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We describe a modular regression framework in which covariate-dependent transformations are composed together and act on probability distributions. This framework is based on group actions of vector spaces, which are computationally convenient families of transformations that are well suited to model building and inference. We quantify covariate contributions to each group action through corresponding linear maps, and these are the only model parameters to be estimated. Algebraic features of group actions—notably, their invariant subsets—are informative about local statistical properties of the regression model. Vector space actions on affine spaces also provide a minimal geometric structure for comparing distributions, with affine transformations characterizing collapsible contrasts. In two substantive data analyses, we illustrate how unconventional models may be expressed as regressions by composition. We exhibit and extend existing nonlinear models for interpolating infant growth curves for individuals, and for producing standard population growth charts. We also use regression by composition to specify and fit a bespoke, mechanistically-motivated binary regression model for antiretroviral therapies in the treatment of HIV.
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