# What’s the order of correlation?

What is expressed by the terms zeroth-, first-, second-, third-, etc. order of correlation?
Thanks!

Here is a nice resource for understanding these issues. It’s excellent; you should read it thoroughly.

However, I will give a quick introduction. Imagine you have 3 variables, $x$, $y$ and $z$. You are primarily interested in the relationship between $x$ and $y$, but you know that $y$ is also related to $z$, and that unfortunately, $z$ is confounded with $x$. If you simply wanted to know the strength of the relationship, Pearson’s product-moment correlation coefficient $r$ is a useful effect size measure.

In this situation, you could simply ignore $z$ and compute the correlation between $x$ and $y$ (this is not really a good idea, as the value would be a biased estimate of the direct correlation). Since you have controlled for nothing, this is a ‘zero-order’ correlation.

You might opt instead for a more conscientious approach and control for the confounding with $z$, by partialling out $z$. (One conceptually clear way to do this, albeit not computationally optimal, is to regress $y$ onto $z$, and $x$ onto $z$, and then compute the correlation between the residuals of the two models.) Because you have controlled for one variable, this would be a ‘first-order’ partial correlation. Another possibility is to partial $z$ out of only one variable, say $y$. For example, you could regress $y$ onto $z$ and correlate those residuals with $x$. This would be a ‘first-order’ semi-partial (or part) correlation*.

I have never seen such a thing in practice, but if you partialled out 17 other variables, you would have a ‘seventeenth-order’ partial correlation. The linked website is very informative, with examples, multiple formulas and diagrams; go read it. To be technical, there isn’t really any such thing as a ‘first-order’ correlation, nor is there such a thing as a ‘zero-order’ partial or semi-partial correlation. There are only ‘zero-order’ correlations, and only ‘first-‘, ‘second-‘, etc., ‘order’ partial and semi-partial correlations.

* Regarding why you might use a partial vs. semi-partial correlation, it depends on the question you want to answer. Often, it may have to do with the pattern of causal connections that people believe creates the pattern of correlations that are seen. For example, a ‘first-order’ partial correlation between $x$ and $y$ controlling for $z$ of $0$ (i.e., $r_{xy|z}=0$) is consistent with the idea that both $x$ and $y$ are effects of $z$ with no direct connection between them. Likewise, someone might want to show that $y$ is correlated with $x$ even after controlling for $z$. Part of what is going on in a Structural Equations Model can be understood as partial and semi-partial correlations.