Estimation and Prediction of Forest Growth Variables Based on Multilevel Nonlinear Mixed Models. Department of Statistics

ABSTRACT. In this article, we describe estimation and prediction methods for nonlinear modeling of forest growth variables that are subject to nested sources of variability. The multilevel nonlinear mixedeffects models that we consider are useful for a variety of forestry applications, but we concentrate on the problem of estimating, and making projections from, growth curves for tree height based on longitudinal data grouped by location. Wolfinger and Lin consider estimating equation approaches to fitting more general nonlinear mixed-effects models, and we adapt their zero-expansion estimating equations to the multilevel case. We develop methods of prediction based on these models that allow predictions of future height both for individual trees and for plot averages. We illustrate these methods by fitting and making predictions from a Chapman-Richards type growth model for tree height data from a loblolly pine spacing study in Putnam County, Georgia. The mean and variance of prediction errors based on our methods are examined by means of cross-validation. We provide a more complete and unified presentation of linearization-based estimation and prediction based on multilevel nonlinear mixed-effects models than has previously appeared in the forestry literature, and we argue that these models lead to substantial advantages in growth and yield prediction over traditional forestry methods. OR MANY PURPOSES IN FORESTRY, it is useful to be able to make accurate future predictions of the mean values of growth variables based on repeated measurements through time made on units that are grouped hierarchically. For example, many forest management decisions are based on yield projections that crucially depend on projections of plot level averages of tree height, basal area, and other morphometric variables. In the forestry literature, Lappi and Bailey (1988) have described the use of one example of a nonlinear mixed effects growth curve model to predict dominant tree height both at the plot level and at the individual tree level. In their model, random effects for plots, and for trees nested within plots enter into a ChapmanRichards-type growth model in a linear manner. Although not originally described in these terms, their model is an example of a multi-level nonlinear mixed effects model (ML-NLMM), because random effects are included in the model corresponding to units of heterogeneity (plots and trees) that have a hierarchical structure with multiple levels. Here we consider a more general version of Lappi and Bailey's model in which multilevel random effects are allowed to enter into the model nonlinearly. We describe methods of fitting such models based on first-order Taylor series linearization techniques that have been developed recently in the rapidly expanding statistical literature on nonlinear mixed-effects models (NLMMs), adapting these methods to the special challenges of the multilevel case. We also develop prediction methods based on these models that allow predictions to be made locally at each level in the model (e.g., to particular trees or plots of interest), as well as globally across the entire population represented in the data. When measurements are available on the units of interest (perhaps from ages prior to the target age), these predictors make use of this relevant information and thus can be ex- Forest Science 47(3) 2001 pected to outperform some other methods of prediction that have appeared in the literature on NLMMs applied to forestry problems (e.g., Schabenberger 1996a, 1996b). The methods we describe apply quite generally to nonlinear modeling of responses that are subject to nested sources of variability. However, we will concentrate on the problem of estimating, and making projections from, growth curves for tree height based on longitudinal data (repeated measures through time) that are grouped by location (typically by plots, but possibly by plots within stands, etc.). In the Example section below, we illustrate these methods by fitting Chapman-Richards growth curves to tree height data from an old-field loblolly pine (Pinus taeda L.) spacing study conducted in Putnam County, Georgia, by the D.B. Warnell School of Forest Resources, University of Georgia (Pienaar and Shiver 1993). Our goals are twofold: (1) to deliver the state-of-the-art in nonlinear mixed-effects modeling methodology to a forest biometrics audience in the context of an important forestry application of these models; and (2) to illustrate the advantages of this approach over traditional methods for growth and yield modeling based on site-index curves