The Magnitude Conversion Problem: Further Insights

Abstract

In 2007, Castellaro and Bormann (2007) studied the performance of various\ntwo-dimensional (2D) regressions between different magnitude scales by mathematical\nsimulations. The study consisted in (1) generating sets of magnitude pairs\nxi; yi with given true slope βtrue following a Gutenberg–Richter distribution with\nb 1 and by adding initial errors ui; ei; (2) studying how far the slopes β obtained\nin those data sets by standard βSR, inverted standard βISR, orthogonal βOR, and generalized\northogonal βGOR regressions were from βtrue. Studies assessing the best\nregression method are important because the misuse of the common standard regression\neasily leads to magnitude conversion errors of 0.2–0.3 units. A different approach\nto the magnitude conversion problem was proposed before Castellaro and Bormann’s\n(2007) work and was based on the χ2 method, which is based on the theory of independent\nand normally distributed errors ui; ei and xi. In this work, we derive\nmathematical explanations for the results of Castellaro and Bormann (2007) in terms\nof the χ2 method and find that results agree for mean initial errors <0:5 magnitude\nunits. Our results demonstrate the importance of knowing and taking into consideration\nthe true initial errors in regression analysis.