APPENDIX E 



DEVELOPMENT OF MORTALITY MODELS 

 Prior Findings 



For uneven-aged stands, noncatastrophic mortality has been found to be correlated to 

 diameter size, severity of cutting, and whether the trees were blackjack or yellow pines. 

 Krauch (1926) reported that percent mortality increased as diameter increased. IVhen percent 

 mortality is plotted over diameter, the curve is U-shaped with the minimum near 20 inches 

 (Pearson 1939, 1950; Pearson and Wads- orth 1941; Wadsworth and Pearson 1943; and Myers and 

 Martin 1963) . 



For severity of cutting, Krauch (1926) discovered that "the percentage of mortality 

 increases in inverse ratio to the degree of cutting." This discovery was supported by Lexen 

 (1935). Wadsworth and Pearson (1943) found that percent mortality of blackjack or yellow pine 

 alone was higher in a cutover stand than in a virgin stand, but that this situation reversed 

 itself when the two were combined. They attributed this reversal to peculiarities of percent- 

 ages. Pearson (1950) also found that a virgin stand had a higher rate of mortality than a 

 moderately cut stand, but that severe cutting increased mortality because of windthrow and 

 lightning losses in the few large trees remaining. Higher mortality rates in yellow pine than 

 in blackjack pine have been reported by Pearson and Wadsworth (1941) and by Wadsworth and 

 Pearson (1943) . 



Mortality should also be related to productivity, total stand density, and time-since-last- 

 cutting. For even-aged stands, Myers and others (1976) found that site and total stand basal 

 area were significant in predicting total stand mortality, and there is no reason to believe 

 this effect is not also significant in uneven-aged stands. 



While it seems intuitive that the longer the time-since-last-cutting, the higher the 

 mortality rate, both Lexen (1935) and Pearson (1939, 1950) have reported that they could find 

 no consistent relationship between mortality and time-since-last-cutting. Their findings 

 could have been clouded by other effects or by the fact that the length of time-since-last- 

 cutting they examined (20 years or less) was not long enough for the effect to manifest itself. 



Definition of Variables and Transformations 



A mortality rate can be expressed in several ways. Ek (1974) and Adams and Ek (1974) 

 expressed it directly as the number of trees dying in a diameter class during the growth 

 period. Lee (1971) expressed mortality as a percentage of the total stand dying per year. 

 Finally, Hamilton (1974), Hamilton and Edwards (1976), and Monserud (1976) all expressed 

 mortality as the probability (or proportion) of a tree dying in a growth period. 



The form of the equations has also differed. Ek (1974) used nonlinear least squares 

 techniques, while Lee (1971) and Adams and Ek (1974) used ordinary, linear least squares 

 techniques. Hamilton (1974), Hamilton and Edwards (1976), and Monserud (1976) all used the 

 logistic function fitted using weighted, nonlinear least squares techniques. 



Expressing mortality as a percentage or proportion has an advantage over expressing it as 

 the number of trees dying. If properly modeled, a proportion (or percentage) is bounded by 

 and 1 (or and 100) . Multiplying this rate by the number of trees in the class will always 

 give a value less than or equal to the number in the class. As a result, predicted number of 

 survivor trees is never negative; a result possible under the other approach. 



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