Covariate Adjustment in Equivalence Classes of Ancestral Graphs
Specialeforsvar ved Christine Winther Bang
Titel: Covariate Adjustment in Equivalence Classes of Ancestral Graphs
Abstract: This thesis is concerned with the issue of identifying causal effects trough covariate adjustment in graphical models. For this we need to recover a valid adjustment set, and at the outset we will be concerned with constructing a graphical criterion for verifying adjustment sets. Well-known graphical criteria for directed acyclic graphs (DAGs) already exist, but these results are not necessarily valid if some variables are unobserved, or if we cannot determine the orientations of all edges in a graph. Recent works have studied covariate adjustment for other types of graphs, and we will explore these results. We will present maximal ancestral graphs (MAGs); a type of graphs that allows for unobserved variables, and we will present the concept of an equivalence class; a class of graphs which are not distinguishable in terms of separation properties. We use a completed partially directed acyclic graph (CPDAG) to represent an equivalence class of DAGs, and a partial ancestral graph (PAG) to represent an equivalence class of MAGs. We will first describe the idea of covariate adjustment as well as the ideas behind previous adjustment criteria. We will then show how one can extend a criterion for DAGs and obtain a necessary and sufficient criterion for covariate adjustment in MAGs, and we will then show how to extend this criterion to include equivalence classes. Hence, we will ultimately obtain a graphical criterion that is necessary and sufficient for covariate adjustment in DAGs, MAGs, CPDAGs and PAGs. Multiple sets may satisfy this criterion, and in closing we will contemplate how to choose a suitable adjustment set. Firstly, we will present a set that will be shown to be a valid adjustment set if, and only if, any valid adjustment set exists. Secondly, we will present a graphical criterion for comparing the asymptotic variance of the estimators given two different adjustment sets. Additionally, we will show how this criterion can be used to construct an algorithm which for a given set can provide the subset with the smallest asymptotic variance. Lastly, we will conduct a simulation study where we examine the applicability of these results.
Vejleder: Niels Richard Hansen
Censor: Lars Nørvang Andersen, Aarhus Universitet