Pseudo-value Based Mean Cumulative Count Regression

Technical Summary: Pseudo-value Based Mean Cumulative Count Regression

Problem Statement
In biomedical applications involving recurrent events (e.g., hospitalizations, disease exacerbations), standard analysis often focuses on event rates or intensities. However, in the presence of right-censoring and terminal events (e.g., death), the scientific question frequently shifts to understanding the cumulative disease burden over a clinically relevant time horizon. Two natural summaries for this burden are the Mean Cumulative Function (MCF), representing the expected number of events accumulated by time $\tau$, and the Area Under the MCF (AUMCF), representing the total event-free time lost.

While nonparametric methods exist for estimating these quantities and comparing treatments, direct regression modeling to assess covariate effects on the MCF and AUMCF is underdeveloped. Existing intensity-based models do not always translate directly into interpretable effects on cumulative burden, particularly when the recurrent event process is truncated by death. Furthermore, recent covariate-adjusted estimators for these functionals have focused on nonparametric treatment comparisons or linearized test statistics rather than providing a general regression framework for modeling covariate effects directly.

Methodology
The authors propose a pseudo-value-based regression approach to estimate covariate effects on the MCF and AUMCF at a fixed truncation time $\tau$.

1. Regression Targets: The method defines regression coefficients ($\beta$) not as parameters of an underlying intensity model, but as the solution to population estimating equations. Specifically, for an identity link, $\beta$ is the unique solution to $E[W \{ \theta(\tau | W) - W^\top \beta \}] = 0$, where $\theta$ represents the conditional MCF or AUMCF given covariates $W$. This separates the inferential target from the assumptions of any specific recurrent-event rate model.
2. Pseudo-values: To estimate these targets, the authors utilize subject-specific pseudo-values as outcomes.
   • Exact Jackknife: The exact pseudo-value for subject $i$ is defined as $\hat{\xi}_i(\tau) = n\hat{\theta}(\tau) - (n-1)\hat{\theta}_{(-i)}(\tau)$, where $\hat{\theta}_{(-i)}$ is the estimator computed after removing subject $i$.
   • Influence Function Approximation: To avoid the computational burden of $n$ leave-one-out recalculations, the authors approximate the pseudo-values using the empirical influence function: $\tilde{\xi}_i(\tau) = \hat{\theta}(\tau) + \hat{\phi}_i(\tau)$. The paper demonstrates that these influence-function-based pseudo-values are asymptotically equivalent to the exact jackknife values.
3. Estimation: Once pseudo-values are generated, regression is performed using standard Generalized Estimating Equation (GEE) machinery. Under an identity link and subject-agnostic working variance, this reduces to Ordinary Least Squares (OLS). Robust variance estimators are used to account for the correlation structure induced by the pseudo-values.

Key Results
The authors evaluated the proposed method through extensive simulation studies and a real-data application:

• Pseudo-value Validation: Simulations confirmed that the influence-function-based approximation is highly concordant with exact jackknife pseudo-values ($R^2 = 1.00$), with a mean absolute difference of only 0.4% of the mean absolute jackknife value.
• Estimation Accuracy and Coverage: Across various sample sizes ($N=50, 100, 200$), truncation times, and recurrent-event rate parameters, the regression estimators were unbiased. Confidence intervals constructed using model-based standard errors achieved nominal 95% coverage.
• Robustness: The method remained robust in scenarios where:
  • Death and recurrent events were dependent via a shared multiplicative frailty.
  • Covariates affected the terminal event process (death rate) rather than the recurrent event rate.
• Statistical Inference: Type I error rates were well-calibrated under the null hypothesis. Power increased with sample size and treatment effect magnitude, with MCF and AUMCF-based tests showing similar performance.
• Data Application: The method was applied to the ORATORIO clinical trial for primary progressive multiple sclerosis (PPMS). Using baseline clinical and biomarker covariates (e.g., EDSS, T2 lesion volume), the authors modeled the AUMCF of disability progression. The analysis identified baseline EDSS and 9HPT as significant prognostic factors. Notably, covariate adjustment reduced the standard error of the treatment effect estimate by 13.6% compared to the unadjusted estimate and by 8.4% compared to covariate adjustment via augmentation.

Significance and Claims
The paper claims that pseudo-value-based regression provides a "simple and interpretable framework" for modeling covariate effects on cumulative recurrent-event burden. Its primary contributions include:

1. Direct Modeling: It enables direct regression of the MCF and AUMCF, bypassing the need to interpret complex intensity model parameters in terms of cumulative burden.
2. Computational Efficiency: By utilizing influence-function-based pseudo-values, the method avoids the computational cost of repeated leave-one-out estimation while maintaining accuracy.
3. Flexibility: The approach leverages standard GEE/OLS software, making it accessible for researchers. It accommodates covariate adjustment to improve precision in randomized trials and allows for the inclusion of interaction terms for subgroup analyses.
4. Interpretability: The resulting coefficients represent the change in expected cumulative event count or event-free time lost per unit change in a covariate, offering clinically meaningful summaries that remain valid in the presence of terminal events.

The authors note a limitation regarding the assumption of non-informative censoring, suggesting that future work could extend the framework to handle covariate-dependent censoring via inverse-probability-of-censoring weights.