## Bounds on Direct Effects in the Presence of Confounded Intermediate Variables (2007)

Citations: | 6 - 0 self |

### BibTeX

@MISC{Cai07boundson,

author = {Zhihong Cai and Manabu Kuroki and Judea Pearl and Jin Tian},

title = {Bounds on Direct Effects in the Presence of Confounded Intermediate Variables},

year = {2007}

}

### OpenURL

### Abstract

Summary. This paper considers the problem of estimating the average con-trolled direct effect (ACDE) of a treatment on an outcome, in the presence of unmeasured confounders between an intermediate variable and the outcome. Such confounders render the direct effect unidentifiable even in cases where the total effect is unconfounded (hence identifiable). Kaufman et al. (2005) applied a linear programming software to find the minimum and maximum possible values of the ACDE for specific numerical data. In this paper, we apply the symbolic Balke-Pearl (1997) linear programming method to derive closed-form formulas for the upper and lower bounds on the ACDE under various assumptions of monotonicity. These universal bounds enable clini-cal experimenters to assess the direct effect of treatment from observed data with minimum computational effort, and they further shed light on the sign of the direct effect and the accuracy of the assessments.

### Citations

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Citation Context ...he average controlled direct effects (ACDEs) are defined as ACDE(z) =pr{y1|do(x1), do(z)}−pr{y1|do(x0), do(z)} (1) for z ∈{z0,z1}, where ’do(·)’ denotes an imposed intervention (Kaufman et al., 2005; =-=Pearl, 2000-=-). pr{y|do(x), do(z)} indicates the probability of Y = y when we set X and Z to specific values x and z respectively by a joint intervention (Pearl, 2000). Using the counterfactual notation of Neyman ... |

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Citation Context ...n Estimation of the direct effect of a treatment on an outcome is a central concern in epidemiological and clinical research (Buyse and Molenberghs, 1998; Kaufman et al., 2005; Petersen et al., 2006; =-=Robins and Greenland, 1992-=-; Rubin, 2004; Taylor et al., 2005; Wang and Taylor, 2002). Pearl (2001) gave a formal definition of the total effect decomposition into direct and indirect effects, and distinguished between the cont... |

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Citation Context ...um values as an estimator of the ACDE. However, they did not provide exact formulas of the bounds on the ACDE. In this paper, we apply the symbolic Balke-Pearl linear programming method (Balke, 1995; =-=Balke and Pearl, 1997-=-) to derive closed-form formulas of the upper and lower bounds on the ACDE under various assumptions of monotonicity. In contrast to the numerical method of Kaufman et al. (2005), these symbolic formu... |

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Citation Context ...int estimator; Potential response type; Stratified analysiss1. Introduction Estimation of the direct effect of a treatment on an outcome is a central concern in epidemiological and clinical research (=-=Buyse and Molenberghs, 1998-=-; Kaufman et al., 2005; Petersen et al., 2006; Robins and Greenland, 1992; Rubin, 2004; Taylor et al., 2005; Wang and Taylor, 2002). Pearl (2001) gave a formal definition of the total effect decomposi... |

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Citation Context ...mum and maximum values as an estimator of the ACDE. However, they did not provide exact formulas of the bounds on the ACDE. In this paper, we apply the symbolic Balke-Pearl linear programming method (=-=Balke, 1995-=-; Balke and Pearl, 1997) to derive closed-form formulas of the upper and lower bounds on the ACDE under various assumptions of monotonicity. In contrast to the numerical method of Kaufman et al. (2005... |

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Citation Context ...onse type; Stratified analysiss1. Introduction Estimation of the direct effect of a treatment on an outcome is a central concern in epidemiological and clinical research (Buyse and Molenberghs, 1998; =-=Kaufman et al., 2005-=-; Petersen et al., 2006; Robins and Greenland, 1992; Rubin, 2004; Taylor et al., 2005; Wang and Taylor, 2002). Pearl (2001) gave a formal definition of the total effect decomposition into direct and i... |

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Citation Context ...eatment on an outcome is a central concern in epidemiological and clinical research (Buyse and Molenberghs, 1998; Kaufman et al., 2005; Petersen et al., 2006; Robins and Greenland, 1992; Rubin, 2004; =-=Taylor et al., 2005-=-; Wang and Taylor, 2002). Pearl (2001) gave a formal definition of the total effect decomposition into direct and indirect effects, and distinguished between the controlled direct effect and the natur... |

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