where 



e 



= Predicted gross International 1/4-inch board foot volume to a 6-inch top 

 = Predicted gross merchantable cubic foot volume to a 6-inch top 

 = Least squares regression coefficient 

 = Residual about regression. 



Next, the necessity for weighting this model was examined through the procedure 

 described by Hann and McKinney (1975) . This resulted in weights being formed of the 

 type: 



Unfortunately, no common set of values e^, 62* ^3 could be found for all sp-NF 

 combinations. The final sp-NF weights were used in weighted least squares regression 

 through the origin to obtain the final slope correction. These slope corrections were 

 then incorporated into the ratio models by multiplication. As a final check, the data 

 points and regression equations were plotted over d.b.h. and total tree height to 

 determine adequacy of fit. 



When considering this approach to modeling International 1/4- inch board foot 

 volume, the basic model form of equation (16) must be kept in mind. If a set of pro- 

 posed ratio equations had been available, then the fitting of equation (16) could have 

 been done by screening the various ratio equations times cubic foot volume to determine 

 which was appropriate for the sp-NF combination. Unfortunately, a proposed set of ratio 

 equations was not available; therefore, the ratio equations had to be developed using 

 ••he same data set. 



There is a tendency to think that what is really being fitted is the model involv- 

 ing the 11 to 15 independent variables that would be formed by multiplying the ratio 

 and merchantable cubic foot volume models together. Using this approach, it was found 

 that some independent variables proved to be insignificant while the signs and 

 magnitude of other regression coefficients changed because of multicol linearity 

 problems. The result was an equation with all of the undesirable properties that 

 were to be avoided. The approach adopted here produces a model that behaves in a rea- 

 sonable and consistent fashion, but sacrifices some statistical niceties. 



VII. Gross International 2/4-Inch Board Foot Volume — Forked Trees 



The method used to determine the ratio correction for forking upon International 

 1/4-inch board foot volume was the same as that used to develop the equations for fork- 

 ing in total stem gross cubic foot volume. All selected equations were tabulated and 

 checked for reasonableness. From this it was discovered that the Douglas-fir equation 

 did not behave as expected. Therefore, the following power model was fitted to Douglas- 



W = (D 2Vj3) 



-1 



C17) 



fir: 



In (R, J = ao+ aiH 



(18) 



where 



R 



= Predicted ratio of actual gross International 1/4-inch board foot volume 

 to a 6-inch top in a forked tree divided by predicted International 1/4- 

 inch board foot volume to a 6-inch top in an unforked tree. 



37 



