Independent component analysis (ICA) is increasingly used to analyze patterns of spontaneous activity in brain imaging. However, there are hardly any methods for answering the fundamental question: are the obtained components statistically significant? Most methods considering the significance of components either consider group-differences or use arbitrary thresholds with weak statistical justification. In previous work, we proposed a statistically principled method for testing if the coefficients in the mixing matrix are similar in different subjects or sessions. In many applications of ICA, however, we would like to test the reliability of the independent components themselves and not the mixing coefficients. Here, we develop a test for such an inter-subject consistency by extending our previous theory. The test is applicable, for example, to the spatial activity patterns obtained by spatial ICA in resting-state fMRI. We further improve both this and the previously proposed testing method by introducing a new way of correcting for multiple testing, new variants of the clustering method, and a computational approximation which greatly reduces the memory and computation required.