United States 

 Department of 

 Agriculture 



Forest Service 



Northeastern Forest 

 Experiment Station 



Research Note NE-345 



Generalized Variance Function 

 Applications In Forestry 



James Alegria 

 Charles T. Scott 



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Abstract 



Adequately predicting the sampling errors of tabular data 

 can reduce printing costs by eliminating the need to publish 

 separate sampling error tables. Two generalized variance 

 functions (GVFs) found in the literature and three GVFs 

 derived for this study were evaluated for their ability to 

 predict the sampling error of tabular forestry estimates. The 

 recommended GVFs for most tables are either a GVF which 

 incorporated the sampling errors of the row and column 

 totals or a nonlinear GVF when the sampling errors are not 

 published. Tables composed with one sampling intensity 

 and containing data from a multinomial distribution can be 

 represented by a simple linear estimator. 



Large surveys such as those conducted by USDA Forest 

 Inventory and Analysis (FIA) Projects generate large 

 amounts of information displayed in tabular form. Most 

 often, the tables are published with few indications of the 

 associated precision of the estimate. The lack of some 

 measure of precision is not an oversight but is due to costs 

 of publishing twice as many tables and the desire to 

 maximize readability of the report. If a measure of precision 

 is presented, it is often the sampling error in percent: 



SE(T,,) = 100(var(T,,))5/T,, 



where T.j is the cell estimate and var(T|,) is the sample 

 variance of the estimate. The sampling error is in the form 



of a simple approximation formula (Nines and Vissage, 

 1988) or a summary table of selected sampling errors. The 

 sampling errors selected for publication are usually the 

 errors of the table totals or subtotals. These can 

 inadvertently lead to the assumption that the sampling 

 errors of the individual cells within the table are of a similar 

 magnitude. While this may be true, it is more common that 

 the sampling error of a row or table total is smaller than the 

 sampling error of an individual cell estimate. The cell 

 sampling error can be many times greater than the 

 sampling error of its row or column total. The objective of 

 this study is to present GVFs as a cost-effective alternative 

 to printing sampling error tables, and to increase the 

 amount of information available in publications that 

 currently do not present sampling errors for each cell in a 

 table. 



Methods 



Models 



The models investigated were selected from the literature or 

 derived. For inclusion in the study, the model had to contain 

 variables that were published regularly or that could easily 

 be added to existing tables. They are the estimated totals of 

 the individual cells, (T,)), the row totals, (T, ), the column 

 totals, (Tj), the grand total (T ), and the sampling errors of 

 the row and column totals, SE(T|) and SE(T| ). The "." 

 signifies the summation over that row or column subscript. 

 The sampling errors for the row and column totals 



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