Compression effects on alpha and beta relaxation process of amorphous drugs are theoretically investigated by developing the elastically collective nonlinear Langevin equation theory. We describe the structural relaxation as a coupling between local and nonlocal activated process. Meanwhile, the secondary beta process is mainly governed by the nearest-neighbor interactions of a molecule. This assumption implies the beta relaxation acts as a precursor of the alpha relaxation. When external pressure is applied, a small displacement of a molecule is additionally exerted by a pressure-induced mechanical work in the dynamic free energy, which quantifies interactions between a molecule with its nearest neighbors. The local dynamics has more restriction and it induces stronger effects of collective motions on single-molecule dynamics. Thus, the alpha and beta relaxation times are significantly slowed down with increasing compression. We apply this approach to determine the temperature and pressure dependence of the alpha and beta relaxation time for curcumin, glibenclamide, and indomethacin, and compare numerical results with prior experimental studies. Both qualitative and quantitative agreement between theoretical calculations and experiments validate our assumptions and reveal their limitations. Our approach would pave the way for the development of the drug formulation process.