The poor results for yellow pine were not totally unexpected. These trees are overmature 

 and frequently afflicted by various damaging agents that influence growth rates. In addition, 

 the relatively small number of yellow pine in the stand often meant that between diameter 

 class differences are actually between individual tree differences. Two mitigating factors, 

 however, argued for retaining the equation. First, the model behaves as expected and does 

 predict somewhat better than the mean. Second, the accurate prediction of yellow pine is not 

 very important in managed stands because, under management, the presence of yellow pine usually 

 would not be a favored condition. 



Check for Normality and Homogeneous Variance 



A check of the final equations revealed that the residuals were highly skewed and lepto- 

 kurtic, which was the same result as first found for the individual plot equations (appendix 

 C) . }iomogeneity of variance was checked by dividing the range of predicted log of basal area 

 growth into intervals, computing a variance for each interval, and then comparing the variances 

 across intervals for trends. Bartlett's chi-square test for homogeneity of variances (Snedecor 

 and Cochran 1967) was not used because the test is highly sensitive to nonnormality . The 

 visual checks for homogeneity indicated no consistent trends, and so weighting was judged 

 unnecessary. 



Log Bias 



The usage of the log model to predict basal area growth introduces a problem because the 

 value of interest for predictive purposes is basal area growth and not the log of basal area 



growth. The problem arises because EXP[In(Y)] is not a "mean-unbiased" estimator of Y; i.e.: 

 EXP[E[i^)]] ^ E[Y] 



Rather, EXP[lr!(Y)] is a "median-unbiased" estimator of Y (Bradu and Mundlak 1970). 



Because the residuals are not normally distributed, the log bias correction factors 

 proposed by Bradu and Mundlak (1970), Oldham (1965), and Baskerville (1972) could not be 

 used. Therefore, I proposed an alternative log bias correction factor that was added to the 

 intercept term of the log of basal area growth model and was computed as the difference between 

 the log of mean basal area growth minus the mean log of basal area growth (see appendix D) . 



Models of Mean Residuals 



Because the residuals are not normally distributed, the mean deviation of the fast, 

 moderate, and slow growers had to be computed empirically. I hypothesized that the range in 

 residuals would increase as the time-since-last-cutting increased because cutting would, if 

 properly applied, homogenize the stand by eliminating rough, cull, suppressed, and damaged 

 trees. The residuals were, therefore, divided into time-since-last-cutting classes, and for 

 each class the means of the upper, middle, and lower one-third residuals were computed. As 

 expected, the means had a tendency to increase as time-since-last-cutting increased. Therefore, 

 the time-since-last-cutting transforms (A^, A2, and A3) were used to develop models that 

 predict mean residuals for the upper and middle thirds as a function of time-since-last-cutting. 

 The lower one-third mean model was then expressed as the negative of the sum of the other two 

 models. This assumed that, for every value of time-since-last -cutting, the means of the three 

 will sum to zero. The final residual models are in table 5. 



9 



