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If we have a linear regression of the form

$$ Y = \beta_0 + \beta_1X_1 + \beta_2X_2 $$

is it valid to interpret the coefficient $\beta_1$ as the associated change in $Y$ when $X_1$ increases by a unit of 1, when $X_2=0$?

mkt
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user321627
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  • The fact that rate of change of (expected) $Y$ wrt $X_i$ is the same no matter what the values of the $X$ variables are is essentially our assumption that the relationship is linear. – John Madden Sep 13 '22 at 22:06

1 Answers1

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Just omit the last caveat and your interpretation is basically accurate, though I will add one small edit based on dipetkov's helpful comment:

$\beta_1$ is the associated change in the expected $Y$ when $X_1$ increases by a unit of 1.

This holds whatever the value of $X_2$ (conditional on the model being accurate, though this caveat is also covered by the addition of 'expected' to the definition). Variation in $X_2$ is irrelevant here because there is no interaction term in your model.

mkt
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    I'd like to be explicit and say "the associated change in the expected Y" or "the expected change in Y associated with". – dipetkov Sep 13 '22 at 05:35
  • Thanks, is what I have a special case or is what I have incorrect as it stands? – user321627 Sep 13 '22 at 06:24
  • @dipetkov Thanks, that is a good idea. I have edited to reflect that. – mkt Sep 13 '22 at 06:49
  • @user321627 I'm not sure what you mean. It's incorrect in so far as your final caveat about X2 was not needed for the model you have specified. And as dipetikov says, 'expected' should be in there to be precise. – mkt Sep 13 '22 at 06:50
  • @mkt Thanks, in the case where I only had $Y = \beta_0 + \beta_1X_1$ (removing the $X_2$, would the interpretation be the same then? – user321627 Sep 13 '22 at 07:00
  • @user321627 Yes, $X_2$ is irrelevant to the interpretation of $\beta_1$ as long as there is no interaction term in the model. – mkt Sep 13 '22 at 07:03
  • @mkt Do we also need to say that $\beta_1$ above is controlling for $X_2$? – user321627 Sep 13 '22 at 07:41
  • @user321627 I'm not sure what is unclear about this statement: X2 is irrelevant to the interpretation of beta1 here. You're reaching for connections that don't exist. – mkt Sep 13 '22 at 07:46
  • @dipetkov Could you please explain what is the meaning of "expected" here? It seems to me that $\beta_1$ is exactly the associated change in $Y$ when $X_1$ increases by 1. What are we taking the expectation of? – Stef Sep 13 '22 at 13:13
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    @Stef: it is a linear regression model - typically not all points fit the equation exactly, but there is assumed to be an error term with zero mean and independent of the $X_i$. – Henry Sep 13 '22 at 13:17
  • @Henry thank you – Stef Sep 13 '22 at 13:20
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    @Stef Regression is a model for the conditional mean E(Y | X = x). To understand this better, consider the Difference between confidence intervals and prediction intervals Also take a look at Section 10.2 in Regression and Other Stories. It's freely available online. – dipetkov Sep 13 '22 at 13:27
  • @dipetkov Awesome, thank you – Stef Sep 13 '22 at 13:34