120 



110 - 



2 4 6 8 10 12 14 16 18 20 



DIAMETER (b.h. inches) 



Figure 4. — Projected height/ 'diameter curves for grand fir on Abies grandis/Pachistima 



habi tat. 



Removing Bias in Logarithmic Form 



When the variable to be predicted is analyzed by making a logarithmic transforma- 

 tion, a bias is introduced when the inverse transformation is used to convert the 

 estimate to the original natural scale. Effect of the bias would be cumulative when 

 these prediction equations are used repetitively to simulate growth of trees and stands 

 through successive periods of time. By removing the bias, repetitive application will 

 produce the same height from the accumulated predictions of increment as would be the 

 result of cumulating observed values of the log-normally distributed height increment. 



Size of the bias depends on the residual variance, which is listed as 0.1853 on 

 the last line of table 4, for the logarithmic model. 



Bradu and Mundlak (1970) devised a correction for the bias that recognizes the 

 effect of sampling error on the parameters of the regression. Unfortunately, their 

 method requires the values of the inverse covariance matrix which is a 41 X 41 array. 

 As an alternative, a simple correction that is conditional on the sample values of the 

 parameters seems adequate. To determine the corrected estimate of the mean on the 

 original scale, the value of one-half the residual variance is added to the estimate on 

 the logarithmic scale before making the transformation back to the natural scale. In 

 this case, the amount to be added is 0.1853 -f 2 = 0.09265. The effect is to multiply 

 the uncorrected estimate of height increment by 1.0971. 



15 



