Wilcoxon signed rank test [Nonparametrics]

posted by Helmut Homepage – Vienna, Austria, 2018-06-19 11:54 (890 d 07:32 ago) – Posting: # 18920
Views: 4,724

Hi libaiyi,

» […] In a 2*2 cross-over BE study. If we want to test the Tmax betweent different treatments. Which test statistic should be used? Wilcoxon signed rank test or Wilcoxon rank sum test?

The former (aka Wilcoxon T test).
The latter (aka Mann–Whitney U test, Mann–Whitney–Wilcoxon test, Wilcoxon–Mann–Whitney test) is for independent samples (parallel design).

» And it is also make me puzzled that the results from SAS and Winnonlin are different (Both are Hodges-Lehmann).

The test gives you only a p value.1 The point estimate (Hodges-Lehmann: x̃ of Walsh averages) and its CI (Moses) are other pieces of magic.2

» Would you please help me to figure the reason out?

Try Detlew’s example with tied data:
subject period sequence treatment  t
  1       1       TR       T      2.0
  1       2       TR       R      2.0
  2       1       RT       R      1.5
  2       2       RT       T      3.0
  3       1       TR       T      2.0
  3       2       TR       R      2.0
  4       1       RT       R      2.0
  4       2       RT       T      2.0
  5       1       TR       T      3.0
  5       2       TR       R      2.0
  6       1       RT       R      3.0
  6       2       RT       T      2.0
  7       1       TR       T      1.5
  7       2       TR       R      2.0
  8       1       RT       R      3.0
  8       2       RT       T      2.0
  9       1       TR       T      2.0
  9       2       TR       R      3.0
 10       1       RT       R      2.0
 10       2       RT       T      1.5
 11       1       TR       T      1.5
 11       2       TR       R      2.0
 12       1       RT       R      2.0
 12       2       RT       T      3.0
 13       1       TR       T      1.5
 13       2       TR       R      3.0
 14       1       RT       R      3.0
 14       2       RT       T      3.0


In Phoenix 8.0 / Crossover object:

T–R    : -0.25
~90% CI: -0.75, +0.25

In R / package coin with some additional coding:

  HL exact       : -0.25  interval midpoint: -0.25
  HL asymptotic  : -0.25  interval midpoint: -0.25
Confidence intervals (CI)
  Exact (90.21%) : -0.75, +0.25
Asymptotic (≥90%): -0.75, +0.25

Both agree with what Detlew got in SAS’ NPAR1WAY (after dividing by two!).
Which setup are you using in SAS?


  1. Koch GG. The Use of None-Parametric Methods in the Statistical Analysis of the Two-Period Change-Over Design. Biometrics. 1972;28(2):577-84. doi:10.2307/2556170.
  2. Hauschke D, Steinijans VW, Diletti E. A distribution-free procedure for the statistical analysis of bioequivalence studies. Int J Clin Pharm Ther Toxicol. 1990;28(2):72–8. PMID 2307548.

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