The RISK computer program (Hamilton 1974) is a nonlinear regression program that 

 efficiently estimates the parameters. This program was used to estimate the 3- for 

 those independent variables selected for inclusion in the model by the data screening 

 procedure . 



Because the sampling probability for each observation in the data set was propor- 

 tional to {DBH)^ , each observation was assigned a weight proportional to the inverse 

 of (DBH)^ in RISK. To model annual mortality rates from a data set that included two 

 unequal length time intervals, the dependent variable associated with each observation 

 was divided by the appropriate time interval. 



A model developed from data measured over a specific time interval produces mor- 

 tality rate estimates that are applicable only to similar time intervals. This could 

 be a severe limitation when we wish to predict the probability of mortality for a range 

 of time intervals. The limitation can be partially overcome by considering mortality 

 rate to behave like compound interest. The model can then be used to predict mortality 

 for any measurement interval, and data sets with unequal remeasurement intervals can 

 be used for model verification. Because we chose a 1-year base for this analysis, 

 mortality functions from RISK will predict annual mortality rate. 



In the following derivation, mortality rate is treated as a negative interest 

 rate. The standard compound interest formula (1) may be used to calculate the number 

 of surviving trees over any desired time interval. 



V = number of trees living after n years ' • 



= number of trees living today 



r = annual mortality rate 



n = number of years in prediction interval. 



This formula may be applied directly when the mortality rate is an annual rate. Alge- 

 braic manipulation of formula (1) produces the following formula that converts an 

 annual mortality rate to an n-year mortality rate: 



V = V il-r) 

 n o 



n 



(1) 



where 



V 



n 



n 



o 



V 



o 



n 



= l-a-r) 



n 



=> 



V 



o 



=> P = l-(l-r) 



n 



When the mortality rate is for a period of n years, the following formula, 

 derived from equations (1) and (2), should be used to calculate mortality: 



C2) 



(3) 



8 



