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I would like to ask the opinion of this community in regard to the following discussion between me and my colleague. The case is this: we have two variables, let's call them Y and X. The AIC of a simple linear regression of Y on X (Y = b1*X + e) is A, and the AIC of the reversed model, regression of X on Y (X = b2*Y + e) is B.

If B is much larger than A, can we say that the first model is preferable? I have never encountered such use of AIC, but I also couldn't find any references that would mention a restriction of the comparison based on AIC to models with same directionality. In other words, how close do the models in comparison have to be for an AIC-based comparison to be valid? Any references are welcome (with preferences for less technical ones, but anything is better than nothing). Thanks!

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    $\begingroup$ As an aside to the excellent answer you already have, note that if one of those models is appropriate, the other cannot be (one has errors on x, the other on y; if the errors are on y and x is without error, then the assumptions of the other regression model can't hold) ... in which case its AIC is meaningless (it's based on assumptions that cannot be true), and so could not be used in a comparison. $\endgroup$
    – Glen_b
    Dec 2 '13 at 18:19
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The likelihoods of the two models are in different units & therefore not comparable, even if AIC had anything to do with the direction of causality, which it doesn't. If that's not obvious once stated, which it should be, use the following R code to demonstrate to your colleague that the "preferable" model would depend on what units you happen to measure x in:

    x <-c (1.1,2.0,2.9,4.2,5.1)
    y <-c (19,50,62,81,100)
    AIC(lm(y~x))
    AIC(lm(x~y))
    X <- 100*x
    AIC(lm(y~X))
    AIC(lm(X~y))

That the response variable must be the same across models compared is said here & in Anderson & Burnham (2002), "Avoiding pitfalls when using information-theoretic methods", J. Wildlife Management, 66, 3.

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  • $\begingroup$ Thank you for the informative and elaborate answer. It solves the question perfectly. $\endgroup$
    – Maxim.K
    Dec 2 '13 at 16:19

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