The role of disorder in the field of three-dimensional time-reversal-invariant topological insulators has become an active field of research recently. However, the computation of ℤ2 invariants for large, disordered systems still poses a considerable challenge. In this paper, we apply and extend a recently proposed method based on the scattering matrix approach, which allows the study of large systems at reasonable computational effort with few-channel leads. By computing the ℤ2 invariant directly for the disordered topological Anderson insulator, we unambiguously identify the topological nature of this phase without resorting to its connection with the clean case. We are able to efficiently compute the ℤ2 phase diagram in the mass-disorder plane. The topological phase boundaries are found to be well described by the self- consistent Born approximation, both for vanishing and finite chemical potentials.