# (Non-)linearity and correlation (Pearson v. Spearman and Kendall)

I am a bit confused about how to interpret correlation coefficient results. I am aware that there are numerous questions about the differences between Pearson, Spearman, and Kendall, but I am more interested in their respective relationship to linearity.

Let's assume that Pearson's r is 0.578 and p is 0.000012. There is a correlation between two variables that is most probably not caused by chance. However, this assumes linearity (and homoscedasticity) and prone to errors when the data contains outliers.

Let's also assume that we draw a scatter plot and find that, indeed, there are outliers in our data. To minimise the effect of outliers, we run a Kendall test. Here we also find a small p and a positive tau. Kendall (and Spearman) do not assume linearity, hence their effectiveness when dealing with outliers. But what are the consequences for trying to fit the data on a line?

If we have normally distributed, linear data (ideal case for Pearson) we can fit all data points on a linear curve (cf. for instance regplot() of the Python package seaborn). But if we have outliers, and Pearson is not a viable option, is there still any assumption for linearity with Kendall or Spearman? Does it still make sense to try and fit the data on a linear curve, or any curve for that matter? Or does the relationship as defined by Kendall or Spearman does not say anything about the fitting of the data, meaning that it does not make sense to try and plot the data on a curve?

(Note that regplot() has a robust option that might be useful here. Related programming question here.)

• Heteroscedasticity would mean that conditional variances say of $y$ given $x$ differ, but in the ideal bivariate Gaussian case Pearson correlation is summarizing a joint distribution, not a possibly varying conditional distribution. You might as well say that it's an assumption of Pearson correlation that there is a perfect linear relationship, meaning no conditional variance at all! If so, then Pearson correlation is measuring the extent to which its "assumptions" are satisfied. I am happy with $y = a + bx$ exactly and either variable not Gaussian as also a perfect case for Pearson. – Nick Cox May 2 '18 at 11:08
• Your main point seems to be wondering what Spearman and Kendall correlations tell you about linearity and the answer is nothing as such, as the ideal cases for either are monotonic relationships. – Nick Cox May 2 '18 at 11:10
• I wouldn't assume that people are familiar with what regplot in R does or want to read about it. Many people here don't use R at all. It's best to explain directly. – Nick Cox May 2 '18 at 11:11