By participating in asynchronous course discussion forums, students can work together to refine their ideas and construct knowledge collaboratively. Typically, some messages simply repeat or paraphrase course content, while others bring in new material, demonstrate reasoning, integrate concepts, and develop solutions. Through the messages they send, students thus display different levels of intellectual engagement with the topic and the course. We refer to this as cognitive quality. The work presented here used two widely studied frameworks for assessing critical discourse and cognitive engagement: the ICAP and Community of Inquiry frameworks. The constructs of the frameworks were used as proxy measures for cognitive quality. Predictive classifiers were trained for both frameworks on the same data in order to discover which attributes of the dialogue were most informative and how those attributes were correlated with framework constructs. We found that longer and more complex messages were associated with indicators of greater quality in both frameworks, and that the threaded reply structure mattered more than chronological order. By including the framework labels as additional model features, we also assessed the links between frameworks. The empirical results provide evidence that the two frameworks measure different aspects of student behaviour relating to cognitive quality.