{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,24]],"date-time":"2026-01-24T16:47:15Z","timestamp":1769273235322,"version":"3.49.0"},"reference-count":24,"publisher":"Wiley","issue":"11","license":[{"start":{"date-parts":[[2005,3,1]],"date-time":"2005-03-01T00:00:00Z","timestamp":1109635200000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Statistics in Medicine"],"published-print":{"date-parts":[[2005,6,15]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Adjusting for a causal intermediate is a common analytic strategy for estimating an average causal direct effect (ACDE). The ACDE is the component of the total exposure effect that is not relayed through the specified intermediate. Even if the total effect is unconfounded, the usual ACDE estimate may be biased when an unmeasured variable affects the intermediate and outcome variables. Using linear programming optimization to compute non\u2010parametric bounds, we develop new ACDE estimators for binary measured variables in this causal structure, and use root mean square confounding bias (RMSB) to compare their performance with the usual stratified estimator in simulated distributions of target populations comprised of the 64 possible potential response types as well as distributions of target populations restricted to subsets of 18 or 12 potential response types defined by monotonicity or no\u2010interactions assumptions of unit\u2010level causal effects. We also consider target population distributions conditioned on fixed outcome risk among the unexposed, or fixed true ACDE in one stratum of the intermediate. Results show that a midpoint estimator constructed from the optimization bounds has consistently lower RMSB than the usual stratified estimator both unconditionally and conditioned on any risk in the unexposed. When conditioning on true ACDE, this midpoint estimator performs more poorly only when conditioned on an extreme true ACDE in one stratum of the intermediate, yet outperforms the stratified estimator in the other stratum when interaction is permitted. An alternate \u2018limit\u2010modified crude\u2019 estimator can never perform less favourably than the stratified estimator, and often has lower RMSB. 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