Linear mixed-effects models to describe individual tree crown width for China-fir in Fujian Province, southeast China

A multiple linear model was developed for individual tree crown width of Cunninghamia lanceolata (Lamb.) Hook in Fujian province, southeast China. Data were obtained from 55 sample plots of pure China-fir plantation stands. An Ordinary Linear Least Squares (OLS) regression was used to establish the crown width model. To adjust for correlations between observations from the same sample plots, we developed one level linear mixed-effects (LME) models based on the multiple linear model, which take into account the random effects of plots. The best random effects combinations for the LME models were determined by the Akaike's information criterion, the Bayesian information criterion and the -2logarithm likelihood. Heteroscedasticity was reduced by three residual variance functions: the power function, the exponential function and the constant plus power function. The spatial correlation was modeled by three correlation structures: the first-order autoregressive structure [AR(1)], a combination of first-order autoregressive and moving average structures [ARMA(1,1)], and the compound symmetry structure (CS). Then, the LME model was compared to the multiple linear model using the absolute mean residual (AMR), the root mean square error (RMSE), and the adjusted coefficient of determination (adj-R2). For individual tree crown width models, the one level LME model showed the best performance. An independent dataset was used to test the performance of the models and to demonstrate the advantage of calibrating LME models.

Nonlinear mixed-effects (NLME) diameter growth models for individual China-Fir (Cunninghamia lanceolata) trees in Southeast China

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.

Non-linear random effects models with continuous time autoregressive errors: a Bayesian approach

Measurements on subjects in longitudinal medical studies are often collected at several different times or under different experimental conditions. Such multiple observations on the same subject generally produce serially correlated outcomes. Traditional regression methods assume that observations within subjects are independent which is not true in longitudinal data. In this paper we develop a Bayesian analysis for the traditional non-linear random effects models with errors that follow a continuous time autoregressive process. In this way, unequally spaced observations do not present a problem in the analysis. Parameter estimation of this model is done via the Gibbs sampling algorithm. The method is illustrated with data coming from a study in pregnant women in Santiago, Chile, that involves the non-linear regression of plasma volume on gestational age.

Estimation of growth parameters from multiple-recapture data

This article develops a method for analysis of growth data with multiple recaptures when the initial ages for all individuals are unknown. The existing approaches either impute the initial ages or model them as random effects. Assumptions about the initial age are not verifiable because all the initial ages are unknown. We present an alternative approach that treats all the lengths including the length at first capture as correlated repeated measures for each individual. Optimal estimating equations are developed using the generalized estimating equations approach that only requires the first two moment assumptions. Explicit expressions for estimation of both mean growth parameters and variance components are given to minimize the computational complexity. Simulation studies indicate that the proposed method works well. Two real data sets are analyzed for illustration, one from whelks (Dicathais aegaota) and the other from southern rock lobster (Jasus edwardsii) in South Australia.

Determination of the unknown age at first capture of western rock lobsters (Panulirus cygnus) by random effects model

We propose a method for fitting growth curves to multiple recapture data of lobsters when the age at first capture is unknown. The von Bertalanffy growth curve is used to model the growth. To account for individual variability, the unknown age in logarithmic scale of a lobster at first capture, the individual asymptotic size, and the individual growth coefficient of its carapace length are modeled as random effects with a trivariate normal distribution. Unlike previously suggested models, the present model permits correlation between the growth coefficient and the age at first capture and can be fitted readily using existing software. The error structures between consecutive recaptures of a lobster are assumed to be a first-order autoregressive process with unequally spaced time points. A comparison between this model and the Fabens growth equation is given. The proposed method is a flexible method and can be applied to fit different growth equations when the age at first capture is unknown.

Estimating equations for parameters in stochastic growth models from tag-recapture data

James (1991, Biometrics 47, 1519-1530) constructed unbiased estimating functions for estimating the two parameters in the von Bertalanffy growth curve from tag-recapture data. This paper provides unbiased estimating functions for a class of growth models that incorporate stochastic components and explanatory variables. A simulation study using seasonal growth models indicates that the proposed method works well while the least-squares methods that are commonly used in the literature may produce substantially biased estimates. The proposed model and method are also applied to real data from tagged rock lobsters to assess the possible seasonal effect on growth.

Rate of decline of chlorfluazuron concentration in the fat of cattle

OBJECTIVE

To determine the rate of decline of chlorfluazuron (CFZ) concentration in the fat of cattle.

DESIGN

A field depletion study.

ANIMALS

Fifteen steers that had become contaminated with CFZ through eating cotton trash or cotton leaf pellets derived from CFZ-treated cotton crops.

PROCEDURE

Fat samples were collected from the cattle at about 3 week intervals according to a schedule where each animal was sampled on four occasions up to 340 days after removal from the contaminated feed source.

RESULTS

When the effects of dilution are removed CFZ concentrations were found to decline slowly for about 200 days. Depletion was minimal between 200 and 340 days.

CONCLUSION

According to this trial, CFZ-contaminated, nonlactating cattle which have finished growing will remain contaminated. Field experience has not supported this conclusion.

Some general estimation methods for nonlinear mixed-effects models

A nonlinear mixed-effects model suitable for characterizing repeated measurement data is described. The model allows dependence of random coefficients on covariate information and accommodates general specifications of a common intraindividual covariance structure, such as models for variance within individuals that depend on individual mean response and autocorrelation. Two classes of procedures for estimation in this model are described, which incorporate estimation of unknown parameters in the assumed intraindividual covariance structure. The procedures are straightforward to implement using standard statistical software. The techniques are illustrated by examples in growth analysis and assay development.