Engagement in Human-Machine Interaction is the process by which entities participating in the interaction establish, maintain, and end their perceived connection. It is essential to monitor the engagement state of patients in various AI-based healthcare paradigms. This includes medical conditions that alter social behavior such as Autism Spectrum Disorder (ASD) or Attention-Deficit/Hyperactivity Disorder (ADHD). Engagement is a multifaceted construct which is composed of behavioral, emotional, and mental components. Previous research has neglected the multi-faceted nature of engagement. In this paper, a system is presented to distinguish these facets using contextual and relational features. This can facilitate further fine-grained analysis. Several machine learning classifiers including traditional and deep learning models are compared for this task. A highest accuracy of 74.57% with an F-Score and mean absolute error of 0.74 and 0.23 respectively was obtained on a balanced dataset of 22242 instances with neural network-based classification.