Discussion on “Instrumental Variable Estimation of the Causal Hazard Ratio,” by Linbo Wang, Eric Tchetgen Tchetgen, Torben Martinussen, and Stijn Vansteelandt

Abstract

We propose and study an augmented variant of the estimator proposed by Wang, Tchetgen Tchetgen, Martinussen, and Vansteelandt.

Rejoinder to Discussions on “Instrumental Variable Estimation of the Causal Hazard Ratio”

“…(2022) is not necessarily the most efficient one within the semiparametric family imposing the proportional hazard assumption, and the augmented estimator by Baer et al. (2022) is not necessarily the most efficient one in the nonparametric family either due to omitting the last two terms in the nonparametric efficient influence function in Proposition 1, their comparison still shed light into the end gain due to the proportional hazards assumption. For example, under correct model specifications, Baer et al.…”

“…Baer et al. (2022) focused on the statistical estimand in Theorem 2 in Wang et al. (2022) and derive the efficient influence function in a nonparametric model for this estimand.…”

“…Motivated by this, Baer et al. (2022) developed an augmented variant of Wang et al. (2022)'s estimator and showed that it is multiply robust.…”

“…Baer et al. (2022)'s results provide insights into a quest by Frandsen (2022), who asks for an analysis of the efficiency gains from imposing the proportional hazard assumption. Although the plug‐in estimator by Wang et al.…”

“…In Section 5, we comment on Baer et al. (2022)'s multiply robust estimator. We respond to the other comments in Section 6.…”