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I have some pairs of datasets (n=200 or thereabouts), of samples which are non-negative and not normally distributed. I think these pairs of variables are related, probably linearly.

Calculating Spearman's rank correlation on these datasets gives some strange results. The correlation coefficients show that the pairs of variables are weakly, positively correlated (e.g. rho of around 0.4), but the p-values are very low (e.g. 4.1e-10).

My vague understanding of this is that the variables are weakly, positively correlated but the probability of unrelated variables producing the same correlation is very low. Does this mean we can be reasonably certain that a positive correlation exists or have I misunderstood?

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    $\begingroup$ 0.4 isn't a weak correlation, In Cohen's terms (often used in social sciences) a large correlation is 0.5, medium is 0.3 and small is 0.1. $\endgroup$
    – Jeremy Miles
    Apr 5, 2013 at 23:47
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    $\begingroup$ WHat you call 'weak' depends on circumstances. Your association is so unlikely to have occurred just by random variation that it would be reasonable to conclude that it must be due to some other cause. If the assumptions hold, you're left with concluding some level of monotonic association is present. $\endgroup$
    – Glen_b
    Apr 6, 2013 at 1:40

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Your understanding of the p-value is correct (well technically it is the probability of seeing the observed correlation or stronger) if no correlation exists.

What is a strong or weak correlation is depends on the context, it is often good to plot your data, or generate random data with a given correlation and plot that to get a feel for the strength of the correlation.

The p-value is determined by the observed correlation and the sample size, so with a large enough sample size a very weak correlation can be significant, meaning that what you saw is likely real and not due to chance, it just may not be very interesting. On the other hand with small sample sizes you can get a very strong correlation that is not statistically significant meaning that chance and no relationship is a plausible explanation (think about 2 points, the correlation will almost always be 1 or -1 even when there is no relationship, so that size can easily be attributed to chance).

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    $\begingroup$ That's very helpful, thanks. With these particular data sets, it makes sense that the data should be correlated and that's a useful result to have. I was a little suspicious because much of the data lies within a segment of the X-axis with a small number of outliers, so I've been plotting the data on a log scale, which makes the line of best fit look a bit odd. $\endgroup$
    – snim2
    Apr 6, 2013 at 10:41
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    $\begingroup$ Is it true that for any given sample size, the weaker the observed correlation, the larger the p-value? $\endgroup$
    – GoodDeeds
    Nov 22, 2021 at 20:11
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    $\begingroup$ @GoodDeeds, for the tests of correlation that I am aware of, with the null being that the true correlation is 0, then yes, for a given sample size, correlation closer to 0 will give a bigger p-value. $\endgroup$
    – Greg Snow
    Nov 23, 2021 at 17:16

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