  In many scientific studies, it becomes increasingly important to delineate
the causal pathways through a large number of mediators, such as genetic and
brain mediators. Structural equation modeling (SEM) is a popular technique to
estimate the pathway effects, commonly expressed as products of coefficients.
However, it becomes unstable to fit such models with high dimensional
mediators, especially for a general setting where all the mediators are
causally dependent but the exact causal relationships between them are unknown.
This paper proposes a sparse mediation model using a regularized SEM approach,
where sparsity here means that a small number of mediators have nonzero
mediation effects between a treatment and an outcome. To address the model
selection challenge, we innovate by introducing a new penalty called Pathway
Lasso. This penalty function is a convex relaxation of the non-convex product
function, and it enables a computationally tractable optimization criterion to
estimate and select many pathway effects simultaneously. We develop a fast
ADMM-type algorithm to compute the model parameters, and we show that the
iterative updates can be expressed in closed form. On both simulated data and a
real fMRI dataset, the proposed approach yields higher pathway selection
accuracy and lower estimation bias than other competing methods.
