  Environmental health studies are increasingly measuring endogenous omics data
($\boldsymbol{M}$) to study intermediary biological pathways by which an
exogenous exposure ($\boldsymbol{A}$) affects a health outcome
($\boldsymbol{Y}$), given confounders ($\boldsymbol{C}$). Mediation analysis is
frequently carried out to understand such mechanisms. If intermediary pathways
are of interest, then there is likely literature establishing statistical and
biological significance of the total effect, defined as the effect of
$\boldsymbol{A}$ on $\boldsymbol{Y}$ given $\boldsymbol{C}$. For mediation
models with continuous outcomes and mediators, we show that leveraging external
summary-level information on the total effect improves estimation efficiency of
the natural direct and indirect effects. Moreover, the efficiency gain depends
on the asymptotic partial $R^2$ between the outcome
($\boldsymbol{Y}\mid\boldsymbol{M},\boldsymbol{A},\boldsymbol{C}$) and total
effect ($\boldsymbol{Y}\mid\boldsymbol{A},\boldsymbol{C}$) models, with smaller
(larger) values benefiting direct (indirect) effect estimation. We robustify
our estimation procedure to incongenial external information by assuming the
total effect follows a random distribution. This framework allows shrinkage
towards the external information if the total effects in the internal and
external populations agree. We illustrate our methodology using data from the
Puerto Rico Testsite for Exploring Contamination Threats, where Cytochrome p450
metabolites are hypothesized to mediate the effect of phthalate exposure on
gestational age at delivery. External information on the total effect comes
from a recently published pooled analysis of 16 studies. The proposed framework
blends mediation analysis with emerging data integration techniques.
