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My statistics professor claims that the word "correlation" applies strictly to linear relationships between variates, whereas the word "association" applies broadly to any type of relationship. In other words, he claims the term "non-linear correlation" is an oxymoron.

From what I can make of this section in the Wikipedia article on "Correlation and dependence", the Pearson correlation coefficient describes the degree of "linearity" in the relationship between two variates. This suggests that the term "correlation" does in fact apply exclusively to linear relationships.

On the other hand, a quick Google search for "non-linear correlation" turns up a number of published papers that use the term.

Is my professor correct, or is "correlation" simply a synonym of "association"?

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    $\begingroup$ Conversely there is also 'linear association'. $\endgroup$ – Bogdanovist Feb 5 '13 at 4:40
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No; correlation is not equivalent to association. However, the meaning of correlation is dependent upon context.

The classical statistics definition is, to quote from Kotz and Johnson's Encyclopedia of Statistical Sciences "a measure of the strength of of the linear relationship between two random variables". In mathematical statistics "correlation" seems to generally have this interpretation.

In applied areas where data is commonly ordinal rather than numeric (e.g., psychometrics and market research) this definition is not so helpful as the concept of linearity assumes data that has interval-scale properties. Consequently, in these fields correlation is instead interpreted as indicating a monotonically increasing or decreasing bivariate pattern or, a correlation of the ranks. A number of non-parametric correlation statistics have been developed specifically for this (e.g., Spearman's correlation and Kendall's tau-b). These are sometimes referred to as "non-linear correlations" because they are correlation statistics that do not assume linearity.

Amongst non-statisticians correlation often means association (sometimes with and sometimes without a causal connotation). Irrespective of the etymology of correlation, the reality is that amongst non-statisticians it has this broader meaning and no amount of chastising them for inappropriate usage is likely to change this. I have done a "google" and it seems that some of the uses of non-linear correlation seem to be of this kind (in particular, it seems that some people use the term to denote a smoothish non-linear relationship between numeric variables).

The context-dependent nature of the term "non-linear correlation" perhaps means it is ambiguous and should not be used. As regards "correlation", you need to work out the context of the person using the term in order to know what they mean.

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    $\begingroup$ +1 A thoughtful and knowledgeable answer. Please consider qualifying the initial "no," because it takes a bit of reading and thought to understand whether it means "no, the professor is not correct" or "no, 'correlation' is not a synonym for 'causation'," or the conjunction of the two. $\endgroup$ – whuber♦ Feb 5 '13 at 12:30
  • $\begingroup$ Thanks for the comment; I've edited my answer accordingly. $\endgroup$ – Tim Feb 6 '13 at 6:01
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    $\begingroup$ This is an excellent answer (and question) that gets some wider issues about terminology, language and communication in general that we all need to take care about. $\endgroup$ – Peter Ellis Feb 12 '13 at 9:53
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    $\begingroup$ So what is association, then? $\endgroup$ – Sheep Jun 6 '16 at 20:51
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I don't see much point in trying to disentangle the terms "correlation" and "association." After all, Pearson himself (and others) developed a measure of nonlinear relationship which they named the "correlation ratio."

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    $\begingroup$ Yes it seems at this point they are quite hard to disentangle, especially given history (e.g., as you mention) and social perception. $\endgroup$ – Behacad Jul 18 '13 at 22:58
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There seems to be misunderstanding of association. Measures of association (effect size) are inherent in quantitative analysis, not qualitative.

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    $\begingroup$ Maybe you should think of turning that into a comment. Answers are usually a bit more extended $\endgroup$ – PhDing Nov 29 '16 at 7:59
  • $\begingroup$ @Alessandro Yup, more needed for an answer, but the OP doesn't have enough reputation (>+50) to comment quite yet. Maybe the moderator can convert it into a comment for him. $\endgroup$ – Carl Nov 29 '16 at 8:15
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I would say that correlation applies to quantitative data and association to qualitative data and both have no obligatory causal relationship.

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  • $\begingroup$ What do you mean by "association to qualitative data"? $\endgroup$ – Randel Sep 5 '15 at 15:55
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The idea that the weight (of a man) is not correlated with the height (because the corresponding function is of 3rd degree, not linear) seems very strange to me. Linear correlation should be treated as a special case of association.

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    $\begingroup$ But which (or whose) idea are you arguing against? Correlation can be calculated here as (b) correlation between observed and predicted values from any combination of terms as well as (presumably) (a) non-zero correlation between weight and height. $\endgroup$ – Nick Cox Dec 10 '17 at 10:44
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Correlation and association are different. Correlation describes the three types of relationship positive, negative and non-correlated. It also describe the magnitude of correlation from 0 to 1, from -1 to 0. The association does not reveals what types of association and how much association.

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  • $\begingroup$ What is the question here? $\endgroup$ – Christoph Hanck Apr 2 '15 at 4:35
  • $\begingroup$ As you don't define association or explain how it differs, it is hard to see why you think you answered the question. This doesn't add to previous answers. $\endgroup$ – Nick Cox Apr 2 '15 at 9:29
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As far as the linearity is concerned the response by Tim and Nick Cox covered it completely. Where I thought I might be able to contribute is a clean way to think about the difference between association and correlation.

Association --- measures how closely related two variables are (i.e. whether they are dependent or independent).

Correlation --- measures in what way two variables are related (i.e. positive or negative).

In the end, I would argue that you can never go wrong treating them distinctly it will help with interpretion and analyses in the long run. Hope this helps.

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    $\begingroup$ I didn't downvote this, and thanks for the positive comment ( I just applied an edit). Unfortunately, it muddies the discussion. Association measures often have nothing to do with which variable if any is dependent. Correlation measures "how closely" too: each definition of correlation is based on a specific definition of the way in which variables are (ideally) related (linearly, monotonicly, etc.) The family examples really don't help, even as analogies: e.g.mother, father, uncle are not quantitative variables in the example. So, sorry, but the distinction you make isn't clean at all. $\endgroup$ – Nick Cox Dec 10 '17 at 9:29
  • $\begingroup$ Also, "shared blood" and "estrangement" are quite different! $\endgroup$ – Nick Cox Dec 10 '17 at 9:37

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