New Results About Difference Scores, Ratios, and Hierarchical Linear Model Parameters as Tools for Comparing Groups

At least since Pearson’s infamous warning in 1897, ratio and difference measures have provoked dispute and misunderstanding, even as they continue to appear in social science research publications. Using differential calculus and hierarchical model decomposition, this article (a) presents a new, streamlined exposition that directs attention to the “effects” of variables rather than correlations between them, (b) explicates a previously unnoticed measurement problem that affects certain ratio measures, (c) shows that hierarchical model decomposition satisfies many or all of the measurement purposes for which ratio and difference measures are used, (d) presents a brief proof that some widely used difference and log-of-ratio measures are mathematically equivalent to hierarchical model measures that do not suffer the problems noted by Pearson, and (e) illustrates these arguments with an analysis of U.S. Census Bureau’s Current Population Survey data on mean earnings, by state, of white, black, American Indian, and Asian men in 2016.