Correction factors for unbiased, efficient estimation and prediction of biomass from log–log allometric models

Abstract

Highlights:• We review nine alternatives for correcting bias in log-log allometrics.• We use simulations to evaluate the ability of these to estimate average biomass.• We evaluate their ability to predict biomass of new trees.• Methods not commonly used in forest science performed best. AbstractAllometric relationships are commonly used to estimate average biomass of trees of a particular size and to predict biomass of individual trees based on an easily measured covariate variable such as stem diameter. They are typically power relationships which, for the purpose of data fitting, are transformed using natural logarithms to convert the model to its linear equivalent.Implementation of these equations to estimate the relationships and to predict biomass of new trees on the natural (i.e., actual) scale requires back-transforming the logarithmic predictions.Because these transformations involve non-linearity, care must be taken during this step to avoid bias. Several correction factors have been proposed in the literature for removing the gross bias in estimates, but their performance as predictors of biomass has not yet been examined. This is a very important problem, and here we review nine such correction factors in terms of their abilities to estimate biomass and predict biomass for new trees. We compare their performance by examining their bias and variability based on large datasets of above-ground biomass and stem diameter for eight species of harvested trees and shrubs in the genera Eucalyptus and Acacia (n = 102-365 individuals per species). We found that good estimates of average biomass turned out to be good predictors of biomass for new trees. The linear model fitted has log of the above-ground biomass as the response variable and log of the stem diameter as the covariate. The only exactly unbiased estimate among those considered was the uniform minimum variance unbiased (UMVU) estimate, which involves evaluating a confluent hypergeometric function to obtain its correction factor. Three alternative correction factors that are easy to compute also performed well. One of these minimises mean squared error and was found to result in low bias, low prediction bias, the lowest mean squared error, and the lowest mean squared prediction error among all correction factors examined.

Guidelines for constructing allometric models for the prediction of woody biomass: How many individuals to harvest?

“…In this study the 'MM' correction factor (Shen and Zhu 2008) was used, as described and recommended in the review by Clifford et al (2013).…”

Aboveground Dendromass Estimation of Juvenile Paulownia Sp.

“…Following the recommendation by Clifford et al (2013), we applied the MM correction factor by Shen & Zhu (2008) to amend the bias of the back-transformed predicted stem biomass. Although it is supposed to remove the bulk of the gross bias and must have superior performance in terms of the mean squared prediction error (Clifford et al, 2013), its estimation requires information of the parametrization data and operations with matrices (Equation 1).…”

“…Although it is supposed to remove the bulk of the gross bias and must have superior performance in terms of the mean squared prediction error (Clifford et al, 2013), its estimation requires information of the parametrization data and operations with matrices (Equation 1). Therefore, a sufficiently reliable and less sophisticated alternative to the MM coefficient could be the ratio correction factor (Snowdon, 1991;Clifford et al, 2013), which is the quotient between the antilogarithms of the mean experimental and the mean predicted values of the dependent variable. We estimated values of the ratio correction factor of 1.025 and 1.024 for models M1 and M4, respectively.…”

“…The MM correction factor was designed to (Shen & Zhu, 2008) and was recommended by Clifford et al (2013) for predicting biomass of new trees. Its estimated values, which differ among the individual trees, are multiplied by the antilogarithm of the predicted values of the log-transformed biomass to produce unbiased estimates.…”

Aboveground dendromass allometry of hybrid black poplars for energy crops

“…To convert the predicted values to arithmetic, untransformed units, additional correction for bias was required (Parresol 1999) and the ratio correction (Clifford et al 2013) was applied. Correction for bias was performed for each of the tree compartments separately, followed by their summation to obtain unbiased estimate of the total lignified biomass.…”