# Significance testing on Pearson's r when the data are non-normal but sample size is large

Hypothetically let's say I'm doing a study in which the data on both variables are likely to be quite positively skewed, but the sample size is potentially quite large.

In this research context using significance tests on Pearson's r is the most common technique and it would be useful if I could use it. It's less desirable for me to use the Spearman correlation because of the audience this study is pitched at.

I know from this question that Pearson's r doesn't "assume normality" per se, but that I may not be able to trust the p-values if I have non-normal data and a small sample size.

Clearly the right way to proceed will depend on how non-normal my data are and how large my sample is. Are there any rules of thumb I can usefully apply?

• It's not clear whether you have a large sample (title) or a small sample (third paragraph). Either way, P-value calculations for correlation can always be supplemented with confidence interval calculations. You could use both the Fisher z method and bootstrapping. The real issue is not so much nonnormality as nonlinearity, so always plot the data and consider transformations. stata-journal.com/sjpdf.html?articlenum=pr0041 may help. Mar 13, 2016 at 6:44
• The third paragraph talks in general terms about the consequences of a small sample size, but the questions is about situations where a large sample has been, or could be, obtained. Basically I'm wondering how quickly the sampling distribution converges to something fairly normal in typical situations, and whether I can use some rule of thumb like "if n > 30 and the data isn't crazily non-normal you're probably OK". Mar 13, 2016 at 6:53
• Such rules of thumb are in my experience useless and in any sense outdated. The bootstrap is nearly 40 years old as a practical way to quantify uncertainty. Clearly, there aren't threshold sample sizes that guarantee anything. If you are interested in the possibility of a linear relationship, fit a regression tailored to the situation. Fields differ but "my correlation is significantly different from zero" is usually a minimal discovery. If that's seriously in doubt, your sample size may be too small for anything much. Mar 13, 2016 at 7:06

## 1 Answer

Choosing a suboptimal estimator because of the audience is in my view not the best practice of statistics. As statisticians we must use the best available methods or invent one that is better than what's available. Our goal is never to give the audience what they want, but to give them what they need. Without knowing more about your problem, and assuming that the only relationship you are interested in is a monotonic one, I would choose Spearman's $\rho$ as a default method. It it robust, powerful, and does not assume linearity. It is heavily used in most disciplines.

If you want to detect and quantify a relationship that is up and down (with one turn) you can use a generalization of Spearman's $\rho$ as implemented in the R Hmisc package spearman2 function. For completely general relationships including a circle you can use Hoeffding's $D$ (see Hmisc hoeffd function).