Logistic 



Regression 



Equations 



stocked plots are estimated, the probability of stocking is used to scale stocked 

 plot attributes to a per-acre basis. As an example, suppose five plots are being 

 projected and the following are predicted for the probability of stocking (PS) 

 and number of trees per stocked plot (TPSP): 



Predicted number 

 of trees per 

 stocked plot 

 (TPSP) 



3 

 1 

 2 

 5 

 1 



Predicted trees 

 per acre = 

 (PS * TPSP * 300)/iV 

 where AT is 5 plots 



108 



15 



96 

 150 



27 



Predicted 

 probability 

 of stocking 

 (PS) 



0.60 

 .25 

 .80 

 .50 

 .45 



Predicted toW 



trees per acre 396 



The third column shows how the probability of stocking is used to scale results 

 of stocked plot analyses to a per-acre basis. The "300" in the third column 

 is the inverse of the plot size used in the study and the "AT' is the number of 

 plots being used to represent the stand. Predicted total trees per acre is 396. 

 Species, size, and other tree attributes are determined during other steps 

 in the model. 



Many of the regeneration data are distributed dichotomously — plots are 

 either stocked with at least one established seedling or they are nonstocked, 

 each of the 10 species was either established on a plot or it was not, and so 

 on. Response surfaces are sigmoid-shaped. 



A nonlinear logistic algorithm called RISK (Hamilton 1974) was used to 

 predict probabilities for dichotomously distributed dependent variables. 

 The logistic equation is: 



P = (l+e-^W)-i (1) 



where 



P = probability 



e = the base of natural logarithms 

 Pi = vector of regression coefficients 

 Xi = vector of independent variables. 



The predicted probability (P) is continuous and bounded within the interval 

 [0,1]. RISK can use up to 30 independent variables. 



Goodness of fit for equations was evaluated at the 0.05 significance level. 

 RISK output includes four types of statistics that can be used to evaluate 

 goodness of fit: Student-i ratios for each regression coefficient, chi-square 

 values for 21 divisions within the probability interval [0,1], total chi-square 

 for the 21 divisions, and an analysis of variance table showing the amount 

 of variation explained by regression. 



A fifth goodness-of-fit criterion was how close the error mean square was to 

 1.0 in the analysis of variance table. As the population size increases, the limit- 

 ing value for the error mean square is 1.0, rather than 0.0 (Hamilton 1974). 

 We found that in successive iterations of the RISK algorithm, the error mesin 

 square would approach 1.0 unless the equation was unstable; thus, error 

 mean squares substantially different from 1.0 usually indicate poor equa- 

 tion formulation. 



6 



