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

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

“…In particular, we argue that the conjecture provided in O'Malley et al. (2022) reflects the variation dependence between the RD and the baseline risk, a parameter that is often used in practice. In comparison, the OR is variation independent of the baseline risk.…”

“…As pointed out by O'Malley et al. (2022), this typically implies effect modification on other scales such as the odds ratio. O'Malley et al.…”

“…O'Malley et al. (2022) argued that for a binary outcome, one would “naturally expect” the RD to be less likely to be homogeneous than other measures such as the odds ratio. They provide a specific example that if the probability $$E[D\mid Z=0, X=x, U=u]$$ was 0.95, then the RD $$E[D\mid Z=1,X,U] - E[D\mid Z=0,X,U]$$ is less likely to be homogeneous compared to if the baseline probability is near 0.5.…”

“…\end{equation}$$Consider the following three statements, where $$\text{RD} (X,U) = E[D\mid Z=1,X,U] - E[D\mid Z=0,X,U]$$, and $$\text{OR}(X,U) = \dfrac{E[D\mid Z=1,X,U](1-E[D\mid Z=0,X,U])}{(1-E[D\mid Z=1,X,U])E[D\mid Z=0,X,U]}:$$ (1)A change in X or U would result in a smaller change in $$\text{RD}(X,U)$$ if the baseline risk was near 0 or 1 compared to near 0.5 (a statement made by O'Malley et al. (2022)); (2)A change in X or U is equally likely to result in a change in $$\text{OR}(X,U)$$ if the baseline risk was near 0 or 1 compared to near 0.5 (our interpretation of O'Malley et al. (2022)'s comment relating to the logistic model); (3)A change in X or U is equally likely to result in a change in $$\text{RD}(X,U)$$ if the OP was near 0 or ∞ compared to near 1. We believe that statements (1) and (2) appearing reasonable is due to the fact that while $$\text{OR}(X,U)$$ is variation independent of $$E[D\mid Z=0,X,U]$$, …”

“…In Section 2, motivated by O'Malley et al. (2022), we discuss whether the risk difference (RD) is less likely to be homogeneous than other measures such as the odds ratio. In Section 3, we discuss the extensions of our identification approach to related causal parameters, including the cumulative distribution functions and the complier hazard ratio.…”