  We derive a sufficient condition for advantage distillation to be secure
against collective attacks in device-independent quantum key distribution
(DIQKD), focusing on the repetition-code protocol. In addition, we describe a
semidefinite programming method to check whether this condition holds for any
probability distribution obtained in a DIQKD protocol. Applying our method to
various probability distributions, we find that advantage distillation is
possible up to depolarising-noise values of $q \approx 9.1\%$ or limited
detector efficiencies of $\eta \approx 89.1\%$ in a 2-input 2-output scenario.
This exceeds the noise thresholds of $q \approx 7.1\%$ and $\eta \approx
90.7\%$ respectively for DIQKD with one-way error correction using the CHSH
inequality, thereby showing that it is possible to distill secret key beyond
those thresholds.
