Height-Diameter Model for Managed Even-aged Stands of Ponderosa Pine for the Western United States Using Hierarchical Nonlinear Mixed-Effects Model

Abstract

Background: A regional individual-tree Height-Diameter (H-D) model was developed for ponderosa pine (P. ponderosa var. ponderosa and var. scopulorum), for the western United States. The dataset came from long-term permanent research plots in even-aged pure stands, both planted and natural on sites capable of the productivity estimated by Meyer [Meyer, W.H., 1938. Yield of Even-Aged Stands of Ponderosa Pine. USDA Technical Bulletin 630]. The database used a common study plan. The study plan divided the range of ponderosa pine trees into five provinces, with installations in Arizona (AZ), northern California (CA), Oregon (OR), Washington (WA), Montana (MT), and South Dakota (SD), USA. Regional H-D models exist for ponderosa pine. However, the study plans, the data collection and the analysis procedures used in developing the models differ. Consequently, comparisons of growth responses that may be due to geographic variation of the species are not possible. Objective: 1) develop a single regional H-D model for ponderosa pine trees that can provide useful guidelines for a variety of management objectives throughout the species range; 2) determine if inclusion of plot-specific auxiliary variables improves prediction accuracy of regional H-D models. Methodology: The dataset consisted of 305 plots and 29,449 trees. The study used 8208 trees that had both height and diameter measurements. The dataset has repeated measurements, which makes the measurements correlated. Additionally, the dataset has location effects and hierarchical structure; consequently, it violated the fundamental regression assumption of independent observations. Nonlinear Mixed-Effects Model (NLMM) approach provided an excellent solution. Specifically, NLMM allows parameter vectors to vary from plot to plot and variance components are simultaneously broken down into a) fixed-effect components that describe the shape of the typical growth curve over the entire population, b) random-effect components that enables the curve to capture the plot-specific and other unit-specific characteristics of the growth pattern. Further, NLMM allows the explicit incorporation of the hierarchical structure of the dataset. Finally, the inclusion of plot-specific auxiliary variables individualized the curve for each plot and location. Conclusions: The Chapman-Richards model form provided the best result. The results demonstrated that irrespective of geographic location, a regional H-D Model with variance components and plot-specific auxiliary variables that individualized the curve to capture the sitespecific, plot-specific, and other unit-specific characteristics of the growth pattern decisively outperformed regional H-D Model with variance components but without plot-specific auxiliary variables. Consequently, the model developed herein should enhance comparisons of H-D relationships that may be due to geographic variation of the species throughout the range of ponderosa pine trees. Therefore, it would provide forest managers and researchers with useful guidelines for a variety of management objectives at the regional scale.