A problem can arise with using ordinary, linear least squares regression for predicting 

 the percent or proportion of trees dying. Linear models can cause the predicted values to 

 exceed zero or one, especially if the prediction falls outside the range of values in the 

 developmental data set (Hamilton 1974; Hamilton and Edwards 1976). However, the nonlinear 

 logistic function, 



y = (1.0 + e ^ -^^ n n ) 



limits the value of y to between zero and one. Also, when the dependent variable is dichotomous 

 (which mortality is) , the logistic function appears to have improved statistical properties 

 over ordinary, linear least squares regression models (Hamilton 1974; Hamilton and Edwards 

 1976) . 



Hamilton (1974) presents a computer program, RISK, for determining the parameters of the 

 logistic function using weighted, nonlinear least squares regression techniques. In using 

 RISK, the dependent variable is assigned a value of zero if the tree survived to the end of 

 the growth period, and a value of one if it did not. By using only diameter class or stand 

 attributes as independent variables, the result is the probability of a tree dying under the 

 specified diameter class and stand conditions. 



The reported U-shaped curve of percent mortality over diameter was for all ponderosa 

 pine. By separating ponderosa pine into the two vigor classes of blackjack pine and yellow 

 pine, the trends will differ from the U-shaped curve. For blackjack pine, it is possible that 

 percent mortality will decline as diameter increases; while for yellow pine, the opposite is 

 expected. Because blackjack pine usually represents the small, vigorous trees and yellow pine 

 the large, overmature trees, it is easy to see why the resulting combined graph could be 

 U-shaped . 



Any transform on diameter class size (D) that causes the transformed value to increase as 

 D increases would allow the desired effect in the logistic function. The two independent 

 variables chosen, D and , both meet this criterion and are also simple and interpretable 

 (D is straightforward, and is analogous to' diameter class basal area). As with the basal 

 area growth equations, vigor was incorporated into the equations by developing separate 

 mortality equations for blackjack pine and for yellow pine. 



Two measures of productivity were tried: site index for the one-half plot (S) , and average 

 5-year growth period rainfall (GRF) . Mortality will decline as productivity increases; 

 therefore, the independent variables chosen were the untransformed values S and GRF. A 

 positive sign on the regression coefficients of these independent variables would provide the 

 desired or expected effect. 



The choice of the independent variables to represent stand density was based, in part, on 

 the results of the growth analysis. The sets of LBA2, MBA2, UBA2, and of IEA2^ , mA2^ , UBA2^, 

 proved best in the growth analysis at representing both the effect of stand density and of 

 diameter class position. Also chosen for analysis in this phase were the independent variables 

 of total stand basal area (BA) and percentile in the basal area distribution (PCT) . 



For time-since-last-cutting, two of the three transforms used in the growth analysis (A^ 

 and A3) were also tried in this analysis. Expressing severity of cutting was a problem. On 

 the plots with cutting, the recording of data did not start until after the cut. Thus, 

 information concerning the amount of trees removed was not available as an index of severity 

 of cutting. While the amount of the residual stand might provide some insight into the severity 

 of cutting, it is not the best measure. Because stand density reflects the amount of the 

 residual stand, it was felt that additional measures were neither justified nor necessary. 



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