  Causal mediation analysis usually requires strong assumptions, such as
ignorability of the mediator, which may not hold in many social and scientific
studies. Motivated by a multilevel randomized treatment experiment using
functional magnetic resonance imaging (fMRI), this paper proposes a multilevel
causal mediation framework for data with hierarchically nested structure, and
this framework provides valid inference even if structured unmeasured
confounding for the mediator and outcome is present. For the first-level data,
we propose a linear structural equation model for a continuous mediator and a
continuous outcome, both of which may contain correlated additive errors. A
likelihood-based approach is proposed to estimate the model coefficients. The
analysis of our estimator characterizes the nonidentifiability issue due to the
correlation parameter. To address the identifiability issue and model the
variability in multilevel data, we propose to incorporate multiple first-level
mediation models across different levels in a unified multilevel mediation
framework. All the model coefficients are estimated simultaneously by our
optimization algorithms, with innovation to estimate the unknown correlation
parameter from data, instead of performing sensitivity analysis. Our asymptotic
analysis shows that the correlation parameter is identifiable, and our
estimates for the mediation effects are consistent with the parametric
convergence rates. Using extensive simulated data and a real fMRI dataset, we
demonstrate the improvement of our approaches over existing methods.
