Spatial independent component analysis (ICA) applied to functional magnetic resonance imaging (fMRI) data identifies functionally connected networks by estimating spatially independent patterns from their linearly mixed fMRI signals. Several multi-subject ICA approaches estimating subject-specific time courses (TCs) and spatial maps (SMs) have been developed, however, there has not yet been a full comparison of the implications of their use. Here, we provide extensive comparisons of four multi-subject ICA approaches in combination with data reduction methods for simulated and fMRI task data. For multi-subject ICA, the data first undergo reduction at the subject and group levels using principal component analysis (PCA). Comparisons of subject-specific, spatial concatenation, and group data mean subject-level reduction strategies using PCA and probabilistic PCA (PPCA) show that computationally intensive PPCA is equivalent to PCA, and that subject-specific and group data mean subject-level PCA are preferred because of well-estimated TCs and SMs. Second, aggregate independent components are estimated using either noise-free ICA or probabilistic ICA (PICA). Third, subject-specific SMs and TCs are estimated using back-reconstruction. We compare several direct group ICA (GICA) back-reconstruction approaches (GICA1-GICA3) and an indirect back-reconstruction approach, spatio-temporal regression (STR, or dual regression). Results show the earlier group ICA (GICA1) approximates STR, however STR has contradictory assumptions and may show mixed-component artifacts in estimated SMs. Our evidence-based recommendation is to use GICA3, introduced here, with subject-specific PCA and noise-free ICA, providing the most robust and accurate estimated SMs and TCs in addition to offering an intuitive interpretation.