Cosmic rays arriving at Earth include the most energetic particles ever observed. The mechanism of their acceleration and their sources are, however, still mostly unknown. Observing astrophysical neutrinos can help solve this problem. Because neutrinos are produced in hadronic interactions and are neither absorbed nor deflected, they will point directly back to their source. This contribution covers continued studies of the high-energy astrophysical neutrino flux observed at the IceCube neutrino observatory, extending them from four to six years of data with a focus on energies above 60 TeV. The spectrum and spatial clustering of the observed neutrinos are discussed.