A new and diverse set of online metrics are emerging to capture the effects of the sharing and discussion of research articles on online platforms. In this paper, we investigate whether altmetrics differ between Open Access (OA) and Non-Open Access (NOA) articles. We define a new metric, the Open Access Altmetric Advantage, and investigate 14 online data sources (Twitter, Facebook, CiteULike, Mendeley, F1000, blogs, mainstream news outlets, Google Plus, Pinterest, Reddit, Sina Weibo, the peer review sites PubPeer and Publons, policy documents, and sites running Stack Exchange (Q&A)). In eight of the data sources investigated, we found that OA articles receive higher altmetrics than NOA articles; however, we found less significant differences when taking into consideration some influential factors such as journal, publication year, and citation count.