Altmetrics have been proposed as a way to assess the societal impact of research. Although altmetrics are already in use as impact or attention metrics in different contexts, it is still not clear whether they really capture or reflect societal impact. This study is based on altmetrics, citation counts, research output and case study data from the UK Research Excellence Framework (REF), and peers' REF assessments of research output and societal impact. We investigated the convergent validity of altmetrics by using two REF datasets: publications submitted as research output (PRO) to the REF and publications referenced in case studies (PCS). Case studies, which are intended to demonstrate societal impact, should cite the most relevant research papers. We used the MHq' indicator for assessing impactan indicator which has been introduced for count data with many zeros. The results of the first part of the analysis show that news media as well as mentions on Facebook, in blogs, in Wikipedia, and in policy-related documents have higher MHq' values for PCS than for PRO. Thus, the altmetric indicators seem to have convergent validity for these data. In the second part of the analysis, altmetrics have been correlated with REF reviewers' average scores on PCS. The negative or close to zero correlations question the convergent validity of altmetrics in thatcontext. We suggest that they may capture a different aspect of societal impact (which can be called unknown attention) to that seen by reviewers (who are interested in the causal link between research and action in society).