Table 7 shows the effects on the general model of varying 

 parameters by species. Since the intercept term in the original 

 model was not significantly different from zero, all subsequent 

 regressions were fitted through the ongin. Varying \r\{H). In(CL), 

 or \n{BA) by species significantly improves the fit. However, 

 inconsistency in signs for the in(CL) or \n(BA) terms occurs in all 

 except model (2). Thus model (2) was chosen for predicting 

 crown width for trees less than 3.5 inches (8.9 cm) d.b.h.: 



ln(ClAO = biln(H) + bzHCL) + bsHBA) 



where the subscript /denotes a different coefficient for the \n(H) 

 term for every species. 



The coefficients in table 6 gave a mean square error of 

 0.06036. The model underpredicts crown width by an average 

 of 2.5 ft (0.8 m) (table 5). A factor of e = 1.031 may be 



applied to the estimate in the natural scale to correct for nega- 

 tive bias. When this is done, crown width is overpredicted by 

 only 0.07 ft (0.02 m). 



Foliage Weight 



The general model for predicting needle weight developed 

 from 114 trees 3.5 inches (8.9 cm) d.b.h. and larger is: 



\n(WT) = bo + biln(D) + b2ln(H) + b^HCL) + 

 b4\n(A) + bsHTPA) + beHDREL) 



The analysis of variance in table 8 is slightly different from that 

 in tables 4 and 7. It was not possible to fit the complete model 

 (species X parameter interactions for seven variables) in the 

 REX regression program because of the dimension of the prob- 

 lem. Instead, the improvement in fit was analyzed between two 

 models at a time by forming an F ratio in which the denominator 

 is the mean square of the model with the most parameters in it. 

 For instance, the general model was fit in (1). Then in (2) the 

 intercept was varied by species, and the reduction in error 

 between (1 ) and (2) associated with 9 d.f . was measured with an 

 F-test. Since this reduction was significant, the species- 

 intercept term was carried through in each successively more 

 complex model, (3) through (8). Beyond fitting the interceptterm 



Table 7. — Analysis of improvement of ln(crown width)^ regression models attainable by varying parameters by species, trees less 

 than 3.5 Inches d.b.h. 



ModeP 



Source 



Remarks 



Degrees Marginal 



of sum of Mean square 

 freedom squares reduction 



(1) b,ln{H) ^ b2ln(CL) - b^lniBA) 



(2) bi^ln(H) - b2ln(CL) + b3ln(e>A) 



(3) hMH) b2,ln(CL) + b3ln(eA) 



(4) biln(H) - b2ln(CL) - bj^HBA) 



(5) bi^ln(H) - b^ln(CL) - bMBA) 



(6) bi^ln(H) ^ bz\n(CL) - b3\n(BA) 



(7) biln(H) - bzHCL) - b^lniSA) 



complete model 

 bi^ln(H) - b2^\n{CL) ~ b^ln(e/^) 



Reduction due 



to model (1 ) 

 Reduction due 

 to model (2) 

 -reduction due 

 to model (1 ) 

 Reduction due 

 to model (3) 

 -reduction due 

 to model (1 ) 

 Reduction due 

 to model (4) 

 -reduction due 

 to model (1 ) 

 Reduction due 

 to model (5) 

 -reduction due 

 to model (2) 

 Reduction due 

 to model (6) 

 -reduction due 

 to model (2) 

 Reduction due 

 to model (7) 

 -reduction due 

 to model (3) 



Error 



Inconsistent 

 signs in CL terms 



Inconsistent 

 signs in BA terms 



Inconsistent 

 signs in CL terms 



3 364.13574 121.37858 2063. 

 10 6.88867 .68887 11.71 



10 



10 



10 



Inconsistent 



signs in BA terms 10 



6.89868 .68987 1 1 .73 



6.77222 .67722 11.51 



1.16358 .11636 1.98 



.81690 .08169 1.39 



inconsistent 



signs in terms 10 1.07251 .10725 1.82 

 Inconsistent signs 



in CL and 64 terms 177 10.41343 .05883 



'Mean In(ClV) = 1 .26648 feet: standard deviation = 46956 

 ^erms varying by species are underlined, 

 ^Tabulated F values: Fj^^oos = 1-63. 



7 



