Analytic bounds on causal risk differences in directed acyclic graphs involving three observed binary variables

Sol Kaufman, Jay S. Kaufman, Richard F. MacLehose

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26 Scopus citations


We apply a linear programming approach which uses the causal risk difference (R DC) as the objective function and provides minimum and maximum values that R DC can achieve under any set of linear constraints on the potential response type distribution. We consider two scenarios involving binary exposure X, covariate Z and outcome Y. In the first, Z is not affected by X, and is a potential confounder of the causal effect of X on Y. In the second, Z is affected by X and intermediate in the causal pathway between X and Y. For each scenario we consider various linear constraints corresponding to the presence or absence of arcs in the associated directed acyclic graph (DAG), monotonicity assumptions, and presence or absence of additive-scale interactions. We also estimate Z-stratum-specific bounds when Z is a potential effect measure modifier and bounds for both controlled and natural direct effects when Z is affected by X. In the absence of any additional constraints deriving from background knowledge, the well-known bounds on R Dc are duplicated: - Pr (Y ≠ X) ≤ R DC ≤ Pr (Y = X). These bounds have unit width, but can be narrowed by background knowledge-based assumptions. We provide and compare bounds and bound widths for various combinations of assumptions in the two scenarios and apply these bounds to real data from two studies.

Original languageEnglish (US)
Pages (from-to)3473-3487
Number of pages15
JournalJournal of Statistical Planning and Inference
Issue number10
StatePublished - Oct 1 2009

Bibliographical note

Funding Information:
This work was supported by Contract R01-HD-39746 from the National Institute of Child Health and Human Development. We are grateful to Alexander A Balke for use of his symbolic LP solutions computer program; to Dr. M. Alan Brookhart, Division of Pharmacoepidemiology and Pharmacoeconomics, Department of Medicine, Harvard Medical School, for access to the data from his article used as an example in Section 4.1 of this paper; and to Dr. E. Jane Costello, Department of Psychiatry and Behavioral Sciences, Duke University Medical Center, for access to the data from the Great Smoke Mountains Study used as an example in Section 4.2 of this paper.


  • Bounding
  • Causality
  • Confounding
  • Counterfactual models
  • Effect decomposition
  • Linear programming
  • Sensitivity analysis


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