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I would like to use Pearson's $r$ to determine the correlation strengths among around $20$ variables. Unfortunately, I had to reject the hypotheses that the data from some (around $5$) comes from normal distributions with Shapiro–Wilk test and $n \in [10, 20]$. What are some good alternatives of correlation measurement to the $r$ for these not-normally-distributed data? Thanks.

P.S. am using SPSS 25.

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Suppose $x = 1, 2, 3, 4, 5$ and $y = 2x$ and so Pearson's $r$ is $1$.

Is that invalid because the joint or marginal distributions aren't normal? The issue raised by non-normality is that P-values may be off. But if you want to assess linearity, stick with Pearson. If you are worried about non-normality, use bootstrapping or a permutation test. (If there is a strong correlation, proceed to regression any way and do your inference with a suitable model.)

What's being premised here is that there is a lurking property "correlation strength" and that if one measure is not ideal, we should use another. It is the other way round. Each measure has its own idea of what a strong correlation is. If it is Pearson $r$, it is a linear relationship. If it is Spearman's or Kendall's, it is a monotonic relationship. And there are other measures that purport to find any kind of relationship (departure from independence).

So, the question raises another: What kind of relationships are you interested in? The distributions need to be looked at, but we have had bootstrapping and permutation testing for several decades now, and worries about non-normality need not drive what you do.

Small samples are problematic whatever you do....

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  • $\begingroup$ Eh my $x$ is fixed and the Shapiro–Wilk test doesn't reject the normality hypothesis, but rejects for some of my $y$s. I look for normality of the data a lot because I want to take independent t-tests with them. Btw I'm not very familiar with bootstrapping or permutation test (esp. in SPSS). Do you have any sources about these? $\endgroup$
    – Yuki.F
    Commented Jan 9, 2020 at 11:21
  • $\begingroup$ Even t tests can work reasonably given some non-normality. Shapiro-Wilk will often reject distributions that aren't really problematic. Sorry, I haven't used SPSS in the current millennium. I imagine it supports those methods. If not, you will need to write your own code or use different software. $\endgroup$
    – Nick Cox
    Commented Jan 9, 2020 at 11:32

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