Affect sensitivity is of the utmost importance for a robot companion to be able to display socially intelligent behaviour, a key requirement for sustaining long-term interactions with humans. This paper explores a naturalistic scenario in which children play chess with the iCat, a robot companion. A person-independent, Bayesian approach to detect the user's engagement with the iCat robot is presented. Our framework models both causes and effects of engagement: features related to the user's non-verbal behaviour, the task and the companion's affective reactions are identified to predict the children's level of engagement. An experiment was carried out to train and validate our model. Results show that our approach based on multimodal integration of task and social interaction-based features outperforms those based solely on non-verbal behaviour or contextual information (94.79 % vs. 93.75 % and 78.13 %).