Instantaneous growth rates can be obtained by taking the differential of 

 equation (1) : 



3ff = b jj W (2) 



or, in its logarithmic form: 



In (3ff) = ln(3Z?) + ln(ff) - ln(Z?) + lntfO (3) 



On the other hand, periodic growth rates expressed as a finite difference derived 

 from (1) would be: 



ln(ff 2 ) - ln(ffi) = b(ln(Z? 2 ) - ln(Z?!)) 



which is the same as: 



ln(ff 2 /ffi) = b ln(Z? 2 /Z?i) (4) 



where the subscripts 1 and 2 designate measured values at the start and end of growth 

 period, respectively. 



Models were derived from (2) , (3) , and (4) by using various transformations of 

 height, the ratio of height to diameter, and crown ratio to estimate the b parameter 

 for each species. The best transformations of these independent variables were selected 

 by combining them in groups of sets for screening overall combinations of the groups 

 by using one set from each group (Grosenbaugh 1967) . Coefficients in these alternative 

 models were estimated by least-squares regression. Goodness-of-fit indices (Furnival 

 1961) for the several transformations of the dependent variable were compared using the 

 best regression for each transformation of the dependent variable. 



For the screening of these alternative models, data from 909 trees of 10 species 

 in the northern Idaho forests were used. 



Differential Model 



The dependent variable in this model is the 10-year periodic height increment, Aff. 

 Some coefficients (ej) in the model may be different for each species; other coeffi- 

 cients may be constant for all 10 species. Those that vary with species have an addi- 

 tional subscript i. 



Aff = f LD[o H * a H ln(Cff) + o 3 f- + ^(f^) 2 + c 5 ln(ff) + c & . jj^] 



The expression in the brackets represents the estimate of b in equation (2) . In compu- 

 tation, was the set of constant terms in the regression model. 



Table 2 is an analysis of variance showing the improvement in the regression sum- 

 of-squares for successively more complex collections of variables. In this table, for 

 example, comparison level 2 indicates that when either the coefficients of RhD/B or 

 the constant terms are allowed to vary by species, then the fit is improved by a sig- 

 nificant amount. Of these two alternatives, the former has a slight advantage. At 

 comparison level 3, varying constant terms by species is of little value so long as the 

 coefficients of HkD/D depend on species. However, the variable ln(ff) is shown to be 

 needed to represent the effect of changing height on the b coefficient. At comparison 

 level 4, crown ratio is still of little use, but addition of hD/D and its square 

 results in a considerable improvement. 



9 



