Methods of adjusted comparison of medians

Consider the problem of making an adjusted comparison of the medians of two populations on an interval type outcome variable. A common method of doing this is through the use of a linear model requiring the residuals to be normally distributed. We describe here two methods based on a linear model after Box-Cox transformation of the outcome variable. The methods require a reference population, which could be either of the populations under study or their aggregate. We compare the new procedures with the comparison of normal means procedure and other procedures proposed for this problem by simulation. It is found that the procedure based on comparison of the predicted values obtained from the observed covariates of the reference population has higher power for testing and smaller mean square error of estimation than the other methods, while maintaining reasonable control of the type I error rate. We illustrate the methods by analyzing the duration of the second stage of labor for women in two large observation studies (Collaborative Perinatal Project and Consortium on Safe Labor) separated by 50 years. We recommend the method based on comparison of the predicted values of the transformed outcomes, with careful attention to how close the resulting residual distribution is to normal.