# Does simple linear regression imply causation?

I know correlation does not imply causation but instead the strength and direction of the relationship. Does simple linear regression imply causation? Or is an inferential (t-test, etc.) statistical test required for that?

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What do you mean by "direction"? Have you read the answers to similar questions stats.stackexchange.com/search?q=causal ? The short answer is no! –  NRH May 11 '11 at 23:41
Neither of your suggestions imply causation (or direction). –  Henry May 11 '11 at 23:43
I think the OP meant "direction" in the sense of positive vs negative correlation, not the direction of any causal relationship between X and Y. –  JMS May 13 '11 at 21:39

The quick answer is, no. You can easily come up with non-related data that when regressed, will pass all sorts of statistical tests. Below is an old picture from Wikipedia (which, for some reason has recently been removed) that has been used to illustrate data-driven "causality".

We need more pirates to cool the planet?

For time series, there is a term called "Granger Causality" that has a very specific meaning.

http://en.wikipedia.org/wiki/Granger_causality

Other than that, "causality" is in the eye of the beholder.

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I meant positive correlation or negative by direction. Thankyou for your response and link to similar questions. –  user4572 May 12 '11 at 0:49
Thats quite a crazy X axis in that picture! (But good example!) –  Andy W May 12 '11 at 1:47
Another.....Cheese, Butter, and Sheep in Bangladesh, versus the S&P500 (R^2=0.99) ...... nerdsonwallstreet.typepad.com/my_weblog/files/… .... –  bill_080 May 12 '11 at 2:02
That graph is obviously outdated. Either that or there is bias due to lack of surveyors available to sample in the Gulf of Aden –  cardinal May 12 '11 at 13:23
That data was before Al Gore became a pirate. –  bill_080 May 12 '11 at 15:01

There is nothing explicit in the mathematics of regression that state causal relationships, and hence one need not explicitly interpret the slope (strength and direction) nor the p-values (i.e. the probability a relation as strong as or stronger would have been observed if the relationship were zero in the population) in a causal manner.

That being said, I would say regression does have a much stronger connotation that one is estimating an explicit directional relationship than does estimating the correlation between two variables. Assuming by correlation you mean Pearson's r, it typically does not have an explicit causal interpretation as the metric is symmetrical (i.e. you can switch which variable is X and which is Y and you will still have the same measure). Also the colloquialism "Correlation does not imply causation" I would suspect is so well known that stating two variables are correlated the assumption is one is not making a causal statement.

Estimated effects in regression analysis are not symetrical though, and so by choosing what variable is on the right hand side versus the left hand side one is making an implicit statement unlike that of the correlation. I suspect one intends to make some causal statement in the vast majority of circumstances in which regression is used (inference vs prediction aside). Even in cases of simply stating correlations I suspect people frequently have some implied goals of causal inference in mind. Given some constraints are met correlation can imply causation!

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From a semantic perspective, an alternative goal is to build evidence for a good predictive model instead of proving causation. A simple procedure for building evidence for the predictive value of a regression model is to divide your data in 2 parts and fit your regression with one part of the data and with the other part of the data test how well it predicts.

The notion of Granger causality is interesting.

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