Session 2: Methodologies
(Moderated by R.L. Bailey)
Contents
Using
Strong Inference to Falsify Differential Equation Models of Tree Height Growth
(Daniel B. Hall and Robert L. Bailey)
Examples
of Practical Methods for Unbiased Parameter Estimation in Self-Referencing
Functions
(M. Harrison, S. Martin, and C.J. Cieszewski)
Using strong inference to falsify differential
equation models of tree height growth
Rolfe Leary
Rolfe Leary & Associates
1382 West Iowa Avenue St. Paul 55108 USA
Abstract
The scientific research strategy called 'strong inference' is widely studied, but is difficult to apply in practice. Here we describe the strong inference steps in the context of selecting mathematical models of tree height growth expressed as differential equations. Three model groups proposed by Zeide (1993) are used as alternatives. These are supplemented with a 4th group -- models formed as second order differential equations, giving four alternative hypotheses that nearly exhaust the possible biological models currently in the literature. The crucial experiment called for in strong inference involves fitting a system of the equations to <height – age> data collected from a cohort of trees felled for stem analysis. Models for the cohort members are identical in right-hand-side algebraic form and share common parameters, but have tree-specific initial height conditions. Results showed the class of models Zeide called logarithm of time decline were extremely sensitive to initial height conditions, and lead to a field experiment based on a modus tollens argument form because initial heights are not that critical to final height attained. The modus tollens argument form is the basis for Popper’s falsification strategy. Conclusions are based on detailed stem analysis data for cohorts of sugar maple, quaking aspen, and northern red oak trees growing in the Lake States, USA.
*Presented by Rolfe A. Leary.
On the Use of Nonlinear Multilevel Mixed Model to
Predict Height Growth in Forest Plantations
Dan Hall
Department of Statistics
University of Georgia
Athens 30606 USA
Robert L. Bailey
Warnell School of Forest Resources
University of Georgia
Athens, GA 30602
Abstract
In forest management it is important to have good statistical models of tree growth. In particular, it is desirable to have models from which accurate predictions of yield at harvest can be made. In this talk we consider the problem of fitting and making predictions from nonlinear multilevel mixed models for tree height; that is, nonlinear models in which random effects occur with a nested structure. Nested random effects structures are natural for modelling forestry data sets in which there are multiple observations on each of many trees within each of several stands. Wolfinger and Lin (1997) consider estimating equation approaches to fitting more general nonlinear mixed effects models, and we adapt their zero-expansion estimating equations to the multi-level case. We develop methods of prediction based on these models that allow prediction of height at harvest both for individual trees and for stand averages. We illustrate these methods by fitting and making predictions from a Richards-type growth curve model for tree height data from the B.F. Grant Spacing Study of loblolly pine. The accuracy of the predictions and their standard errors is examined by means of cross-validation.
*Presented
by Daniel Hall.
Examples of Practical Methods for Unbiased Parameter
Estimation in Self-Referencing Functions
Mike Harrison, Stacey Martin, and Chris J.
Cieszewski
School of Forest Resources
University of Georgia
Athens USA
Abstract
Since its inception in 1976, the primary objective of the Plantation Management Research Cooperative (PMRC) at the Warnell School of Forest Resources has been the development of growth and yield models for southern pines. Recent advances in computer and software technology have enabled the PMRC to explore the most computationally intensive data analysis methods. So-called traditional estimation techniques for dominant height projection equations involve the arbitrary choice of observed growth intervals and base ages, where each combination results in a different set of parameter estimates. Bailey and Clutter (1974) introduced the concept of base age invariant site index equations. In this paper, we present two practical approaches to this estimation technique and compare their results to those obtained from traditional methodologies. The base age invariant techniques produced identical results regardless of the choice of base ages or applied algorithm. The traditional methods were notably affected by the choice of base age and measurement intervals.
*Presented by Micheal Harrison.
Mechanisms Causing Bias in Parameter Estimates for
Site Index Models and other Self-Referencing Functions
Chris J. Cieszewski
School of Forest Resources
University of Georgia
Athens 30602 USA
Abstract
The traditional methods of parameter estimation for site index models, and other self-referencing equations, rely on the assumption that the site index measurements do not contain errors. This assumption and methods affect resulting estimates and biases in the parameters' estimates may result. This, in turn, may lead to erroneous predictions of forest developments. Others have discussed this problem in the literature, pointing at potential reasons for the biases in parameter estimates, and have offered informal solutions. I give a new perspective on this problem and explain some of the bias-generating mechanisms, which have not been previously discussed in the literature. I present arguments that are applicable to most of the common practices applied in development of site index equations and other self-referencing growth and yield models. The main problem with the traditional parameter estimation methods relates to the restrictive nature of the implicitly defined equations, which are required for the definition of the self-referencing models. In models based on such equations, the curves are forced through the exact values of the measurements' errors whenever the models' initial conditions assume values containing these errors. This phenomenon is the main cause of bias in the parameter estimates. The solution to this problem is in rejecting the traditional assumption that the site index measurements do not contain errors. Such rejection leads to applications of methods that estimate the model parameters simultaneously with the estimation of measurement errors in the initial conditions.
*Presented
by Chris J. Cieszewski.
Last revised: April 6, 2000