title: Directed acyclic graphs with edge-specific bounds.
abstract: We give a definition of a bounded edge within the causal directed acyclic graph framework. A bounded edge generalizes the notion of a signed edge and is defined in terms of bounds on a ratio of survivor probabilities. We derive rules concerning the propagation of bounds. Bounds on causal effects in the presence of unmeasured confounding are also derived using bounds related to specific edges on a graph. We illustrate the theory developed by an example concerning estimating the effect of antihistamine treatment on asthma in the presence of unmeasured confounding.
authors.affiliation: Department of Epidemiology, Harvard School of Public Health, 677 Huntington Avenue, Boston, Massachusetts 02115, U.S.A.,  tvanderw@hsph.harvard.edu.
authors.forename: Tyler J
authors.initials: TJ
authors.lastname: Vanderweele
authors.forename: Zhiqiang
authors.initials: Z
authors.lastname: Tan