Causal inference under directed acyclic graphs

Directed acyclic graph (DAG) is used to describe the relationships among variables \nin causal structures according to some priori assumptions. This study mainly \nfocuses on an application area of DAG for causal inference in genetics. In genetic \nassociation studies, an observed effect of a genetic marker on a target phenotype can \nbe caused by a direct genetic link and an indirect non-genetic link through an intermediate \nphenotype which is influenced by the same marker. We consider methods \nto estimate and test the direct effect of the genetic marker on the continuous target \nphenotypic variable which is either completely observed or subject to censoring. The \ntraditional standard regression methods may lead to biased direct genetic effect estimates. \nTherefore, Vansteelandt et al. [2009] proposed a two-stage estimation method \nusing the principle of the sequential G-estimation for direct effects in linear models \n(Goetgeluk, Vansteelandt and Goetghebeur, 2009). In the first stage, the effect of the \nintermediate phenotype is estimated and an adjusted target phenotype is obtained by removing the effect of the intermediate phenotype. In the second stage, the direct \ngenetic effect of the genetic marker on the target phenotype is estimated by regressing \nthe genetic marker on the adjusted target phenotype. The two-stage estimation \nmethod works well when outcomes are completely observed. In this study, we show \nthat the extension of the two-stage estimation method proposed by Lipman et al. \n[2011] for analysis of a target time-to-event phenotype which is subject to censoring \ndoes not work, and we propose a novel three-stage estimation method to estimate and \ntest the direct genetic effect for censored outcomes under the accelerated failure time \nmodel. In order to address the issue in the adjustment procedure caused by survival \noutcomes which are subject to censoring, in the first stage, we estimate the true values \nof underlying observations and adjust the target phenotype for censoring. Then, \nwe follow the two-stage estimation method proposed by Vansteelandt et al. [2009] \nto estimate the direct genetic effect. The test statistic proposed by Vansteelandt et \nal. [2009] cannot be directly used due to the adjustment for censoring conducted \nin the first stage; therefore, we propose to use a Wald-type test statistic to test the \nabsence of the direct effect of the genetic marker on the target time-to-event phenotype. \nConsidering the variability due to the estimation in the previous stages, we \npropose a nonparametric bootstrap procedure to estimate the standard error of the \nthree-stage estimate of the direct effect. We show that the new three-stage estimation method and the Wald-type test statistic can be effectively used to make inference on \nthe direct genetic effect for both uncensored and censored outcomes. \nFinally, we address the real situation in which the causal association between different \nphenotypes is not consistent with investigators’ assumptions, and models used \nto make inference for the direct genetic effect are misspecified. We show that in genetic \nassociation studies, simply using a wrong model without having enough evidence on \nwhich model is correct will lead to wrong conclusions if the causal relationship among \nphenotypes is unknown.

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