Crown Width 



For 160 trees 3.5 inches (8.9 cm) d.b.h. and larger, the 

 general model is: 



\n{CW) = bo + biln(0) + b2ln(H) + bglnCC/.) 



In this and all subsequent models, a log-log model form was 

 chosen in order to linearize the geometric relationships and 

 stabilize the variance structure commonly encountered when 

 working with a wide range of data (for instance, when variance 

 is proportional to tree size). In this and subsequent models, the 

 second assumption is borne out by the uniform random patterns 

 shown in scatterplots of residuals versus predicted values in the 

 log scale. 



It is interesting to note that in this model no stand density term 

 is included as a measure of competition. However, if the log 

 model were inversely transformed to the natural scale, the 

 combination of the CL term with its positive coefficient, and H 

 with its negative coefficient (table 3) could be interpreted as 

 crown ratio. The effect of competition on predicted crown width 

 would be represented by crown ratio instead of stand density. It 

 is assumed that this effect would be positive, that is. crown width 

 would increase with increasing crown ratio. 



Table 4 is an analysis of variance showing the improvement 

 in the regression sum of squares when variables in the general 

 model are allowed to vary by species. For example, varying the 



Table 3. — Coefficients for estimating In (crown width) of trees 3.5 

 inches d.b.h. and larger: 

 \n{CW) = bo, + biln(D) + b^ln(H) + b3ln(CL) 



intercept term by species improves the regression by a signifi- 

 cant amount. Then, when either ln(D) or \r\(H) are allowed to 

 vary by species, the fit is further significantly improved. Of these 

 two alternatives, the latter has a slight advantage. Although 

 fitting In(CL) by species, or any more successively complex 

 model also improves the regression, the signs of the coeffi- 

 cients for the H and CL terms become inconsistent (some 

 species with positive coefficients and some with negative coeffi- 

 cients for a single parameter). The inconsistencies of sign 

 among species could not be logically explained, for example, as 

 a pattern indicating ranking of species by tolerance. Therefore it 

 is assumed that the terms change sign because the model is 

 overfitting the particular data set. Acceptable models were thus 

 constrained to those with consistent coefficients. The final mod- 

 el chosen is that in which the intercept and height terms vary by 

 species (indicated by the subscript / on the parameter): 



In(ClV) = bo, + biln(D) + b2ln(H) + bMCL) 



The coefficients for this model, shown in table 3, gave a mean 

 square error of 0.04898. The equation overpredicts crown width 

 in the natural scale for four species and underpredicts for seven 

 species, as shown in table 5. The mean crown width is under- 

 estimated by 5.7 ft (1 .7 m). 



Negative bias (mean residual deviation from zero), intro- 

 duced when the inverse logarithmic transformation is used to 

 convert log-normally distributed estimates to the original natural 

 scale, can be approximately corrected by adding one-half the 

 residual variance to the estimate on the log scale (Baskerville 

 1972). This amounts to multiplying the estimate of crown width 

 in the natural scale by exp[V2MSE]. Thus a factor of e ' ^' = 

 1.025 may be applied to the estimate in the natural scale to 

 correct for underprediction. After "bias adjustment," mean 

 crown width is overestimated slightly, 1.2 ft (0.4 m). 



Species^ 



Variable coefficients 



Intercept 



ln(H) 



bj 



BP 



-0.91984 



-0.07299 



WP 



4.30800 



- 1 .37265 



WC 



2.79784 



- .89666 



WL 



2.31359 



- .80919 



WH 



1 .32772 



- .52554 



AF 



1 .74558 



- .73972 



LP 



1 .06804 



- .55987 



GF 



2.20611 



- .76936 



ES 



3.76535 



-1.18257 



DF 



3.02271 



- 1 .00486 



PP 



1 .62365 



- .68098 



Variables 



Variable coefficients 





ln(D) 



bi = 1.08137 





In(CL) 



ba = 0.29786 





Regression sum of squares/d.f. 

 Error sum of squares/d.f. 

 Mean square error 



= 33.37909/23 

 = 6.66189/136 

 = 0.04898 

 = 0.8336 



'See text page 2 for species list. 



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