Intepreting p-value of a regression coefficient 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?
 A: 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.
