Independent censoring is often plausible only after conditioning on observed history. In treatment studies, one must additionally distinguish descriptive conditional hazards from causal survival contrasts. Inverse-probability weighting, outcome regression, and efficient influence functions connect these two problems through explicit observation and treatment mechanisms.
Conditional independent censoring
Definition. Let \(H(t)\) be observed history immediately before \(t\), and let \[ G(t\mid H)=P(C\ge t\mid H(t)) \] denote conditional censoring survival. Conditional independent censoring means that, given \(H(t)\) and being event-free and uncensored at \(t\), the censoring increment carries no additional information about the future event increment.
Proposition (inverse-censoring identity). For any integrable function \(h(T,Z)\) and fixed \(\tau\), \[ E\left[ \frac{\Delta\,1\{X\le\tau\}} {G(X-\mid H)} h(X,Z) \right] =E\{1(T\le\tau)h(T,Z)\}, \] provided \(G>0\) on the relevant support and censoring is conditionally independent.
Proof. Conditional on the full event time and observed history, \[ E\{\Delta\mid T,H\} =P(C\ge T\mid T,H) =G(T-\mid H). \] Therefore the conditional expectation of the weighted term is \[ 1\{T\le\tau\}h(T,Z) \frac{G(T-\mid H)}{G(T-\mid H)}, \] which proves the identity after taking expectation.
Weighted counting processes
Definition. Define the censoring weight \[ W_i(t) =\frac{1\{C_i\ge t\}} {G_i(t-\mid H_i)}. \] A weighted estimating equation for a hazard regression is \[ U_W(\beta) =\sum_i\int_0^\tau W_i(t)\{Z_i(t)-\bar Z_W(\beta,t)\} \,\mathrm{d}N_i(t), \] where \[ \bar Z_W(\beta,t) =\frac{\sum_iW_i(t)Y_i(t)e^{\beta^\top Z_i(t)}Z_i(t)} {\sum_iW_i(t)Y_i(t)e^{\beta^\top Z_i(t)}}. \]
Proposition. If \(G\) is correct and positive, weighting restores the full-data risk-set moment: \[ E\{W(t)1(C\ge t)Y^T(t)f(H(t))\} =E\{Y^T(t)f(H(t))\}, \] where \(Y^T(t)=1\{T\ge t\}\).
Proof. Conditional on full event and covariate history, \[ E\left[ \frac{1(C\ge t)}{G(t-\mid H)} \middle|T,H \right]=1. \] Multiply by the full-data predictable quantity \(Y^T(t)f(H(t))\) and take expectations.
Estimated weights add nuisance-estimation terms to the influence function. Extreme weights reveal practical positivity failure and inflate variance; truncating them changes the estimand and introduces bias. Both the untruncated and sensitivity analyses should be reported.
Potential event times and causal assumptions
Definition. For treatment \(A\in\{0,1\}\), let \(T^a\) be the event time under intervention \(A=a\). Causal survival and restricted-mean targets are \[ S_a(t)=P(T^a>t), \] \[ \Delta_S(t)=S_1(t)-S_0(t), \] \[ \mu_a(\tau)=E(T^a\wedge\tau) =\int_0^\tau S_a(t)\,\mathrm{d}t, \qquad \Delta_{\mathrm{RMST}}(\tau)=\mu_1(\tau)-\mu_0(\tau). \]
Identification assumptions.
- Consistency: \(T=T^A\).
- Conditional exchangeability: \((T^0,T^1)\perp A\mid L\).
- Treatment positivity: \(0<P(A=a\mid L)<1\) almost surely.
- Conditional independent censoring: \(T\perp C\mid(A,L)\), or its longitudinal-history analogue.
- Censoring positivity: \(P(C\ge t\mid A,L)>0\) wherever \(P(T\ge t\mid A,L)>0\).
Theorem (g-formula identification). Under consistency and conditional exchangeability, \[ S_a(t) =E\{P(T>t\mid A=a,L)\}. \] With conditionally independent censoring, the inner conditional survival is identified from observed censored data.
Proof. By iterated expectation and exchangeability, \[ P(T^a>t) =E\{P(T^a>t\mid L)\} =E\{P(T^a>t\mid A=a,L)\}. \] Consistency replaces \(T^a\) by \(T\) among those with \(A=a\). Conditional independent censoring identifies the resulting conditional event-time law through risk-set or likelihood methods.
Inverse-probability identification
Theorem. Let \[ e_a(L)=P(A=a\mid L),\qquad G_a(t\mid L)=P(C\ge t\mid A=a,L). \] Under the identification assumptions, \[ S_a(t) =E\left[ \frac{1\{A=a\}1\{X>t\}} {e_a(L)G_a(t\mid L)} \right]. \]
Proof. Conditional on \((A,L,T)\), \[ E\left[ \frac{1(C>t)}{G_a(t\mid L)} \middle|A=a,L,T \right]=1 \] on \(\{T>t\}\). Thus the expectation reduces to \[ E\left[ \frac{1(A=a)1(T>t)}{e_a(L)} \right]. \] Conditioning on \(L,T^a\), exchangeability and consistency give \[ E\{1(A=a)/e_a(L)\mid L,T^a\}=1, \] leaving \(P(T^a>t)\).
Augmented estimation
Definition. Let \[ m_a(t,L)=P(T>t\mid A=a,L). \] Ignoring the additional longitudinal augmentation needed when censoring occurs before \(t\), the treatment-augmented full-data influence form is \[ \phi_{a,t}^{\mathrm{full}} =\frac{1(A=a)}{e_a(L)} \{1(T>t)-m_a(t,L)\} +m_a(t,L)-S_a(t). \] Observed-data efficient influence functions replace the unavailable \(1(T>t)\) component by a censoring-martingale augmentation.
Proposition (double robustness in full data). The estimating function obtained by substituting estimators \(\widehat e_a,\widehat m_a\) into \(\phi_{a,t}^{\mathrm{full}}\) has mean zero if either \(e_a\) or \(m_a\) is correctly specified.
Proof. If \(m_a\) is correct, then \[ E\{1(T>t)-m_a(t,L)\mid A=a,L\}=0, \] so the weighted residual term vanishes regardless of \(\widehat e_a\), while the remaining mean is zero. If \(e_a\) is correct, conditional expectation of the weighted outcome term equals \(E\{m_a^{0}(t,L)\}\), where \(m_a^{0}\) is the true outcome regression; the subtraction of the possibly misspecified \(\widehat m_a\) and its plug-in mean cancels algebraically.
With censoring, genuine double robustness usually means consistency under a specified union of treatment, censoring, and outcome-model conditions. The exact union depends on the influence function; “doubly robust” should never be stated without naming which nuisance components may be wrong.
Cross-fitting and rate conditions
Theorem (orthogonal one-step template). Let \(\eta\) collect treatment, censoring, and outcome nuisances, and let \(\phi(O;\psi,\eta)\) be a Neyman-orthogonal influence function for a scalar survival target \(\psi\). Estimate \(\eta\) on training folds and solve \[ \mathbb P_n\phi(O;\widehat\psi,\widehat\eta_{-k(i)})=0. \] If positivity and moment conditions hold and the relevant nuisance errors satisfy a product rate \[ \|\widehat\eta_1-\eta_{01}\| \|\widehat\eta_2-\eta_{02}\| =o_P(n^{-1/2}), \] then \[ \sqrt n(\widehat\psi-\psi_0) =\frac1{\sqrt n}\sum_{i=1}^n \phi(O_i;\psi_0,\eta_0)+o_P(1). \]
Proof outline. Expand the cross-fitted estimating equation at \((\psi_0,\eta_0)\). Orthogonality removes the first derivative in nuisance directions, leaving a second-order product remainder. Sample splitting makes the empirical-process term conditionally centered without requiring the fitted nuisance class itself to be Donsker. The product-rate condition makes the remainder \(o_P(n^{-1/2})\), and the ordinary CLT applies to the true influence function.
Hazard ratios and causal interpretation
Remark. Even in a randomized trial, a hazard ratio at time \(t\) compares treatment groups among different sets of survivors. If treatment affects earlier survival, conditioning on survival can induce selection, so a marginal causal hazard ratio is difficult to interpret as an individual-level treatment effect. Survival differences, risk ratios at a fixed horizon, and RMST differences are often more transparent causal targets.
Competing events under interventions
For competing risks, the total effect on \[ F_k^a(t)=P(T^a\le t,J^a=k) \] is identified under ordinary treatment and censoring assumptions. A “direct effect eliminating the competing event” invokes a different intervention and generally requires cross-world or separable-effect assumptions. Treating the competing event as censoring silently changes from an observable total-effect target to a hypothetical latent event-time target that is not nonparametrically identified.
The broader potential-outcome, longitudinal-treatment, and efficient-influence-function theory is developed in causal inference.