Is there formal test of non-linearity in linear regression? In logistic regression there is Box-Tidwell but I know of nothing like that in linear regression. I use partial residual plots to look for this, a graphical feature, but would love to find a formal test (in honesty I doubt you can do a formal test of this, but I could be wrong).
 A: The best formal tests come from relaxing the linearity assumption, then seeing if removing the nonlinearities damages the explained variation in Y.  For example you can expand X using a regression spline and test the nonlinear components.  My RMS Course Notes goes into details.
But once you've allowed for the possibility of nonlinearity, you distort statistical inference by removing the nonlinear terms.  The real numerator degrees of freedom for the regression are the number of chances to give the model, which must take into account the nonlinear terms.  So the best advice overall is to allow effects not known to be linear to be be nonlinear and be done with it.  This will preserve confidence interval coverage, etc.
A: Box-Tidwell was developed for ordinary least squares regression models.
So if you were inclined to use Box-Tidwell for this, that's actually what it's designed for.
It's not the only possible approach, but it sounds like an approach you're already familiar with.
However, I'm not convinced that (most times it's used) a formal test is appropriate - I believe it usually answers the wrong question, while the diagnostic plots you've been looking at come closer to answering a useful question. [I have a similar opinion of many other tests of regression assumptions]
A: Fit a non-linear regression (e.g. spline model like GAM) and then compare it to the linear model using AIC or likelihood ratio test. This is a simple and intuitive method of testing non-linearity. If the test rejects, or if AIC prefers the GAM, then conclude there are non-linearities. 
