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arxiv: 2606.24024 · v1 · pith:O7BANEJDnew · submitted 2026-06-23 · 📊 stat.ME · stat.AP

Pseudo-value Based Mean Cumulative Count Regression

Zachary R. McCaw , Alex Ocampo , Enrico Giudice , FengQi Song , Jessica Gronsbell This is my paper

Pith reviewed 2026-06-25 23:25 UTC · model grok-4.3

classification 📊 stat.ME stat.AP
keywords pseudo-valuesmean cumulative functionrecurrent eventsregression analysisgeneralized estimating equationscovariate adjustment
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The pith

Pseudo-value regression estimates covariate effects on recurrent event accumulation using standard tools.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a method that converts the mean cumulative function and its area into pseudo-values derived from influence functions. These pseudo-values then serve as outcomes in a regression model fitted by generalized estimating equations or ordinary least squares. This setup allows direct estimation of how covariates influence the total burden of recurrent events up to a fixed time, while accounting for censoring and terminal events. Simulations confirm the approach's performance, and it is applied to trial data on multiple sclerosis treatment.

Core claim

Influence-function-based pseudo-values can be used as regression outcomes to estimate covariate effects on the mean cumulative function and the area under the mean cumulative function at a fixed truncation time, with estimation performed via standard generalized estimating equation machinery or ordinary least squares under an identity link.

What carries the argument

Influence-function-based pseudo-values constructed for the MCF and AUMCF, used directly as outcomes in a regression model.

If this is right

  • Standard regression software can be applied to recurrent event data without custom implementations.
  • Covariate effects on cumulative event burden can be estimated at any fixed time horizon.
  • Both the mean cumulative count and its integrated area can be modeled in the same framework.
  • Type I error and coverage properties hold under the assumed censoring conditions in simulations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method may extend naturally to time-varying covariates if the pseudo-value construction can be adapted.
  • Similar pseudo-value approaches could apply to other recurrent event summaries beyond the MCF.
  • Applications in other fields with recurrent events, such as reliability engineering, become more accessible with standard tools.

Load-bearing premise

The influence function used to create the pseudo-values assumes censoring and terminal events are independent of the event process conditional on the covariates.

What would settle it

Observing bias in the regression coefficients when the censoring mechanism depends on unobserved factors that also affect the event rate.

Figures

Figures reproduced from arXiv: 2606.24024 by Alex Ocampo, Enrico Giudice, Fengqi Song, Jessica Gronsbell, Zachary R. McCaw.

Figure 1
Figure 1. Figure 1: Jackknife and influence function-based pseudo-values are concordant. Across various sample sizes and truncation times, pseudo-values were calculated either exactly, via the leave-one-subject-out jackknife, or approximately, via the influence function. by the population estimating equations rather than on recovery of the rate-model parameters themselves. The target parameters were estimated via linear regre… view at source ↗
Figure 2
Figure 2. Figure 2: Estimation of target regression parameters is unbiased. Estimation error is shown for the MCF regression coefficients β and AUMCF regression coefficients γ across sample sizes, truncation times, and recurrent-event rate parameters. Rate parameters were varied one at a time while the remaining parameters were fixed at zero. 3.2.3 Frailty study To investigate the impact of dependence between the death and re… view at source ↗
Figure 3
Figure 3. Figure 3: Confidence intervals exhibit nominal coverage. Empirical coverage of 95% confi￾dence intervals is shown for the MCF regression coefficients β and AUMCF regression coefficients γ across sample sizes, truncation times, and recurrent-event rate parameters. Confidence intervals were constructed using model-based standard errors. 3.2.5 Statistical inference For type I error simulations, data were generated in t… view at source ↗
Figure 4
Figure 4. Figure 4: Estimation accuracy and coverage are robust to dependence between death and recurrent events. Bias and empirical 95% confidence interval coverage are shown for the treatment effects on the MCF and AUMCF across sample sizes, truncation times, and shared-frailty variances. Larger frailty variances induce stronger dependence between the death and recurrent￾event rates. 12 [PITH_FULL_IMAGE:figures/full_fig_p0… view at source ↗
Figure 5
Figure 5. Figure 5: Type I error is calibrated under the null. Uniform quantile-quantile plots are shown for Wald-test p-values under the null hypotheses of no treatment effect on the MCF and AUMCF regression targets. Simulations were conducted in the absence of treatment effects on the recurrent-event rate, either directly or through interaction with the baseline covariate. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Power increases with sample size and treatment effect magnitude. Empirical power is shown for Wald tests of the treatment effects on the MCF and AUMCF regression targets across sample sizes, truncation times, and recurrent-event treatment-effect parameters. Power was defined as the proportion of simulations in which the null hypothesis of no treatment effect was rejected. 14 [PITH_FULL_IMAGE:figures/full_… view at source ↗
Figure 7
Figure 7. Figure 7: Forest plot of pseudo-value based regression estimate for the covariate effects. The covariates are standardized before putting them into the model, that is, the effects in the plot correspond to one standard deviation increase in the unit of the covariate. The label ∗∗∗ corresponds to a p-value < 0.001,∗∗ < 0.01, and ∗ < 0.05. The covariate effects are visualized in a forest plot in [PITH_FULL_IMAGE:figu… view at source ↗
read the original abstract

The mean cumulative function (MCF) summarizes how events accumulate over time for a recurrent or multi-component endpoint. The MCF, and its integral over a given time horizon, the area under the MCF (AUMCF), provide interpretable summaries of recurrent-event burden in the presence of right-censoring and terminal events. Existing approaches for these estimands have focused primarily on nonparametric treatment comparisons, covariate-adjusted augmentation, and linearized test statistics. Herein, we propose a pseudo-value-based regression approach for estimating covariate effects on the MCF and AUMCF at a fixed truncation time. The proposed method uses influence-function-based pseudo-values as regression outcomes, allowing estimation with standard generalized estimating equation machinery and, under an identity link, ordinary least squares. Through simulation studies, we evaluate estimation accuracy, confidence interval coverage, type I error control, and power across a range of recurrent-event settings. We demonstrate the utility of the proposed covariate adjustment procedure through an application to the ORATORIO clinical trial, evaluating the safety and efficacy of ocrelizumab for the treatment of primary progressive multiple sclerosis. Overall, pseudo-value-based regression provides a simple and interpretable framework for modeling covariate effects on cumulative recurrent-event burden over time.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a pseudo-value-based regression approach for estimating covariate effects on the mean cumulative function (MCF) and area under the MCF (AUMCF) at a fixed truncation time. Influence-function-based pseudo-values are constructed from the nonparametric MCF estimator and used as regression outcomes in GEE machinery (or OLS under identity link), with performance evaluated in simulations for accuracy, coverage, type I error, and power, plus an application to the ORATORIO trial.

Significance. If the central consistency result holds, the approach supplies a straightforward, software-friendly route to covariate-adjusted inference on recurrent-event burden summaries. Credit is due for the simulation battery (accuracy, coverage, type I error, power) and the real-data demonstration on a phase-III multiple-sclerosis trial; these elements make the practical contribution concrete.

major comments (2)
  1. [Pseudo-value construction and consistency argument] The consistency claim for the OLS/GEE estimator rests on E[pseudo-value_i | X_i] equaling the true conditional MCF given X_i. This equality holds only when the influence function is derived under censoring and terminal-event mechanisms that are independent of the recurrent process conditional on the covariates entering the regression (see abstract description of pseudo-value construction and its direct use as outcomes). The manuscript does not state this conditional-independence requirement explicitly or provide a proof/relaxation.
  2. [Simulation studies section] Simulation design evaluates performance only under data-generating processes that satisfy the independent-censoring assumption implicit in the influence-function derivation. Scenarios with covariate-dependent censoring or omitted censoring covariates would directly test whether bias appears in the regression coefficients when the weakest assumption is violated.
minor comments (2)
  1. [Notation and definitions] Define AUMCF explicitly as the integral of the MCF up to the fixed truncation time and state whether the pseudo-value construction is applied directly to the integrated functional or obtained by integrating the MCF pseudo-values.
  2. [Simulation result tables] In tables reporting simulation results, add a column or footnote indicating the link function and whether GEE or OLS was used for each row.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major comment below, indicating where we will revise the manuscript to improve clarity and completeness.

read point-by-point responses

  1. Referee: [Pseudo-value construction and consistency argument] The consistency claim for the OLS/GEE estimator rests on E[pseudo-value_i | X_i] equaling the true conditional MCF given X_i. This equality holds only when the influence function is derived under censoring and terminal-event mechanisms that are independent of the recurrent process conditional on the covariates entering the regression (see abstract description of pseudo-value construction and its direct use as outcomes). The manuscript does not state this conditional-independence requirement explicitly or provide a proof/relaxation.

    Authors: We agree that the conditional independence of censoring and terminal events given the covariates must be stated explicitly for the consistency result to hold. In the revised manuscript we will add a dedicated paragraph in the Methods section stating this assumption and its role in ensuring E[pseudo-value_i | X_i] equals the target conditional MCF. While we do not provide a self-contained proof (as the result follows from standard properties of influence-function pseudo-values under independent censoring, as in the cited pseudo-value literature), we will include a brief justification with references to the relevant theory and note that the GEE/OLS step inherits consistency from the unbiased pseudo-values. revision: partial

  2. Referee: [Simulation studies section] Simulation design evaluates performance only under data-generating processes that satisfy the independent-censoring assumption implicit in the influence-function derivation. Scenarios with covariate-dependent censoring or omitted censoring covariates would directly test whether bias appears in the regression coefficients when the weakest assumption is violated.

    Authors: The referee correctly identifies that our current simulations maintain the independent-censoring assumption. We will add a new simulation scenario in which censoring depends on covariates (and an additional scenario with an omitted censoring covariate) to quantify the resulting bias in the regression coefficients. These results will be reported in a revised Table or supplementary figure, together with a short discussion of when the method remains reliable versus when the assumption is critical. revision: yes

Circularity Check

0 steps flagged

No circularity; standard application of influence-function pseudo-values to GEE/OLS

full rationale

The paper derives its estimator by first obtaining influence-function pseudo-values from the nonparametric MCF estimator at a fixed truncation time (standard construction under independent censoring), then treating those pseudo-values as outcomes in off-the-shelf GEE or OLS regression. No equation reduces the target regression parameters to a fitted quantity by construction, no self-citation chain is load-bearing for the central claim, and the method does not rename or smuggle an ansatz. The derivation is self-contained against external benchmarks of influence-function theory and GEE consistency.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of influence-function pseudo-values for the MCF under right-censoring and terminal events; no free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Influence function for the MCF yields unbiased pseudo-values when censoring is independent conditional on covariates.
    Invoked by the abstract's description of pseudo-value construction and direct regression use.

pith-pipeline@v0.9.1-grok · 5749 in / 1310 out tokens · 14511 ms · 2026-06-25T23:25:52.146591+00:00 · methodology

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Reference graph

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