Linear Regression Linear regression was used to estimate heights of best trees. Goodness of 

 Equations fit was evaluated at the 0.05 significance level. The best transformation of 



the dependent variable was found to be the natural logarithm of tree height. 

 Various transformations of independent variables were also explored and used. 



In some steps of the regeneration model, it is important to mimic the dis- 

 tribution of the dependent variable. An example is the number of seedlings 

 established on stocked plots, which has a reversed J-shape distribution. For 

 these situations, the distribution is modeled with Weibull cumulative den- 

 sity distributions (Bailey and Dell 1973). The form of the equation for a two 

 parameter Weibull cumulative density distribution is: 



Fix) = l-EXP[-(x/Br] (2) 



where 



Fix) = cumulative density 



X = dependent variable being modeled 

 B = Weibull scale parameter 

 C = Weibull shape parameter. 



The area bounded by the Weibull equation is within the interval [0,1]. 

 Equation 2 can be solved for x as shown by Bailey and Dell (1973): 



a: = B[-ln(l-F(a:))Fc (3) 



A uniformly distributed pseudo-random number in the interval [0,1] can 

 be substituted for Fix) in equation 3, resulting in an unbiased choice for x. 

 Plot replication in the regeneration model results in enough random draws 

 that the distribution is approximated. 



Bailey and Dell (1973) also discuss coefficients for the C (shape) parameter. 

 If C is less than 1, the distribution has a reversed J shape. If C equals 1, 

 the distribution is exponential. If C is between 1 and 3.6, the distribution is 

 mound-shaped and positively skewed. When C is approximately 3.6, a nor- 

 mal distribution results. And when C is greater than 3.6, the distribution is 

 mound-shaped and negatively skewed. 



Version 1 of the regeneration model predicted regeneration on four 

 broadly defined habitat types using the habitat type classification system of 

 Daubenmire and Daubenmire (1968). Since then, classification of habitat 

 types has been refined for northern Idaho (Cooper and others 1991). We were 

 fortunate that Neiman, one of the authors of the 1991 northern Idaho classi- 

 fication, was available to update data collected during 1975-76 to the 1991 

 classification. Each plot sampled during the 1975-76 study was reclassified 

 by reviewing the original field sheets on which field crews had listed species 

 of shrubs, forbs, and grasses by plot. 



Thus, for northern Idaho we used the habitat type classification of Cooper 

 and others (1991), for Montana we used the classification of Pfister and others 

 (1977), and for central Idaho we used the classification of Steele and others 

 (1981). This resulted in 93 different habitat types and phases being sampled. 



It was necessary to combine similar habitat types into groups to reduce 

 independent variables to a reasonable number. This was done before fitting 

 equations. Groupings were made after statistical analyses by habitat types 

 that compared the percentage of stocked plots, trees per stocked plot, and 

 species per stocked plot. These analyses provided ideas for preliminary 

 groupings. Preliminary groupings were reviewed by forest ecologists, and 

 their suggestions were used to develop the final groupings. 



WeibuU 

 Distributions 



Habitat Type 

 Groupings 



7 



