Taylor Series Approximation for Accurate Generalized Confidence Intervals of Ratios of Log-Normal Standard Deviations for Meta-Analysis Using Means and Standard Deviations in Time Scale

With contemporary anesthetic drugs, the efficacy of general anesthesia is assured. Health-economic and clinical objectives are related to reductions in the variability in dosing, variability in recovery, etc. Consequently, meta-analyses for anesthesiology research would benefit from quantification of ratios of standard deviations of log-normally distributed variables (e.g., surgical duration). Generalized confidence intervals can be used, once sample means and standard deviations in the raw, time, scale, for each study and group have been used to estimate the mean and standard deviation of the logarithms of the times (i.e., "log-scale"). We examine the matching of the first two moments versus also using higher-order terms, following Higgins et al. 2008 and Friedrich et al. 2012. Monte Carlo simulations revealed that using the first two moments 95% confidence intervals had coverage 92%-95%, with small bias. Use of higher-order moments worsened confidence interval coverage for the log ratios, especially for coefficients of variation in the time scale of 50% and for larger $\left(n=50\right)$ sample sizes per group, resulting in 88% coverage. We recommend that for calculating confidence intervals for ratios of standard deviations based on generalized pivotal quantities and log-normal distributions, when relying on transformation of sample statistics from time to log scale, use the first two moments, not the higher order terms.