DISTRIBUTION WITHIN DIAMETER CLASS AND DIAMETER CLASS SIZE 



The distribution of the trees within a diameter class can influence the number of trees 

 advancing out of the class, given an average class diameter growth rate (Wahlenberg 1941; Meyer 

 1942). The easiest distribution to use, and the one assumed by Moser (1974), is the uniform 

 distribution. If this distribution is valid, then the following method can be used for comput- 

 ing upgrowth (Wahlenberg 1941; Meyer 1942). The quotient found by dividing diameter growth by 

 diameter class size is computed. The integer portion of this quotient is then the number of 

 diameter classes in which all trees will advance, and the fractional portion of the quotient is 

 the proportion of the trees that will advance one diameter class further. (For example, if the 

 quotient was 2.29, then all trees will advance two diameter classes, and 29 percent of them 

 will advance three diameter classes.) 



Because it is the easiest to apply, I first tested the appropriateness of the uniform 

 distribution through use of the chi-square "goodness-of-f it" test (see appendix B for details) . 

 A total of 714 diameter classes were tested at the 99 percent level of significance, and of 

 these, only seven were found to be significantly different from the uniform distribution. 

 These findings were further substantiated by visually checking the distributions of many of 

 these classes. Therefore, the assumption of within-class uniformity could not be rejected in 

 the even- and uneven-aged stands of the Fort Valley Experimental Forest. 



DIAMETER GROWTH 



The final component necessary for the calculation of upgrowth is an estimate of the diam- 

 eter growth rate for each class. The following summarizes procedures used to develop the 

 appropriate diameter growth models. A more complete discussion is in appendix C. 



Initial Review and Modeling Decisions 



A review of the literature for southwestern ponderosa pine helped identify factors histor- 

 ically recognized as related to diameter growth. The significant factors identified included: 

 site index, rainfall, total stand density, diameter class size, tree vigor as indicated by bark 

 color (in other words, "yellow" pine and "blackjack" pine), and the structure of the competition 

 as indicated by the position of the diameter class in the stand. 



Given the general factors that can affect diameter growth, it was then necessary to 

 choose: the general structure of the model, the form of the dependent variable, the method for 

 handling the error structure of the model during prediction, and the appropriate transforma- 

 tions of the independent variables. After careful consideration, I decided to predict the 

 natural log of diameter class basal area growth and then convert that value to a diameter 

 growth rate. Three basic reasons for this choice are: 



1. A nonlinear, multiplicative error model form probably best represented the interaction 

 of the independent variables with themselves and their effect upon diameter growth. 



2. Basal area growth is often nearly linear over short time periods, which simplifies 

 the extrapolation of growth rates to growth periods different from that originally used in 

 equation development. 



5. The residuals of log of basal area growth more often approach normality and homo- 

 geneous variance than other dependent variables, which is advantageous in model development and 

 significance testing. 



Several ways of using information about the error structure of the model are available when 

 using the finished model for predictive purposes. In deterministic modeling, the random error 

 element of the model is simply ignored for predictive purposes. If the model is properly 

 constructed, predictions from a deterministic model represent the "expected" value of the 

 output. In stochastic modeling, the random error element is considered in prediction. To 

 obtain an expected value of the output from a stochastic model, however, requires a "Monte 

 Carlo analysis," which can be time consuming and expensive. 



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