General Model Form 



When modeling diameter growth, one of the first decisions is the general form that the 

 model should take. The three most common forms suggested for growth modeling are the following: 



Y = -x + £ (1) 



Y = f(a,x) + e: (2) 



Y = fib,x) • e (3) 



where 



Y - the dependent growth variable 



X = an array of independent variables that may themselves be functions of basic growth 

 factors 



§, b = arrays of model parameters to be estimated in the appropriate fashions 

 f(,) = a nonlinear function 

 E = a random error. 



Model form (1) assumes that growth is a linear, additive function of transformed independent 

 variables. The estimation of the parameters of the model is done through ordinary least 

 squares regression techniques. Vuokila (1965) used this form when he modeled percent diameter 

 growth of individual trees. 



Model form (2) assumes that growth is a nonlinear function of the independent variables. 

 Because the error term, e, is additive, the appropriate technique for estimating the model 

 parameters is nonlinear, least squares regression (Kmenta 1971). 



Model form (3) differs from (2) in how the error term is introduced. In model (3), the 

 error term is multiplicative instead of additive. This allows the error structure model to be 

 linearized through the use of natural logarithms. For this model form to work, it is necessary 

 that the parameters (b) of the nonlinear function can also be linearized through the log 

 transforming process. The resulting model is of the form: 



(4) 



In (Y) = b' • g(x) + e* 

 where 



g(x ) = an array of appropriately transformed independent variables to linearize the 

 parameters of the the function f(b,x) 



e* = in(e) 



If these assumptions are met, then estimation of the parameters in model (4) can be done 

 by ordinary least squares regression techniques. Model form (4) has been used by Lemmon and 

 Schumacher (1962) to model individual tree periodic radial increment, and by Cole and Stage 

 (1972) and Stage (1973) to model individual tree periodic basal area increment. 



Another choice faced by the modeler of diameter growth is the form of the dependent 

 variable. Following are some of the choices: radial growth, diameter growth, basal area 

 growth, or any of the previous three expressed as a percentage. Cole and Stage (1972) and 

 Stage (1973) settled upon the usage of basal area growth for two reasons: 



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