Phase 4. Ridge regression was then used as a means of reducing the problems 



of multicollinearity . I hoped the minimization of multicollinearity 

 might reverse the problem with sign on the site index transform, but 

 the resulting analysis did not help. Because of the desire to have 

 a model applicable over the range site indices naturally found in 

 the habitat type, I decided to force a reasonable value of the site 

 transform upon the model. This was done by defining a new dependent 

 variable--the natural log of the quotient basal area growth divided 

 by site index. By using this dependent variable, I assumed, therefore, 

 that basal area growth would increase in a direct proportion to an 

 increase in site index. 



Phase 5. A final set of screening runs was then made with the new dependent 



variable using the same methods described in the third phase of the 

 analysis. From these screening runs, I selected promising models 

 and made additional runs to determine their regression coefficients. 



Phase 6. Ridge regression was then used to selectively remove several of the 



time-since-last-cutting independent variables in order to minimize 

 multicollinearity. Parameters to the final models were then deter- 

 mined by ordinary, least squares regression. 



Log of Basal Area Growth Equations 



The finished log of basal area growth equations is in table 2. The two blackjack pine 

 equations differ because of the presence or absence of the even-aged data when they were 

 developed. The number of observations, the relative mean square residual (RMSQR) , ^ and the 

 coefficient of determination (R^ ) of each equation are found in table 3. 



Table 2. --Log of basal area growth equations 



Blackjack pine using all data: 



InCBasal Area Growth] = -8.51836897 + 1 . 16754330 (in (D) ) - 4 . 00970143E-02 (D) 



- 3.84298771E-03(LBA2) - 7 . 15483662E-03 (MBA2) - 1 . 58234269E-02 (UBA2) 



- 3.26097273E-01(AiIn(D) j + 8 . 80676713E-01 (A3) + 1.0(ln(S)) 



Blackjack pine using uneven-aged data: 



In(Basal Area Growth) = -8,45357088 + 1.18165715(1/7(0)) - 4 . 77068027E-02 (D) 



- 1 .25420926E-03(LBA2) - 7 . 693001 13E-03 (MBA2) - 1 . 74056839E-02 (UBA2) 



- 3.13247572E-01(AiIn(D)) + 9 . 10604169E-01 (A3) + 1.0(in(S)) 



Yellow pine: 



in(Basal Area Growth) = -15.2464932 + 4 . 27656656 (in (D) ) - 1 . 83161626E-01 (D) 



- 7.27361567E-05(LBA2)2 - 9 . 11165626E-04 (MBA2) ^ - 2 .41462106E-04 (UBA2) ^ 



- 1 .05776062(Ai) + 1.0(in(S)) 



where 



D 

 S 



MBA2 



LBA2 

 UBA2 

 Ai 

 A3 

 TIME 



diameter class size 

 Minor's (1964) site index 



basal area in the given diameter class plus the two adjoining 

 larger and smaller diameter classes 



total basal area below the smallest diameter class in MBA2 

 area above the largest diameter class in MBA2 

 1.244178*EXP(-(1. 176471 - . 01 9607*TIME) 3) 

 1.000203*EXP(-(1. 428571 - . 023809*TIME) &) 

 year growth periods since last cutting 



total basal 

 -0.244178 + 

 -0.000203 + 

 number of 5 



^Relative mean square residual is the mean square residual for the model divided by the 

 corrected variance of the dependent variable. See appendix C for more detail. 



7 



