Using SAEM to estimate parameters of models of response to applied fertilizer

Many nonlinear functions have been used to predict crop yield response to applied fertilizer. Accurate estimates of the parameter values are required for the formulation of satisfactory fertilizer dose recommendations. In fertilizer trials, several yield measurements for different fertilizer doses are usually made at the same site and in the same year. Model errors are therefore unlikely to be independent for the same site-year. Nonlinear mixed effects models can be used for yield response data. We evaluated the performance of a recently proposed stochastic approximation of the EM algorithm (SAEM) for estimating the parameters of nonlinear mixed effects models. We used the SAEM method to estimate the parameters of four different nonlinear models, using a real dataset including 37 site-years of yield measurements. We then carried out simulations, to determine the bias and the root mean squared errors of the estimators. We compared the results obtained with SAEM with those obtained using a first-order conditional method. SAEM gave better results in most cases. The estimates produced by SAEM were less biased and less affected by initial values.     

Statistical inference for calibration points in nonlinear mixed effects models

Many disciplines conduct studies in which the primary objectives depend on inference based on a nonlinear relationship between the treatment and response. In particular, interest often focuses on calibration—that is, estimating the best treatment level to achieve a particular result. Often, data for such calibration come from experiments with split-plots or other features that result in multiple error terms or other nontrivial error structures. One such example is the time-of-weed-removal study in weed science, designed to estimate the critical period of weed control. Calibration, or inverse prediction, is not a trivial problem with simple linear regression, and the complexities of experiments such as the time-of-weed-removal study further complicate the procedure. In this article, we extend existing calibration techniques to nonlinear mixed effects models, and illustrate the procedure using data from a time-of-weed-removal study.     

Regression spline mixed models: A forestry example

In this article, regression splines are used inside linear mixed models to explore nonlinear longitudinal data. The regression spline bases are generated using a single knot chosen using biological information—a knot position supported by an automated knot selection procedure. A variety of inferential procedures are compared. The variance in the data was closely modeled using a flexible model-based covariance structure, a robust method and the nonparametric bootstrap, while the variance was underestimated when independent random effects were assumed.     

An Approximate Method for Population Toxicokinetic Analysis With Aggregated Data

Standard statistical models for analyzing inter-individual variability in clinical pharmacokinetics (nonlinear mixed effects; hierarchical Bayesian) require individual data. However, for environmental or occupational toxicants only aggregated data are usually available, so toxicokinetic analyses typically ignore population variability. We propose a hierarchical Bayesian approach to estimate inter-individual variability from the observed mean and variance at each time point, using a bivariate normal (or lognormal) approximation to their joint likelihood. Through analysis of both simulated data and real toxicokinetic data from 1,3-butadiene exposures, we conclude that given information on the form of the individual-level model, useful information on inter-individual variability may be obtainable from aggregated data, but that additional sensitivity and identifiability checks are recommended.     

The simultaneous analysis of mixed discrete and continuous outcomes using nonlinear threshold models

Mixed discrete and continuous outcomes are commonly measured on each experimental unit in dose-response studies in toxicology. The dose-response relationships for these outcomes often have dose thresholds and nonlinear patterns. In addition, the endpoints are typically correlated, and a statistical analysis that incorporates the association may result in improved precision. We propose an extension of the generalized estimating equation (GEE) methodology to simultaneously analyze binary, count, and continuous outcomes with nonlinear threshold models that incorporates the intra-subject correlation. The methodology uses a quasi-likelihood framework and a working correlation matrix, and is appropriate when the marginal expectation of each outcome is of primary interest and the correlation between endpoints is a nuisance parameter. Because the derivatives of threshold models are not continuous at each point of the parameter space, we describe the necessary modifications that result in asymptotically normal and consistent estimators. Using dose-response data from a neurotoxicity experiment, the methodology is illustrated by analyzing five outcomes of mixed type with nonlinear threshold models. In this example, the incorporation of the intra-subject correlation resulted in decreased standard errors for the threshold parameters.     

Using a nonlinear mixed effects model to characterize cholinesterase activity in rats exposed to Aldicarb

A statistical mixed effects model was applied to data from a two-phased experiment in which cholinesterase activity (CA) in red blood cells (RBCs) of rats was monitored following single-and double-gavagead ministrations of the carbamate pesticide, Aldicarb. Our goals were to develop and estimate a nonlinear mixed effects model for describing the inhibition and recovery pattern of CA as a continuous function of time and Aldicarb dose, and to use the results to characterize both the typical-animal CA response and its variability across animals. The Phase 1 experiment involved adult male CD® rats randomly assigned to a control and two dose groups. Blood samples were taken via jugular cannula prior to exposure and at 11 post-dose time points (over six hours) for determination of RBC CA. The Phase II experiment was similar, but involved a second administration approximately 4.5 hours after the first. An open, one-compartment model with first-order kinetics appeared to provide an adequate structural basis for the statistical nonlinear mixed effects model, which was fit to the RBC data of the 26 treatment-group rats from both phases. For the “typical” animal, the model predicts that about a 20% (40%) maximal reduction in activity will occur for the low (high) dose group and that recovery to 90% of the baseline level will occur in 105 minutes (179 minutes) for the low (high) dose group. Based on simulated distributions derived from the model’s variance-covariance matrix of random effects, we estimated that about 99% of animals are expected to require less than 143 minutes (267 minutes) to recover to 90% of their baseline level for the low (high) dose group. Because virtually all animals had returned to near pre-exposure, levels by the time of the second administration, the level of the first dosing had little impact on the CA patterns following the second dose.     

Power analysis based on spatial effects mixed models: A tool for comparing design and analysis strategies in the presence of spatial variability

Spatial variability among experimental units is known to exist in many designed experiments. Agronomic field trials are a particularly well-known example, but there are others. Historically, spatial variability has been dealt with in one of two ways: either though design, by blocking to account for spatial effects, or though analysis, by nearest neighbor adjustment. More recently, mixed models with spatial covariance structures such as those used in geostatistics have been proposed. These mixed model procedures have tempted some to conclude—to the dismay of many consulting statisticians—that design principles may be bypassed, since spatial covariance models can recover any lost information. Although design principles clearly should not be ignored, spatial procedures do raise questions. Are traditional notions of appropriate design affected? If so, how? How do spatial effects mixed models compare to conventional analysis of variance used in conjunction with blocked designs? This article presents mixed model methods to assess power and precision of proposed designs in the presence of spatial variability and to compare competing design and analysis strategies. The main conclusion is that, if anything, spatial models reinforce the need for sound design principles, particularly the use of incomplete block designs.     

Effect of solutes and matrix structure on water mobility in glycerol-agar-water gel systems: a nuclear magnetic resonance approach

Nuclear magnetic resonance spectroscopy (NMR) has been widely used to determine water molecular mobility in food systems. This study aimed to examine the effects of matrix structure and solutes on the dynamics of water molecules in model mixed systems, glycerol-agar-water gels, using low- and high-resolution NMR. Simple models to explain water relaxation rates and self-diffusion coefficients in mixed systems were developed using the experimental values obtained for the individual binary systems (glycerol-water solutions and agar-water gels). The spin-lattice relaxation of mixed systems was influenced by interactions of both glycerol and agar with water, while the spin-spin relaxation of mixed systems was dominated by the interaction of agar with water. Water diffusion was influenced by not only molecular interactions between all components but also the gel matrix structure. These models are able to differentiate the effect of solutes from that of matrix structure on water molecular dynamics.     

嘉陵江上游低山暴雨区不同水土保持林结构模式水源涵养效益研究

水土保持林的水源涵养效益是研究森林防止水土流失的一个热点问题。以20世纪80年代末期营造和自然恢复的湿地松林、桤柏混交林、刺槐(灌木)林 、刺槐林 和桤木林为研究对象,选择林冠截留、土壤容重、土壤孔隙度、土壤贮水能力、土壤渗透性和枯落物持水性,作为研究不同水土保持林结构模式水源涵养效益的指标,对不同水土保持林的水源涵养机理和水源涵养效益进行了较深入的研究。结果表明,各水土保持林的水源涵养能力均明显高于农耕地;不同水土保持林结构模式的水源涵养效益也有较大差异。最后提出了桤柏混交林、湿地松林、桤木林等优良水土保持林结构模式,为当地的生态环境建设,特别是当前正在进行的退耕还林工程的模式选择提供了依据。     

Statistical models for exposure-biomarker relationships with measurement error and censoring

Measurements of biomarkers have been used, in conjunction with those of external exposure, to provide information about the rates of human uptake, bioactivation, and detoxification of chemicals. Direct knowledge of these processes is valuable because it can reduce reliance on animal models for making extrapolations about disease risks in humans. In practice, however, valid and precise quantification of exposure—biomarker relationships has been hampered by a number of problems. These include errors in the measurement of true mean exposure and true mean biomarker levels, nonlinear exposure—biomarker relationships, background biomarker levels in unexposed individuals, and response levels that fall below analytic limits of detection. In this article we develop nonlinear mixed models and corresponding likelihood expressions that can address these concerns, and we describe maximum likelihood estimation techniques that permit valid statistical inferences to be made using standard software. The models and methods are applied to both experimental and simulated exposure—biomarker data to illustrate their utility and limitations.     

Analysis of unbalanced mixed model data: A case study comparison of ANOVA versus REML/GLS

Major transition has occurred in recent years in statistical methods for analysis of linear mixed model data from analysis of variance (ANOVA) to likelihood-based methods. Prior to the early 1990s, most applications used some version of analysis of variance because computer software was either not available or not easy to use for likelihood-based methods. ANOVA is based on ordinary least squares computations, with adoptions for mixed models. Computer programs for such methodology were plagued with technical problems of estimability, weighting, and handling missing data. Likelihood-based methods mainly use a combination of residual maximum likelihood (REML) estimation of covariance parameters and generalized least squares (GLS) estimation of mean parameters. Software for REML/GLS methods became readily available early in the 1990s, but the methodology still is not universally embraced. Although many of the computational inadequacies have been overcome, conceptual problems remain. Also, technical problems with REML/GLS have emerged, such as the need for adjustments for effects due to estimating covariance parameters. This article attempts to identify the major problems with ANOVA, describe the problems which remain with REML/GLS, and discuss new problems with REML/GLS.     

Comparison of linear and nonlinear regression for modeling the first-order degradation of pest-control substances in soil

First-order kinetic models are often used to profile the degradation of pest-control compounds in soil. This approach is based on enzyme theory and is often favored due to its simplicity and its requirement by regulatory agencies. Here, linear and nonlinear regression approaches to modeling first-order decay are compared. Composite residual plots of many soil degradation data sets are presented on a normalized scale. These plots illustrate the general error structure for the data and are useful for detecting common mis-specifications of the models. Results indicate that a nonlinear regression approach to modeling first-order decay of compounds in soil more accurately describes most data sets when compared with a linear approach. Specifically, the observed error structure does not support the broad use of a logarithmic transformation to stabilize the variance. In addition, models generated using the linear approach generally exhibit more dramatic systematic deviations from the observations as compared with models generated using the nonlinear approach. The analysis methods described here may be useful for comparing alternative models in this and other research areas.     

工程车辆车桥位移谱统计分布建模及分步参数识别

针对非公路用车的车桥实测位移谱统计分布建模中模型选择、参数识别的初值选取主观性大和计算效率低等难题,该文以实测的车桥位移信号为研究对象,分别进行时域分析、频域功率谱分析,对信号进行分组,统计频数,获得统计直方图和累计概率分布曲线。分别采用正态分布、双峰正态分布、威布尔分布和双峰威布尔分布模型对位移谱进行建模,提出分步参数识别方法。引入灰色关联度目标函数,以人工鱼群算法获得的参数作为模型参数的初始值,采用迭代非线性最小二乘法levenberg-marquardt (LM)算法进行精确参数识别,使用相关系数和kolmogorov-smirnov(KS)检验对各模型的拟合优度进行比较。结果表明,混合威布尔分布与统计直方图的相关系数为(0.9800,0.9908,0.9867,0.9665),混合正态分布为(0.9793,0.9904,0.9783,0.9661),威布尔模型为(0.8613,0.9113,0.8618,0.8854),正态模型为(0.8611,0.9127,0.8624,0.8869),混合威布尔模型可以对车桥位移谱进行高精度拟合,而所提出的分步参数识别法可以高效、准确地进行模型的参数识别。研究结果可为车辆疲劳载荷谱的编制和台架试验提供参考。     

Gaussian Copula Mixed Models for Clustered Mixed Outcomes,With Application in Developmental Toxicology

This paper is concerned with the analysis of clustered data from developmental toxicity studies with mixed responses, i.e., where each member of the cluster has binary and continuous outcomes. A copula-based random effects model is proposed that accounts for associations between binary and/or continuous outcomes within clusters, including the intrinsic association between the mixed outcomes for the same subject. The approach allows the adoption of flexible distributions for the mixed outcomes as well as for the random effects. The model includes the correlated probit model of Gueorguieva and Agresti (2001) and the generalized linear mixed models of Faes et al. (2008), and Faes, Geys, and Catalano (2009) as special cases. Maximum likelihood estimation of our model parameters is implemented using standard software such as PROC NLMIXED in SAS. The proposed methodology is motivated by and illustrated using a developmental toxicity study of ethylene glycol in mice. This article has supplementary material online.     

Formulating mixed models for experiments, including longitudinal experiments

Mixed models have become important in analyzing the results of experiments, particularly those that require more complicated models (e.g., those that involve longitudinal data). This article describes a method for deriving the terms in a mixed model. Our approach extends an earlier method by Brien and Bailey to explicitly identify terms for which autocorrelation and smooth trend arising from longitudinal observations need to be incorporated in the model. At the same time we retain the principle that the model used should include, at least, all the terms that are justified by the randomization. This is done by dividing the factors into sets, called tiers, based on the randomization and determining the crossing and nesting relationships between factors. The method is applied to formulate mixed models for a wide range of examples. We also describe the mixed model analysis of data from a three-phase experiment to investigate the effect of time of refinement on Eucalyptus pulp from four different sources. Cubic smoothing splines are used to describe differences in the trend over time and unstructured covariance matrices between times are found to be necessary.     

不同坡度下掺沙对黑土坡面径流侵蚀特征的影响

黑土区野外考察发现,当地农民通过土壤掺沙的方法改良土壤质地,以提高作物产量。鉴于此,基于人工模拟降雨试验,探究东北典型坡度(3°,5°和7°)条件下掺沙比例(0,10%,20%和30%)对黑土坡面径流侵蚀特征的影响,以期为东北农业生产和土壤侵蚀防治提供科学依据。结果表明:掺沙比例对坡面径流特征的影响较坡度更加明显;随着坡度的增加,掺沙比例对径流特征的影响增大。随着掺沙比例的增加,平均径流流速从5.2～5.3 cm/s减小到2.7～3.4 cm/s,产流率和径流量逐渐减小,掺沙30%时达到最小值。产沙率随掺沙比例和坡度的增大呈先增加再减小的变化。侵蚀量随掺沙比例增大变化显著,呈先增加后减小的变化,在掺沙10%和20%时较大。可见,土壤掺沙(20%)在减小坡面径流量的同时,亦能增加坡面土壤侵蚀量。从水土保持角度分析,掺沙处理不利于防治黑土坡面土壤流失,但是综合考虑其生态经济效益,适度掺沙能够作为一种黑土区坡耕地农业技术措施。     

Application of an efficient nonlinear regression technique for sewer cost modelling

This article investigates the effectiveness of a nonlinear regression technique for estimating the empirical parameters contained in a separable and a generalized cost functions. The regression technique needs only to optimize the values of the nonlinear parameters and thus makes it considerably easier for a user to employ it for modelling of a sewer cost function. The results of this study indicated that the regression technique is capable of providing a set of reasonably good estimates for the empirical parameters contained in the cost models under a wide range of initial guessed solutions and under a wide level of incorporated noise. In addition, the two types of cost models investigated were found to be rather insensitive to some minor errors in the estimated values of its model parameters     

Randomly Truncated Nonlinear Mixed-Effects Models

A class of truncated nonlinear mixed-effects models is constructed by assuming that the variable of interest follows a truncated distribution parametrized by a location and a scale parameter. The location parameter of the responses is associated with a nonlinear continuous function of covariates and unknown parameters, and with unobserved random effects. We also assume the scale parameter of the responses to be characterized by a known continuous function of covariates and unknown parameters. Maximum likelihood estimator of the parameters is obtained by direct maximization of the log-likelihood function via an iterative procedure, and diagnostic analysis tools are considered to check for model adequacy. A data set consisting of observations on soil-water retention from a soil profile from the Buriti Vermelho River Basin database is analyzed using the proposed methodology.