Designing Learning Intervention Studies: Identifiability of Heterogeneous Hidden Markov Models

Hidden Markov models (HMMs) are popular for modeling complex, longitudinal data. Existing identifiability theory for conventional HMMs assume emission probabilities are constant over time and the Markov chain governing transitions among the hidden states is irreducible, which are assumptions that may not be applicable in all educational and psychological research settings. We generalize existing conditions on homogeneous HMMs by considering heterogeneous HMMs with time-varying emission probabilities and the potential for absorbing states. Researchers are investigating a family of models known as restricted HMMs (RHMMs), which combine HMMs and restricted latent class models (RLCMs) to provide fine-grained classification of educationally and psychologically relevant attribute profiles over time. These RHMMs leverage the benefits of RLCMs and HMMs to understand changes in attribute profiles within longitudinal designs. The identifiability of RHMM parameters is a critical issue for ensuring successful applications and accurate statistical inference regarding factors that impact outcomes in intervention studies. We establish identifiability conditions for RHMMs. The new identifiability conditions for heterogeneous HMMs and RHMMs provide researchers insights for designing interventions. We discuss different types of assessment designs and the implications for practice. We present an application of a heterogeneous HMM to daily measures of positive and negative affect.