Statistical considerations regarding the use of ratios to adjust data

Abstract

Objective: The use of ratios to adjust data (i.e. 'index' variables) is common in obesity and related research. The rationale for the use of ratios often seems to be the desire to control or eliminate the influence of the variable in the denominator. The purpose of this paper is to gain a greater appreciation of the statistical assumptions underlying ratios and their impact on data interpretation.

Results: We demonstrate the limitations of the indiscriminant use of ratios to adjust data. Specifically, we show that: (1) given linearity, a zero intercept between the numerator and denominator are necessary and sufficient conditions for a ratio to remove the confounding effects of the denominator; (2) seemingly minor departures from a zero intercept can have major consequences on the ratio's ability to control for the denominator; (3) the ratio of two normally distributed variables cannot be normally distributed, and this may violate the assumptions of subsequent parametric statistical analyses; (4) the use of ratios affects the error distribution of the data which may also violate the assumptions of subsequent parametric statistical analyses; (5) the use of ratios cannot easily take nonlinear effects between the numerator and denominator into account; (6) the use of ratios can introduce spurious correlations among the ratios and other variables; (7) the use of ratios can create interpretive difficulties. We also clarify that the mean of ratios is not necessarily equivalent to the ratio of the means of the numerator and denominator. Finally, we present and discuss formulae for the reliability of ratios and residuals.

Conclusion: Because of the above issues, we question the indiscriminant use of ratios and advocate that investigators consider regression-based approaches as alternatives.