# 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

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 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.

I have never seen such a thing in practice, but if you partialled out 17 other vairables, you would have a 'seventeenth-order' partial correlation. The linked website is very informative, with examples, multiple formulas and diagrams; go read it.

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Thanks gung, that was very helpful! –  Mika Jan 20 '12 at 10:32
Whether the link is only available to you or whether it is no longer available at all, I don't know, but that link doesn't work for me. Still, thanks for the info =] "You don't have permission to access /~mbrannic/files/regression/Partial.html on this server." –  Chris NH Street May 15 at 13:03
Welcome to the site, @Chris. This is not an answer to the OP's question. Please only use the "Your Answer" field to provide answers. I know it's frustrating, but you will be able to comment anywhere when your reputation reaches 50. Since you're new here, you should read our FAQ, which contains info like this about the site. –  gung May 15 at 13:33
Update: tragically, the link may now be dead & I can't find (via Google) that the content was moved somewhere else. –  gung May 15 at 13:37