An alternative to R^2 to characterize the quality of fit for linear least squares using offsets of variable orientation related to uncertainties in the data.

It is not appropriate to use the determination coefficient, R^2, to characterize the quality of fit for a least squares fitted line. In this paper, the maximum of R^2 is found as a function of the rotation angle of the data and gives the quality of fit for the line found by linear least squares with perpendicular offsets. The same rotation method is used to derive the perpendicular offset fit to the data, which yields two possible solutions where the correct root can be identified by a simple discriminant. These results are then generalized for any arbitrarily oriented offset, bringing about a new measure for the quality fit of a line, Q^2. Unlike the determination coefficient, R^2, this quality of fit measure is invariant to rotational transformations of the data and is specific to the offset’s orientation, which is directly related to the uncertainties in x- or y-data. Finally, this paper provides a method to determine the slope and intercept of a fitted line, as well as its quality of fit, given any estimate of the uncertainty ratio.