A Inverse Probability Censoring Weights
In the transplantation literature, survival on the liver waiting list is regularly modeled with Cox proportional hazards models adjusting for biomarkers reported at listing. Such a model can consistently estimate the parameters of the Cox proportional hazards model under the assumption of conditionally independent censoring. However, this independent censoring assumption is implausible for liver waiting list survival, because expected waiting list survival is continuously monitored using MELD scores. Throughout part I of the thesis, we use inverse probability censoring weighting (IPCW) to correct for such dependent censoring, based on an approach originally proposed by Gong and Schaubel (2013) [57]. In this technical supplement, we explain how these IPCW weights are defined in the context of the calendar-time cross-sections, defined in Chapter 3. They may be readily adapted to a “from registration” approach.
Definition of IPCW weights
A graphical summary of how the inverse probability censoring weights are defined is shown in Figure A.1. Let \(R_{i}\) denote the registration date in calendar time for patient \(i\), and \(r\) denote the time elapsed since patient \(i\)’s registration time \(R_{i}\). Each patient has a waiting list death time (\(D_{i}\)), removal or censoring time \(( C_{i} )\), and transplantation time \(T_{i},\) all defined relative to the time origin \(R_{i}\). In general, only one of these events is observable to us per patient, i.e., we observe \(X_{i} = \min( D_{i},C_{i},T_{i} )\).
Figure A.1: Figure demonstrating how inverse probability censoring weights (IPCW) are calculated for a subject \(i\). Subject \(i\) is registered at \(R_{i}\), is active at cross-section date \(CS_{k}\), and experiences an event at time \(X_{i}\). Since subject \(i\) has an active registration at \(CS_{k}\), this subject contributes an observation to the data set. To correct for dependent censoring, the spell is weighted by the inverse probability that patient \(i\) is transplanted between \(CS_{k}\) and \(X_{i}\), controlling for time-varying \(Z( r )\) (type A weight). To this end, cumulative hazards treatment hazards are estimated from registration \(R_{i}\) to \(CS_{k}\), and \(R_{i}\) to \(X_{i}\). The type A weight is the inverse probability of being transplanted before \(X_{i}\), conditional on not being transplanted up to \(CS_{k}\). It is thus strictly greater than 1, thereby unstabilized. Gong and Schaubel propose to normalize the type A weight by the conditional probability of comparable subjects in cross-section \(k\) experiencing an event between \(CS_{k}\) and \(X_{i}\), conditional on the time-frozen covariate information \(Z( S_{ik} )\).
Note that candidates for transplantation may become (temporarily) non-transplantable. To account for this, let \(A_{i}( r )\) denote whether patient \(i\) has an active registration \(r\) time units after registration, i.e., \(A_{i}( r )\) = 1 only if patient \(i\) is eligible for transplantation at calendar time \(R_{i} + r.\) In addition, updated covariate information (for instance, MELD scores) may be reported for patient \(i\). Denote with \(Z_{i}( r )\) all covariate history reported up to \(r\) time units after registration for patient \({i}\). Note that this covariate history can consist of observed covariates and other summaries of treatment eligibility history (\(A_{i}( r )\)). The key idea in Gong and Schaubel (2013) is to use a series of cross-section dates (\(CS_{1},\ldots CS_{K})\), and model the mortality hazard from each cross-section onwards for patients actively waiting at cross-section date \(CS_k\). These mortality hazard models are partly conditional, which means they adjust only for covariate history observed prior to \(CS_{k}\). The timescale used is the time elapsed since the cross-section date \(CS_{k}\), which we denote by \(\text{s}\). For notational convenience, it helps to define the time registered for patient \(i\) until cross-section \(k\) by \(S_{ik}\), i.e. \(S_{ik} = CS_{k} - R_{i}.\) Gong and Schaubel’s approach can then be represented with the following hazard model
\[ \lambda_{ik}^{D}( s ) = A_{i}( S_{ik} )\lambda_{0k}^{D}( s )\exp\left\{ \mathbf{\beta}_{\mathbf{0}}^{'}\mathbf{Z}_{\mathbf{i}}( S_{ik} ) \right\},\quad s > 0 \]
where \(A_{i}( S_{ik} )\) indicates patients are active at the cross-section, \(\lambda_{0k}^{D}( s )\) is a baseline hazard stratified by cross-section, and \(Z_{i}( S_{ik} )\) is patient \(i\)’s covariate history observed before cross-section date \(CS_{k}\).
Direct estimation of Equation A1 through Cox regression results in biased \(\widehat{\beta_{0}}\), since covariate information (e.g., MELD) reported after cross-section date \(CS_{K}\) may still affect the probability of transplantation and waiting list mortality after \(CS_{k}\). To correct for this, Gong and Schaubel propose weighing spells observed from cross section \(CS_{k}\) to time \(r\) by the inverse conditional probability of remaining on the waiting list up to time \(r\), i.e. \[W_{\text{ik}}\left( r \right) = \left\lbrack P\left( T_{i} > r \middle| T_{i} > S_{\text{ik}},Z_{i}\left( t \right),t \leq r \right) \right\rbrack^{- 1} = \left\lbrack \frac{P\left( T_{i} > r \middle| Z_{i}\left( t \right),t \leq r \right)}{P\left( T_{i} > S_{\text{ik}} \middle| Z_{i}\left( t \right),t \leq S_{\text{ik}} \right)} \right\rbrack^{- 1}.\]
Gong and Schaubel refer to this weight as the “type A” weight. Note that this weight is only defined as if the conditional probability of being transplanted between the cross-section and \(r\) is strictly larger than 0. This assumption is known as positivity. If we additionally assume that there is no unmeasured confounding of the relation between transplantation and survival, IPCW can be used to construct a “pseudo-population”, which would have been observed if transplantation had not existed. This means that under these assumptions, we can consistently estimate \(\beta_0\) through Cox regression on the weighted population.
To construct this pseudo-population, we have to estimate these IPCW weights. For this, Gong and Schaubel propose the following treatment hazard model:
\[\lambda_{i}^{T}\left( r \middle| Z_{i}\left( r \right) \right) = A_{i}\left( r \right)\lambda_{0}^{T}\left( r \right)\exp\left\{ \mathbf{\theta}_{\mathbf{0}}^{'}\mathbf{Z}_{\mathbf{i}}\left( r \right) \right\}.\]
This treatment hazard model use time since registration (\(r)\ \)as the timescale, and adjusts for time-varying covariate information (\(Z_{i}( r )\)). Using the definition of the hazard rate, one can show that the type A weight reduces to
\[ \begin{aligned} W_{ik}(r) &= \left[ \frac{P\!\bigl(T_i > r \mid Z_i(t), t \le r\bigr)} {P\!\bigl(T_i > S_{ik} \mid Z_i(t), t \le S_{ik}\bigr)} \right]^{-1}\\ &= \exp\!\Bigl[ \int_{S_{ik}}^{r} A_i(u)\,\lambda_0^T(u)\, \exp\{\theta_0' Z_i(u)\}\,du \Bigr],\\ &= \exp\!\bigl[\Lambda_i^T(r) \;-\;\Lambda_i^T\bigl(S_{ik}\bigr)\bigr]. \end{aligned} \]
where \(\Lambda_{i}^{T}\left( r \right) = \int_{0}^{r}{\lambda_{i}^{T}\left( u \middle| Z\left( u \right) \right)\text{du}}\) is the cumulative hazard of transplantation.
The type A weight allows for unbiased estimation of \(\beta_{0}\) under no unmeasured confounding and positivity. However, since \(W_{ik}( r )\) is an inverse probability weight, it is greater than or equal to 1 for all individuals and cross-sections. This can result in instabilities when conditional probabilities become small. To avoid this, Gong and Schaubel also propose to stabilize the type A weight by a partial conditional estimate of the conditional probability of being transplanted, i.e., stabilize \(W_{ik}( r )\) with
\[P\left( T_{i} > r \middle| Z_{i}\left( S_{\text{ik}} \right),t \leq r \right).\]
Gong and Schaubel attain an estimate of this probability using the following partly conditional treatment hazard model,
\[\lambda_{ik}^{T}( s ) = A_{ik}( s )\lambda_{0k}^{T}( s )\exp\left\{ \theta_{0}^{'}Z_{i}( S_{ik} ) \right\}.\]
Note that this model is partly conditional and uses time since cross-section (\(s)\) as the timescale. Gong and Schaubel confirm with simulations that empirically the type B weight results in smaller standard errors than the type A weight. Also note that IPCW weights can be calculated both for the chance of obtaining a transplantation, as well as for the chance of being removed from the waiting list. Under the assumption that waiting list removal and transplantation are conditionally independent, a joint weight can be obtained which is the product of IPCW weights for transplantation and IPCW weights for delisting. Throughout part I of the thesis, we use these joint type B weights to correct for dependent censoring by transplantation and delisting.
For the treatment and delisting hazard models, we adjust for a broad set of confounders since IPCW relies on a no-unmeasured confounding assumption. Patient factors adjusted for are sex, blood group, weight, listing country, and age at listing. Clinical variables adjusted for are whether the patient has a downgraded MELD, is simultaneously listed for a kidney, and the percentage of time a patient has been non-transplantable (too good/too bad/other). We directly adjust for MELD rather than MELD components, since Eurotransplant allocates based on MELD. Since allocation is a national affair, we also interact MELD with the patient country.