The mathematical relationship among different forms of responsiveness coefficients

Background Little consensus exists regarding the most appropriate measure of responsiveness. While most indices are variants on Cohen's effect size, the mathematical relationships among these indices have not been elucidated. Consequently, the health-related quality of life (HRQL) literature contains many publications in which a variety of different indices are computed and differences among them noted. These differences are completely predictable when the underlying analytical form of each coefficient is explicated. Methods In this paper, we begin with a mathematical analysis of the variance components underlying an observed change score. From this, we determine analytically the relationships among the more commonly used indices of responsiveness. Conclusions Based on this analysis, we conclude that Cohen's effect size and the Standardized Response Mean are the two most appropriate measures, as each provides unique information and each best captures an important relation between treatment effect and variability in response. However, the latter should be interpreted with caution, as under some circumstances, any measure based on variability in change scores can give misleading information. On this basis, we recommend that future analysis of responsiveness be restricted to the Cohen effect size to ensure interpretability and comparability with treatment effects in other domains.