Error propagation in calculated ratios

Background:Calculated quantities that combine results of multiple laboratory tests have become popular for screening, risk evaluation, and ongoing care in medicine. Many of these are ratios. In this paper, we address the specific issue of propagated random analytical error in calculated ratios.Methods:Standard error propagation theory is applied to develop an approximate formula for the mean, standard deviation (SD), and coefficient of variation (CV) of the ratio of two independent, normally distributed random variables. A method of mathematically modeling the problem by random simulations to validate these formulas is proposed and applied. Comparisons are made with the commonly quoted formula for the CV of a ratio.Results:The approximation formula for the CV of a ratio R = X/Y of independent Gaussian random variables developed herein has an absolute percentage error less than 4% for CVs of less than 20% in Y. In contrast the commonly quoted formula has a percentage error of up to 16% for CVs of less than 20% in Y.Conclusion:The usual formula for the CV of a ratio functions well when the CV of the denominator is less than 10% but for larger CVs, the formula proposed here is more accurate. Random analytical error in calculated ratios may be larger than clinicians and laboratorians are aware. The magnitude of the propagated error needs to be considered when interpreting calculated ratios in the clinical laboratory, especially near medical decision limits where its effect may lead to erroneous conclusions.