A practical measure for determining if diameter (D) and height (H) should be combined into D2H in allometric biomass models

Tree diameter at breast height (D) and tree height (H) are often used as predictors of individual tree biomass. Because D and H are correlated, the combined variable D2H is frequently used in regression models instead of two separate independent variables, to avoid collinearity related issues. The justification for D2H is that above-ground biomass is proportional to the volume of a cylinder of diameter, D, and height, H. However, the D2H predictor constrains the model to produce parameter estimates for D and H that have a fixed ratio, in this case, 2.0. In this paper we investigate the degree to which the D2H predictor reduces prediction accuracy relative to D and H separately and propose a practical measure, Q-ratio, to guide the decision as to whether D and H should or should not be combined into D2H. Using five training biomass datasets and two fitting approaches, weighted nonlinear regression and linear regression following logarithmic transformations, we showed that the D2H predictor becomes less efficient in predicting aboveground biomass as the Q-ratio deviates from 2.0. Because of the model constraint, the D2H-based model performed less well than the separate variable model by as much as 12 per cent with regard to mean absolute percentage residual and as much as 18 per cent with regard to sum of squares of log accuracy ratios. For the analysed datasets, we observed a wide variation in Q-ratios, ranging from 2.5 to 5.1, and a large decrease in efficiency for the combined variable model. Therefore, we recommend using the Q-ratio as a measure to guide the decision as to whether D and H may be combined further into D2H without the adverse effects of loss in biomass prediction accuracy.