Refinement checking plays an important role in system verification. This means that the correctness of the system is established by showing a refinement relation between two models; one for the implementation and one for the specification. In [21], Wang et al. describe an algorithm based on antichains for efficiently deciding stable failures refinement and failures-divergences refinement. We identify several issues pertaining to the correctness and performance in these algorithms and propose new, correct, antichain-based algorithms. Using a number of experiments we show that our algorithms outperform the original ones in terms of running time and memory usage.