Exceedance Probabilities of Log-normal Distributions for One Group, Two Groups, and Meta-analysis of Multiple Two-group Studies, With Application to Analyses of Prolonged Times to Tracheal Extubation

Many randomized trials measure means and standard deviations of anesthesia recovery time (e.g., times to tracheal extubation). We show how to use generalized pivotal methods to compare the probabilities of exceeding a tolerance limit (e.g., > 15 min, prolonged times to tracheal extubation). The topic is important because the economic benefits of faster anesthesia emergence depend on reducing variability, not means, especially prevention of very long recovery times. Generalized pivotal methods are applied using computer simulation (e.g., using two Excel formulas for one group and three formulas for two group comparisons). The endpoint for each study with two groups is the ratio between groups of the probabilities of times exceeding a threshold or the ratio of the standard deviations. Confidence intervals and variances for the incremental risk ratio of the exceedance probabilities and for ratios of standard deviations are calculated using studies’ sample sizes, sample means in the time scale of recovery times, and sample standard deviations in the time scale. Ratios are combined among studies using the DerSimonian-Laird estimate of the heterogeneity variance estimate, with Knapp-Hartung adjustment for the relatively small (N = 15) numbers of studies in the meta-analysis. We show larger absolute variability among studies’ results when analyzed based on exceedance probabilities rather than standard deviations. Therefore, if an investigator’s primary goal is to quantify reductions in the variability of recovery times (e.g., times until patients are ready for post-anesthesia care unit discharge), we recommend analyzing the standard deviations. When exceedance probabilities themselves are relevant, they can be analyzed from the original studies’ summary measures.