This study introduces a novel mechanistic modeling and statistical framework for analyzing motion energy dynamics within psychotherapy sessions. We transform raw motion energy data into an interpretable narrative of therapist-patient interactions, thereby revealing unique insights into the nature of these dynamics. Our methodology is established through three detailed case studies, each shedding light on the complexities of dyadic interactions. A key component of our approach is an analysis spanning four years of one therapist's sessions, allowing us to distinguish between trait-like and state-like dynamics. This research represents a significant advancement in the quantitative understanding of motion dynamics in psychotherapy, with the potential to substantially influence both future research and therapeutic practice.

Ordinary differential equations (ODEs) are foundational in modeling intricate dynamics across a gamut of scientific disciplines. Yet, a possibility to represent a single phenomenon through multiple ODE models, driven by different understandings of nuances in internal mechanisms or abstraction levels, presents a model selection challenge. This study introduces a testing-based approach for ODE model selection amidst statistical noise. Rooted in the model misspecification framework, we adapt foundational insights from classical statistical paradigms (Vuong and Hotelling) to the ODE context, allowing for the comparison and ranking of diverse causal explanations without the constraints of nested models. Our simulation studies validate the theoretical robustness of our proposed test, revealing its consistent size and power. Real-world data examples further underscore the algorithm's applicability in practice. To foster accessibility and encourage real-world applications, we provide a user-friendly Python implementation of our model selection algorithm, bridging theoretical advancements with hands-on tools for the scientific community.