One expects that the β-functions of a perturbatively renormalizable quantum field theory receive contributions only from one-particle-irreducible (1PI) diagrams; surprisingly, for a completely general renormalization scheme this is not the case. Using consistency conditions derived from a function conjectured to satisfy the strong a-theorem, we review our conclusions for 6D φ^3 theory where this discovery was first made, followed by the analogous 4D case involving scalars and chiral fermions. We conclude by demonstrating how the existence of one-particle-reducible (1PR) contributions necessitates a redefinition of the fields in the theory, in addition to the usual redefinition of the couplings that links any two consistent renormalization schemes.