A comparison among general orthogonal regression methods applied to earthquake magnitude conversions

Until a decade ago, regression analyses for conversions between different types of magnitude were using only the ordinary least squares method,which assumes that the independent variable is error free, or the simple orthogonal regression method,which assumes equal uncertainties for the two variables. The recent literature became aware of the inadequacy of such approaches and proposes the use of general orthogonal regression methods that account for different uncertainties of the two regression variables. Under the common assumption that only the variance ratio eta between the dependent and independent variables is known, we compared three of such general orthogonal regression methods that have been applied to magnitude conversions: the chi-square regression, the general orthogonal regression, and the weighted total least squares. Although their formulations might appear quite different, we show that, under appropriate conditions, they all compute almost exactly the same regression coefficients and very similar (albeit slightly different) formal uncertainties. The latter are in most cases smaller than those estimated by bootstrap simulation but the amount of the deviation depends on the data set and on the assumed variance ratio.