An effect is attributable to treatment if it would not have been observed had the individual been exposed to control instead. Extending earlier results on attributable effects in unmatched groups, a method of exact randomization inference and sensitivity analysis is developed for case-referent, case-crossover, and cohort studies with matched sets, and a large sample approximation to the exact inference is given. The unmatched case, considered previously, has certain symmetries that the matched case, considered here, does not have. As a result, approximation for the matched case requires the use of the recently developed method of asymptotic separability, which was not needed in the unmatched case. Several examples are presented, including a case-referent study of Helicobactre pylori infection as a cause of myocardial infarction, a case-crossover study of alcohol as a cause of injury, a cohort study of women who gave birth at home, and a study of the effects of cadmium exposure with a continuous outcome measur ing kidney function. Unlike tests of no effect, inference about attributable effects has a different form in case-referent and cohort studies.
KEY WORDS: Asymptotic separability; Attributable effect; Displacement effect; Randomization inference; Sensitivity analysis.
1. ATTRIBUTABLE EFFECTS IN EXPERIMENTS AND OBSERVATIONAL STUDIES
1.1 How Does Randomization Affect Inference?
In his 1935 book Design of Experiments, Fisher carefully argued that the random assignment of treatments in experiments justifies certain inferences about the effects caused by those treatments--that randomization forms the "reasoned basis for inference" in experiments--and that these same inferences would not be justified by identical data obtained in a nonrandomized study. In Fisher's view, causal inference depends partly on the observed data, but also partly on how the data were obtained. This view has two desirable consequences. First, it serves to encourage randomized experimentation when randomization is ethical and feasible. Second, it forces analyses of nonrandomized or observational studies of treatment effects to explicitly acknowledge, as part of the quantitative findings, greater uncertainty about causal effects than would be present had random assignment been used.
A limitation is that many randomization tests of the hypothesis of no treatment effect are not paired with confidence intervals. If the treatment has an additive effect, [tau], then a randomization test of no effect can be inverted to yield confidence intervals and point estimates for [tau] (see, e.g., Hodges and Lehmann 1963; Moses 1965; Lehmann 1963, 1975; Rosenbaum 1995a sec. 2). The model of an additive effect is useful in many settings but is inapplicable in many others- for instance, for binary responses, where inferences typically invoke distributional assumptions not derived from random assignment. This article extends a line of reasoning begun in earlier work (Rosenbaum 2001), in which attributable effects are used to substantially expand the collection of randomization tests that may be inverted to yield confidence intervals. The approach also yields sensitivity analyses that measure the added uncertainty present when treatments are not randomly assigned.
An attributable effect describes how treated subjects would have responded had they been spared exposure to the treatment. As a consequence, the attributable effect depends in part on the identity of the subjects exposed to treatment, and so different randomizations typically produce different attributable effects. In this sense, the attributable effect is not a parameter, but rather is an unobserved random variable. Nonetheless, inference proceeds along a relatively conventional path. Before embarking on that path, I present a useful illustration.
1.2 Example: Infection as a Possible Cause of Early Onset Myocardial Infarction