Multiple linear regression for hypothesis testing I am familiar with using multiple linear regressions to create models of various variables. However, I was curious if regression tests are ever used to do any sort of basic hypothesis testing. If so, what would those scenarios/hypotheses look like?
 A: The essential test in regression models is the Full-Reduced test.  This is where you are comparing 2 regression models, the Full model has all the terms in it and the Reduced test has a subset of those terms (the Reduced model needs to be nested in the Full model).  The test then tests the null hypothesis that the reduced model fits just as well as the full model and any difference is due to chance.
Common printouts from statistical software include an overall F test, this is just the Full-Reduced test where the reduced test is an intercept only model.  They also often print a p-value for each individual predictor, this is just a series of Full-Reduced model tests, in each one the reduced model does not include that specific term.  There are many ways to use these tests to answer questions of interest.  In fact pretty much every test taught in a introductory stats course can be computed using regression models and the Full-Reduced test and the results will be identical in many cases and a very close approximation in the few others.
