Additive nonlinear biomass equations: A likelihood-based approach

Since Parresol’s (Can. J. For. Res. 31:865– 878, 2001) seminal article on the topic, it has become standard to develop nonlinear tree biomass equations to ensure compatibility among total and component predictions and to fit these equations using multistep generalized least-squares methods. In particular, many studies have specified equations for total tree biomass by aggregating the expectations of M component biomass models and have fit all M + 1 equations jointly using seemingly unrelated regression. More recently, an alternative approach has been used wherein compatibility is ensured by deriving multiplicative component expectations, with estimation carried out using instrumental variables and generalized least squares. Yet neither of these strategies considers the fundamental additivity of biomass data themselves nor the implied stochastic constraints necessary for maximum likelihood (ML) estimation. For model selection based on information criteria, stochastic simulation, Bayesian inference, or estimation with missing data, it is important to base estimation and inference on valid probabilistic models. Here, we show how aggregative and disaggregative nonlinear equations can be specified within a probabilistic framework and fit using Gaussian ML with open-source software. We use Parresol’s slash pine (Pinus elliottii Engelm. var. elliottii) data to contrast model forms and predictions. We also show how the ML approach can accommodate unobserved or aggregated component biomass data and can thus be useful for integrating felled-tree data collected under different protocols.