Introduction to Causal Modelling and Inference

The concepts of cause and effect are central to most areas of scientific inquiry.Still, in spite of centuries of debate among philosophers and scientists there is little consensus about basic definitions and formal methods for inference. The formal statistical treatment of the topic dates back to early works on designed experiments by Neyman and Fisher.The so-called ‘Rubin Causal Model’ was introduced in the mid-1970s and translates arguments from the experimental to the observational setting. Recently statisticians have addressed issues of causality and causal modelling with renewed interest and intensity.Competing schools have put forward their respective formalisms, and we have seen profound disagreement between some of the best thinkers in our field.Attitudes range from ‘much ado about nothing’ to ‘a revolution in statistical modelling’, the latter expecting to affect the way in which statistics is taught and practised.Whichever view will prevail, the debate is stimulating.It touches on central principles for extracting new knowledge from empirical observations, and as such it touches the very roots of statistical science. The two discussion papers at the 19th Nordic Conference on Mathematical Statistics, Stockholm, June 2002 (NORDSTAT 2002) aim at introducing to a broader Nordic audience the concepts and some of the controversies related to causal modelling and inference.Indeed, the invited papers by Rubin and Arjas seem at first sight to have little in common.Rubin draws on the concept of potential outcomes and his principles for inference follow from randomization assumptions about treatment assignment mechanisms.Arjas explicitly avoids potential variables and he elaborates around Bayesian ideas and techniques for time-dynamic processes. Rubin is concerned with relationships between treatment A (he calls it W), covariate X and response Y, and his focus is on an intermediate variable S (a surrogate) in the pathway between A and Y.His time ordering is essentially X fi A fi S fi Y and he is concerned with the definition, computation and inference for direct (not mediated through S) and indirect (mediated through S) effects of A on Y.The causal effect is defined as a contrast between the outcomes for a single unit under different possibilities for the exposure and intermediate variables.Only one of the potential outcomes can be observed on any one unit, and the analysis translates to a special type of latent variable analysis. Arjas describes a probabilistic framework for how treatment process A and covariate process X relate to a right-censored response event Y in the presence of a potential (unobserved) confounder process U.Using marked point process theory coupled with Bayesian inference he defines confounding and postulates conditional independence assumptions that lead to causal statements about effects of treatment A on response Y.In Arjas’ setting the local time ordering is U fi X fi A fi Y and no intermediate variable features between A and Y. In the invited discussion Lauritzen addresses some of the provocative remarks made by Rubin against the use of graphical models.Graphical models and notions of conditional independencies provide yet another formalism for defining and dealing with casual statements. The graphical model language has obvious links to Arjas’ probabilistic causal framework.