Abstract: From their inception, causal systems models (more commonly known as structural-equations models) have been accompanied by graphical representations or path diagrams that provide compact summaries of qualitative assumptions made by the models. These diagrams can be reinterpreted as probability models, enabling use of graph theory in probabilistic inference, and allowing easy deduction of independence conditions implied by the assumptions. They can also be used as a formal tool for causal inference, such as predicting the effects of external interventions. Given that the diagram is correct, one can see whether the causal effects of interest (target effects, or causal estimands) can be estimated from available data, or what additional observations are needed to validly estimate those effects. One can also see how to represent the effects as familiar standardized effect measures. The present article gives an overview of: (1) components of causal graph theory; (2) probability interpretations of graphical models; and (3) methodologic implications of the causal and probability structures encoded in the graph, such as sources of bias and the data needed for their control.

By expressing the conditional causal risk difference as a sum of products of stratum specific risk differences and conditional probabilities, it is possible to give a classification of the types of causal relationships that can give rise to effect modification on the risk difference scale. Directed acyclic graphs make clear the necessary causal relationships for a particular variable to serve as an effect modifier for the causal risk diference concerning two other variables. The directed acyclic graph causal framework thereby gives rise to a four-fold classification for effect modi…cation: direct effect modification, indirect effect modification, effect modification by proxy and efect modification by a common cause. Brief discussion is given to the case of multiple effect modification relationships and multiple effect modifiers as well as measures of effect other than that of the causal risk difference.

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It is difficult, nowadays, to open a popular science magazine, or a leading science journal, without reading about complexity, the

Background The counterfactual or potential outcome model has become increasingly standard for causal inference in epidemiological and medical studies. Discussion This paper provides an overview on the counterfactual and related approaches. A variety of conceptual as well as practical issues when estimating causal effects are reviewed. These include causal interactions, imperfect experiments, adjustment for confounding, time-varying exposures, competing risks and the probability of causation. It is argued that the counterfactual model of causal effects captures the main aspects of causality in health sciences and relates to many statistical procedures. Summary Counterfactuals are the basis of causal inference in medicine and epidemiology. Nevertheless, the estimation of counterfactual differences pose several difficulties, primarily in observational studies. These problems, however, reflect fundamental barriers only when learning from observations, and this does not invalidate the counterfactual concept.

Various concepts of interaction are reconsidered in light of a sufficient-component cause framework. Conditions and statistical tests are derived for the presence of synergism within sufficient causes. The conditions derived are sufficient but not necessary for the presence of synergism. In the context of monotonic effects, but not in general, the conditions which are derived are closely related but not identical to effect modification on the risk difference scale.

Various concepts of interaction are reconsidered in light of a sufficient-component-cause framework. Conditions and statistical tests are derived for the presence of synergism within sufficient causes. The conditions derived are sufficient but not necessary for the presence of synergism. In the context of monotonic effects, the conditions derived are closely related to effect modification on the risk difference scale; however, this is not the case without the assumption of monotonic effects.

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