variance functions. Most of ttie models were able to predict 

 well thie estimated cell variances for ihe area tables. The 

 area tables are derived from information on thie entire plot 

 with each attribute recorded only once per plot. This is in 

 contrast to the other tables where data are recorded on 

 individual trees in various combinations of plot designs. The 

 problem is especially acute when the table is separated into 

 diameter classes. When more than one fixed plot size is 

 used, the decision whether to record information from a tree 

 is based on the tree's distance from the plot center and the 

 diameter. So the diameter classes in a table also 

 correspond to plot size. The relationship between the cell 

 totals and the sampling errors is not a "smooth curve" but 

 actually two or more curves because sampling error 

 increases as plot size decreases. 



The problem of estimating a table composed of more than 

 one fixed plot size becomes apparent when the table is 

 divided into diameter classes. The table for number of live 

 trees per acre contains three fixed plot sizes and the 

 columns are divided into diameter classes. Deciding which 

 plot a tree belongs to depends on the diameter of the stem 

 as well as its distance from the plot center, so the column 

 headings divided the tables in plot sizes as well as diameter 

 classes. Three separate equations, one for each plot size, 

 was fitted to the table using Model 3; the improved from 

 0.53 to 0.82, indicating that the GVF differs for each of the 

 three diameter-class groups. 



A model's ability to predict the sampling errors of "mixed" 

 sampling intensities depends on the amount of information 

 about the table within the model and its degree of flexibility. 

 Model 5 included information concerning the sampling 

 errors of T, and T, and consequently performed better than 

 the other models. Model 4 was second because of its ability 

 to adjust the power of the exponents to better estimate a 

 compromise coefficient through the collection of points 

 produced by the different sampling intensities. 



Conclusion 



Models 2 through 5 performed well for the area tables. If 

 their performance can be classified as about equal, then 

 two factors remain in deciding which model to use. The first 

 is ease of use for the reader — the fewer the number of 

 coefficients to multiply and the fewer variables to locate in 



the table, the better. The second factor is ease in 

 implementing the regression procedure into existing 

 software and any maintenance of the procedure that may be 

 required by the producers of the tables. Nonlinear models 

 are more difficult to implement because of the initial 

 estimates of the parameters for each table or groups of 

 tables and the possibility of changing the initial values from 

 one data set to another. Changing the estimates can 

 become burdensome, especially when many tables are 

 produced as a part of a larger program requiring 

 considerable computer resources. On the basis of these 

 factors, Model 3 is recommended. 



For tables constructed from more than one sampling 

 intensity, Model 5 is recommended if SE(Tj) and SE(Tj ) are 

 published. Otherwise, Model 4 is recommended so long as 

 the problems of implementing a nonlinear subroutine are 

 not considered significant. Model 3 is a consideration 

 should Models 4 and 5 be deemed unsuitable. 



Including generalized variance functions with tables has 

 been shown to be feasible even with tables composed of 

 varying sampling intensities. GVF's have the advantage of 

 saving costs as compared to publishing separate tables of 

 sampling errors, adding utility to those publications that 

 currently do not publish sampling errors for all tables, and 

 maintaining a high degree of readability in the final product. 



Literature Cited 



Hines, F. Dee; Vissage, John S. 1988. Forest statistics for 

 Arkansas counties — 1988. Resour. Bull. SO-141. New 

 Orleans, LA: U.S. Department of Agriculture, Forest 

 Service, Southern Forest Experiment Station. 68 p. 



Valliant, R. 1987. Generalized variance functions in 

 stratified two-stage sampling. Journal of the American 

 Statistical Association. 82: 499-508. 



JAMES ALEGRIA is a forester, Northeastern Forest 

 Experiment Station, Radnor, Pennsylvania 19087. 



CHARLES T. SCOTT is a research forester and Project 

 Leader, Northeastern Forest Experiment Station, Delaware, 

 Ohio 43015. 



MANUSCRIPT RECEIVED FOR PUBLICATION 18 MARCH 1991 



Northeastern Forest Experiment Station 

 5 Radnor Corporate Center 

 100 Matsonford Road, Suite 200 

 P.O. Box 6775 

 Radnor, Pennsylvania 19087 



April 1991 



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