model, t-statistics which test the significance of the regression coefficients from 

 a value of 0, "...and a chi-square table that evaluates goodness-of-f it over the range 

 of predictions" (Hamilton 1974). Of the three, Hamilton suggests use of the chi-square 

 statistic for screening alternative models, but he also states that the final choice 

 "... is left to the discretion of the user." 



The dichotomous dependent variable is defined to be 1 if the tree has any cull in 

 its total stem and if not. The resulting equations for cubic foot volume, however, 

 are to be applied to both total stem and merchantable cubic foot net volume estimation. 

 It was felt that, in most cases, if the tree had cull in the total stem it would also 

 have cull in the merchantable portion. In those cases where removal of the top also 

 removed the cull portion, the fraction cull in the tree would then be 0, and this is a 

 legitimate value in the final fraction call model for merchantable cubic foot volume. 



For each sp-NF combination, 10 models were fitted with d.b.h., total tree height, 

 and number of tips as the tree attributes used for independent variables. Five models 

 had one independent variable (H,D,T,DH, or D^H) and the rest had two independent varia- 

 bles CH,D;H,T;D,T;DH,T; or D^H.T). Generally, the final runs picked were the ones that 

 minimized the chi-square statistic while maintaining the highest significance level 

 attained by the F-statistic for the one- or two-independent variable models (that is, 

 if some of the one- or two-variable models were significant at the 99 percent level, 

 then the final model was picked from those that minimized the chi-square statistic). 

 In some cases where the chi-square statistics were close, a model with a slightly higher 

 chi-square statistic was picked if the F- and t-statistics were considerably larger. 



One- and two-variable models are presented in those instances where the second 

 variable improved the model. The added variable, number of tips in all cases, was not 

 always statistically significant, but the resulting model behaved as expected (that is, 

 as the number of tips increased, so did the probability of cull). 



Some data sets were combined to gain data strength. Before combining, however, 

 the preliminary models and mean values of PrC^ were examined to assure compatibility. 

 Engelmann spruce and corkbark fir were separated because it was found that their cull 

 structures were very different. 



Damage information was available on the data sets from the Santa Fe and Carson 

 National Forests, Therefore, in those cases where enough common damage was present to 

 justify the effort, the previously picked models were fitted with dichotomous damage 

 variables added. These damage variables did not prove to be significant. 



All final equations were plotted and checked for reasonableness. 



XI. Fraation Cull in Total Stem Cubic Foot Volume Given the Tree Is Unsound — Unforked 

 and Forked Trees 



Like the probability of a tree being unsound, fraction cull takes on a value that 

 ranges from to 1 . To provide some control over the value of fraction cull, the 

 following model was used: 



FC^ = ^^(1.0 - e^^) (27) 



where 



FC^ = Predicted fraction cull in total stem cubic foot volume given the tree is 

 unsound 



= A function of the tree's attributes 

 h, = A final constant value. 



40 



