  The quark-connected leading-order hadronic contributions to the running of
the electromagnetic fine structure constant, $\alpha_{\rm QED}$, and the weak
mixing angle, $\theta_W$, are determined by a four-flavour lattice QCD
computation with twisted mass fermions. Full agreement of the results with a
phenomenological analysis is observed with an even comparable statistical
uncertainty. We show that the uncertainty of the lattice calculation is
dominated by systematic effects which then leads to significantly larger errors
than obtained by the phenomenological analysis.
