A graph of the data (fig. 3) indicates a simple linear equation may be appropriate.
The sums, sums of squares, cross-products, and averages needed for least squares computations are (letting Y = DBH and X = DS)
Least squares estimates of the coefficients are then calculated using [1] and [2] as
Summary statistics for the fit are given by equations [3] and [4] as
A graph of studentized residuals versus DS is given in figure 4. No trends seem apparent and all points fall within a horizontal band centered at zero. The least squares fit to the data appears appropriate. The magnitude of R2 indicates a strong linear relationship between DBH and DS. On average the data points fall about 1.03 (square root of 1.0534) inches from the fitted line.
One possible use of the fitted equation is the prediction of DBH from an observed DS. For example, it may be necessary to predict DBH from the DS measurements of trees illegally taken from an area. A local volume equation (volume as a function of DBH) could then be used to estimate volume removed. Of course, it is important that consideration be given to the applicability of the equation to the particular situation. In the example, questions such as: Are the subject trees white oaks? Did the subject trees grow on sites similar to Virginia coves? Are the subject trees in the size range of the trees used to fit the equation? Were the subject trees cut to an approximate one-foot stump? would need to be addressed.
To illustrate the calculations consider two trees: one with DS = 10.0 and one with DS = 25.0. Predicted DBHs are
with variances (equation [5])
Standard errors of the predictions are thus 1.04 inches and 1.15 inches, respectively. Here we should be fairly confident in both predictions though the latter is in some question, being outside the range of the sample data.