The contribution of heavy fermions (muons) to the electron anomalous moment in quantum electrodynamics is considered within the context of effective field theories, correct to order ..cap alpha../sup 2/m/sup 2//M/sup 2/. Although no violation of decoupling is found, it is shown that the leading contribution of the muon to the electron magnetic moment is incorrectly given by the local effective Lagrangian obtained from functional integration over heavy fields. It is also shown that this same local effective Lagrangian, with all terms retained of arbitrary order in (1/M/sup 2/), has the wrong analyticity in this variable in that the electron anomalous moment has a term of O(..cap alpha../sup 2/(m/sup 4//M/sup 4/)ln(M/sup 2//m/sup 2/)) which cannot be obtained from this local effective Lagrangian. The correct calculation of these effects requires calculation of mixed light-heavy contributions, an issue previously discussed for contributions to operators of dimension < or =4.