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Nonparametric Estimation of Event-Free Survival for Data with Left-Truncated Death and Intermittently Assessed Nonfatal Events

This paper proposes a novel nonparametric kernel smoothing method to estimate event-free survival probabilities and restricted mean times for data characterized by left truncation, right-censored death, and interval-censored nonfatal events, while leveraging supplemental data sources and demonstrating the approach through simulations and an application to the ARIC Study.

Original authors: Han Lu (Division of Biostatistics and Health Data Science, School of Public Health, University of Minnesota), Xianghua Luo (Division of Biostatistics and Health Data Science, School of Public Health

Published 2026-09-14

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CC BY 4.0

Original authors: Han Lu (Division of Biostatistics and Health Data Science, School of Public Health, University of Minnesota), Xianghua Luo (Division of Biostatistics and Health Data Science, School of Public Health, University of Minnesota), Yifei Sun (Department of Biostatistics, Mailman School of Public Health, Columbia University), Wendy Wang (Department of Epidemiology, Peter O'Donnell Jr School of Public Health, University of Texas, Department of Internal Medicine, University of Texas Southwestern Medical Center), Thomas Mosley (The MIND Center, University of Mississippi Medical Center), Priya Palta (Department of Neurology, School of Medicine, University of North Carolina), Elsayed Z. Soliman (Epidemiological Cardiology Research Center, Department of Cardiovascular Medicine, Wake Forest University School of Medicine), Lin Yee Chen (Department of Medicine, Medical School, University of Minnesota), Anne Eaton (Division of Biostatistics and Health Data Science, School of Public Health, University of Minnesota)

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of medical research, scientists often track how long patients remain free from a specific illness or complication. This measure, known as event-free survival, is a powerful tool because it captures the full story of a patient's health, not just whether they are alive. It answers a more nuanced question: how long can a person live without suffering a serious setback, such as a stroke or a diagnosis of dementia? To find the answer, researchers must follow people over time, watching for two things: the moment a non-fatal health event occurs, and the moment of death. However, the path to this knowledge is rarely a straight line. In many studies, patients join the research after their disease has already begun, meaning their early history is missing from the record. Furthermore, while death is recorded the moment it happens, other health events are often only discovered during periodic check-ups. This creates a gap in the timeline; researchers know an event happened sometime between two visits, but they do not know the exact day. This uncertainty, combined with the fact that patients entered the study at different stages of their illness, makes calculating accurate survival times a difficult statistical puzzle.

A team of researchers has developed a new way to solve this puzzle, allowing them to estimate these survival times more accurately than before. The challenge they addressed arises when a study includes patients who have already been living with a disease for some time before joining the research, a situation known as left truncation. In these cases, the standard methods used to calculate survival often fail because they assume everyone started being watched from the very beginning of their illness. Additionally, the researchers had to account for the fact that non-fatal events, like a diagnosis of dementia, are only found when a patient visits a clinic, creating a "fuzzy" window of time rather than a precise moment. The team proposed a new mathematical approach that smooths out these gaps and corrects for the missing early history. By separating the calculation into two parts—one for the risk of death and another for the risk of the non-fatal event—they created a method that can handle data where patients enter the study late and where events are found only intermittently.

To test their new method, the researchers ran thousands of computer simulations. They created virtual populations of patients with different patterns of illness and visit schedules. In these simulations, they compared their new approach against older, standard methods. The results showed that the traditional methods often produced biased answers, either overestimating or underestimating how long patients would stay healthy. The new method, however, stayed remarkably close to the true values, even when the data was messy or incomplete. The researchers also discovered that their method could be improved by adding extra information. If a study had data on patients who were followed only for death, without tracking the non-fatal events, this information could be combined with the main group to make the estimates even more precise. Similarly, adding data from patients who joined the study right when their illness began helped fill in the gaps for those who had been living with the disease for years.

The team then applied their new technique to real-world data from the Atherosclerosis Risk in Communities study, a long-running project that has followed thousands of Americans since 1987. They focused on people who had suffered a heart attack and wanted to know how long they could expect to live without developing dementia. The data presented a perfect storm of the difficulties the researchers aimed to solve: many participants had their heart attacks years before they were first assessed for dementia, and the dementia checks happened only at specific intervals over many years. Using their new method, the researchers estimated that, on average, a person who survives a heart attack can expect to live about 2.87 years without dementia in the five years following the event. Over a longer period of twenty years, that number rises to about 5.72 years. When they compared their results to those from older methods, the traditional approaches suggested that people would remain dementia-free for much longer, likely because they failed to account for the delays in detecting the condition and the fact that many patients had already been living with the heart attack for years before the study began.

The study also looked at whether these survival times differed based on race, sex, or age. The researchers found no significant differences between men and women, or between Black and white participants, in the short or long term. However, age made a clear difference in the long run. While younger and older patients had similar survival times in the first five years, those who had their heart attack before age 74 were expected to live about 3.51 years longer without dementia over the next twenty years compared to those who were older at the time of their heart attack. This suggests that the timing of the initial heart attack plays a crucial role in long-term brain health outcomes.

The researchers emphasize that their work is not just a theoretical exercise but a practical tool for making sense of complex medical records. In the real world, data is often imperfect; patients miss appointments, doctors record events at different times, and people join studies at various stages of life. The new method provides a way to extract reliable answers from this imperfect data without forcing it into a shape it does not fit. By acknowledging the gaps in the timeline and the specific way patients enter the study, the method offers a clearer picture of how diseases progress. The authors note that while their approach works well, it relies on the assumption that the timing of a patient's visits is not directly caused by the symptoms of the disease itself. If patients only go to the doctor because they feel sick, the method might need further adjustment. Nevertheless, this new approach opens the door to using vast amounts of existing medical data to understand disease progression more accurately, potentially leading to better insights into how to keep patients healthy for longer.

Technical Summary: Nonparametric Estimation of Event-Free Survival for Data with Left-Truncated Death and Intermittently Assessed Nonfatal Events

Problem Statement
In clinical and epidemiological research, Event-Free Survival (EFS) is a composite endpoint defined as the time from a disease onset (index event) until the occurrence of either a serious nonfatal event or death, whichever happens first. Estimating EFS probability is complicated by two specific data challenges often present simultaneously:

Component-wise Censoring: The nonfatal event is often assessed only at periodic intervals (e.g., clinic visits), leading to interval censoring, while death is typically right-censored. Standard methods like the Kaplan-Meier (KM) estimator, which assume right-censoring for all components, introduce bias when applied to such data by treating the detection date as the event date.
Left Truncation: In prevalent cohort studies (or studies using registry data), individuals enroll after the index event has already occurred. This results in left-truncated data, where individuals who experienced the event or died before the study entry are not observed.

Existing methods for component-wise censoring (e.g., Eaton et al., 2022) are designed for incident cohorts and fail to handle left truncation. Conversely, standard product-limit estimators for left-truncated data (e.g., Lynden-Bell, 1971; Tsai et al., 1987) require treating the composite endpoint as right-censored, thereby ignoring the interval nature of the nonfatal event. There is a lack of nonparametric methods capable of simultaneously addressing left-truncated death and component-wise censored nonfatal events.

Methodology
The authors propose a nonparametric kernel smoothing method to estimate EFS probability and the Restricted Mean Event-Free Survival time (EF-RMST) in the presence of left truncation and component-wise censoring. The approach decomposes the EFS probability $p(t)$ into two components:
$p(t) = r(t) \times S_{D_0}(t)$
where $S_{D_0}(t)$ is the probability of being alive at time $t$ , and $r(t)$ is the probability of being free of the nonfatal event conditional on being alive.

Estimation of Survival ( $S_{D_0}(t)$ ): The authors propose using a Left-Truncated, Right-Censored Product-Limit (LT-RC-PL) estimator (based on Tsai et al., 1987) to estimate the survival function. This accounts for the delayed entry of participants.
Estimation of Event-Free Probability ( $r(t)$ ): A kernel estimator is proposed for $r(t)$ , adapted from Eaton et al. (2022) but modified to handle the risk sets defined by left truncation. This estimator utilizes a second-order modified kernel function to correct for boundary effects.
Composite Estimator: The final EFS estimator is the product of the two components: $\hat{p}(t) = \hat{S}_D(t)\hat{r}_h(t)$ .
EF-RMST: The Restricted Mean Event-Free Survival time up to time $t$ is estimated by integrating the estimated survival curve: $\hat{\mu}(t) = \int_0^t \hat{p}(u) du$ .
Supplemental Data Integration: The method allows for the incorporation of two types of supplemental data to improve precision, particularly in early follow-up times where risk sets are small:
- Incident Cohort Data: Participants enrolled at the time of the index event (left truncation time = 0).
- Death-Only Data: Participants followed only for death (no nonfatal event assessment). These data contribute to the estimation of $S_{D_0}(t)$ but not $r(t)$ .

Theoretical properties, including asymptotic normality and variance estimation, are derived for the proposed estimators. Bandwidth selection is addressed via a data-adaptive procedure using leave-one-out cross-validation.

Key Results

Simulation Studies: The proposed LT-Kernel estimators demonstrated minimal bias and the lowest Mean Squared Error (MSE) compared to the KM estimator, the LT-RC-PL estimator, and the original kernel method (Eaton et al., 2022) across scenarios with truncated normal and Poisson visit schedules.
- The KM and LT-RC-PL estimators showed significant bias due to their failure to account for component-wise censoring or left truncation, respectively.
- Incorporating incident cohort data significantly reduced the variance of the estimator, particularly for early time points.
- Adding death-only data also improved precision, though to a lesser extent than adding full incident data, by enlarging the risk set for the survival component.
Real Data Application (ARIC Study): The method was applied to the Atherosclerosis Risk in Communities (ARIC) study to estimate dementia-free survival following a myocardial infarction (MI).
- The dataset included prevalent cases (MI before Visit 5) and incident cases (MI after Visit 5), with dementia assessed intermittently.
- The proposed method yielded lower (and likely less biased) estimates of dementia-free survival compared to standard KM and LT-RC-PL approaches.
- Subgroup analysis revealed that while short-term (5-year) dementia-free survival differences by sex, race, or age were not statistically significant, older participants (MI onset $\ge$ 74) had significantly lower long-term (20-year) dementia-free survival compared to younger participants.

Significance and Claims
The paper claims to provide the first nonparametric solution for estimating EFS in prevalent cohorts where the composite endpoint involves left-truncated death and interval-censored nonfatal events. The significance of the work lies in:

Addressing a Methodological Gap: It bridges the gap between methods for left-truncated data and methods for component-wise censored data, which previously could not be handled simultaneously.
Robustness: The method does not require assumptions about the dependence between the interval-censored nonfatal event and the right-censored death event.
Data Efficiency: It demonstrates how to leverage existing real-world data structures, including prevalent cohorts, incident cohorts, and death-only registries, to improve estimation precision without requiring new data collection.
Clinical Relevance: By applying the method to the ARIC study, the authors show its utility in estimating complex survival outcomes (dementia-free survival post-MI) where standard methods would yield biased results due to the specific censoring and truncation mechanisms inherent in the study design.

The authors note that while the method is robust, it relies on the assumption of conditional independence between the visit process and the event times given the truncation time. They acknowledge that in real-world data (unlike clinical trials), visit schedules may be event-driven, potentially violating this assumption, and identify the development of semiparametric regression models and robustness checks as future directions.

Technical Summary: Nonparametric Estimation of Event-Free Survival for Data with Left-Truncated Death and Intermittently Assessed Nonfatal Events

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