1. Basal area growth is often nearly linear over short time periods, and this makes 

 extrapolation of growth rates easier for growth periods different from that originally used in 

 equation development. 



2. The log of basal area growth is often found to be more normally distributed with 

 homogeneous variance than other dependent variables. 



The latter finding may also indicate that the residuals of the log of basal area growth 

 are additive. If the assumption of normality, homogeneity of variance and additive, indepen- 

 dent residuals can be met, then the ordinary least squares estimators, b, are the maximum 

 likelihood estimators and the UMVUE's (Uniformly Minimum- Variance Unbiased Estimator), and 

 various procedures for testing the significance of the model and its parameters can be validly 

 applied (Kmenta 1971; Draper and Smith 1966). 



Given these various choices of model forms and dependent variables, I decided that the 

 log of basal area growth would be used because: 



1. Previous experiences with modeling diameter growth in ponderosa pine'^ indicate that a 

 nonlinear, multiplicative model best represents the interaction of the independent variables 

 with themselves and their effect upon diameter growth. 



2. Cole and Stage (1972) and Stage (1973) have shown that basal area growth is often 

 lognormally distributed with a multiplicative error term. This fits the requirements of model 

 (3). 



The Random Error Component 



The two approaches for handling the random error component are stochastic modeling and 

 deterministic modeling. In stochastic modeling, each component of the model with an error 

 element is randomly assigned an error value from the proper distribution. The prediction from 

 this component is then modified by this random disturbance, and this "randomized" prediction 

 interacts with the other model components to produce a randomized estimate from the model 

 (usually with an unknov\m error structure). This process is then repeated a number of times, 

 each time using a new set of random errors, and the results of these numerous trials can be 

 averaged to get an "expected" model estimate. This repetitive stochastic process is often 

 called a "Monte Carlo" process. One disadvantage of this approach is the sometimes large 

 number of computations needed to find the "expected" estimates. 



The deterministic approach assigns each component of the model its expected value. There 

 is no random element. Each component interacts with other components in the appropriate 

 fashion to produce model estimates. The advantage of deterministic models is that "expected" 

 estimates are provided directly. 



Stage (1973) describes and uses a method in his prognosis model that maintains the relative 

 simplicity of the deterministic approach while introducing a stochastic element. He calls the 

 approach a Monte Carlo "swindle." He says, "The purpose is to produce a prognosis that over- 

 all is the result of averaging many replications of the random process without actually having 

 to carry out the replication." The random element of this "swindle" is applied to the log of 

 basal area growth equations in one of two ways. The particular method is dependent upon the 

 number of trees sampled in the stand. 



^David W. Hann. Diameter growth equations developed for Mora County, New Mexico, and the 

 Black Hills, South Dakota, inventories, on file with the Forest and Range Resources Evaluation 

 Project, USDA Forest Service, Intermountain Forest and Range Experiment Station, Ogden, Utah. 



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