  Effects of strong interactions on the two loop electroweak radiative
corrections to the muon anomalous magnetic moment, $a_\mu=(g_\mu-2)/2$, are
examined. Short-distance logs are shown to be unaffected. Computation of
long-distance contributions is improved by use of an effective field theory
approach that preserves the chiral properties of QCD and accounts for
constraints from the operator product expansion. Small, previously neglected,
two loop contributions, suppressed by a $1-4\sin^2\theta_W$ factor, are
computed and the complete three loop leading short-distance logs are
reevaluated. These refinements lead to a reduction in uncertainties and a
slight shift in the total electroweak contribution to $a_\mu^{\rm EW} =
154(1)(2)\times 10^{-11}$ where the first error corresponds to hadronic
uncertainties and the second is primarily due to the allowed Higgs mass range.
