‘Does God toss logistic coins?’ and other questions that motivate regression by composition

Abstract Regression by composition is a new and flexible toolkit for building and understanding statistical models. Focusing here on regression models for a binary outcome conditional on a binary treatment and other covariates, we motivate the need for regression by composition. We do this first by exhibiting—using L’Abbé plots—the families of relationships between untreated and treated conditional outcome risks that emerge from generalized linear models for many different link functions. These are compared with the relationships (between untreated and treated risks) that arise from mechanistic sufficient component cause models, which are first principles causal models for binary outcomes. By considering mechanistic models that allow for non-monotone causal effects and by allowing sufficient causes to be associated, we expand upon similar discussions in the recent literature. We discuss conditions under which commonly used statistical models for binary data, such as logistic regression, arise from mechanistic models where the sufficient causes are associated in a particular way, as well as other situations in which the statistical models arising do not correspond to a generalized linear model but can be naturally expressed as a regression by composition model.