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From here and here I see that we cannot use Mann-Whitney test if symmetry assumption is violated. Which test(s) can we use instead of Mann-Whitney test for non-parametric continues data if symmetry assumption is violated? I want to compare two continues independent variables which are non-normally distributed which violates the assumption for t-test. Null hypothesis: median/mean (depending on which of them the test can for test; I think median is better in that case) of one group is similar to another. Alternative hypothesis: median/mean of groups are different.

I do not have actual data yet. I may have it soon and I need to be prepared in case I will not be able to use Mann-Whitney test if symmetry assumption is violated.

vasili111
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    What are you actually testing? What are your null and alternative hypotheses? – jbowman Jan 14 '20 at 18:16
  • @jbowman Added that information to question. – vasili111 Jan 14 '20 at 18:32
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    "Non-parametric" can be a property of a *probability model* but it's not a characteristic of a variable. If you would like reasonable advice, then please supply some information about (a) the empirical distributions of these variables and (b) the null hypothesis you wish to test. – whuber Jan 14 '20 at 18:35
  • @whuber I updated the question and I think now it looks better. If it looks good then please reopen it or please let me know what else can be added. Thank you. – vasili111 Jan 14 '20 at 18:45
  • Thank you: you're halfway there. You also need to describe your data. – whuber Jan 14 '20 at 18:48
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    @whuber I do not have actual data yet. I may have it soon and I need to be prepared in case I will not be able to use Mann-Whitney test if symmetry assumption is violated. I understand that you want to make question as best as possible and I really appropriate it (no irony of any kind here). Of course it will be better to add data description to it too but as I said I do not have it and even in this state I think my question is currently valid. – vasili111 Jan 15 '20 at 00:31
  • That's good enough (+1). – whuber Jan 15 '20 at 14:06
  • Note that the term "Mann-Whitney" is often used for a test comparing two independent samples. This does *not* assume symmetry, only the test for paired samples does, and it doesn't assume symmetry of the raw distributions but rather of the *differences* between the two values in a pair. – Christian Hennig Jan 15 '20 at 14:18
  • Also see my answer here: https://stats.stackexchange.com/questions/332469/symmetry-assumption-in-wilcoxons-signed-rank-test/444022#444022 – Christian Hennig Jan 15 '20 at 14:19
  • My previous remark refers to the paired/signed Wilcoxon test of which I don't know whether it is also referred to as Mann-Whitney. Anyway, as opposed to standard Mann-Whitney, that one has a symmetry assumption for the null hypothesis. – Christian Hennig Jan 15 '20 at 14:27
  • The answer from @Lewian should reassure you on one account. That said, the two-sample unpaired Mann-Whitney (or Wilcoxon) test does _not_ provide a test on medians or means. Rather, it tests whether the probability is 50% that the value of a random draw from one of the distributions will be larger than the value of a random draw from the other. See [this answer](https://stats.stackexchange.com/a/35432/28500) for an example in which medians are the same but Mann-Whitney detects a (both statistically and visually) significant difference between the two distributions. – EdM Jan 15 '20 at 15:22
  • By the way, the t-test is approximately valid for many non-normal distributions including asymmetric ones if the sample size is large enough, due to the Central Limit Theorem. Extreme outliers or extreme skewness could be a problem though. – Christian Hennig Jan 15 '20 at 15:40

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The Mann-Whitney two-sample test does not require symmetry, and actually the two links that you give don't claim that it does. The first reference addresses the question when Mann-Whitney is a powerful test for equality of medians, but this doesn't mean you can't use it otherwise. The second link doesn't mention the term "symmetry" at all.

Christian Hennig
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