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 (Cramers V = 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. – Nick Cox Sep 5 '14 at 10:38
• I meant to say Pearson Chi square, I've edited the post. – Grace Carroll Sep 5 '14 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. – Nick Cox Sep 5 '14 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...'? – gung - Reinstate Monica Sep 5 '14 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. – RoyalTS Sep 5 '14 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? – Grace Carroll Sep 5 '14 at 10:43
• It's not big, but the sample is large enough to make it clear it's different from zero. – Glen_b -Reinstate Monica Sep 5 '14 at 10:58