Confounding-adjustment methods for the difference in medians

With continuous outcomes, the average causal effect is typically defined using a contrast of mean potential outcomes. However, for skewed outcome data the mean may no longer be a meaningful summary statistic and the definition of the causal effect should be considered more closely. In practice the typical approach is to either "ignore or transform" - ignore the skewness in the data entirely or transform the outcome to obtain a more symmetric distribution. In many practical settings neither approach is entirely satisfactory. Alternatively, the causal effect could be defined using a contrast of median potential outcomes, although discussion or availability of confounding-adjustment methods to estimate this parameter is currently limited. To address this gap, we described and evaluated confounding-adjustment methods to estimate the causal difference in medians, specifically multivariable quantile regression, an inverse probability weighted (IPW) estimator, weighted quantile regression and adaptations of the g-computation approach. Performance of these methods was assessed within a simulation study under varying degrees of skewness in the outcome, and within an empirical study motivated by the Longitudinal Study of Australian Children. Results indicated the IPW estimator and weighted quantile regression were the best performing across all simulation settings if the propensity score model is correctly specified. Other methods had similar or higher bias than an unadjusted analysis. Application to the empirical study yielded more consistent estimates across methods. The methods presented provide appealing alternatives to the common "ignore or transform" approach, enhancing our capability to obtain meaningful causal effect estimates with skewed outcome data.