Much statistical analysis seeks to identify associations between exposures and outcomes. The population attributable fraction (PAF) is an epidemiologic measure widely used to assess the public health impact of exposures in populations. PAF is defined as the fraction of all cases of a particular disease or other adverse condition in a population that is attributable to a specific exposure; PAF equals (O − E)/O, where O and E refer to the observed number of cases and the expected number of cases under no exposure, respectively. The term “attributable” has a causal interpretation: PAF is the estimated fraction of all cases that would not have occurred if there had been no exposure.1 As an example, in early 1950,2 Doll derived O = 11189 and E = 1875 using the Doll and Hill case-control study of smoking and lung cancer deaths throughout England and Wales,3 so the smoking PAF for lung cancer deaths was (11189 − 1875)/11189 = 83%. Using a cohort study, following Miettinen, we can estimate the PAF from …