The proportion of research outputs published in open access journals or made available on other freely-accessible platforms has increased over the past two decades, driven largely by funder mandates, institutional policies, grass-roots advocacy, and changing attitudes in the research community. However, the relative effectiveness of these different interventions has remained largely unexplored. Here we present a robust, transparent and updateable method for analysing how these interventions affect the open access performance of individual institutes. We studied 1,207 institutions from across the world, and found that, in 2017, the top-performing universities published around 80--90% of their research open access. The analysis also showed that publisher-mediated (gold) open access was popular in Latin American and African universities, whereas the growth of open access in Europe and North America has mostly been driven by repositories.