The behavior of amplitudes and cross sections in the soft and collinear limits has played an essential role in our understanding of perturbative calculations in gauge theories. While often studied at leading power in the soft or collinear expansion, the structure of power corrections is of both formal and phenomenological interest. In this talk I will show how techniques from effective field theory can be used to systematically study subleading power corrections, and subleading power factorization. As an application, I will use these techniques to analytically compute the dominant power corrections for NNLO subtraction schemes based on physical resolution variables, in particular N-jettiness. I will discuss in detail the case of Drell-Yan like color singlet production, and show how the analytic calculation of the leading power corrections, and an improved definition of the observable, each lead to an order of magnitude improvement of the subtractions.