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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 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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