Scaling up uncertainties in allometric models: How to see the forest, not the trees

Abstract

Quantifying uncertainty in forest assessments is challenging because of the number of sources of error and the many possible approaches to quantify and propagate them. The uncertainty in allometric equations has sometimes been represented by propagating uncertainty only in the prediction of individuals, but at large scales with large numbers of trees uncertainty in model fit is more important than uncertainty in individuals. We compared four different approaches to representing model uncertainty: a formula for the confidence interval, Monte Carlo sampling of the slope and intercept of the regression, bootstrap resampling of the allometric data, and a Bayesian approach. We applied these approaches to propagating model uncertainty at four different scales of tree inventory (10 to 10,000 trees) for four study sites with varying allometry and model fit statistics, ranging from a monocultural plantation to a multi-species shrubland with multi-stemmed trees. We found that the four approaches to quantifying uncertainty in model fit were in good agreement, except that bootstrapping resulted in higher uncertainty at the site with the fewest trees in the allometric data set (48), because outliers could be represented multiple times or not at all in each sample. The uncertainty in model fit did not vary with the number of trees in the inventory to which it was applied. In contrast, the uncertainty in predicting individuals was higher than model fit uncertainty when applied to small numbers of trees, but became negligible with 10,000 trees. The importance of this uncertainty source varied with the forest type, being largest for the shrubland, where the model fit was most poor. Low uncertainties were observed where model fit was high, as was the case in the monoculture plantation and in the subtropical jungle where hundreds of trees contributed to the allometric model. In all cases, propagating uncertainty only in the prediction of individuals would underestimate allometric uncertainty. It will always be most correct to include both uncertainty in predicting individuals and uncertainty in model fit, but when large numbers of individuals are involved, as in the case of national forest inventories, the contribution of uncertainty in predicting individuals can be ignored. When the number of trees is small, as may be the case in forest manipulation studies, both sources of allometric uncertainty are likely important and should be accounted for.