Independent Component Analysis (ICA) is one of the most popular techniques for the analysis of resting state FMRI data because it has several advantageous properties when compared with other techniques. Most notably, in contrast to a conventional seed-based correlation analysis, it is model-free and multivariate, thus switching the focus from evaluating the functional connectivity of single brain regions identified a priori to evaluating brain connectivity in terms of all brain resting state networks (RSNs) that simultaneously engage in oscillatory activity. Furthermore, typical seed-based analysis characterizes RSNs in terms of spatially distributed patterns of correlation (typically by means of simple Pearson's coefficients) and thereby confounds together amplitude information of oscillatory activity and noise. ICA and other regression techniques, on the other hand, retain magnitude information and therefore can be sensitive to both changes in the spatially distributed nature of correlations (differences in the spatial pattern or "shape") as well as the amplitude of the network activity. Furthermore, motion can mimic amplitude effects so it is crucial to use a technique that retains such information to ensure that connectivity differences are accurately localized. In this work, we investigate the dual regression approach that is frequently applied with group ICA to assess group differences in resting state functional connectivity of brain networks. We show how ignoring amplitude effects and how excessive motion corrupts connectivity maps and results in spurious connectivity differences. We also show how to implement the dual regression to retain amplitude information and how to use dual regression outputs to identify potential motion effects. Two key findings are that using a technique that retains magnitude information, e.g., dual regression, and using strict motion criteria are crucial for controlling both network amplitude and motion-related amplitude effects, respectively, in resting state connectivity analyses. We illustrate these concepts using realistic simulated resting state FMRI data and in vivo data acquired in healthy subjects and patients with bipolar disorder and schizophrenia.