Some perspectives on models and their parametrisations

Two broad positions within statistics define a treatment effect, on the one hand, as a parameter of a statistical model, and on the other, as an appropriate population-level difference in outcomes or counterfactual outcomes under the different treatment regimes. I will start by presenting some simple but consequential perspectives on the two formulations, contrasting the answers under a fictitious idealisation that isolates key issues in an illuminating form.

The most common objection to model-based formulations is the possibility of misspecification, and cautious agnosticism often entails the introduction of a large number of parameters. These may be considered in different ways, depending on context, e.g. as fixed arbitrary constants, as iid random variables, or as determined by potentially relevant intrinsic variables. I will discuss some of the implications of each formulation with reference to work with Nancy Reid on the role of parametrisation in models with a misspecified nuisance component, and with David Cox on confidence sets of models.