With this in mind, two models were hypothesized for trial: 



Rj^^ = ao - aiH/D - a2H/ (D^) (13) 



where 



Rj ,^ = Ratio of actual gross International 1/4-inch board foot volume to a 6-inch 

 top divided by predicted gross merchantable cubic foot volume to a 6-inch 

 top in an unforked tree. 



The signs on the coefficients a^, a2, and bi, 2?2 must be positive for the model to 

 behave as expected. 



Before these models were tried, the ratio values were formed for all trees 6.1 

 inches and larger and plotted for the sp-NF combinations previously delineated. An 

 examination of these plots resulted in the decision to combine blackjack pine data sets 

 with the yellow pine data sets and to combine both Douglas-fir data sets. The plots 

 also showed a few outliers that were eliminated from the appropriate data sets. 



The two models, (13) and (14), were then fitted to the data sets to determine which 

 of the two minimized RMSQR. The model with tree height (13) proved to be the worst and, 

 therefore, was eliminated. The fits-for model (14) were reasonable for only the 

 ponderosa pine and the Douglas-fir data sets. The other data sets either had a negative 

 sign on bi or 2>2 or the plots of data sets versus the predicted curves indicated a poor 

 fit. The ponderosa pine model was then "forced" onto the white pine and the white fir 

 on the Lincoln National Forest data sets with good results. This was done by scaling 

 the ponderosa pine model through an overall slope correction using least squares regres- 

 sion. Techniques similar to those described by Jensen (1964, 1973, and 1976) and Jensen 

 and Homeyer (1970 and 1971) were used to develop better behaved (within a reasonable 

 usage range) ratio models for Engelmann spruce-corkbark fir, aspen, and white fir on 

 the Santa Fe and Carson National Forests data sets. 



The resulting model form for white fir was the same as model (14) but, of course, 

 these parameter estimates are not least squares estimates in this case. The form of 

 the models for aspen and Engelmann spruce-corkbark fir was: 



^I/C " ^° " ^1°"^ " ^^^'^ ~ ^3D"^ (15) 



For both species groups, at least one of the parameters was negative which resulted in 

 an undesirable "peak" and "valley" in the ratio equation outside the available data 

 range. For aspen, the undesirable points also occurred outside the reasonable usage 

 range, but for Engelmann spruce-corkbark fir, the points did occur within the reasonable 

 usage range. Fortunately, the magnitudes of the "peak" and "valley" were small and it 

 was felt that they would not cause serious problems when predicting International 1/4- 

 inch board foot volume. 



The appropriate ratio models were then multiplied by their predicted merchantable 

 cubic foot volume to a 6-inch top for each sp-NF combination and then fitted to actual 

 International 1/4-inch board foot volume to a 6-inch top, using least squares regression 

 through the origin to provide a slope correction. The resulting basic model form and 

 its error structure, therefore, is the following: 



Vj = ^1 X (Rj/^) X (V^^^„^) . e (16) 



36 



