Heather Battey’s contribution to the Discussion of ‘Parameterizing and simulating from causal models’ by Evans and Didelez

原文摘要

My brief comment relates to the inferential discussion of Section 5. The sample-space factorization of the joint probability from the marginal structural model induces a parameter-space factorization of the likelihood function, with the implication that θZX is orthogonal to (⁠θY|X*,ϕYZ|X*⁠) in the sense of, e.g. Cox and Reid (1987), and leading to the simplified asymptotic covariance matrix of Theorem 5.1. The parameter cut discussed at the beginning of Section 5 also suggests inference by a version of partial likelihood (Cox, 1975), in which the part of the likelihood function involving the nuisance parameter pX|Z (or θZX⁠) is discarded. This raises the question of whether the propensity score can be bypassed in an analogous way that partial likelihood evades the baseline hazard function in the proportional hazards model; both are in principle infinite-dimensional nuisance parameters. There may well be practical difficulties in the present context. Evans and Didelez note (second paragraph of Section 5) that Theorem 5.1 allows the propensity score model pX|Z to be misspecified thanks to the parameter cut. Recent work (Battey & Reid, 2024) has explored structure in parametric statistical models that guarantees consistency of the maximum likelihood estimator for a parameter of interest in spite of arbitrary misspecification of the nuisance part of the model, the interest parameter being common to both the true and the fitted models. The structure exploited in the paper under discussion is an example case, with pX|Z the nuisance component. The parameter cut is not a necessary condition for consistency; the latter can be ensured under one of two weaker conditions presented in Propositions 1.1 and 1.2 of Battey and Reid (2024) alongside example cases. This may shed some light on Evans and Didelez’s statement below Theorem 5.1 that “if the model is misspecified there is no guarantee that the estimator will be consistent or even close to the true value”. Although Theorem 5.1 looks to obviate the sandwich formula, I(θ*) is the Fisher information under pZXY⁠, which is assumed (first paragraph of Section 5) to be correctly specified even if pX|Z is not. The implication, I think, is that I(θ*) is not explicitly calculable when pX|Z is misspecified. I enjoyed reading the authors’ thought-provoking work.

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