Altmetric measurements derived from the social web are increasingly advocated and used as early indicators of article impact and usefulness. Nevertheless, there is a lack of systematic scientific evidence that altmetrics are valid proxies of either impact or utility although a few case studies have reported medium correlations between specific altmetrics and citation rates for individual journals or fields. To fill this gap, this study compares 11 altmetrics with Web of Science citations for 76 to 208,739 PubMed articles with at least one altmetric mention in each case and up to 1,891 journals per metric. It also introduces a simple sign test to overcome biases caused by different citation and usage windows. Statistically significant associations were found between higher metric scores and higher citations for articles with positive altmetric scores in all cases with sufficient evidence (Twitter, Facebook wall posts, research highlights, blogs, mainstream media and forums) except perhaps for Google+ posts. Evidence was insufficient for LinkedIn, Pinterest, question and answer sites, and Reddit, and no conclusions should be drawn about articles with zero altmetric scores or the strength of any correlation between altmetrics and citations. Nevertheless, comparisons between citations and metric values for articles published at different times, even within the same year, can remove or reverse this association and so publishers and scientometricians should consider the effect of time when using altmetrics to rank articles. Finally, the coverage of all the altmetrics except for Twitter seems to be low and so it is not clear if they are prevalent enough to be useful in practice.