Understanding of linear regression coefficients I know that commonly it is said that the regression weights (i.e. the estimated "beta") is the relevance of the associated independent variable when all other variables are held constant. I have two questions with regards to this aspect:


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*Why isn't it that the beta weight is the relevant of the associated independent variable when everything else is set to zero (instead of held constant)? I am thinking of a line, where we see the slope of the regression weight of interest when we set the other slopes to zero..

*Is this relevant always as compared to the regression "baseline" (i.e. the intercept)?
Thanks
 A: The other predictor variables in the model have to be set to plausible values. 
As an example, let's say you have a model of the form:
Cholesterol_Level = beta0 + beta1*Blood_Pressure + beta2*Body_Weight + error (1)

If the model concerns adults, it doesn't make sense to set Body Weight to 0 kg and then - after fitting the model to the data - say things like: 
Among all adults with a Body Weight of 0 kg, we found a significant  
positive linear relationship between Blood Pressure and Cholesterol  
Level.  

Usually, what we do say is something like this:
Among all adults with *the same* Body Weight, we found a significant   
positive linear relationship between Blood Pressure and Cholesterol  
Level.

with the tacit understanding that Body Weight is set to a plausible value given the data (e.g., among all adults with a weight of 65kg). It doesn't really matter what that value is (as long as it is plausible given the data) since the relationship between Blood Pressure and Cholesterol Level is the same regardless of the set value for Body Weight. 
Comment: 
If we want to ignore the effect of Body Weight when estimating the effect of Blood Pressure on Cholesterol Level, all we have to do is fit the following model: 
 Cholesterol_Level = beta0 + beta1*Blood_Pressure + error (2)

This model is obtained by setting beta2 to zero in model (1).  Note that we set the unknown parameter beta2 to zero, NOT the estimated value of this parameter obtained from the data. 
Assuming that (i) the estimated value of beta1 is positive after fitting both models to the data and (ii) the test of the hypotheses Ho: beta1 = 0 versus Ha: beta1 != 0 produces a significant p-value, then: 
Conclusion for Model 1:
We found a significant positive linear relationship between Blood 
Pressure and Cholesterol Level among all adults represented by our 
study *who have the same Body Weight*.

Conclusion for Model 2:
We found a significant positive linear relationship between Blood 
Pressure and Cholesterol Level among all adults represented by our 
study, *irrespective of their Body Weight*.

