Estimating the uncertainty associated with Widmark's equation as commonly applied in forensic toxicology

Abstract

No computation is performed more frequently by forensic toxicologists than that involving Widmark's equation. The equation is employed to estimate either the number of drinks consumed or the corresponding blood or breath alcohol concentration. Despite the wide use of Widmark's equation, rarely is an uncertainty estimate also provided. Estimates from Widmark's equation involve at least seven uncertain random variables. Uncertainty estimates are presented that rely on methods of general error propagation compared to a method developed by Widmark. Assuming reasonable variable and uncertainty estimates, the error propagation method yielded for N = 10.4 drinks, a combined uncertainty (standard deviation) of 1.3 drinks (CV = 12.3%). Similarly, estimating the blood alcohol concentration yielded for 0.120 g/100 ml, an uncertainty of 0.0255 g/100 ml (CV = 21.2%). Widmark's uncertainty method yielded 1.6 drinks (CV = 15.4%). The derivation of Widmark's uncertainty estimate is also presented, showing that he considered only ρ and β to be uncertain. Widmark estimates for the number of drinks should include a 2CV estimate of approximately 25% while the blood alcohol concentration estimate should include a 2CV estimate of approximately 42%. Including valid estimates of uncertainty should enhance the legal admissibility and confidence for Widmark estimations.