Confounding-adjustment methods for the difference in medians

With continuous outcomes, the average causal effect is typically defined using a contrast of expected potential outcomes. However, in the presence of skewed outcome data, the expectation may no longer be meaningful and the definition of the causal effect should be considered more closely. When faced with this challenge 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 for which the expectation is interpretable as the central value. However, neither approach is entirely satisfactory. An appealing alternative is to define the causal effect using a contrast of median potential outcomes, although there is limited discussion or availability of confounding-adjustment methods to estimate this parameter. Within this study, we described and compared confounding-adjustment methods to estimate the causal difference in medians, addressing this gap. The methods considered were multivariable quantile regression, an inverse probability weighted (IPW) estimator, weighted quantile regression and two possible, little-known implementations of g-computation. The methods were evaluated within a simulation study under varying degrees of skewness in the outcome and applied to an empirical study. Results indicated that the IPW estimator, weighted quantile regression and g-computation implementations minimised bias across all simulation settings, if the corresponding model was correctly specified, with g-computation additionally minimising the variance in estimates. The methods presented within this paper provide appealing alternatives to the common "ignore or transform" approach, enhancing our capability to obtain meaningful causal effect estimates with skewed outcome data.