Next, the resulting slopes (b coefficients) were plotted over basal diameter minus 

 minimum top diameter limit. Basal diameter minus minimum top diameter limit was used 

 because of the second constraint mentioned previously. Matchacurve techniques (Jensen 

 1964, Jensen and Homeyer 1970, 1971, Jensen 1973, 1976) were used to describe the trends 

 of the slopes in relation to basal diameter minus minimum top diameter for each minimum 

 top diameter class. For pinyon, the power function of the form 



7 ^ a/" 



was a satisfactory descriptor with a and n to be a function of minimum top diameter 

 alone. For juniper, a single power function did not provide enough upward curvature 

 for the larger basal diameters; therefore, a multiple component model having the follow- 

 ing form was used: 



where the a-^J^"^ component becomes essentially zero for small diameters. 



As in step 1, each model component was fitted through the origin by weighted 

 (number of observations) least squares techniques. 



For juniper, the slopes were also a function of number of stems. The number-of- 

 stems effect was allowed to asymptote toward zero at 20 stems. 



Appropriate power functions for the minimum top diameter limit effect were then 

 determined. The same procedure as used in step 2 was employed, except this time the 

 independent variable was minimum top diameter limit. 



The components were combined for each model and the latter were each fitted to 

 their respective data set by least squares techniques. The simple linear model, 

 Y - h (model), was used, forcing the model through the origin. The resulting coeffi- 

 cients were very close to 1, indicating that the models as developed fitted the overall 

 data trends very well. 



Next the residuals were plotted over estimated volume. The plotted residuals 

 showed increasing residuals with increasing estimated volume, indicating a need to 

 weight the estimate of b in the final models. Draper and Smith (1966) and Cunia (1964) 

 recommend weighting the dependent variable by the reciprocal of the variance where 

 variance is unequal. An estimate of the variance can be obtained from the residuals 

 squared. 



The next step was to screen possible variables for correlation with residuals 

 squared. The LOG of the residuals squared was screened against all possible combina- 

 tions of the log's of the following variables: 



1. Estimated cubic volume 



2. Basal diameter minus minimum top diameter limit 



3. Minimum top diameter limit 



4. Height 



5. Number of stems (juniper only). 



The screen indicated the estimated cubic volume was the simplest, best overall 

 predictor of residuals squared. 



