Solved – How to describe the differences in skewed data with same median but statistically different distribution

I am comparing length of stay after laparoscopic and open appendectomy in over 160000 patients. LOS is typically a skewed variable so I use the median and interquartile range and ranksum test to describe the result, which is 2, IQR 2-3 and 2, IQR 2-4, p<0.0001. So there is a highly significant difference but it does not show that well in the median and IQR.

The difference in length of stay is very small. This finding is a negative finding but still interesting as it goes against what most people think about LOS after laparoscopic appendectomy. That is why I want to report on the difference in LOS which is not evident if I only report median and IQR with p-value.

The arithmetic mean (3.1 vs 2.8 and a difference of 0.3 days) is not recommended as the distribution is skewed. I wonder if any other measure could be used like, geometric mean (2.5 vs 2.3, difference 0.2 days) or harmonic mean (2.1 vs 1.9 days, difference 0.2 days)? When can I use geometric mean or harmonic mean?

I am inclined to use the harmonic mean which comes closest to the median and gives a difference of 0.2 days. Is this incorrect?

If you want to describe the difference in distribution, perhaps something like a Q-Q plot, or a pair of kernel densities would be common tools, though I assume you have quite discrete looking distributions, in which case that may make something like a pair of barcharts/histograms or a pair of ECDFs better (if possible on a single display)

If you do go with looking at the difference in ECDF, the two-sample Kolmogorov-Smirnov statistic would be the ideal "measure" of significance to go with that, since it's based on the biggest (vertical) difference in ECDF.

If you compare barplots(/histograms) for a discrete distribution, you might like to look at the kinds of measures of discrepancy indicated by smooth tests of goodness of fit (Neyman-Barton type tests), but where the statistic is adapted to two samples. This would in effect correspond to partitioning a chi-square into components associated with orthogonal polynomials, the first of which would correspond to location (a linear term), the second to variance (a quadratic term), and so on (typically you'd look out to order 4 or 6). Books and numerous articles by Rayner and Best (and some others) cover some of that territory.

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