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This is probably a weird question for this forum, but you may have some insight or an answer. My question is simply this: Is it possible to have a three-way correlation? And is it possible to have a meaningless three-way correlation where meaningful two-way correlations exist for all combinations? What I mean by a three-way correlation is perhaps best explained by an example. And it is entirely possible that I am getting the terminology all mixed up. So this is my example:

Any word has an probability of occurring within a sentence. However, by writing a word, the probability of the next (adjacent) word changes. Some words are highly correlated, i.e. you have a high probability of seeing them one after the other. For example, the words "goes to" are highly correlated. You see "He goes to the shop" or "She goes to the airport" etc. You may also see "He goes with his dog" so the probability of the two words "goes" and "to" occurring together is less than one.

Now, there can also be words that often occur in threes. For example, "three-way tie" or "is not going". These have meaning when linked together and have a finite probability of occurring in threes. The last of these also has meaning with two-way combinations of all three word, i.e. "is not", "is going" and "not going" each have a non-zero probability of occurring outside of the three-way combination. However, two-way combinations are more dubious for "three-way tie" ("three-way" is common, but "way tie" without a third numeric modifier before "way" seems odd, and it would be rare for the "three tie" combination to have meaning).

But then there is the following example. "Duck fat" would have a fairly high correlation. As would "fat mouth". "Duck mouth" while not highly probable, certainly has meaning in the following example "A bill is a duck mouth." (Yes, I know, "A bill is a duck's mouth" is better English, but I am looking at both options to cover the entire probability space.) However, "duck fat mouth" has no meaning, and I suggest, a zero probability of occurring together.

So I suppose I am asking whether people have studied the above type of question and whether they use correlation in anything more than between two variables? And if not, what terminology do they use?

Brendan
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    In principle you could generalize [the definition of covariance](https://en.wikipedia.org/wiki/Covariance) to three variables. But I've never encountered this, and have no idea whether it might be useful for anything. For your example, I suggest reading about [n-grams](https://en.wikipedia.org/wiki/N-gram). – Dan Hicks Feb 12 '18 at 05:00
  • Closely related: https://stats.stackexchange.com/questions/67240/pearson-correlation-of-3-variables/267021#267021. – whuber Feb 12 '18 at 14:59
  • Maybe ideas from https://stats.stackexchange.com/questions/88915/is-there-an-intuitive-characterization-of-distance-correlation – kjetil b halvorsen Jan 16 '19 at 15:39

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