Andreas Buja, Richard A. Berk, Arun K. Kuchibhotla, Linda Zhao and Ed George’s Contribution to the Discussion of ‘Assumption-Lean Inference for Generalised Linear Model Parameters’ by Vansteelandt and Dukes

原文摘要

We thank the authors for their stimulating article. At the highest level, it agrees with our thinking as expressed in the second of two articles in Statistical Science (Buja et al., 2019). The guiding principle is to focus on quantities of interest that are meaningful statistical functionals, beyond the statistical models that motivate them, even when those models are fully misspecified. In some form, this focus has existed for decades in the semi-parametric literature. We suggest below a reinterpretation of the authors’ approach and a modification of their third principle. 1.Interpretation: The authors’ formulation can be reinterpreted as follows: The authors’ proposal (5) is the solution to the minimization problem (1), which shows that it is a projection functional based on a non-standard loss function. When the model is 1st order correct, gE[Y|A,L]=βA+ωL⁠, it holds that βw is the same for all weight functions w(L): βw ≡ β. When the model is 1st order misspecified, there exist weight functions w1(L) and w2(L) such that βw1≠βw2⁠, which is equivalent to ‘effect heterogeneity’, i.e., β | L is not the same (a.s.) for all confounder strata L. This lends itself to a ‘well-specification’ diagnostic for regression functionals, as elaborated in Buja et al. (2019). It specializes here to a diagnostic for effect heterogeneity.

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