# How do you interpret a significant but weak correlation?

I have carried out a number of Pearson Chi Square tests on the relationship between tail lesions and abscessation in pigs. I have found a significant (p = 0.005) but weak (r = 0.288) association. How do I interpret this?

For example, does this mean that there is a clear relationship between the two but that it is just on a small scale?

• If it's a Pearson correlation, why do you cite Cramér's [NB] $V$? Alternatively, if it's not a Pearson correlation, please edit accordingly. Sep 5, 2014 at 10:38
• I meant to say Pearson Chi square, I've edited the post. Sep 5, 2014 at 10:41
• Thanks for the edit. For your thesis/paper/report, note accent and apostrophe as at en.wikipedia.org/wiki/Cram%C3%A9r%27s_V Some software is very lax on either or both. However, "correlation" really should now be "association" in your title and question. Sep 5, 2014 at 10:47

More meaningful in this case is the $\text{R}^2$ which explains the proportion of variation in your observations accounted by the association. For example if your $R$ was 0.1 (p= 0.005) due to the large sample size, it means 1% of the variation in tail lesions in pigs is accounted for by abscesses. In a multifactorial situation such associations though informative may not be very meaningful. Again be cautious since correlation does not imply causation.

• There may be some typos in your post, by "...if your $R$ was 0.1... it means 1% of the variation...", did you mean '...if your $R^2$ was 0.1... it means 10% of the variation...'? Sep 5, 2014 at 15:17
• No, he's correct. R here is the correlation coefficient and R^2 is, as its name implies the square of the correlation coefficient. It's also the share of the variation in one variable that is explained by the other. Sep 5, 2014 at 16:56

You could say it like this:

The association is small, but not zero.

However, I don't know that I'd call a V of 0.288 "small".

Don't confuse "statistically significant" with important. Statistical significance is very different from practical importance.

• Thanks, can you explain what the actual meaning of this finding it? Why is it significant but weak? Sep 5, 2014 at 10:43
• It's not big, but the sample is large enough to make it clear it's different from zero. Sep 5, 2014 at 10:58

As he said, it is significant but weak because the sample size is large enough to make a small effect significant. But now you need a way of showing the effect size. V is one such way, but it isn't intuitive to many and is not as well known as some other measures.

From your question, it seems like you have a 2x2 table of lesions (yes/no) and abscess (yes/no). You could then show that table; you could give the odds ratio or any of a number of other measures for 2x2 tables.

• n-thing the Odds ratio. I know nothing about pig's tails, but the most informative question here seems to be: how much more likely is a pig to have an abscessation, given it has a tail lesion? (Include a CI, too.)
– jona
Sep 5, 2014 at 13:00

I like to comment on the perceived usefulness of the relationship. For instance:

The association is statistically significant but not practically relevant.

That being said, your association doesn't seem to be that weak.