The final class of independent variables is those indicating the position of the diameter 

 class within the stand. For this purpose, Stage (1973) developed the independent variable 

 "percentile in the basal area distribution" (PCT) . Stage defined this variable as the basal 

 area in all trees equal to or smaller in diameter than a given tree, divided by total stand 

 basal area. In this study, PCT was defined in a similar fashion, substituting "diameter 

 class" for "tree." Transforms of this variable include the following: In(PCT), PCT (Stage 



1975) , and PCT". 



Another "position" variable that Cole and Stage (1972) and Stage (1973) have used is 

 diameter of the tree divided by the diameter of the tree of mean basal area (DCP) . This 

 independent variable can also adapt to the diameter class situation by dividing diameter class 

 size by the diameter of the tree of mean basal area. Potentially, useful transforms of this 

 variable include DCP and IniDCP) . 



The final method of representing the position of the diameter class within the stand 

 consists of separating total stand basal area into three components. The middle basal area 

 component is defined as that falling in a specified diameter class (MBA^); or within the 

 specified diameter class plus the adjoining larger and smaller diameter classes, if they 

 exist (MBA2) ; or within the specified diameter class plus the adjoining two larger and two 

 smaller diameter classes, if they exist (MBA3) . The lower basal area component (LBA^, LBA2, 

 or LBA3) is the total basal area existing below the smallest diameter class in MBA^, and the 



upper basal area component (UBA| , UBA2, or UBA3) is the total basal area existing above the 

 largest diameter class in MBA^ . 



In this fashion, three sets containing three independent variables were derived: LBAj , 

 MBAi, and UBA^ ; LBA2, MBA2, and UBA2; and LBA3, MBA3 , and UBA3 . These variables should behave 

 in the same fashion as the total stand density variables, and therefore the only additional 

 transformation used was to also square each value. This resulted in six sets, each containing 

 three independent variables. For convenience, the definition of all independent variables and 

 their abbreviations can be found in table 23. 



Due to the desire to divide the residuals about the final models into fast, average, and 

 slow growers, the appropriate dependent variable is the log of basal area growth for each 

 tree. While the dependent variable is an individual tree value, it must be remembered that 

 all independent variables are based only on diameter class information. 



Development of Equations 



Equations were developed for both blackjack pine and yellow pine using the least square 

 regression program REX (Grosenbaugh 1967) . REX is a powerful screening tool because it is 

 designed to fit all combinations of independent variables following specified combinatorial 

 rules. The screening statistic used is the relative mean square residual (RMSQR) , which is 

 defined as the mean square error about regression divided by the variance of the dependent 

 variable. Therefore, a perfect fit results in an RMSQR value of zero, while an RMSQR value of 

 one would indicate that the regression equation is no better at reducing squared residuals 

 about regression than a simple mean. The advantage of the RMSQR over the more commonly used 

 coefficient of determination (R^) is that the former reflects the reduction in the degrees of 

 freedom caused by adding another independent variable. Therefore, it is easier to compare 

 models with a different number of independent variables. 



55 



