Additivity of component biomass regression equations when the underlying model is linear

Abstract

Results obtained by Kozak (A. Kozak. 1970. For. Chron. 46(5): 402–404.) concerning conditions for additivity of component biomass regression equations are formalized and extended. More specifically Kozak demonstrated, using multiple linear regression equations to model three biomass components (bole, bark, and crown) for individual trees, that corresponding total biomass can be determined as the sum of the component regression equations, provided that the same independent variables are used in each component equation. Clearly, Kozak's result can be extended to the case of k (≥2) component equations and we use this case to give more general conditions for the additivity problem. Results are also given for the estimation and inference problems associated with fitting the total biomass model using the additivity result. Additionally, by using the principle of fitting subject to constraints or what has been termed "conditional fitting," it is demonstrated in this paper that additivity of the component equations can be assured even when different independent variables are used in different component equations subject to certain assumptions being met. This principle is then used to generalize the additivity of component regression equations problem and, finally, an example is given to illustrate the application of this generalized additivity theory.