual trees representing 116 species, and measured from 52 0.25 ha sample plots. A K-means clustering was used to classify the species into three groups based on height-diameter ratios. The DLA models were trained for each species-group in which diameter at beast height, quadratic mean diameter and number of trees per ha were used as input variables. Predictions by the DLA models were compared with those developed by nonlinear least squares (NLS) and nonlinear mixed-effects (NLME) using different evaluation statistics and equivalence test. In addition, the predicted heights by the models were used to estimate aboveground biomass. The results showed that the DLA models with 100 neurons in 6 hidden layers, 100 neurons in 9 hidden layers and 100 neurons in 7 hidden layers for groups 1, 2, and 3, respectively, outperformed the NLS and NLME models. The root mean square error for the DLA models ranged from 1.939 to 3.887 m. The results also showed that using height predicted by the DLA models for aboveground biomass estimation brought about more than 30% reduction in error relative to NLS and NLME. Consequently, minimal errors were created in aboveground biomass estimation compared to those of the classical methods.

Bayesian analysis of minimal model under the insulin-modified IVGTT

A Bayesian analysis of the minimal model was proposed where both glucose and insulin were analyzed simultaneously under the insulin-modified intravenous glucose tolerance test (IVGTT). The resulting model was implemented with a nonlinear mixed-effects modeling setup using ordinary differential equations (ODEs), which leads to precise estimation of population parameters by separating the inter- and intra-individual variability. The results indicated that the Bayesian method applied to the glucose-insulin minimal model provided a satisfactory solution with accurate parameter estimates which were numerically stable since the Bayesian method did not require approximation by linearization.

Stand basal area modelling for Chinese fir plantations using an artificial neural network model

Artificial neural network models are a popular estimation tool for fitting nonlinear relationships because they require no assumptions about the form of the fitting function,non-Gaussian distributions,multicollinearity,outliers and noise in the data.The problems of backpropagation models using artificial neural networks include determination of the structure of the network and overlearning courses.According to data from 1981 to 2008 from 15 permanent sample plots on Dagangshan Mountain in Jiangxi Province,a back-propagation artificial neural network model(BPANN)and a support vector machine model(SVM)for basal area of Chinese fir(Cunninghamia lanceolata)plantations were constructed using four kinds of prediction factors,including stand age,site index,surviving stem numbers and quadratic mean diameters.Artificial intelligence methods,especially SVM,could be effective in describing stand basal area growth of Chinese fir under different growth conditions with higher simulation precision than traditional regression models.SVM and the Chapman–Richards nonlinear mixed-effects model had less systematic bias than the BPANN.

Associations of short-term ambient temperature exposure with lung function in middle-aged and elderly people:A longitudinal study in China

The short-term associations of ambient temperature exposure with lung function in middle-aged and elderly Chinese remain obscure.The study included 19,128 participants from the Dongfeng-Tongji cohort's first(2013)and second(2018)follow-ups.The lung function for each subject was determined between April and December 2013 and re-assessed in 2018,with three parameters(forced vital capacity[FVC],forced expiratory volume in 1 s[FEV1],and peak expiratory flow[PEF])selected.The China Meteorological Data Sharing Service Center provided temperature data during the study period.In the two follow-ups,a total of 25,511 records(average age:first,64.57;second,65.80)were evaluated,including 10,604 males(41.57%).The inversely J-shaped associations between moving average temperatures(lag01–lag07)and FVC,FEV1,and PEF were observed,and the optimum temperatures at lag04 were 16.5C,18.7C,and 16.2C,respectively.At lag04,every 1C increase in temperature was associated with 14.07 mL,9.78 mL,and 62.72 mL/s increase in FVC,FEV1,and PEF in the lowtemperature zone(<the optimum temperatures),whereas 5.72 mL,2.01 mL,and 11.64 mL/s decrease in the high-temperature zone(the optimum temperatures),respectively(all P<0.05).We observed significant effect modifications of gender,age,body mass index,body surface area,smoking status,drinking status,and physical activity on the associations(all Pmodification<0.05).Non-optimal temperatures may cause lung function decline.Several individual characters and lifestyles have effect modification on the temperature effects.

Using the response-effect trait framework to disentangle the effects of environmental change on the ecosystem services

Functional traits play a vital role in mediating the responses of ecosystem services to environmental changes and in predicting the functioning of the ecosystem.However,the connection between functional traits and ecosystem services has become increasingly intricate due to climate change and human activities for degraded ecosystems.To investigate this relationship,we selected 27 sampling sites in the Yanhe River Basin of the Chinese Loess Plateau,each containing two types of vegetation ecosystems:natural vegetation and artificial vegetation ecosystems.At each sampling site,we measured ecosystem services and calculated the composition index of community traits.We established a response–effect trait framework that included environmental factors such as climate,elevation and human activities.Our results showed that leaf tissue density(LTD)was the overlapping response and effect trait when responding to climate change.LTD is positively correlated with mean annual temperature and negatively correlated with supporting services.Under the influence of human activities,leaf nitrogen content and leaf dry matter content were carriers of environmental change.Comparing the two vegetation ecosystems,the relationship between functional traits and ecosystem services showed divergent patterns,indicating that human activities increased the uncertainty of the relationship between functional traits and ecosystem services.Trait-based ecology holds promise for enhancing predictions of ecosystem services responses to environmental changes.However,the predictive ability is influenced by the complexity of environmental changes.In conclusion,our study highlights the importance of understanding the complex connection between functional traits and ecosystem services in response to climate changes and human activities.

Lower clearance of sodium tanshinone IIA sulfonate in coronary heart disease patients and the effect of total bilirubin: a population pharmacokinetics analysis

This study developed a population pharmacokinetic model for sodium tanshinone IIA sulfonate(STS) in healthy volunteers and coronary heart disease(CHD) patients in order to identify significant covariates for the pharmacokinetics of STS. Blood samples were obtained by intense sampling approach from 10 healthy volunteers and sparse sampling from 25 CHD patients, and a population pharmacokinetic analysis was performed by nonlinear mixed-effect modeling. The final model was evaluated by bootstrap and visual predictive check. A total of 230 plasma concentrations were included, 137 from healthy volunteers and 93 from CHD patients. It was a two-compartment model with first-order elimination. The typical value of the apparent clearance(CL) of STS in CHD patients with total bilirubin(TBIL) level of 10 μmol×L^(–1) was 48.7 L×h^(–1) with inter individual variability of 27.4%, whereas that in healthy volunteers with the same TBIL level was 63.1 L×h^(–1). Residual variability was described by a proportional error model and estimated at 5.2%. The CL of STS in CHD patients was lower than that in healthy volunteers and decreased when TBIL levels increased. The bootstrap and visual predictive check confirmed the stability and validity of the final model. These results suggested that STS dosage adjustment might be considered based on TBIL levels in CHD patients.