Date of Award
Master of Science
John C. Rennie
Mary Sue Younger, Glendon W. Smalley
Foresters have treated measurements at successive points on tree stems as independent observations when fitting tree taper models. From a statistical viewpoint, considering related measurements as independent is a problem, although there has been no viable alternative. Existing taper equations use total height as a variable. However, merchantable height is often measured in lieu of total height in hardwoods because many hardwood species exhibit a decurrent form.
This study presents a practical solution to the problem of correlated measurements and devises a way to use merchantable height rather than total height in taper equations. Dendrometry data from blackgum (Nyssa sylvatica Marsh.), black oak (Quercus velutina Lam), chestnut oak (Quercus prinus L.), hickory (Carya spp), northern red oak (Quercus rubra L.), Scarlet oak (Quercus coccinea Muenchh.), white oak (Quercus alba L.), and yellow-poplar (Liriodendron tulipifera L.) in Tennessee were used with the modified taper equation to predict ratios of squared diameters (upper stem diameter squared) / (diameter breast height squared) and diameters inside bark.
Nonindependence of variables was overcome by: 1) creating a data subset by randomly choosing one measurement from each tree, 2) computing a regression coefficient for this subset with the ordinary least squares method (standard method), 3) repeating steps one and two 1300 times, and 4) averaging the resulting coefficients to obtain an applicable coefficient.
Since taper is the relative rate of change in stem diameter with increasing tree height, the diameter at merchantable height can be subtracted from all diameters down the stem without changing the relative taper. The diameter at merchantable height becomes zero, just as the actual diameter at the top of the tree. Using merchantable height in place of total height, transformed diameters at various heists in place of measured diameters at various heights, and transformed diameter breast height in place of measured diameter breast height, the hardwood tree data are then compatible with existing taper equations.
Using the coefficients generated by the iterative method, a taper model developed by Kbzak et al. (1969) was modified and evaluated for its ability to predict ratios of squared diameters and diameters inside bark. Standard errors of the estimates and biases suggest a good fit. Plotting differences in predicted and actual ratios of squared diameters and differences in diameters inside bark, against relative heights (where diameter measurements were taken) to merchantable height also indicated an acceptable fit. However, the plottings showed that the model tended to overestimate ratios of squared diameters and diameters inside bark on the lower stem, and slightly underestimate them at 50 to 75 percent of merchantable height.
Coefficients were also obtained after fitting the data to the modified model while ignoring correlated measurements. As expected these coefficients were similar to those obtained by the iterative method. Standard errors of the coefficients calculated according to the iterative method, unlike the standard method as reported in literature, did not underestimate the true standard error of the estimated regression coefficients. The iterative method for generating coefficients and the technique using merchantable height and transformed diameters are practical solutions to the problems of nonindependence of variables and of using merchantable height in taper equations.
Fowler, John H., "Use of merchantable height and correlated measurements in taper equations. " Master's Thesis, University of Tennessee, 1986.