Bootstrap-Aggregated Method-of-Moments Estimation of the Copula Correlation Parameter for Marginal Survival Inference under Dependent Censoring

Abstract: In dependently censored survival data, the usual assumption of independent censoring or an incorrect specification of the correlation between the event and censoring times can bias marginal survival inference. Likelihood-based estimation of this dependence can be numerically unstable with large variance, and practical alternatives are limited. The proposed method uses generalized method-of-moments to estimate the copula correlation parameter of a Normal, Clayton, Gumbel, or Frank copula that links exponential, Weibull, or log-normal marginal survival times. Bootstrap-aggregation of simulated annealing is employed over candidate correlation ranges to obtain stable estimates. Simulations assess accuracy and uncertainty via mean absolute error, bootstrap confidence intervals, and empirical coverage. The method is applied to a double-blind randomized clinical trial with dependent censoring from early patient dropouts, where accurate marginal survival inference is needed to estimate the effect of a treatment on patient survival.

• The paper introduces a bootstrap-aggregated GMM estimator that accurately recovers the copula correlation parameter between event and censoring times.
• It employs copula families and bagging techniques within a generalized method-of-moments framework to achieve numerically stable estimates across various survival distributions.
• Simulation studies and an application to HIV trials demonstrate the estimator’s robustness, precision, and its impact on unbiased marginal survival and treatment effect inference.

Bootstrap-Aggregated Method-of-Moments Estimation for Copula Correlation in Marginal Survival under Dependent Censoring

Background and Motivation

Classical survival analysis typically presumes independent censoring to permit unbiased estimation of marginal survival probabilities. This assumption is often violated in clinical and biomedical contexts, notably in randomized clinical trials (RCTs) where early patient withdrawal tends to correlate positively with deteriorating health and the primary event of interest. Standard approaches such as Kaplan-Meier (K-M) yield inflated survival estimates under informative (dependent) censoring, exposing the necessity for robust inference procedures that accommodate dependence between event and censoring times.

Additionally, dependent censoring frequently arises in competing risks scenarios, wherein occurrence of a competing event censors the event of interest. Parametric identifiability in such models is nontrivial due to the fundamental non-identifiability result established under nonparametric assumptions; only the observed minimum of the event and competing risk times can be used, never both. Parametric copula-based approaches have emerged to address this gap, with copulas modeling dependency structures independently from marginal distributions.

Methodological Framework

The central contribution is a bootstrap-aggregated, generalized method-of-moments (GMM) estimator for the copula correlation parameter linking event and censoring times. The procedure is agnostic to marginal distributions, accommodating exponential, Weibull, and log-normal survival times. Dependence is specified via four copula families: Normal, Clayton, Gumbel, and Frank.

Observed data are $X_i = \min(T_i, C_i)$ and $\delta_i = \mathbb{I}\{T_i \leq C_i\}$ for $n$ subjects. The joint distribution is parameterized as $H(t, c) = C(F_T(t), F_C(c))$, allowing the survival function of $X_i$ to be written as $S_X(x) = C(S_T(x), S_C(x))$. Moments of $X$ and $\delta$ (including conditional means and variances for event/censoring times) are computed to form GMM estimating equations. The theoretical moments for the bivariate normal/log-normal marginals are derived analytically, utilizing truncated normal distributions and standard identities.

The search space for marginal parameters is regularized using copula-graphic estimators, restricting optimization via empirical quartile bounds from representative values of the correlation parameter (Kendall's tau). Bootstrap aggregation (bagging) is used alongside global stochastic optimization (GenSA) to robustly select plausible correlation ranges, followed by local gradient-based refinement.

The proposed method yields a five-dimensional estimator—marginal parameters plus copula correlation—minimizing an objective function weighted by inverse bootstrap variances. Asymptotic properties are inherited from general GMM theory, with consistent and asymptotically normal estimators under standard regularity conditions.

Simulation Results

Simulations under varied copula families and marginal distributions demonstrate accurate estimation for the copula correlation parameter, with mean absolute errors (MAE) and empirical coverage probabilities (CP) near nominal levels. The method distinctly resolves none, low, moderate, and high dependence cases (Kendall's tau of 0, 0.3, 0.5, 0.8).

Comparative analysis with maximum likelihood estimation (MLE) reveals that the GMM approach is numerically stable, yielding narrower confidence intervals and lower standard errors, particularly for bivariate Weibull marginals. Notably, MLE exhibits substantial bias, instability, and inflated confidence intervals except for the Clayton copula, where both methods converge but GMM retains precision advantages.

The method supports inference beyond just copula estimation, enabling accurate recovery of marginal treatment effects in simulated RCTs, thereby correcting biases induced by dependent censoring.

Real-World Application: HIV Clinical Trial

Application to the AIDS Clinical Trials Group (ACTG) Study 175 dataset involved estimation of the correlation between primary endpoint time and early withdrawal/administrative censoring. Marginal survival curves under the estimated correlation ($\widehat{\tau} \approx 0.3$) deviated markedly from curves assuming independence or strong dependence ($\tau = 0.8$), demonstrating the critical importance of correct dependence modeling on treatment effectiveness analyses.

Regression coefficients for combination therapy versus monotherapy were substantially different under the three correlation scenarios: assuming independence overestimates efficacy, assuming high dependency underestimates, and estimation via the proposed method yields intermediate, more plausible effect sizes.

Implications and Future Directions

Practically, the method is adaptable to diverse clinical settings, facilitating unbiased marginal survival inference and valid treatment effect estimation where informative censoring or competing risks distort classical approaches. Theoretically, it strengthens the parametric identifiability paradigm for censored data, providing rigorous estimation when only the minimum of dependent survival times is observable.

The method does not require informative covariates linking event and censoring times, distinguishes itself from cross-validation or covariate-dependent methods, and achieves stable estimation for moderate sample sizes. However, precise estimation for intermediate correlations necessitates larger samples (n ∼ 1000), a trait shared by other parametric identification strategies given the inherent limitations posed by non-identifiability.

Potential extensions include generalization to multivariate survival times, incorporation of semi-parametric models, and application to high-dimensional dependence structures via generative modeling or deep learning paradigms, as alluded to recent literature (2604.04032).

Conclusion

In sum, bootstrap-aggregated GMM for copula correlation estimation enables valid marginal survival inference under dependent censoring, overcoming numerical instabilities of likelihood-based approaches and circumventing the non-identifiability barrier via parametric assumptions. The approach holds promise for broad biostatistical applications requiring robust handling of informative censoring and competing risks, with extensions to more complex dependency structures anticipated.