Functional structural equation modeling with latent variables

Abstract: Handling latent variables in Structural Equation Models (SEMs) in a case where both the latent variables and their corresponding indicators in the measurement error part of the model are random curves presents significant challenges, especially with sparse data. In this paper, we develop a novel family of Functional Structural Equation Models (FSEMs) that incorporate latent variables modeled as Gaussian Processes (GPs). The introduced FSEMs are built upon functional regression models having response variables modeled as underlying GPs. The model flexibly adapts to cases when the random curves' realizations are observed only over a sparse subset of the domain, and the inferential framework is based on a restricted maximum likelihood approach. The advantage of this framework lies in its ability and flexibility in handling various data scenarios, including regularly and irregularly spaced points and thus missing data. To extract smooth estimates for the functional parameters, we employ a penalized likelihood approach that selects the smoothing parameters using a cross-validation method. We evaluate the performance of the proposed model using simulation studies and a real data example, which suggests that our model performs well in practice. The uncertainty associated with the estimates of the functional coefficients is also assessed by constructing confidence regions for each estimate. The goodness of fit indices that are commonly used to evaluate the fit of SEMs are developed for the FSEMs introduced in this paper. Overall, the proposed method is a promising approach for modeling functional data in SEMs with functional latent variables.