One penalty for using constrained regressions is that there is no widely accepted 

 simple quantity that reflects the degree of agreement between the equation and the data, 

 in the way that the "coefficient of determination," called r 2 , does for unconstrained 

 regressions. If y. represents a bulk depth data point to be fitted by regression, y. , 

 the value of the bulk depth predicted by the regression expression, and y the average 

 of the data points to be fitted, then, by statistical theory the expression for the 

 coefficient of determination for a constrained regression should be 



r 2 = 1 - E(y. - yi ) 2 /E(y. 2 ). 



This quantity is, however, a poor measure of the "goodness" of the regression equation. 

 So we have used, in this presentation, a nonrigorous but intuitively more appealing 

 measure of the suitability of the regression description: 



s - 1 - E(y. - y i ) 2 /E(y i - y) 2 . 



This parameter compares the variance about the regression to the variance about the mean, 

 as does "r 2 " for an unconstrained regression, but it is not limited numerically to the 

 range to 1 . It is restricted only to be less than unity, and can be negative. 



RESULTS AND DISCUSSION 



Depth Versus Intercept Counts 



Regression results for all of the skidding, species, and age combinations obtained 

 in the study are shown in table 3. The relationships for skyline and helicopter 

 skidding were similar; so they were combined and called "high-lead." The high values 

 of the fit descriptor, s, are largely due to the manner of aggregating data. Examples 

 of fit are shown in figures 1, 2, and 3. Because of the data aggregation, tests for 

 differences among the slash groups seem irrelevant. However, recognizing the large 

 amount of variation among sampling points and the similarity of some of the regressions 

 in table 3, the equations in figure 4 are recommended for application to fire modeling. 

 To obtain the equations for initial depth shown in figure 4, some of the results in 

 table 3 had to be adjusted to age 1 year. 



Merchantable top diameter limits and d.b.h. of trees thinned precommercial ly 

 appear to significantly influence the depth-intercept count relationship. We were 

 unable to include a wide range of conditions for these factors; thus, application of 

 our results should be restricted to conditions similar to ours. Applicable conditions 

 include merchantable top diameters of 4 to 6 inches for pine and 5 to 6 inches for other 

 species, and precommercial thinning of trees 2 inches and greater in d.b.h. 



s 



