Appendix C. Description of the reduction of error and product means tests.
The equation used to calculate the reduction of error (RE; Eq. C1) can be expressed in terms of the ŷ_{i} estimates and y_{i} predictand that are expressed as departures from the dependent period mean value:
(C.1) |
The term on the right-hand side of Eq. C.1 is the ratio of the total squared error obtained with the regression estimates and the total squared error obtained using the dependent period mean as the only estimate. This average becomes a standard against which the regression estimate is compared. If the reconstruction is better than the average of the dependent period, then the total error of the regression estimate is lower, the ratio is less than one, and the RE is positive.
The product means test (PM; Eq. C.2) calculates the products of the deviations and collects the positive and negative products in two separate groups based on their sign. The values of the products in each group are summed, and the means computed. The difference between the absolute values of the two means M_{+} – M_{-} can be tested for significance using the t statistics:
(C.2) |
where n^{+} and n^{-} are the number of positive and negative products and S_{+}^{2} and S_{-}^{2} are the corresponding variances.