Hodges-Lehmann Shift Estimate for Two Samples

The Hodges-Lehmann shift estimate for two samples is the median of all pairwise differences between observations of the second and first sample, with a distribution-free confidence interval from the ordered differences.

두 표본 Hodges-Lehmann 이동량은 두 번째 표본과 첫 번째 표본 관측값 사이의 모든 짝 차이의 중앙값이며, 정렬한 차이로 분포 무관 신뢰구간을 구합니다.

My Mann-Whitney test was significant; how large is the difference between the two groups and what is its confidence interval?

Mann-Whitney 검정이 유의하게 나왔는데 두 집단 차이가 얼마나 되고 신뢰구간은 어떻게 되나요?

The Hodges-Lehmann estimator takes every difference between an observation in one group and an observation in the other and reports their median as the location shift. Its confidence interval uses order statistics of those differences, chosen from the exact Mann-Whitney distribution in small samples without ties and from a normal approximation otherwise.

When to use Hodges-Lehmann Shift Estimate for Two Samples

• You report a Mann-Whitney test and need an effect size in the original units.
• The outcome is skewed or has outliers, so a mean difference would be misleading.
• You need a confidence interval for the difference in location between two independent groups.

When to use something else

• The two groups differ mainly in spread or shape rather than location; use the k-sample Anderson-Darling or Kolmogorov-Smirnov test.
• The samples are paired; use the one-sample Hodges-Lehmann estimate of the paired differences with the Wilcoxon signed-rank test.

Assumptions and requirements

• The two samples are independent.
• The two distributions have the same shape and differ only by a location shift.
• The outcome is at least ordinal and measured on an interval scale for the shift to be meaningful.

What the analysis reports

• Shift estimate (group 2 minus group 1)
• Confidence interval with the method used (exact or normal approximation) and achieved confidence when exact
• Mann-Whitney U and p value
• Group medians and sizes

How to report it

The Hodges-Lehmann estimate of the shift in [outcome] between [group 2] and [group 1] was [value], 95% CI [[lower], [upper]], Mann-Whitney U = [value], p = [value].

Related methods

• Mann-Whitney U Test
• Robust Location and Scale Estimation
• Wilcoxon Signed-Rank Test

Run Hodges-Lehmann Shift Estimate for Two Samples on Quorant