Bias in the mean tree model as a consequence of Jensen's Inequality

Stand growth models keep track of only mean tree properties for projections of growth, because tree lists are not maintained. These properties are typically calculated as simple averages of easily-measurable tree dimensions. An example of this type of model is 3-PG, which uses only the mean diameter at breast height (D) of the stand to describe stand growth. The mean tree approach can lead to serious bias, as a consequence of Jensen's Inequality. This type of model is therefore applicable only to even-aged stands that are relatively homogenous, although it is unclear how homogenous stands must be to avoid serious error. We investigate the magnitude of the bias for the allornetric equation that predicts stem mass from stem diameter, a very common function in process-based growth models. We compare an approximation of this bias, based on a Taylor expansion, with a stochastic simulation. The two methods generally agree well, especially when the coefficient of variation (CV) is low. For the average CV from a large data set of managed stands, and an allometric exponent of 2.3, the bias would be 11.7%. An example with forest inventory data shows that the bias is often much larger. We show how the bias due to Jensen's inequality can be approximated with a very simple expression that is a function of only the CV and the exponent in the allometric equation. (C) 2003 Elsevier B.V. All rights reserved.