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Say I have some data on people where I have a measure of their general health (some score out of 1000), the number of apples they eat in a year, and the number of oranges they eat in a year.

Then I do a multiple regression and I get something like:

$\hat Y = 442.22 + 1.1a + 0.7o$

where $a$ is the apple variable, and $o$ is the orange variable.

Is it legitimate for me to then say to someone who eats 1 apple and 1 orange a year, "Your predicted health score is $442.22 + 1.1 + 0.7 = 444.02$?"

I've been taught that I can only say things like "for each extra apple you eat your health is predicted to increase by 1.1 points, holding constant scores on the other variables". Did my interpretation in the previous paragraph violate this "holding constant" rule?

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There are already many answer; so as an aside I would just like to notice that I find the "holding constant" rule very dangerous in practice when people start interpreting their coefficient causally at the same moment. For example, regression earnings over gender and many other factors and concluding that gender has no significant influence; even though gender also may have tremendous influence on other significant predictors. It's not really possible to hold other factors constant in practice. – Erik Apr 8 '14 at 15:45
See also @gung's answer to the very similar question at…. – user1205901 May 4 '15 at 12:27
up vote 5 down vote accepted

The score you calculated will be correct. The rule doesn't get violated since all you're doing is scoring an entity.

The "holding constant" rule essentially helps you identify how much influence each variable has on the outcome. In the presence of multiple factors, the only way for us to estimate impact of one variable is to control for the others.

Interpreting 1.1 to be the contribution of apples would only hold if you compare two people who both eat the same number of oranges.

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It is legitimate to do so. You are estimating the expected/mean health score, given the number of apples and oranges, which is exactly what the regression is able to do.

You can also, in this case, directly compare the effect of 1 apple ($+1.1$) with that of one orange ($+0.7$) upon the health score. They are on the same scale (number of things), and there is no interaction present. The caveat about keeping other things fixed arises not (as another answer here suggests) when comparing these effects in a model like yours, with strictly additive terms. The problem would be if, for instance, your model contained an interaction between apples and oranges, which would mean that the effect of apples is dependent upon the number of oranges (and, equivalently, vice-versa). Then you could only talk about the apple effect for a specific "orange level".

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Your interpretation is fine, but I wonder about your regression. I would imagine that apples and oranges have nonlinear effects on health. Eating an apple a day is good. Eating an apple a year - not so good. But, eating 10 apples a day, probably also not so good.

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+1 – We must test this hypothesis immediately! I've always enjoyed fresh fruit, so I'll volunteer to provide an observation at $a=o=10$. Someone give me 30 apples and 30 oranges; I will report back in a few days. – Nick Stauner Apr 8 '14 at 12:49

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