*Annals of Applied Statistics*11, no. 1: 340-361, available at https://projecteuclid.org/euclid.aoas/1491616884.

Prediction of aboveground biomass, particularly at large spatial scales, is necessary for estimating global-scale carbon sequestration. Since biomass can be measured only by sacrificing trees, total biomass on plots is never observed. Rather, allometric equations are used to convert individual tree diameter to individual biomass, perhaps with noise. The values for all trees on a plot are then summed to obtain a derived total biomass for the plot. Then, with derived total biomasses for a collection of plots, regression models, using appropriate environmental covariates, are employed to attempt explanation and prediction. Not surprisingly, when out-of-sample validation is examined, such a model will predict total biomass well for holdout data because it is obtained using exactly the same derived approach.
Apart from the somewhat circular nature of the regression approach, it also fails to employ the actual observed plot level response data. At each plot, we observe a random number of trees, each with an associated diameter, producing a sample of diameters. A model based on this random number of tree diameters provides understanding of how environmental regressors explain abundance of individuals, which in turn explains individual diameters.
We incorporate density dependence because the distribution of tree diameters over a plot of fixed size depends upon the number of trees on the plot. After fitting this model, we can obtain predictive distributions for individual-level biomass and plot-level total biomass. We show that predictive distributions for plot-level biomass obtained from a density-dependent model for diameters will be much different from predictive distributions using the regression approach. Moreover, they can be more informative for capturing uncertainty than those obtained from modeling derived plot-level biomass directly.
We develop a density-dependent diameter distribution model and illustrate with data from the national Forest Inventory and Analysis (FIA) database. We also describe how to scale predictions to larger spatial regions. Our predictions agree (in magnitude) with available wisdom on mean and variation in biomass at the hectare scale.