Robust confidence intervals for Hodges–Lehmann median difference

The cendif module is part of the somersd package, and calculates confidence intervals for the Hodges–Lehmann median difference between values of a variable in two subpopulations. The traditional Lehmann formula, unlike the formula used by cendif, assumes that the two subpopulation distributions are different only in location, and that the subpopulations are therefore equally variable. The cendif formula therefore contrasts with the Lehmann formula as the unequal-variance t-test contrasts with the equal-variance t-test. In a simulation study, designed to test cendif to destruction, the performance of cendif was compared to that of the Lehmann formula, using coverage probabilities and median confidence interval width ratios. The simulations involved sampling from pairs of Normal or Cauchy distributions, with subsample sizes ranging from 5 to 40, and between-subpopulation variability scale ratios ranging from 1 to 4. If the sample numbers were equal, then both methods gave coverage probabilities close to the advertized confidence level. However, if the sample numbers were unequal, then the Lehmann coverage probabilities were over-conservative if the smaller sample was from the less variable population, and over-liberal if the smaller sample was from the more variable population. The cendif coverage probability was usually closer to the advertized level, if the smaller sample was not very small. However, if the sample sizes were 5 and 40, and the two populations were equally variable, then the Lehmann coverage probability was close to its advertised level, while the cendif coverage probability was over-liberal. The cendif confidence interval, in its present form, is therefore robust both to non-Normality and to unequal variablity, but may be less robust to the possibility that the smaller sample size is very small. Possibilities for improvement are discussed.