APPENDIX I 



EXAMINATION OF STATISTICAL TESTS USED IN VALIDATION 



To pursue possible statistical procedures for testing differences between actual and 

 predicted diameter distributions, I conducted a literature review of the validation techniques 

 used in operations research. Of those methods described by Mihram (1972a, 1972b), Van Horn 

 (1971), and Kleijnen (1974), two seemed appropriate for additional evaluation--the chi-square 

 "goodn^ss-of-f it" test and ordinary least squares regression. 



The chi-square "goodness-of -f it" test statistic can be computed as: 



K 2 

 2 I - "^i^ 



' ^ = 1 



where 



= computed test statistic 



n - total number of observations 



N. = number of observations that fell into the ith class 



p. - theoretical probablity of an observation falling into the ith class 



k = total number of classes. 



This statistic is normally used to test the frequency distribution of test data against a 

 theoretical distribution (Snedecor and Cochran 1967; Kmenta 1971). Therefore, it seemed to be 

 a natural statistic for testing an observed diameter distribution against a theoretical (or 

 predicted) diameter distribution. An examination of the assumptions underlying this test, 

 however, revealed problems that render the usage of the chi-square "goodness-of-fit" statistic 

 for testing diameter distributions questionable, if not inappropriate. 



The foundation of the chi-square test is the multinomial distribution. If the assumptions 

 of multinomial distribution can be met, then the limiting distribution of the chi-square 

 "goodness-of-fit" statistic is the chi-square distribution. The two assumptions of the multi- 

 nomial distribution of interest are: (1) the probability of an observation being in the ith 

 class, p., is fixed; and (2) the number of observed values in each class must sum to n (Mood 

 and others 1974; Kendall and Stuart 1973). 



Translating these assumptions to the problem at hand, the first assumption would state 

 that the probability of a tree being in a specified diameter class is fixed. Miile this 

 assumption appears to be unreasonable, it may be possible to define the ' s as being condi- 

 tional upon site and stand conditions. Therefore, p. would be the probability of a tree being 

 in the ith diameter class at the end of the growth period given the present diameter distri- 

 bution and the specified site index and time-since-last-cutting. 



The second assumption states that the total number of trees in the observed diameter 

 distribution must equal the total number of trees in the predicted diameter distribution. To 

 achieve this condition, upgrowth, ingrowth, mortality, and conversion from blackjack to yellow 

 pine must be predicted exactly--and this is highly unlikely. Therefore, while the problem 

 with the first assumption may be avoidable, violation of the second assumption seems unavoid- 

 able. As a result, the use of the chi-square "goodness-of-fit" statistic for testing differences 

 between predicted and actual diameter distributions is judged to be inappropriate. 



89 



