The general structure of evidence factors is examined in terms of the knit product of two permutation groups. An observational or nonrandomized study of treatment effects has two evidence factors if it permits two (nearly) independent tests of the null hypothesis of no treatment effect and two (nearly) independent sensitivity analyses for those tests. Either of the two tests may be biased by nonrandom treatment assignment, but certain biases that would invalidate one test would have no impact on the other, so if the two tests concur, then some aspects of biased treatment assignment have been partially addressed. Expressed in terms of the knit product of two permutation groups, the structure of evidence factors is simpler and less cluttered, but at the same time more general and easier to apply in a new context. The issues are exemplified by an observational study of cigarette smoking as a cause of periodontal disease.
"The General Structure of Evidence Factors in Observational Studies." Statist. Sci. 32 (4) 514 - 530, November 2017. https://doi.org/10.1214/17-STS621