raised to the -1 .5 power as the weight (Schreuder and 

 Anderson 1984). 



Several steps were used to obtain final volume model 

 parameters. After the first regression, Cook's-D influence 

 statistics (Cook 1977) were examined. Using criteria sug- 

 gested by Belsley and others (1980, p. 29), Cook's-D val- 

 ues corresponding to each tree's volume were arrayed by 

 species group. Trees above gaps in the upper tails of the 

 distributions were identified. Further examination of 

 these data showed inconsistencies in visual volume seg- 

 mentation application for about half the cases, suggesting 

 some data should be deleted. 



Additional parameter estimates were then obtained 

 after deleting data exceeding a given Cook's-D value. 

 After several comparison trials, a Cook's-D value of 0.10 

 was chosen as a reasonable cut-off point, resulting in 

 deletion of 20 trees. Reducing the Cook's-D cut-off point 

 below 0.10, and deleting more trees, had a diminishing 

 effect on parameter estimation. Final model parameters 

 and tabulated volume predictions are given in the appen- 

 dix in tables 3, 4, 5, and 6. 



MODEL RELIABILITY 



Results of regression goodness-of-fit analyses are sum- 

 marized in table 2. The coefficient of determination (R 2 ) 

 was computed using the regression weights and computed 

 again without regression weights. Neither computation 

 method was completely satisfactory because the weighted 

 R 2 was based on a rescaled sum of squares that cannot be 

 scaled back to the original data, and the unweighted R 2 



was based on a sum of squares that violates the homogene- 

 ous variance assumption of least squares regression. But 

 for those who subscribe to J? 2 -values, those in table 2 show 

 a reasonable data fit to the model. 



Confidence intervals, another method to assess 

 goodness-of-fit, were examined. For each volume equation, 

 confidence intervals were computed for two DRC classes 

 corresponding to trees above and below the point of change 

 (XA The median size tree was selected in each DRC class 

 for actual computation. Then 95 percent confidence inter- 

 vals were computed for predicting mean tree volume from 

 varying numbers of sample trees. For example, if five 

 multiple-stem junipers about 9 inches DRC (see first line of 

 table 2) were measured for volume prediction, the expected 

 true volume would lie in an interval ±36 percent of 2.5 ft 3 

 (1.6 to 3.4 ft 3 ). If 10 junipers of 9 inches DRC were 

 sampled, the expected true volume would lie in a smaller 

 interval of ±26 percent of 2.5 ft 3 (1.9 to 3.2 ft 3 ). A sample 

 size of 20 junipers would further reduce the confidence 

 intervals to 19 percent of 2.5 ft 3 (2.0 to 3.0 ft 3 ). Even 

 larger sample sizes resulted in still smaller confidence 

 intervals, but the interval reduction in relation to sample 

 size diminished with the larger sample sizes. 



The confidence intervals were computed as follows: 



CI = 1.96 J MSE 0-fWTIn + H) (5) 

 where 



CI = 95 percent confidence interval for a predicted 

 mean 



Table 2 — Regression statistics from volume modeling 



Median statistics' 



Species 

 group 



DRC 

 class 



Basal 

 stems 



Number 

 of trees 





R 2 



Predicted 

 volume 



Median 

 DRC 





95 percent mean CM 



Weighted' Unweighted* 



n=5 



n=10 



n=20 



n=50 n: 



=100 





Inches 











Ff 3 



Inches 







Percent 







Juniper 



<20 



Multiple 



203 



0.91 



0.82 



2.5 



9 



36 



26 



19 



13 



10 





>20 



Multiple 



56 



.91 



.79 



27.6 



26 



16 



12 



8 



5 



4 





<20 



Single 



262 



.92 



.88 



1.4 



6 



34 



25 



18 



13 



11 





>20 



Single 



24 



.92 



.81 



28.1 



20 



12 



9 



6 



4 



3 



Pinyon 



<12 



Single 



203 



.89 



.82 



.9 



7 



39 



30 



23 



19 



17 





>12 



Single 



23 



.89 



.96 



12.1 



13 



16 



12 



8 



6 



4 



Oak 



<16 



Multiple 



130 



.91 



.87 



2.3 



10 



27 



20 



15 



11 



9 





>16 



Multiple 



9 



.91 



.77 



11.9 



18 



15 



11 



8 



6 



4 





<16 



Single 



193 



.90 



.82 



1.4 



7 



35 



26 



20 



15 



13 





>16 



Single 



18 



.90 



.89 



21.7 



19 



14 



10 



7 



5 



4 



Mesquite 



<12 



Multiple 



168 



.86 



.81 



1.3 



8 



38 



28 



21 



16 



14 





>12 



Multiple 



22 



.86 



.89 



9.7 



14 



18 



13 



9 



6 



5 





<12 



Single 



123 



.89 



.88 



.5 



5 



48 



39 



34 



30 



29 





>12 



Single 



8 



.89 



.92 



11.6 



14 



13 



9 



7 



6 



5 



'Statistics based on the median of the DRC class. 



Confidence intervals (CI) for mean predicted volume for several sample sizes for the median level of DSQH (roughly median DRC) in each DRC 

 class. CI's are expressed as a percentage of predicted volume for each DRC class. 

 Computed from weighted regression sum of squares with all tree sizes combined. 

 "Recomputed from data without consideration of regression weights. 



4 



