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The results from the ALMA proposal peer review process in Cycles 0-6 are analyzed to identify any systematics in the scientific rankings that may signify bias. Proposal rankings are analyzed with respect to the experience level of a Principal Investigator (PI) in submitting ALMA proposals, regional affiliation (Chile, East Asia, Europe, North America, or Other), and gender. The analysis was conducted for both the Stage 1 rankings, which are based on the preliminary scores from the reviewers, and the Stage 2 rankings, which are based on the final scores from the reviewers after participating in a face-to-face panel discussion. Analysis of the Stage 1 results shows that PIs who submit an ALMA proposal in multiple cycles have systematically better proposal ranks than PIs who have submitted proposals for the first time. In terms of regional affiliation, PIs from Europe and North America have better Stage 1 rankings than PIs from Chile and East Asia. Consistent with Lonsdale et al. (2016), proposals led by men have better Stage 1 rankings than women when averaged over all cycles. This trend was most noticeably present in Cycle 3, but no discernible differences in the Stage 1 rankings are present in recent cycles. Nonetheless, in each cycle to date, women have had a lower proposal acceptance rate than men even after differences in demographics are considered. Comparison of the Stage 1 and Stage 2 rankings reveal no significant changes in the distribution of proposal ranks by experience level, regional affiliation, or gender as a result of the panel discussions, although the proposal ranks for East Asian PIs show a marginally significant improvement from Stage 1 to Stage 2 when averaged over all cycles. Thus any systematics in the proposal rankings are introduced primarily in the Stage 1 process and not from the face-to-face discussions., 26 pages, 14 figures, accepted for publication in PASP