Independent component analysis (ICA) has proven quite useful for the analysis of functional magnetic resonance imaging (fMRI) data. However, stability of ICA decompositions is an issue in ICA of fMRI analysis primarily due to the noisy nature of fMRI data and the iterative nature of algorithms. In this work, we present an approach that utilizes an objective criterion and that is particularly suitable for image analysis to select the best of multiple ICA runs to use for further analysis and inference. In addition, a growing number of studies are focusing on the decomposition of single subject data and/or using high ICA model order, which both require an effective way to align components obtained from different ICA runs. In this paper, while presenting a method that provides superior performance in selecting the best run and interpreting the statistical reliability of ICA estimates, we also address the component sorting issue. Both simulated and real fMRI results show that our method selects more useful ICA runs than those selected by the widely used ICASSO software and that it is a more objective and better motivated approach to evaluate results and hence a promising tool for ICA analysis of fMRI data.