Engagement is essential to meaningful social interaction between humans. Understanding the mechanisms by which we detect engagement of other humans can help us understand how we can build robots that interact socially with humans. However, there is currently a lack of measurable engagement constructs on which to build an artificial system that can reliably support social interaction between humans and robots. This paper proposes a definition, based on motivation theories, and outlines a framework to explore the idea that engagement can be seen as specific behaviors and their attached magnitude or intensity. This is done by the use of data from multiple sources such as observer ratings, kinematic data, audio and outcomes of interactions. We use the domain of human-robot interaction in order to illustrate the application of this approach. The framework further suggests a method to gather and aggregate this data. If certain behaviors and their attached intensities co-occur with various levels of judged engagement, then engagement could be assessed by this framework consequently making it accessible to a robotic platform. This framework could improve the social capabilities of interactive agents by adding the ability to notice when and why an agent becomes disengaged, thereby providing the interactive agent with an ability to reengage him or her. We illustrate and propose validation of our framework with an example from robot-assisted therapy for children with autism spectrum disorder. The framework also represents a general approach that can be applied to other social interactive settings between humans and robots, such as interactions with elderly people.