Interpreting and using CPDAGs with background knowledge

We develop terminology and methods for working with maximally oriented partially directed acyclic graphs (maximal PDAGs). Maximal PDAGs arise from imposing restrictions on a Markov equivalence class of directed acyclic graphs, or equivalently on its graphical representation as a completed partially directed acyclic graph (CPDAG), for example when adding background knowledge about certain edge orientations. Although maximal PDAGs often arise in practice, causal methods have been mostly developed for CPDAGs. In this paper, we extend such methodology to maximal PDAGs. In particular, we develop methodology to read off possible ancestral relationships, we introduce a graphical criterion for covariate adjustment to estimate total causal effects, and we adapt the IDA and joint-IDA frameworks to estimate multi-sets of possible causal effects. We also present a simulation study that illustrates the gain in identifiability of total causal effects as the background knowledge increases. All methods are implemented in the R package pcalg.Comment: 17 pages, 6 figures, UAI 201

Conditional Adjustment in a Markov Equivalence Class

We consider the problem of identifying a conditional causal effect through covariate adjustment. We focus on the setting where the causal graph is known up to one of two types of graphs: a maximally oriented partially directed acyclic graph (MPDAG) or a partial ancestral graph (PAG). Both MPDAGs and PAGs represent equivalence classes of possible underlying causal models. After defining adjustment sets in this setting, we provide a necessary and sufficient graphical criterion -- the conditional adjustment criterion -- for finding these sets under conditioning on variables unaffected by treatment. We further provide explicit sets from the graph that satisfy the conditional adjustment criterion, and therefore, can be used as adjustment sets for conditional causal effect identification.Comment: 29 pages, 6 figure

Complete Graphical Characterization and Construction of Adjustment Sets in Markov Equivalence Classes of Ancestral Graphs

We present a graphical criterion for covariate adjustment that is sound and complete for four different classes of causal graphical models: directed acyclic graphs (DAGs), maximum ancestral graphs (MAGs), completed partially directed acyclic graphs (CPDAGs), and partial ancestral graphs (PAGs). Our criterion unifies covariate adjustment for a large set of graph classes. Moreover, we define an explicit set that satisfies our criterion, if there is any set that satisfies our criterion. We also give efficient algorithms for constructing all sets that fulfill our criterion, implemented in the R package dagitty. Finally, we discuss the relationship between our criterion and other criteria for adjustment, and we provide new soundness and completeness proofs for the adjustment criterion for DAGs.Comment: 58 pages, 12 figures, to appear in JML

A Complete Generalized Adjustment Criterion

Covariate adjustment is a widely used approach to estimate total causal effects from observational data. Several graphical criteria have been developed in recent years to identify valid covariates for adjustment from graphical causal models. These criteria can handle multiple causes, latent confounding, or partial knowledge of the causal structure; however, their diversity is confusing and some of them are only sufficient, but not necessary. In this paper, we present a criterion that is necessary and sufficient for four different classes of graphical causal models: directed acyclic graphs (DAGs), maximum ancestral graphs (MAGs), completed partially directed acyclic graphs (CPDAGs), and partial ancestral graphs (PAGs). Our criterion subsumes the existing ones and in this way unifies adjustment set construction for a large set of graph classes.Comment: 10 pages, 6 figures, To appear in Proceedings of the 31st Conference on Uncertainty in Artificial Intelligence (UAI2015

Graphical Criteria for Efficient Total Effect Estimation via Adjustment in Causal Linear Models

Covariate adjustment is a commonly used method for total causal effect estimation. In recent years, graphical criteria have been developed to identify all valid adjustment sets, that is, all covariate sets that can be used for this purpose. Different valid adjustment sets typically provide total effect estimates of varying accuracies. Restricting ourselves to causal linear models, we introduce a graphical criterion to compare the asymptotic variances provided by certain valid adjustment sets. We employ this result to develop two further graphical tools. First, we introduce a simple variance reducing pruning procedure for any given valid adjustment set. Second, we give a graphical characterization of a valid adjustment set that provides the optimal asymptotic variance among all valid adjustment sets. Our results depend only on the graphical structure and not on the specific error variances or edge coefficients of the underlying causal linear model. They can be applied to directed acyclic graphs (DAGs), completed partially directed acyclic graphs (CPDAGs) and maximally oriented partially directed acyclic graphs (maximal PDAGs). We present simulations and a real data example to support our results and show their practical applicability.Comment: 63 pages, 17 figure

Graphical Criteria for Efficient Total Effect Estimation via Adjustment in Causal Linear Models

Covariate adjustment is a commonly used method for total causal effect estimation. In recent years, graphical criteria have been developed to identify all valid adjustment sets, that is, all covariate sets that can be used for this purpose. Different valid adjustment sets typically provide total effect estimates of varying accuracies. Restricting ourselves to causal linear models, we introduce a graphical criterion to compare the asymptotic variances provided by certain valid adjustment sets. We employ this result to develop two further graphical tools. First, we introduce a simple variance reducing pruning procedure for any given valid adjustment set. Second, we give a graphical characterization of a valid adjustment set that provides the optimal asymptotic variance among all valid adjustment sets. Our results depend only on the graphical structure and not on the specific error variances or edge coefficients of the underlying causal linear model. They can be applied to directed acyclic graphs (DAGs), completed partially directed acyclic graphs (CPDAGs) and maximally oriented partially directed acyclic graphs (maximal PDAGs). We present simulations and a real data example to support our results and show their practical applicability.ISSN:1369-7412ISSN:0035-9246ISSN:1467-986