One of the primary goals of Human-Robot Interaction (HRI) research is to develop robots that can interpret human behavior and adapt their responses accordingly. Adaptive learning models, such as continual and reinforcement learning, play a crucial role in improving robots' ability to interact effectively in real-world settings. However, these models face significant challenges due to the limited availability of real-world data, particularly in sensitive domains like healthcare and well-being. This data scarcity can hinder a robot's ability to adapt to new situations. To address these challenges, causality provides a structured framework for understanding and modeling the underlying relationships between actions, events, and outcomes. By moving beyond mere pattern recognition, causality enables robots to make more explainable and generalizable decisions. This paper presents an exploratory causality-based analysis through a case study of an adaptive robotic coach delivering positive psychology exercises over four weeks in a workplace setting. The robotic coach autonomously adapts to multimodal human behaviors, such as facial valence and speech duration. By conducting both macro- and micro-level causal analyses, this study aims to gain deeper insights into how adaptability can enhance well-being during interactions. Ultimately, this research seeks to advance our understanding of how causality can help overcome challenges in HRI, particularly in real-world applications.