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Apologies for an awkward title.

I am looking for concrete examples of how it is important to make the distinction "holding all else fixed", when talking about coefficients of predictors in multivariate regression. Examples that would be relatable, for people looking at regression output but not familiar with how regression works.

I want to be able to preempt things when they say "this variable should not be negatively correlated with the response, not positively correlated; the model is worthless!" (exaggerated, of course)

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    $\begingroup$ Could you describe what an "example" would consist of? $\endgroup$ – whuber Jul 26 '17 at 21:33
  • $\begingroup$ I guess I am thinking of an example in reality where intuitively, we know x1 is negatively correlated with y. But when we include x2, it also becomes intuitive that when x2 is fixed, x1 becomes positively correlated with y. $\endgroup$ – spinodal Jul 26 '17 at 21:39
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    $\begingroup$ Here you go: stats.stackexchange.com/… $\endgroup$ – whuber Jul 26 '17 at 23:19
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The easiest example I can think of is when economists use the term ceteris paribus when describing a supply-demand model.

It is easy to say that holding everything else constant, generally the price of a good should decrease as the supply increases. We can think of price $P$ as the output, supply $S$ as a predictor.

But this definitely doesn't hold true once other things come into play. Simply introducing any other predictor complicates the model - and then you have multivariate regression. Tadah!

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