Reply to Discussion

I want to begin by thanking the editorial board of Scandinavian Journal of Statistics for assembling this interesting package of materials on causal inference. Although I find much to agree with in the discussions of Aalen and Lauritzen, I also find points of disagreement or at least puzzlement. First, I agree with Professor Aalen about the crucial role statistics has to play in many areas of empirical investigation into causal effects, including medicine. Also, I agree on the criticality of substantive knowledge in this endeavour, as I hope I indicated in my very brief description of the kinds of assumptions needed in the anthrax vaccine experiments, as acknowledged by Professor Lauritzen. In fact, I have been personally involved in real applications (in psychology, medicine, economics, astrophysics, pharmacology, etc.) for four decades, and have only worked on statistical topics that were stimulated by real problems. Perhaps my article should have emphasized this more clearly to avoid the implicit criticism in Professor Aalen’s final concluding paragraph, where he emphasizes the need for statistics to interact with substantive fields. Secondly, comments on modes of inference are relevant to both discussions. Although completely compatible, the randomization-based frequentist approaches to causal inference, due initially to Fisher (1926) and Neyman (1923), are distinctly different from the predictive Bayesian (model-based) approach (Rubin, 1978). The former base inferences solely on a model for the assignment mechanism, Pr(W|Y(0), Y(1), X), where the potential outcomes Y(0), Y(1) and covariates X are treated as fixed, possibly unknown, quantities without any distribution (or notions such as independence) assumed, and W indicates the treatment assignments. The assignment mechanism models what actions we take to try to learn about casual effects of the treatment on the potential outcomes at different values of the covariates. The Bayesian approach supplements the model for the assignment mechanism with a model for the ‘science’ Pr(Y(0), Y(1), X) 1⁄4 Pr(Y(0), Y(1)|X)Pr(X) – about which we hope to learn by comparing Y(1) with Y(0) at various values of X. With respect to Aalen’s discussion, it is puzzling to me to read Aalen’s comment that ‘It is the impression of this discussant that the distinction between Bayesian and frequentist analysis is rapidly disappearing, at least in a conceptual sense. ’ Perhaps Aalen includes within ‘frequentist analysis’ all varieties of model-based, hierarchical random parameter approaches and excludes the classical randomization-based approaches to causal inference of Fisher and Neyman. If so, I still find that the conceptual differences between the approaches are important to understand for practice. The realization that so much can be accomplished from the randomization-based perspective alone is, in my view, responsible for the great advances in experimental design (e.g. as reflected in the classic texts by Cochran & Cox, 1950; Kempthorne, 1952; Cox, 1958) and more modest advances in the design of observational studies (e.g. Rosenbaum & Rubin, 1983; Rosenbaum, 1995; Rubin, 2002). With respect to Lauritzen’s discussion, the critical distinction between what we can do to learn about causal effects through design, the assignment mechanism, and how the gods have aligned the world (the science) seems to be largely obfuscated in the graphical approach. How does a graph represent the classical Fisher–Neyman perspective where the only random quantity is the assignment indicator, W, or represent classical designs such as splitplots with their levels of randomization? Evidently, missing arrows indicate both (a) the implications of randomization (actual or assumed) on conditional distributions and 196 D. B. Rubin Scand J Statist 31