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Targeted maximum likelihood estimation for longitudinal two-stage designs with outcome subsampling

This paper proposes and evaluates two efficient estimators, IPCW-LTMLE and a novel plug-in LTMLE that avoids inverse weighting, for causal inference in longitudinal two-stage designs with outcome subsampling, demonstrating significant variance reductions over standard weighted Kaplan-Meier methods and establishing the necessity of cross-fitted variance estimation for valid statistical inference.

Original authors: Kirsten E. Landsiedel, Maya L. Petersen, Mark J. van der Laan

Published 2026-07-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Kirsten E. Landsiedel, Maya L. Petersen, Mark J. van der Laan

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

Imagine you are trying to figure out how long people in a specific community are living. You have a list of everyone you are tracking, but there's a big problem: many people stop showing up to the clinic. In the medical world, this is called being "lost to follow-up."

Usually, if someone stops showing up, researchers just assume they are still alive until proven otherwise, or they throw that person's data away. But in reality, people who stop showing up are often sicker and might have passed away. If you ignore them, your math says people are living longer than they actually are.

To fix this, researchers use a "resampling" strategy. Think of it like a detective agency: if a person goes missing, the agency goes back and tries to find them (maybe by calling their family or checking hospital records) to see if they are alive or dead. They only do this for a subset of the missing people because finding everyone is too expensive and time-consuming.

The Problem with the Old Way
The standard way to analyze this data is like a "weighted average."

If you know someone's status from the clinic, you count them normally.
If you found a missing person through your detective work, you count them "more" to represent all the other missing people you didn't find.
If you never found a missing person, you ignore them completely.

The paper argues this old method is wasteful. It throws away a goldmine of information: the long history of medical visits, test results, and behaviors that every participant had before they disappeared. It's like trying to guess the weather tomorrow by only looking at the sky right now, ignoring all the barometer readings and wind patterns you recorded for the last week.

The New Solution: Two Smart Tools
The authors, statisticians from UC Berkeley, propose two new, smarter ways to crunch the numbers. They realized that "resampling" is just a specific type of a broader puzzle called a "two-stage design" (Stage 1: get basic info on everyone; Stage 2: get detailed info on a few).

Here are their two new tools, explained with analogies:

1. The "Smart Weight" Tool (IPCW-LTMLE)
The old method uses fixed weights (like a static recipe). The new tool is like a dynamic chef.

Instead of just using the known probability of finding a missing person, this tool learns from the data to predict who was likely to be found.
It then "tweaks" (or targets) these predictions to make sure the final math is perfectly balanced.
The Result: It uses the rich history of all participants to make the "missing person" weights more accurate. In their tests, this reduced the "wobble" (variance) in the results by up to 36% compared to the old method.

2. The "Plug-and-Play" Tool (LTMLE)
The authors realized that even the "Smart Weight" tool is a bit clunky because it relies on "inverse weighting" (dividing by small numbers), which can be unstable.

They built a new tool that doesn't use weights at all. Instead, it treats the act of "being found" as just another step in a chain of events, like a stop in a relay race.
Imagine a relay race where the baton is passed from "Baseline Health" -> "Visits" -> "Did they get found?" -> "Final Outcome."
This tool runs a simulation (a "counterfactual") asking: "What would the outcome be if everyone had been found?" It uses the full history of everyone in the race, even those who were never found, to predict the result.
The Result: This is the most efficient tool. In their tests, it reduced the "wobble" by up to 73% compared to the old standard. It's like using a high-tech GPS instead of a paper map.

The "Overconfidence" Trap
The paper also found a hidden danger. When researchers use these fancy new tools, the computer can get "overconfident." It thinks it knows the answer better than it actually does, leading to confidence intervals (the margin of error) that are too narrow. It's like a student who studies hard but forgets to check their work, thinking they got a 100% when they actually got a 76%.

To fix this, the authors introduced "Cross-Fitting."

The Analogy: Imagine a teacher grading a test. If the teacher writes the questions and grades the answers, they might subconsciously grade their own questions too easily.
The Fix: You split the class into groups. Group A writes the questions, and Group B grades them. Then you swap. This ensures the grading is honest and the "margin of error" is calculated correctly.
Without this step, the new tools gave false confidence (coverage as low as 76%). With cross-fitting, they hit the correct target (95%).

The Bottom Line
The paper shows that by treating "resampling" as a general "two-stage" problem and using these new, sophisticated statistical tools, researchers can get much more accurate answers about survival rates. They use all the available data, not just the easy-to-find parts, and they do it in a way that doesn't trick themselves into thinking they are more precise than they really are.

In short: They turned a messy, incomplete puzzle into a clear picture by using the whole picture, not just the pieces that were easy to find.

Technical Summary: Targeted Maximum Likelihood Estimation for Longitudinal Two-Stage Designs with Outcome Subsampling

Problem Statement
The paper addresses the challenge of efficiently estimating general causal parameters in longitudinal two-stage designs featuring outcome subsampling. This design is motivated by longitudinal survival studies in resource-limited settings, particularly those investigating HIV-related mortality. In these studies, a significant proportion of participants are lost to follow-up (LTFU) before death is recorded or right-censoring occurs. Standard practice involves retrospectively tracing a subset of these LTFU individuals to ascertain their vital status (resampling).

The dominant analytical approach, the inverse-probability-weighted Kaplan-Meier (wKM) estimator, assigns weights based on known resampling probabilities. However, the authors argue that wKM estimators suffer from substantial efficiency losses because they discard the rich longitudinal covariate history available for all participants, relying instead on inverse weighting which discards data from non-sampled individuals. Furthermore, standard variance estimation methods often fail to provide valid inference when flexible, data-adaptive nuisance estimators are employed, leading to confidence interval coverage well below nominal levels (as low as 76%).

Methodological Framework
The authors formalize resampling designs as a specific instance of the broader class of two-stage designs with outcome subsampling. In this framework:

Stage 1: Data $V$ (covariates and censoring times) are collected for all participants.
Stage 2: Outcome information $X$ is collected only for a selected subset indicated by $\Delta=1$ .
Full Data: Defined as $X = (W, \tau, \bar{L}(\tau), \bar{Y}(\tau))$ , where $\tau$ is administrative censoring and $\bar{Y}(\tau)$ is the survival process.

The paper proposes two novel estimators based on Targeted Maximum Likelihood Estimation (TMLE) to address the inefficiencies of wKM:

IPCW Longitudinal TMLE (IPCW-LTMLE):
- This is an extension of the inverse probability of censoring weighted TMLE (IPCW-TMLE) for two-stage designs to the longitudinal setting.
- It applies an LTMLE to the "full data" (observed only for $\Delta=1$ ) and uses inverse probability weights ( $\Delta/\Pi_\Delta$ ) to account for missingness.
- Innovation: The authors demonstrate that even when sampling probabilities are known by design, estimating and targeting these probabilities yields significant variance reductions (up to 36%). This is achieved by adding a fluctuation step to enforce a mean-zero influence curve condition, effectively incorporating information from all participants rather than just the sampled subset.
LTMLE with Sampling as an Intervention Node:
- Recognizing that inverse weighting inherently sacrifices efficiency relative to plug-in estimation, the authors propose a second estimator that avoids inverse weighting entirely.
- This method incorporates the second-stage sampling indicator $\Delta$ directly as an additional intervention node in the sequential regression framework.
- The estimator targets a counterfactual outcome under a joint intervention that prevents censoring and sets $\Delta=1$ for all participants.
- This approach leverages the full longitudinal covariate history of all participants (regardless of sampling status) and adheres to plug-in estimation principles.

Variance Estimation and Cross-Fitting
A critical contribution of the paper is the development of cross-fitted variance estimators for both proposed methods. The authors identify that standard influence-curve-based variance estimation yields invalid inference (undercoverage) when flexible nuisance estimators (e.g., Super Learner) are used due to overfitting.

They propose a hybrid approach: the point estimate is computed on the full data to preserve stability and efficiency, while the variance estimation step utilizes sample splitting (cross-fitting).
For IPCW-LTMLE, this involves cross-fitting both the full-data influence curve component and the projection term onto the sampling mechanism tangent space.
This procedure is shown to be essential for restoring confidence interval coverage to nominal levels.

Simulation Results
The authors evaluate the estimators via simulations across sample sizes $N \in \{500, 1000, 3000\}$ in a resampling setting with bivariate censoring (administrative and sampling).

Efficiency: The plug-in LTMLE (without inverse weighting) consistently achieves the lowest variance, outperforming wKM with known weights by up to 73% at smaller sample sizes and early time points. IPCW-LTMLE with targeted weights also shows consistent gains of 20–35% over standard wKM.
Bias: All proposed estimators exhibit negligible empirical bias across settings.
Coverage: Standard variance estimators yield coverage as low as 76% for IPCW-LTMLE and 80% for Hazard TMLE at $N=500$ . The proposed cross-fitted variance estimators consistently restore coverage to near-nominal levels (approx. 95%).
Finite Sample Behavior: At $N=500$ and early time points ( $t=1$ ), all estimators face challenges due to low event counts and high degrees of deterministic information, leading to non-normal sampling distributions. However, coverage improves significantly as sample size increases.

Significance and Claims
The paper claims to provide a systematic extension of semiparametric efficiency theory to longitudinal two-stage designs with outcome subsampling.

Formalization: It establishes the connection between resampling designs and the broader class of two-stage designs with outcome subsampling, allowing for the application of advanced causal inference tools.
Efficiency: The proposed estimators offer highly efficient closed-form alternatives in settings where fully efficient closed-form estimators do not exist due to bivariate censoring.
Practical Utility: By demonstrating that targeting known sampling weights and avoiding inverse weighting entirely can yield substantial variance reductions, the paper offers a more efficient alternative to the widely used wKM estimator.
Robust Inference: The work emphasizes that cross-fitted variance estimation is not merely an optional improvement but an essential practice for valid inference when using data-adaptive nuisance estimation in these complex designs.

The authors conclude that these methods expand the methodological toolkit for analyzing two-stage designs, offering estimators that are more efficient, robust, and reliable than existing approaches, particularly in the context of longitudinal survival studies with outcome subsampling.

Technical Summary: Targeted Maximum Likelihood Estimation for Longitudinal Two-Stage Designs with Outcome Subsampling

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