Wilcoxon Signed Rank Symmetry Assumption

Question

The assumption of symmetricity for signed rank test (and its relevance) is becoming extremely confusing for me. I am hypothesizing that sub-population A (before treatment) and sub-population B (after treatment) come from same population (no effect of treatment). Does my paired difference need to conform to the assumption of symmetricity?

jbowman noted in his response here that "Note that under the typical null hypothesis, if you assume sub-populations A and B are drawn from the same distribution, symmetry of the paired differences between A and B is assured regardless of the lack of symmetry of the underlying distribution." Whereas texts elsewhere say "If you are testing the null hypothesis that the mean (= median) of the paired differences is zero, then the paired differences must all come from a continuous symmetrical distribution. "

Answer 1

Although on the surface the two statements above may appear contradictory, they aren't. The Wilcoxon Signed Rank test does require that the paired differences come from a continuous symmetric distribution (under the null hypothesis, as Michael Chernick points out in comments.) In the special case when the two sub-populations $A$ and $B$ from which the paired samples will be drawn (one each from $A$ and $B$) have the same (continuous) distribution, it is guaranteed that the pairwise differences between the samples $a_i,b_i$ will come from a continuous symmetric distribution.

You can see this by observing that if the two samples come from the same distribution, $p(a_i = x, b_i = y) = p(a_i = y, b_i = x)$. In the former case, the paired difference $\delta_{i,1} = x-y$, and in the latter case the paired difference $\delta_{i,2} = y-x = -\delta_{i,1}$. Since the probabilities of the two cases are equal, it follows that $p(\delta_i) = p(-\delta_i)$, i.e., that the distribution is symmetric around 0.

Therefore, if you can make the assumption that the two sub-populations have the same continuous distribution under the null hypothesis, you've satisfied the Wilcoxon Signed Rank assumption requirements. Often it is easier to see why this assumption might be true than to see why the more general "pairwise differences come from a continuous symmetric distribution" might be true, hence its occasional use.

Answer 2

Note that the samples may tell you nothing about the suitability of the assumption that is required for the null. If the null is false, you don't necessarily require symmetry (and it's easy to demonstrate examples where everything works as desired without needing symmetry under the alternative).

People seem to spend a deal of effort worrying about testing an assumption using data that may not be relevant to the question of whether the assumption is reasonable. The assumption might better be dealt with by considering its plausibility before even seeing the data (jbowman's excellent answer covers this in some detail; under the assumption that the treatment doesn't change the distribution at all, it should follow immediately).

[This doesn't mean that the situation under the alternative is arbitrary (it's a test sensitive to particular kinds of deviations from symmetry about zero in pair-differences and it's insensitive to other kinds of deviations) - you are still looking for a tendency for differences to be typically larger than 0 or typically smaller than 0]

Additionally, testing the assumption (and then choosing what you do based on whether or not you reject the null on that prior test) will impact the properties of the subsequent procedures you're considering (p-values will no longer be uniform under the null, for example).

In some cases it may be convenient to assume a location shift alternative (certainly it's possible to estimate its size and even give a confidence interval for it). It would for example, make it simpler to discuss the treatment effect. However, that doesn't make it an assumption of the test itself, it makes it an additional assumption we've made for other reasons. In this situation, the data may be relevant to assessing whether that assumption was suitable, but we still have the problem that if we choose a test on the basis of the data, we no longer have our desired significance level (there are also impacts on power).

Answer 3

This is less an answer and more a request for further clarification from respondents, and maybe a clarification itself.

Here are a couple of examples from textbooks showing how the symmetry assumption for the signed-rank test is sometimes handled. On my bookshelf I couldn't find anything more clear or more explanatory.

There are also numerous examples from the internet and from Cross Validated where the symmetry assumption is stated simply and without further explication.

The Wilcoxon [signed-rank] test assumes that the sampled population is symmetric (in which case the median and mean are identical and this procedure becomes a hypothesis test about the mean as well as about the median, but the one-sample t test is typically a more powerful test about the mean).

-- Zar, Biostatitics, 5th, 2010, § 7.9

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The important difference between the sign test and this [Wilcoxon signed-rank] test is an additional assumption of symmetry of the distribution of the differences.

...

Assumptions:

1. The distribution of each Di [difference in paired observations] is symmetric.
2. The Dis are mutually independent.
3. The Dis all have the same mean.
4. The measurement scale of the Dis is at least interval.

-- Conover, Practical Nonparametric Statistics, 3rd, 1999, § 5.7

However, I think the responses by @jbowman and @Glen_b explain this assumption in a different light.

If the null is false, you don't necessarily require symmetry.

...

In some cases it may be convenient to assume a location shift alternative... However, that doesn't make it an assumption of the test, it makes it an additional assumption we've made for other reasons.

-- @Glen_b

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The Wilcoxon Signed Rank test does require that the paired differences come from a continuous symmetric distribution (under the null hypothesis, as Michael Chernick points out in comments.)

-- @jbowman

One way I might be able to sum up my understanding is as follows. The null hypothesis for the signed-rank test is that the differences are symmetrically distributed about a value. There are two ways this hypothesis could be wrong. 1) The differences could be symmetrical about a different value. 2) The differences could be not symmetrical. If #1 is the case, then the differences have a different location than in the null hypothesis, which is a useful result, and easy to interpret. If #2 is the case, then the data are skewed in a way that is a useful result. This may be a little less easy to interpret that in #1, but an examination of a histogram of the differences should reveal useful information about the distribution.

With this understanding, I think there is no assumption about the distribution of the original data, the differences, or the ranks of the differences.

How does this sound?

Answer 4

I'm late to the party here, but I was looking into this question and wanted to add something (hopefully) helpful.

In Nonparametric Statistical Inference fifth edition by Gibbons and Chakraborti I found this:

The Wilcoxon signed-rank statistics can also be used as tests for symmetry if the only assumption made is that the random sample is drawn from a continuous distribution...If the null hypothesis is accepted, we can conclude that the population is symmetric and has median M₀. But if the null hypothesis is rejected, we cannot tell which portion (or all) of the composite statement [null and alternative] is not consistent with the sample outcome. With the two-sided alternative, for example, we must conclude that either the population is symmetric with median not equal to M₀, or the population is asymmetric with median equal to M₀, or the population is asymmetric with median not equal to M₀. Such a broad conclusion is generally not satisfactory, and this is why in most cases the assumptions that justify a test procedure are separated from the statement of the null hypothesis.

I think this helps reinforce the other answers here to explain why the symmetry assumption is considered as plausible by researchers using this method, but not necessarily desiring a formal test of it based on the sample data.