INFLUENCE FUNCTION BASED COVARIANCE
MATRIX FOR SUMMARY STATISTICS OF
SURVIVAL DATA

Szilárd Nemes

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Szilárd Nemes. INFLUENCE FUNCTION BASED COVARIANCE MATRIX FOR SUMMARY STATISTICS OF SURVIVAL DATA. 2026. hal-05510734

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INFLUENCE FUNCTION BASED COVARIANCE MATRIX FOR
SUMMARY STATISTICS OF SURVIVAL DATA

Szilárd Nemes

Institute of Clinical Sciences

University of Gothenburg

Gothenburg

szilard.nemes@outlook.com

ABSTRACT

When multiple survival summaries are reported from the same clinical trial, joint uncertainty and agreement between estimators depend on their within-trial covariance structure. This note revisits covariance estimation for a vector comprising a fixed-time survival difference, an RMST difference, and a Cox log-hazard ratio under proportional hazards. Using asymptotic linearity and influence functions, we derive plug-in covariance estimators as empirical second moments of estimated influence values and express them in standard event-time (Greenwood-type) form. We then map these expressions to the stochastic-integral representation used by Ristl et al. and show that the resulting plug-in estimators coincide after matching notation and conventions, with remaining differences attributable to implementation choices such as tie handling and numerical integration of RMST.

**Keywords** Influence functions; Covariance estimation; Restricted mean survival time; Focused information criterion

# 1 Introduction

Time-to-event endpoints are central in many clinical trials, but the summary used to report a treatment effect is rarely unique. Alongside the Cox hazard ratio, practitioners often report fixed-time survival differences $S(t)$, restricted mean survival time (RMST) contrasts $\mu(\tau)$, or several characteristics simultaneously. When multiple summaries are reported from the same dataset, it is useful to quantify their agreement—for example, whether different summaries tend to support the same qualitative conclusion, and how strongly they co-move across repeated samples. This requires estimators of the joint large-sample variance-covariance structure (or equivalently, correlations) for a vector of estimators.

There are multiple principled ways to obtain covariance estimators for the same targets. Ristl et al. Ristl et al. [2024] derive a general framework for simultaneous inference on multiple survival characteristics, using a counting-process/ martingale representation to obtain asymptotic linear expansions and corresponding plug-in covariance estimators. In parallel, the general machinery of *M*-estimation provides a direct route to the same goal: once an estimator admits an influence-function representation, the asymptotic covariance matrix can be estimated by empirical second moments of estimated subject-level influence contributions. In practice, both approaches can be implemented with standard survival- analysis ingredients and yield consistent covariance estimators—yet they may differ in notation, in the intermediate objects they compute, and (depending on implementation details) in finite-sample behavior.

This technical note is written primarily as a learning exercise. The aim is (i) to re-derive covariance estimators for common survival summaries using a straightforward influence-function/*M*-estimation route, (ii) to restate the construction of Ristl et al. in comparable notation, and (iii) to compare the resulting covariance estimators both theoretically (by mapping the two linearizations) and numerically via simulation. The simulations are designed to highlight when the two approaches agree closely and when practical choices (e.g., tie handling, step-function conventions, or the specific survival estimator used) can produce visible differences in finite samples.
## 2 Notation, data structure, and estimands

We consider i.i.d. right-censored time-to-event data, motivated by two-arm clinical trials but applicable to general two-sample settings. Let $T \ge 0$ denote the event time of interest and $C \ge 0$ a right-censoring time. For each subject we observe the follow-up time $\bar{X} = \min(T, C)$ and the event indicator $\delta = I(T \le C)$, together with a treatment/exposure indicator $Z$ (for a two-arm trial, $Z \in \{0, 1\}$). The observed data for subject $i$ are $O_i = (X_i, \delta_i, Z_i)$, $i = 1, \dots, n$. Write $n_a = \sum_{i=1}^n I(Z_i = a)$ and assume $\Pr(Z = 1) = p \in (0, 1)$ with $n_a/n \to p_a$.

Throughout we assume independent censoring within exposure strata, $T \perp C | Z$, and positivity of follow-up at the horizons of interest: $\Pr(C \ge u | Z = a) > 0$ for all $u \in [0, \max(t, \tau)]$ and $a \in \{0, 1\}$. All estimands are defined at prespecified horizons: a fixed time point $t$ and an RMST horizon $\tau$.

Within arm $a \in \{0, 1\}$ we define the survival function $S_a(t) = \Pr(T > t | Z = a)$ and the restricted mean survival time (RMST) $\mu_a(\tau) = \mathbb{E}\{\min(T, \tau) | Z = a\} = \int_0^\tau S_a(u) du$. Treatment effects are summarized by contrasts $\Delta S(t) = S_1(t) - S_0(t)$ and $\Delta \mu(\tau) = \mu_1(\tau) - \mu_0(\tau)$.

In addition, we consider the Cox proportional hazards parameter $\beta$ from $\lambda(t | Z) = \lambda_0(t) \exp(\beta Z)$, with hazard ratio $HR = \exp(\beta)$. For the Cox component we assume the proportional hazards model is correctly specified, so that $\beta$ is the usual log-hazard ratio parameter. The Cox partial likelihood is computed using a prespecified tie handling rule (e.g., Breslow).

For joint inference we focus on

$$ \theta(t, \tau) = (\Delta S(t), \Delta \mu(\tau), \beta)^{\top}. \quad (1) $$

Let $\hat{S}_a(t)$ denote the Kaplan-Meier estimator within arm $a$ and define the plug-in RMST estimator $\hat{\mu}_a(\tau) = \int_0^\tau \hat{S}_a(u) du$. The empirical contrasts are $\Delta \hat{S}(t) = \hat{S}_1(t) - \hat{S}_0(t)$ and $\Delta \hat{\mu}(\tau) = \hat{\mu}_1(\tau) - \hat{\mu}_0(\tau)$. Let $\hat{\beta}$ denote the Cox partial-likelihood estimator and $\widehat{HR} = \exp(\hat{\beta})$.

Our primary target is the within-trial covariance matrix of $\hat{\theta}(t, \tau) = (\Delta \hat{S}(t), \Delta \hat{\mu}(\tau), \hat{\beta})^{\top}$, including the cross-covariances that govern joint uncertainty.

## 3 Asymptotic linearity and influence-function framework

Let $\theta = \theta(P)$ denote a (possibly vector-valued) parameter defined as a functional of the data-generating distribution $P$. For a regular asymptotically linear (RAL) estimator, the influence function $\varphi_\theta$ is the mean-zero function that characterizes the canonical first-order behavior, and can be defined via the appropriate Gâteaux derivative of $\theta(\cdot)$ at $P$; see, e.g., Reid [1981], Reid and Crépeau [1985].

Specifically, an estimator $\hat{\theta}$ is said to be asymptotically linear if it admits the representation

$$ \sqrt{n}(\hat{\theta} - \theta) = \frac{1}{\sqrt{n}} \sum_{i=1}^{n} \varphi_{\theta}(O_i) + o_p(1), \quad (2) $$

where $\mathbb{E}\{\varphi_\theta(O)\} = 0$ and $\mathbb{E}\{\varphi_\theta(O)^2\} < \infty$. For a vector $\hat{\theta} = (\hat{\theta}_1, \dots, \hat{\theta}_K)^\top$ with componentwise representations of the form (2), the multivariate central limit theorem implies

$$ \sqrt{n}(\hat{\theta} - \theta) \Rightarrow N(0, \Sigma), \quad \Sigma_{kl} = \mathbb{E}\{\varphi_{\theta_k}(O) \varphi_{\theta_l}(O)\}. \quad (3) $$

A consistent plug-in estimator is obtained by empirical averaging of estimated influence values,

$$ \hat{\Sigma} = \frac{1}{n} \sum_{i=1}^{n} \hat{\varphi}(O_i) \hat{\varphi}(O_i)^{\top}, \quad (4) $$

where $\hat{\varphi}(O_i) = (\hat{\varphi}_{\theta_1}(O_i), \dots, \hat{\varphi}_{\theta_K}(O_i))^\top$.

Equation (3) is the organizing principle for the remainder of the paper. Once influence functions are identified for each component estimand (Kaplan-Meier survival, RMST, and the Cox log-hazard ratio), the full asymptotic covariance matrix follows from inner products of influence functions. For estimands defined by contrasts of arm-specific functionals, the corresponding influence functions inherit the usual $I(Z = a)/p_a$ weighting from the two-sample decomposition.
In practice, influence functions for Kaplan-Meier/Nelson-Aalen functionals follow from standard martingale representations under independent censoring, influence functions for Cox regression follow from the score and information (yielding the usual model-based form under proportional hazards), and smooth transformations (such as RMST) are handled by the functional delta method; see, e.g., Jullum and Hjort [2019] for a concise treatment in semiparametric survival settings.

# 4 Asymptotic covariance structure

This section specifies the asymptotic covariance structure for the joint vector $\theta(t, \tau) = (\Delta S(t), \Delta \mu(\tau), \beta)^{\top}$ using the influence-function linearizations summarized in Section 3. The development follows standard martingale arguments for censored survival functionals and proportional hazards regression [Reid, 1981, Reid and Crépeau, 1985], combined with the generic joint-CLT template for asymptotically linear estimators [Jullum and Hjort, 2019].

## 4.1 Within-arm building blocks: $\hat{S}_a(t)$ and $\hat{\mu}_a(\tau)$

Fix an arm $a \in \{0, 1\}$. Let $S_a$ denote the arm-specific survival function and define the RMST $\mu_a(\tau) = \int_0^\tau S_a(u) du$. The corresponding estimators $\hat{S}_a(t)$ (Kaplan-Meier) and $\hat{\mu}_a(\tau) = \int_0^\tau \hat{S}_a(u) du$ admit influence-function representations based on the standard martingale formulation for right-censored data [Reid, 1981]. Although $\hat{S}_a$ is the product-limit estimator, its first-order linearization is conveniently expressed in terms of Nelson-Aalen increments [Reid, 1981].

Write $Y_a(u) = \Pr(X \ge u | Z = a)$ for the arm-specific at-risk probability and $\lambda_a(u)$ for the event hazard in arm a. Let

$$R_a(u; \tau) = \int_u^\tau S_a(v) dv \qquad (5)$$

denote the RMST tail area. The within-arm $\sqrt{n_a}$-scaled asymptotic variances are

$$\mathrm{AVar}(\sqrt{n_a}\{\hat{S}_a(t) - S_a(t)\}) = S_a(t)^2 \int_0^t \frac{\lambda_a(u)}{Y_a(u)} du, \qquad (6)$$

$$\mathrm{AVar}(\sqrt{n_a}\{\hat{\mu}_a(\tau) - \mu_a(\tau)\}) = \int_0^\tau R_a(u; \tau)^2 \frac{\lambda_a(u)}{Y_a(u)} du. \qquad (7)$$

A single application of the martingale isometry yields the asymptotic cross-covariance between $\hat{S}_a(t)$ and $\hat{\mu}_a(\tau)$:

$$\mathrm{ACov}(\sqrt{n_a}\{\hat{S}_a(t) - S_a(t)\}, \sqrt{n_a}\{\hat{\mu}_a(\tau) - \mu_a(\tau)\}) = S_a(t) \int_0^{t \wedge \tau} R_a(u; \tau) \frac{\lambda_a(u)}{Y_a(u)} du, \qquad (8)$$

where $t \wedge \tau = \min\{t, \tau\}$. For later use, let $\Sigma_a(t, \tau)$ denote the $2 \times 2$ asymptotic covariance matrix of $(\hat{S}_a(t), \hat{\mu}_a(\tau))^\top$ under $\sqrt{n_a}$ scaling,

$$\Sigma_a(t, \tau) = \begin{pmatrix} (6) & (8) \\ \cdot & (7) \end{pmatrix}. \qquad (9)$$

## 4.2 Treatment contrasts: scaling and armwise combination

Define contrasts $\Delta S(t) = S_1(t) - S_0(t)$ and $\Delta \mu(\tau) = \mu_1(\tau) - \mu_0(\tau)$ with estimators $\Delta \hat{S}(t) = \hat{S}_1(t) - \hat{S}_0(t)$ and $\Delta \hat{\mu}(\tau) = \hat{\mu}_1(\tau) - \hat{\mu}_0(\tau)$. Under independent sampling across arms and a fixed allocation proportion $p = \lim n_1/n \in (0, 1)$, standard two-sample scaling yields

$$\sqrt{n} \left( \frac{\Delta \hat{S}(t) - \Delta S(t)}{\Delta \hat{\mu}(\tau) - \Delta \mu(\tau)} \right) \Rightarrow N \left( 0, \frac{1}{p} \Sigma_1(t, \tau) + \frac{1}{1-p} \Sigma_0(t, \tau) \right). \qquad (10)$$

## 4.3 Covariances with the Cox log-hazard ratio

Assume the Cox proportional hazards model holds, $\lambda(t | Z) = \lambda_0(t) \exp(\beta Z)$, and let $\hat{\beta}$ denote the Cox partial- likelihood estimator of the log-hazard ratio parameter $\beta$. Its asymptotic linearity and influence-function representation are standard; see Reid and Crépeau [1985]. Write $\mathcal{I}(\beta)$ for the usual (per-subject) partial-likelihood information, so that

$$\mathrm{AVar}\{\sqrt{n}(\hat{\beta} - \beta)\} = \mathcal{I}(\beta)^{-1}. \qquad (11)$$
Let $\bar{z}(u)$ denote the Cox risk-set weighted mean exposure at time $u$ (at the true $\beta$),

$$ \bar{z}(u) = \frac{\mathbb{E}\{Y(u) Z e^{\beta Z}\}}{\mathbb{E}\{Y(u) e^{\beta Z}\}}, \quad Y(u) = I(X \ge u). \qquad (12) $$

Combining the influence function of $\hat{\beta}$ with those of $\hat{S}_a(t)$ and $\hat{\mu}_a(\tau)$ yields explicit cross-covariances through the identity $\Sigma_{kl} = \mathbb{E}\{\varphi_{\theta_k}(O)\varphi_{\theta_l}(O)\}$ from Section 3. In particular, the covariance with the RMST contrast can be written as

$$ \mathrm{ACov}(\sqrt{n}\{\hat{\beta} - \beta\}, \sqrt{n}\{\Delta\hat{\mu}(\tau) - \Delta\mu(\tau)\}) = -\mathcal{I}(\beta)^{-1} \left[ \frac{1}{p} \int_0^\tau \{1 - \bar{z}(u)\} R_1(u; \tau) \frac{\lambda_1(u)}{Y_1(u)} du \right. \\ \left. + \frac{1}{1-p} \int_0^\tau \{0 - \bar{z}(u)\} R_0(u; \tau) \frac{\lambda_0(u)}{Y_0(u)} du \right]. \quad (13) $$

Similarly, the covariance with the survival contrast at fixed time $t$ is obtained by replacing $R_a(u; \tau)$ by $S_a(t) I(u \le t)$:

$$ \mathrm{ACov}(\sqrt{n}\{\hat{\beta} - \beta\}, \sqrt{n}\{\Delta\hat{S}(t) - \Delta S(t)\}) = -\mathcal{I}(\beta)^{-1} \left[ \frac{1}{p} \int_0^t \{1 - \bar{z}(u)\} S_1(t) \frac{\lambda_1(u)}{Y_1(u)} du \right. \\ \left. + \frac{1}{1-p} \int_0^t \{0 - \bar{z}(u)\} S_0(t) \frac{\lambda_0(u)}{Y_0(u)} du \right]. \quad (14) $$

Equations (10), (13), and (14), together with $\mathrm{AVar}\{\sqrt{n}(\hat{\beta} - \beta)\} = \mathcal{I}(\beta)^{-1}$, define the full $3 \times 3$ asymptotic covariance matrix for $(\Delta\hat{S}(t), \Delta\hat{\mu}(\tau), \hat{\beta})^\top$.

# 5 Estimation and plug-in covariance estimators

The asymptotic covariance structure in Section 4 is determined by influence functions and, equivalently, by first-order linearizations of the corresponding estimators [Reid, 1981, Reid and Crépeau, 1985, Jullum and Hjort, 2019]. This section describes a practical plug-in strategy for estimating the full covariance matrix of $(\Delta\hat{S}(t), \Delta\hat{\mu}(\tau), \hat{\beta})^\top$ by constructing empirical influence values and taking their sample second moments.

## 5.1 Event-time notation and empirical martingale increments

Fix an arm $a \in \{0, 1\}$ when working armwise. Let $t_{a1} < \dots < t_{aD_a}$ denote the ordered distinct event times in arm $a$. For subject $i$ in arm $a$ define

$$ dN_{ai}(t_{aj}) = I(X_i = t_{aj}, \delta_i = 1), \qquad (15) $$

$$ Y_{ai}(t_{aj}) = I(X_i \ge t_{aj}), \qquad (16) $$

with aggregated counts $d_{aj} = \sum_{i:Z_i=a} dN_{ai}(t_{aj})$ and $Y_{aj} = \sum_{i:Z_i=a} Y_{ai}(t_{aj})$. The Nelson–Aalen increment estimator is $d\hat{\Lambda}_a(t_{aj}) = d_{aj}/Y_{aj}$. The Kaplan–Meier estimator is the product-limit

$$ \hat{S}_a(t) = \prod_{t_{aj} \le t} \left(1 - \frac{d_{aj}}{Y_{aj}}\right). \qquad (17) $$

Following Reid [1981], we express the first-order linearization of $\hat{S}_a$ in terms of the Nelson–Aalen increments and the associated empirical martingale increments

$$ d\hat{M}_{ai}(t_{aj}) = dN_{ai}(t_{aj}) - Y_{ai}(t_{aj}) d\hat{\Lambda}_a(t_{aj}) = dN_{ai}(t_{aj}) - Y_{ai}(t_{aj}) \frac{d_{aj}}{Y_{aj}}. \qquad (18) $$

This discrete-time representation underlies Greenwood-type plug-in estimators for Kaplan–Meier functionals under independent censoring [Reid, 1981].
## 5.2 Empirical influence values for within-arm Kaplan-Meier and RMST

Let $\hat{S}_a(\cdot)$ denote the Kaplan-Meier estimator in arm a. For a prespecified horizon $\tau$, define the plug-in RMST $\hat{\mu}_a(\tau) = \int_0^\tau \hat{S}_a(u) du$ using the usual right-continuous step-function convention. Define the empirical tail area at event times

$$ \hat{R}_a(t_{aj}; \tau) = \int_{t_{aj}}^{\tau} \hat{S}_a(u) du. \qquad (19) $$

Equivalently, letting $t_{a,D_a+1} = \tau$ and using the right-continuous convention,

$$ \hat{R}_a(t_{aj}; \tau) = \sum_{m:j \le m \le D_a} \hat{S}_a(t_{am})((t_{a,m+1} \wedge \tau) - (t_{am} \wedge \tau)). \qquad (20) $$

Discrete analogues of the martingale influence representations [Reid, 1981] yield the empirical influence values:

* Survival at a fixed time t.

$$ \hat{\varphi}_{S_a(t),i} = -\hat{S}_a(t) \sum_{t_{aj} \le t} \frac{d\hat{M}_{ai}(t_{aj})}{Y_{aj}}. \qquad (21) $$

* RMST at horizon $\tau$.

$$ \hat{\varphi}_{\mu_a(\tau),i} = - \sum_{t_{aj} \le \tau} \frac{\hat{R}_a(t_{aj}; \tau)}{Y_{aj}} d\hat{M}_{ai}(t_{aj}). \qquad (22) $$

The sample second moments of (21)–(22) provide consistent plug-in estimators for the within-arm asymptotic variances and cross-covariance. For example,

$$ \widehat{\mathrm{Cov}}\{\hat{S}_a(t), \hat{\mu}_a(\tau)\} = \frac{1}{n_a} \sum_{i:Z_i=a} \hat{\varphi}_{S_a(t),i} \hat{\varphi}_{\mu_a(\tau),i} = \hat{S}_a(t) \sum_{t_{aj} \le t \wedge \tau} \hat{R}_a(t_{aj}; \tau) \frac{d_{aj}}{Y_{aj}^2}, \qquad (23) $$

and analogous event-time expressions hold for $\widehat{\mathrm{Var}}\{\hat{S}_a(t)\}$ and $\widehat{\mathrm{Var}}\{\hat{\mu}_a(\tau)\}$. With ties, $Y_{aj}^{-2}$ in these plug-in expressions may be replaced by the usual exact correction $\sum_{\ell=0}^{d_{aj}-1} (Y_{aj} - \ell)^{-2}$.

## 5.3 Empirical influence values for treatment contrasts

To obtain joint inference on the $\sqrt{n}$ scale, the arm-specific influence values are combined with the usual two-sample weights. Let $p = n_1/n$ and $1-p = n_0/n$. For each subject i, define

$$ \hat{\varphi}_{\Delta S(t),i} = I(Z_i = 1) \frac{\hat{\varphi}_{S_1(t),i}}{p} - I(Z_i = 0) \frac{\hat{\varphi}_{S_0(t),i}}{1-p}, \qquad (24) $$

$$ \hat{\varphi}_{\Delta\mu(\tau),i} = I(Z_i = 1) \frac{\hat{\varphi}_{\mu_1(\tau),i}}{p} - I(Z_i = 0) \frac{\hat{\varphi}_{\mu_0(\tau),i}}{1-p}. \qquad (25) $$

These definitions match the inflation factors in the contrast-level limit covariance (10).

## 5.4 Empirical influence values for the Cox log-hazard ratio

Assume the Cox proportional hazards model holds. Let $\hat{\beta}$ denote the Cox partial-likelihood estimator. Under standard regularity conditions, $\hat{\beta}$ is asymptotically linear with influence function given by the inverse observed information times an individual score contribution [Reid and Crépeau, 1985]. Let $U_n(\beta) = \sum_{i=1}^n U_i(\beta)$ denote the partial-likelihood score and $J_n(\beta) = -\partial U_n(\beta)/\partial \beta$ the observed information. Define the empirical influence value

$$ \hat{\varphi}_{\beta,i} = J_n(\hat{\beta})^{-1} U_i(\hat{\beta}). \qquad (26) $$

Under correct proportional hazards specification, the model-based variance of $\hat{\beta}$ is $J_n(\hat{\beta})^{-1}/n$.
## 5.5 Plug-in covariance matrix and correlations

Collect the empirical influence values into the subject-level vector

$$ \hat{\varphi}_i(t, \tau) = (\hat{\varphi}_{\Delta S(t),i}, \hat{\varphi}_{\Delta \mu(\tau),i}, \hat{\varphi}_{\beta,i})^{\top}. \quad (27) $$

A consistent estimator of the CLT covariance matrix $\Sigma(t, \tau)$ in (3) is

$$ \hat{\Sigma}(t, \tau) = \frac{1}{n} \sum_{i=1}^{n} \hat{\varphi}_i(t, \tau) \hat{\varphi}_i(t, \tau)^{\top}, \quad (28) $$

optionally with centering by the empirical mean of $\hat{\varphi}_i(t, \tau)$ for finite-sample stability. The estimated covariance matrix of $\hat{\theta}(t, \tau) = (\Delta \hat{S}(t), \Delta \hat{\mu}(\tau), \hat{\beta})^{\top}$ is $\hat{\Sigma}(t, \tau)/n$, and estimated standard errors and correlations follow from the diagonal entries and the usual normalization of covariance entries.

**Remark (model-based Cox variance).** If one wishes to enforce the model-based Cox variance under proportional hazards in finite samples, the $\beta\beta$ entry of $\hat{\Sigma}(t, \tau)$ in (28) may be replaced by $J_n(\hat{\beta})^{-1}$, leaving the cross-covariances with $\Delta \hat{S}(t)$ and $\Delta \hat{\mu}(\tau)$ unchanged.

## 5.6 Notation mapping to Ristl et al.

Ristl et al. [2024] consider functionals $\theta_{k,a}$ (within arm $a \in \{0, 1\}$) whose estimators admit a stochastic-integral linearization of the form

$$ \sqrt{n_a} (\hat{\theta}_{k,a} - \theta_{k,a}) = a_{k,a} \int_0^{t_{k,a}} H_{k,a}(u) \frac{1}{Y_a(u)} dM_a(u) + o_p(1), \quad (29) $$

with $a_{k,a}$ a constant and $H_{k,a}$ predictable with respect to the martingale $M_a$. Their plug-in covariance estimator for two components $(k, \ell)$ in the same arm is

$$ \widehat{\mathrm{Cov}}_{\mathrm{Ristl}}(\hat{\theta}_{k,a}, \hat{\theta}_{\ell,a}) = \hat{a}_{k,a} \hat{a}_{\ell,a} \sum_{t_{aj} \le t_{k,a} \wedge t_{\ell,a}} \hat{H}_{k,a}(t_{aj}) \hat{H}_{\ell,a}(t_{aj}) \frac{d_{aj}}{Y_{aj}^2}, \quad (30) $$

where $t_{a1} < \dots < t_{aD_a}$ are the distinct event times in arm $a$, $d_{aj}$ is the number of events at $t_{aj}$, and $Y_{aj}$ is the risk-set size just prior to $t_{aj}$. With tied event times, the factor $d_{aj}/Y_{aj}^2$ can be replaced by $\sum_{\ell=0}^{d_{aj}-1} (Y_{aj} - \ell)^{-2}$.

Table 1 lists the choices of $(a_{k,a}, H_{k,a}, t_{k,a})$ corresponding to the Kaplan–Meier survival $\hat{S}_a(t)$ and RMST $\hat{\mu}_a(\tau)$ used in this note. With these choices, (30) coincides algebraically with the influence-function outer-product estimator

$$ \widehat{\mathrm{Cov}}_{\mathrm{IF}}(\hat{\theta}_{k,a}, \hat{\theta}_{\ell,a}) = \frac{1}{n_a} \sum_{i:Z_i=a} \hat{\varphi}_{\theta_{k,a},i} \hat{\varphi}_{\theta_{\ell,a},i} $$

from Section 5. Under proportional hazards, an analogous correspondence holds for the Cox log-hazard ratio estimator $\hat{\beta}$, since its first-order expansion can also be written as a martingale stochastic integral (via the Cox score process) or equivalently as an influence-function sum.

# 6 Conclusion

This note compared two routes to within-trial covariance estimation for commonly reported survival summaries. The influence-function/M-estimation route leads to simple plug-in covariance estimators obtained as empirical second moments of estimated influence values. After translating notation, these estimators coincide with the stochastic-integral plug-in expressions of Ristl et al. [2024], with any remaining discrepancies attributable to implementation details (e.g., tie handling or RMST integration). For the Cox log-hazard ratio component, proportional hazards is required for the usual model-based interpretation, but the same influence-function machinery provides the joint covariance structure once the relevant linear representation is specified.

More broadly, the influence-function perspective is a viable and practical alternative because it provides a modular recipe: identify an asymptotic linear representation for the estimators of interest and form covariances by inner products (or their empirical analogues). This recipe applies well outside survival analysis.

It is applied in decision tools that require joint uncertainty, namely the Focused Information Criterion (FIC) calculations for model selection, where covariance estimation is a key input Jullum and Hjort [2017], Nemes [2024]. In that sense, the value of the approach is not the survival-specific formulas, but the reusable framework for constructing and implementing covariance estimators for arbitrary statistics.
<table><caption>Table 1: Dictionary between the stochastic-integral representation of Ristl et al. and the influence-function plug-in estimators used here (within a single arm a).</caption><thead><tr><th>Quantity in arm a</th><th>Choice of (a<sub>k,a</sub>, H<sub>k,a</sub>, t<sub>k,a</sub>) in (29)</th><th>Resulting plug-in covariance (Ristl = IF)</th></tr></thead><tbody><tr><td>S&#770;<sub>a</sub>(t)</td><td>a<sub>S,a</sub> = -S<sub>a</sub>(t); H<sub>S,a</sub>(u; t) = I(u &le; t);<br/>t<sub>S,a</sub> = t.</td><td>V&#770;ar{S&#770;<sub>a</sub>(t)} = S&#770;<sub>a</sub>(t)<sup>2</sup> &sum;<sub>t<sub>aj</sub> &le; t</sub> <sup>d<sub>aj</sub></sup>&frasl;<sub>Y<sub>aj</sub><sup>2</sup></sub>.</td></tr><tr><td>&mu;&#770;<sub>a</sub>(&tau;) = &int;<sub>0</sub><sup>&tau;</sup> S&#770;<sub>a</sub>(u) du</td><td>a<sub>&mu;,a</sub> = -1; H<sub>&mu;,a</sub>(u; &tau;) = R<sub>a</sub>(u; &tau;)I(u &le; &tau;);<br/>t<sub>&mu;,a</sub> = &tau;.</td><td>V&#770;ar{&mu;&#770;<sub>a</sub>(&tau;)} = &sum;<sub>t<sub>aj</sub> &le; &tau;</sub> R&#770;<sub>a</sub>(t<sub>aj</sub>; &tau;)<sup>2</sup> <sup>d<sub>aj</sub></sup>&frasl;<sub>Y<sub>aj</sub><sup>2</sup></sub>.</td></tr><tr><td>Cov{S&#770;<sub>a</sub>(t), &mu;&#770;<sub>a</sub>(&tau;)}</td><td>Use (a<sub>S,a</sub>, H<sub>S,a</sub>) and (a<sub>&mu;,a</sub>, H<sub>&mu;,a</sub>) above.</td><td>Cov{S&#770;<sub>a</sub>(t), &mu;&#770;<sub>a</sub>(&tau;)} =<br/>S&#770;<sub>a</sub>(t) &sum;<sub>t<sub>aj</sub> &le; t &and; &tau;</sub> R&#770;<sub>a</sub>(t<sub>aj</sub>; &tau;)<sup>d<sub>aj</sub></sup>&frasl;<sub>Y<sub>aj</sub><sup>2</sup></sub>.</td></tr><tr><td>Contrasts &Delta;S&#770;(t), &Delta;&mu;&#770;(&tau;)</td><td>Armwise independence and two-sample scaling<br/>with p = lim n<sub>1</sub>/n.</td><td>&Sigma;&#770;<sub>&Delta;</sub>(t, &tau;) = p<sup>-1</sup> &Sigma;&#770;<sub>1</sub>(t, &tau;) + (1 &minus; p)<sup>-1</sup> &Sigma;&#770;<sub>0</sub>(t, &tau;).</td></tr></tbody></table>

Here $R_a(u; \tau) = \int_u^\tau S_a(v) dv$ and $\hat{R}_a(t_{aj}; \tau) = \int_{t_{aj}}^\tau S_a(u) du$. With tied event times, replace $d_{aj}/Y_{aj}^2$ by $\sum_{\ell=0}^{d_{aj}-1} (Y_{aj} - \ell)^{-2}$.

## References

* Martin Jullum and Nils Lid Hjort. Parametric or nonparametric: The fic approach. *Statistica Sinica*, pages 951–981, 2017.

* Martin Jullum and Nils Lid Hjort. What price semiparametric cox regression? *Lifetime data analysis*, 25(3):406–438, 2019.

* Szilárd Nemes. Parametric analysis and model selection for economic evaluation of survival data. *Model Assisted Statistics and Applications*, 19(2):123–131, 2024.

* Nancy Reid. Influence Functions for Censored Data. *The Annals of Statistics*, 9(1):78–92, 1981. doi: 10.1214/aos/1176345334.

* Nancy Reid and Helene Crépeau. Influence functions for proportional hazards regression. *Biometrika*, 72(1):1–9, 1985.

* Robin Ristl, Heiko Götte, Armin Schüler, Martin Posch, and Franz König. Simultaneous inference procedures for the comparison of multiple characteristics of two survival functions. *Statistical Methods in Medical Research*, 33(4): 589–610, 2024.