Appendix

This appendix contains technical details of the DOR methodology outlined in Main.

Asymptotic distribution of \hat S_D(t)

Using the functional delta method on the estimator \hat{G}_C(t) within \hat S_D(t), we can show that, as n\to\infty, \hat S_{D}(t)\to_p S_D(t) and n^{1/2}\{\hat{S}_{D}(t)-S_D(t)\}\to_d N\{0,\sigma^2_{D}(t)\}, with \sigma^2_{D}(t) consistently estimated by \hat\sigma^2_{D}(t)=n^{-1}\sum_{i=1}^n\left\{ \frac{\Delta_{2i}}{\hat G_C(X_{2i})}I(\tilde D_i>t)-\hat S_{D}(t) +\int_0^\tau \frac{\hat\zeta(u,t)}{\hat\pi_2(u)}{\rm d}\hat M_i^C(u) \right\}^2, where \hat\zeta(u,t)= n^{-1}\sum_{i=1}^n\Delta_{2i}I(\tilde D_i>t,X_{2i}\geq u)/\hat G_C(X_{2i}), \hat\pi_2(u)=n^{-1}\sum_{i=1}^nI(X_{2i}\geq u), \hat M_i^C(u)=I(X_{2i}\leq u, \Delta_{2i}=0)-\hat\Lambda_C(X_{2i}\wedge u), and \hat\Lambda_C(\cdot) is the Nelsen-Aalen estimator for the cumulative hazard function of C.

Inference on the median

Let \mathcal M(S_D)=\inf\{t: S_D(t)\leq 0.5\} denote the median operator based on the survival function S_D(\cdot). Then we can re-write the estimator for median DOR as \mathcal M(\hat S_D). To make inference, we need to quantify the randomness in \hat S_D(t) as a Gaussian process. To do so, note that the previous section suggests that the influence function of \hat S_D(t) is estimated by

\hat\eta_i(t)\equiv \frac{\Delta_{2i}}{\hat G_C(X_{2i})}I(\tilde D_i>t)-\hat S_{D}(t) +\int_0^\tau \frac{\hat\zeta(u,t)}{\hat\pi_2(u)}{\rm d}\hat M_i^C(u). This means that the asymptotic distribution of \hat S_D(t) is approximated by the Gaussian process

\underbrace{S_D(t)}_{\rm central\, term} + \underbrace{n^{-1}\hat{\boldsymbol\eta}(t)^{\rm T}\boldsymbol Z}_{\rm perturbation\, term}, where {\boldsymbol\eta}(t) = \{\hat\eta_1(t),\ldots, \hat\eta_n(t)\}^{\rm T} and \boldsymbol Z = (Z_1, \ldots, Z_n)^{\rm T} is a vector of i.i.d. standard normal random variables. Then we can approximate the asymptotic distribution of \mathcal M(\hat S_D) by that of \mathcal M\left\{\hat S_D(\cdot) + n^{-1}\hat{\boldsymbol\eta}(\cdot)^{\rm T}\boldsymbol Z\right\} under repeatedly generated \boldsymbol Z (the default is N = 1000 replicates).