# What's the order of correlation?

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

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Hi. You're more likely to get a useful answer if you indicate what efforts you've made to solve the problem yourself (eg which definitions have you looked at), what is puzzling you about them (so we can help!), and what the context is (eg regression, time series, multivariate analysis). – Peter Ellis Jan 19 '12 at 18:39

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 biased). 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.