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I was wondering how you interpret the p-value of a coefficient for a multivariate linear regression. Is the p-value intepreted only when everything else is fixed? Is there a way to incorporate the dependencies of other variables?

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This answer should really be in your textbook, any decent textbook that includes multiple regression.

The p-value associated with a single term in a multiple regression (there should also be an overall p-value) is the adjusted test, i.e. testing that term adjusting for all the other terms in the model. The null hypothesis is that the given term has 0 slope, or does not contribute linearly to the prediction, after accounting for all the other terms in the model.

For example if my response (y) variable is a persons weight in pounds and I have 2 potential predictors, x1 is height in inches (rounded) and x2 is height in centimeters (rounded), then I would expect to see significant p-values when x1 or x2 is included, but not both. On the other hand for a model with both x1 and x2 I would expect to see non-significant p-values because while I expect a relationship between height and weight, having the height in inches is redundant (does not add anything meaningful) if height in centimeters is already in the model, and vice versa.

This p-value already incorporates dependencies between the predictor variables and their relationship with the response variable.

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    $\begingroup$ +1. I would probably say half-decent, not even necessarily decent textbook. To that extend a very nice free e-book I came across lately was OpenIntro Statistics, it includes some comments about $p$-values on linear analysis (and a general treatment of $p$-values in Hypothesis testing obviously). $\endgroup$
    – usεr11852
    Commented May 17, 2014 at 0:04
  • $\begingroup$ @usεr11852, I will second OpenIntro as a great resource here, especially given the price. I thought of it when writing my answer, but did not remember if it covered multiple regression (and was too lazy to double check). Thanks for bringing it up. $\endgroup$
    – Greg Snow
    Commented May 19, 2014 at 14:00

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