class, change. As percent defect becomes greater, the probability of mortality in- 

 creases. If the tree's crown class changes to a dominant position, its probability of 

 mortality will decrease, while a change to an intermediate or suppressed crown class 

 will increase its probability of mortality. 



Model Verification 



The models discussed must be verified before being employed to describe mortality 

 in a stand over time. Verification tests whether the proposed model does more than 

 reproduce peculiarities of a specific set of data. 



"Goodness of fit" of the models was tested by means of the chi-square statistic 



n 



= E (0. - E.)2 /E. 

 ^=l 



where 



0^ = observed number of mortality trees falling in the tth class 

 = expected number of mortality trees falling in the ith. class 



n = number of classes 

 = chi-square statistic 



A computer program was written for verifying the proposed models with the chi- 

 square statistic. The population is divided into twenty-one 2-inch diameter classes 

 that range from 0-2 inches to >40 inches. The verification program uses the nonlinear 

 regression model developed with the aid of RISK to predict the probability of mortality 

 for each observation in the population being used for validation. Goodness of fit of 

 the model is determined by using the chi-square statistic to compare the observed 

 number of dead trees in each 2-inch diameter class with the expected (predicted) number 

 of dead trees in the class. 



The set of data for verifying the models from RISK has a remeasurement interval 

 of 6 years. Thus, it was necessary to use the compound interest formula to convert 

 the annual probability of mortality predicted by the model to the probability of 

 mortality over the measurement interval. 



The results of model verification of the grand fir model developed by RISK are 

 presented in table 1. We will use line 3 of table 1 to explain the output. Class 3 

 contains all trees from 4 to 6 inches d.b.h. This class contains 73 trees (col. 4) or 

 4.74 percent (col. 3) of the population. The probability of mortality for each tree 

 in the class is summed to obtain the expected number of trees that will die. Of the 

 73 trees in class 3, 7.2 trees (col. 6) are expected to die. Eight of these trees 

 (col. 5) actually died. The chi-square statistic comparing these values is reported 

 in column 7. The sum of the chi-square statistics for all diameter classes is the 

 measure of goodness of fit for the model. At the 95 percent confidence level, the 

 tabulated chi-square statistic for 21 degrees of freedom is 32.7. Since the calculated 

 chi-square value of 26.73 is less than the tabulated value, we cannot reject the model 

 at the 95 percent confidence level. 



10 



