#### Abstract

Equations to predict the volume of an individual tree bole between stump height and the height at which its diameter has tapered to a specified minimum are common in forestry. When fitting such a regression equation, a sample of trees which span the range of sizes needed for eventual application of the equation is selected. Bole diameter is measured at ascending heights on the bole. Each tree, therefore, contributes multiple measurements to the data fitted to the equation. In contrast to past practice, we model these data in a manner which accounts for the likely spatial correlation among measurements within a tree. The resulting mixed-effects nonlinear model is fitted by REML and also by generalized estimating equations (GEE). Results from the two approaches are nearly identical, which suggests that the computationally less demanding GEE may be acceptable as a routine alternative to a fully parameterized approach.

#### Keywords

Nonlinear modeling, mixed-effects models, REML, GEE

#### Creative Commons License

This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License.

FITTING BOLE-VOLUME EQUATIONS TO SPATIALLY CORRELATED WITHIN-TREE DATA

Equations to predict the volume of an individual tree bole between stump height and the height at which its diameter has tapered to a specified minimum are common in forestry. When fitting such a regression equation, a sample of trees which span the range of sizes needed for eventual application of the equation is selected. Bole diameter is measured at ascending heights on the bole. Each tree, therefore, contributes multiple measurements to the data fitted to the equation. In contrast to past practice, we model these data in a manner which accounts for the likely spatial correlation among measurements within a tree. The resulting mixed-effects nonlinear model is fitted by REML and also by generalized estimating equations (GEE). Results from the two approaches are nearly identical, which suggests that the computationally less demanding GEE may be acceptable as a routine alternative to a fully parameterized approach.