An elaborate theory of predictions of a causal hypothesis consists of several falsifiable statements derived from the causal hypothesis. Statistical tests for the various pieces of the elaborate theory help to clarify how much the causal hypothesis is corroborated. In practice, the degree of corroboration of the causal hypothesis has been assessed by a verbal description of which of the several tests provides evidence for which of the several predictions. This verbal approach can miss quantitative patterns. In this paper, we develop a quantitative approach. We first decompose these various tests of the predictions into independent factors with different sources of potential biases. Support for the causal hypothesis is enhanced when many of these evidence factors support the predictions. A sensitivity analysis is used to assess the potential bias that could make the finding of the tests spurious. Along with this multiparameter sensitivity analysis, we consider the partial conjunctions of the tests. These partial conjunctions quantify the evidence supporting various fractions of the collection of predictions. A partial conjunction test involves combining tests of the components in the partial conjunction. We find the asymptotically optimal combination of tests in the context of a sensitivity analysis. Our analysis of an elaborate theory of a causal hypothesis controls for the familywise error rate.
"Assessment of the extent of corroboration of an elaborate theory of a causal hypothesis using partial conjunctions of evidence factors." Ann. Statist. 48 (6) 3283 - 3311, December 2020. https://doi.org/10.1214/19-AOS1929