Instrumental Variable Estimation of the Causal Hazard Ratio

Abstract

Cox's proportional hazards model is one of the most popular statistical models to evaluate associations of exposure with a censored failure time outcome. When confounding factors are not fully observed, the exposure hazard ratio estimated using a Cox model is subject to unmeasured confounding bias. To address this, we propose a novel approach for the identification and estimation of the causal hazard ratio in the presence of unmeasured confounding factors. Our approach is based on a binary instrumental variable, and an additional no‐interaction assumption in a first‐stage regression of the treatment on the IV and unmeasured confounders. We propose, to the best of our knowledge, the first consistent estimator of the (population) causal hazard ratio within an instrumental variable framework. A version of our estimator admits a closed‐form representation. We derive the asymptotic distribution of our estimator and provide a consistent estimator for its asymptotic variance. Our approach is illustrated via simulation studies and a data application.

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

“…We have doubts about the appropriateness of A5 in Wang et al. (2022), which we restate: $$$$\begin{eqnarray} \delta ^{D}(X,U) &=& E[D \mid Z=1, X,U] - E[D \mid Z=0, X,U]\nonumber \\ &=& E[D \mid Z=1, X] - E[D \mid Z=0, X]\nonumber \\ &=& \delta ^{D}(X), \end{eqnarray}$$$$As noted in Remark 3 of Wang et al. (2022), Equation (2) implies that the treatment selection mechanism cannot follow the logistic or other nonlinear form, except in the case that $$E[D \mid Z, X,U]$$ is independent of U , which would imply that U is not a confounder.…”

“…The causal model assumed in Wang et al. (2022) is given by the marginal structural model $$$$\begin{equation} \lambda _{d}^{T}(t)=\lambda _{0}^{T}(t)\exp (\psi d), \end{equation}$$$$where $$\lambda _{d}^{T}(t)=-[S_{d}^{T}(t)]^{\prime }/S_{d}^{T}(t)$$ is the potential hazard function and $$S_{d}^{T}(t)$$ the potential survival function at follow‐up time t . The target parameter is ψ, the log causal hazard ratio.…”

“…(2022), which we restate: $$$$\begin{eqnarray} \delta ^{D}(X,U) &=& E[D \mid Z=1, X,U] - E[D \mid Z=0, X,U]\nonumber \\ &=& E[D \mid Z=1, X] - E[D \mid Z=0, X]\nonumber \\ &=& \delta ^{D}(X), \end{eqnarray}$$$$As noted in Remark 3 of Wang et al. (2022), Equation (2) implies that the treatment selection mechanism cannot follow the logistic or other nonlinear form, except in the case that $$E[D \mid Z, X,U]$$ is independent of U , which would imply that U is not a confounder. (Tangentially, we note that in the highly simplified case in which there are no measured covariates ($$X=\emptyset$$) and U is binary, the logistic regression model could satisfy Equation (2) as one can solve for the model parameters so that $$E[D \mid Z=1,U=1] - E[D \mid Z=0,U=1] = E[D \mid Z=1,U=0] - E[D \mid Z=0,U=0]$$.…”

“…In Wang et al. (2022), the inclusion of measured covariates is accommodated in their time‐to‐event model for binary treatments by using inverse weighting by the propensity of the binary instrument; that is, $$f(Z \mid X)$$. If there are no measured covariates the weight function they employ, $$w_0(Z,X)=(2Z-1)/\lbrace f(Z|X)\delta ^D(X)\rbrace$$, is proportional to $$w_0(Z)=(2Z-1)/f(Z)$$.…”

Instrumental variable estimation of complier causal treatment effect with interval‐censored data

“…Given the noncollapsibility issue of hazard ratio and odds ratio (Martinussen and Vansteelandt, 2013; Wang et al. , 2018), we generally expect that β 2 is not the same as the marginal local causal treatment effect that is only conditional on the complier subgroup. In addition, in the one‐sided compliance case, where subjects with $$A=0$$ have no access to treatment (i.e., $$P(D^0=0\mid {\bm {X}})=1$$), we can show that for $$d=1$$ or 0, $$ P(T^d > t \mid D=1, \bm { X})=\exp \lbrace -G_2[\Lambda (t) \exp (\beta _2 d + \gamma _2^{\top } \bm { X})]\rbrace$$.…”