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I'm trying to learn about rank tests, and having doubts abouut how I should a result from a Wilcoxon Signed Rank test. Suppose, we are given: Z = -2.201, r = - 0.845, P < 0.05, then how should I interprete these three numbers and describe them in words?

Error404
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2 Answers2

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BBR Section 7.2 and especially 7.2.1 cover this. It shows how to scale the signed rank statistic to [0,1] representing the probability that a randomly chosen pair of observations sum to a positive number.

Don't state P < 0.05. Give the P-value, and a proper conclusion is not reject/accept $H_0$ but rather, when P is small, that there is evidence against the supposition of no difference.

Frank Harrell
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  • Thank you for sharing the detailed pdf. I will go through it and hopefully have my doubts resolved. Also, I saw in some of the papers that their Wilcoxon Signed Rank test result was documented as P < 'some constant value' instead of a fixed P-value. This was another reason why I got confused. – Error404 Mar 28 '22 at 16:05
  • A proper conclusion is found/failed to find evidence for $H_{a}$, never "accept $H_0$". Rejection decision is reject $H_0$ or fail to reject $H_0$. – Alexis Mar 28 '22 at 16:16
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It is not that simple to explain what the signed-rank test actually tests for. My best advice is read a few respected textbooks covering the test, and pay attention to how they state the null and alternative hypotheses. However, be ware that some texts may simplify the language for these hypotheses, or may pull in additional assumptions to make interpreting the test easier.

Essentially, the signed rank test assesses whether the differences in pairs ---- or of the one-sample observations ---- is symmetric about zero. Except that the test uses the ranks after taking the differences. You might say that it tests if the paired differences are systematically different than zero.

Various plots may be helpful. A histogram of the paired differences. Plotting group 1 vs. group two and superimposing a 1:1 line.

Beyond that, z and p are commonly reported statistics in the analysis of experiments. r in this case is likely the z value divided by the number of pairs, and is an effect size statistic. It more or less ranges from -1 to 1. A negative r suggests that the second group has larger values than the first group. (But double check this, as software can reverse this !). An absolute value of 0.8 for this statistic is likely a large value.

Sal Mangiafico
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    As stated earlier it tests precisely whether the probability that a randomly chosen pair of values sums to a positive number is equal to 0.5 or not. But a difference in ranks procedure may work better: https://bpspsychub.onlinelibrary.wiley.com/doi/abs/10.1111/j.2044-8317.1990.tb00939.x – Frank Harrell Mar 28 '22 at 17:15
  • @FrankHarrell , wouldn't that be the hypothesis tests for the sign test, and not for the Wilcoxon signed rank test ? For example, for the following one-sample data, the p-value for the sign test is 1, whereas the p-value for the signed rank test is c. 0.02. The probability of A being positive from this sample is 0.5. But because rank(A) ⋅ sign(A) is not symmetric about zero, the signed rank test identifies this asymmetry. A = (-15, -14, -13, -12,-11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115) – Sal Mangiafico Mar 29 '22 at 13:06
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    No. The sign test test $H_{0}: \Pr(Y > 0) = \frac{1}{2}$. The signed rank test tests $H_{0}: \Pr(Y_{i} + Y_{j} > 0) = \frac{1}{2}$. – Frank Harrell Mar 30 '22 at 21:53
  • Thanks, @FrankHarrell . – Sal Mangiafico Mar 31 '22 at 11:47
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    Here we upvote instead of thanks. Thanks. – Frank Harrell Mar 31 '22 at 15:42