The data used in the generalized least squares analysis in the text to develop a model to relate DBH (diameter at breast height, 54 inches) to diameters at various stump heights, Example 12.6. The numbers in the table are
ij., where
ijk
and Xj
are defined in the text. The estimated variance–covariance matrix is shown in equation 12.47.
(a) Use a matrix computer program to do the generalized least squares analysis on these data as outlined in Example 12.6. Notice that the model contains a zero intercept. Give the regression equation, the standard error of the regression coefficient, and the analysis of variance summary. (Your answers may differ slightly from those in Example 12.6 unless the variance–covariance matrix is rounded as described.)
(b) It would appear reasonable to simplify the variance–covariance matrix, equation 12.46, by assuming homogeneous variances and common covariances. Average the appropriate elements of B to obtain a common variance and a common covariance. Redo the generalized regression with B redefined in this way. Compare the results with the results in (a) and the unweighted regression results given in Example 12.6.
Example 12.6
The example used to illustrate weighted and generalized least squares Example 12.6 comes from an effort to develop a prediction equation for tree diameter at 54 inches above the ground (DBH ) based on data from diameters at various stump heights. The objective was to predict amount of timber illegally removed from a tract of land and DBH was one of the measurements needed. Diameter at 54 inches (DBH ) and stump diameters (SD) at stump heights (SHt) of 2, 4, 6, 8, 10, and 12 inches above ground were measured on 100 standing trees in an adjacent, similar stand. The trees were grouped into 2-inch DBH classes. There were n = 4, 16, 42, 26, 9, and 3 trees in DBH classes 6, 8, 10, 12, 14, and 16 inches, respectively. It was argued that the ratio of DBH to the stump diameter at a particular height should be a monotonically decreasing function approaching one as the stump height approached 54 inches. This relationship has the form of an exponential decay function but with much sharper curvature than the exponential function allows. These considerations led to a model in which the dependent variable was defined as
where i is the DBH class (i = 1,..., 6); j is the stump height class j = 1,..., 6); k is the tree within each DBH class (k = 1,...,ni); and ln(SDijk) and ln(DBHik) are the logarithms of stump diameters and DBH. The averages of Yijk
over k for each DBH –stump height category are given in Table 12.3. The exponent c, applied to the stump heights, was used to straighten the relationship (on the logarithmic scale) and was chosen by finding the value c = 0.1 that minimized the residual sum of squares for the linear relationship. Thus, the model is