arXiv subjects

Mark J. van der Laan

Publications and source records attributed to Mark J. van der Laan.

At least 19 recordsLinked to original sources

arXiv2026

Targeted Highly Adaptive Lasso Minimum Loss Estimation of Target Functions

We propose a Targeted Highly Adaptive Lasso for estimation of non-pathwise differentiable functional parameters such as the dose-response curve (DRC) for continuous exposure. We assume the target function lies in the $k$-th order smoothness class used to define the $k$-th order Highly Adaptive Lasso (HAL), which can be well approximated by linear spans of $k$-th order spline basis functions. We construct a projection of the true target function onto a large finite dimensional working model spanned by an initial set of $k$-th order spline basis functions, which defines a pathwise differentiable approximation of the target functional parameter. A standard TMLE is then applied with a data-adaptive initial fit, replacing the MLE targeting step with a LASSO step over HAL spline basis functions that span the target function. We prove that the resulting Targeted HAL-MLE is pointwise asymptotically normally distributed and achieves a convergence rate determined solely by the dimension and smoothness of the target function, giving dimension free rates up till $\log n$-factors. Through a simulation study for the DRC, we show that the Targeted HAL outperforms a HAL plug-in estimator in terms of bias and mean squared error. Targeted HAL offers a fully data-adaptive approach to inference on functional parameters without requiring sieve specification or parametric assumptions.

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arXiv2026

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

We consider efficient estimation of causal parameters in longitudinal two-stage designs with outcome subsampling, motivated by resampling designs in HIV-related mortality studies. In these studies, many participants become lost to follow-up; resampling designs address this by tracing a subset of lost individuals to ascertain their outcomes. Analyses often use inverse-probability-weighted Kaplan-Meier (wKM) estimators that discard longitudinal covariate information and suffer from efficiency losses. We note that resampling designs are an instance of a broader class: two-stage designs with outcome subsampling, in which a first stage collects some data on all participants and a second stage collects outcome information on a selected subset. This connection motivates two estimators. First, drawing on inverse probability of censoring weighted targeted maximum likelihood estimation (IPCW-TMLE) for two-stage designs, we develop its longitudinal extension, IPCW longitudinal TMLE (IPCW-LTMLE) and show that estimating and targeting the known second-stage sampling weights yields variance reductions of up to 36% over the use of known sampling probabilities. Second, given that inverse weighting sacrifices efficiency, we propose an LTMLE that incorporates the second-stage sampling indicator as an intervention node in the sequential regression framework, returning to plug-in estimation and avoiding inverse weighting entirely. Simulations across sample sizes show that LTMLE achieves up to 73% lower variance than wKM with known sampling weights, with reductions of 30-50% common across settings, while IPCW-LTMLE achieves consistent gains of 20-35%. We further demonstrate that cross-fitted variance estimation is essential for valid inference: standard variance estimators yield confidence interval coverage as low as 76%, while our cross-fitted variants consistently restore coverage to nominal levels.

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arXiv2026

Higher-Order Efficient Estimators: A Review and Simulation-Based Benchmark Study

Higher-order efficient estimators extend standard first-order semiparametric estimators by replacing second-order residuals with third- or higher-order terms, potentially enabling asymptotic efficiency under slower nuisance function convergence rates and improving finite-sample performance. Existing methods achieve higher-order expansions through structurally different approximation strategies, including basis truncation, kernel smoothing, and highly adaptive lasso (HAL) representations, making direct theoretical and practical comparison difficult. In this manuscript, we provide a focused review and a simulation-based empirical benchmark for second-order efficient estimators, using treatment-specific mean estimation as a canonical causal inference and missing data problem. We compare how higher-order influence function (HOIF) estimators, kernel-based higher-order targeted minimum loss-based estimator (HOTMLE), and HAL-based HOTMLE construct higher-order expansions and the approximation or regularization burdens they introduce. The asymptotic and numerical study evaluates first-order and empirical second-order estimators under controlled nuisance errors with constant or increasing sectional variation complexity. Results show that higher-order debiasing can substantially reduce first-order estimation bias; however, gains depend strongly on stability of the approximation or regularization required for higher-order correction. Empirical HAL-based HOTMLE shows relatively stable performance, while empirical HOIF remains sensitive to basis truncation and tuning choices. Overall, this manuscript clarifies when higher-order asymptotic improvements are attained in theory, when they may be practically visible, and when implementation instability may offset theoretical advantages.

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arXiv2026

Causal inference via implied interventions

In the context of having an instrumental variable, the standard practice in causal inference begins by targeting an effect of interest and proceeds by formulating assumptions enabling its identification. We turn this around by adhering to the interventions the observational distribution allows to identify, rather than starting with a desired causal estimand and imposing untestable conditions. The randomization of an instrument and its exclusion restriction define a class of auxiliary stochastic interventions on the treatment that are implied by stochastic interventions on the instrument. This mapping characterizes the identifiable causal effects of the treatment on the outcome given the observable distribution. The identified effect is the impact of a stochastic encouragement by the instrument that propagates through the unaltered treatment selection mechanism, rather than the effect of a hypothetical intervention that overrides how treatment is naturally chosen. Alternatively, searching for an intervention on the instrument whose implied one best approximates a desired target naturally leads to a projection representing the closest identifiable treatment effect. The generality of this projection allows to select different norms and indexing functional sets that give rise to diverse estimation problems, some of which we address using Expectation-Maximization and the Highly Adaptive Lasso.

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arXiv2026

Estimation of treatment effects in presence of differential use of post-randomization concomitant medication with time-to-event outcomes

In placebo-controlled randomized trials, the post-randomization use of concomitant medications may be higher in the placebo arm than in the treatment arm. This may dilute the full benefits of the randomized drug as estimated by the intention-to-treat analysis. We focus on cardiovascular outcomes trials in type-2 diabetes patients of glucose-lowering treatments where patients in the placebo arm are more likely to add other glucose-lowering agents with established cardio-protective properties. As a supplement to the intention-to-treat analysis, we propose a class of estimands within a causal framework that isolates the specific impact of the treatment being studied from that of concomitant treatment use. These estimands are defined under time-dependent treatment interventions to balance exposure to additional medications across intervention arms. We advocate for specific stochastic interventions to achieve this balance while minimizing positivity violations, which arise when certain treatment combinations or characteristics are not sufficiently represented in the data. We employ targeted minimum loss-based estimation (TMLE) to optimize the estimation procedure for our estimands while allowing for flexible adjustments for time-dependent covariates from follow-up visits. Finally, we demonstrate the application of the methods through a simulation study and a real-world example from the LEADER cardiovascular outcomes trial, which assessed cardiovascular risk for liraglutide versus placebo.

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arXiv2026

Adaptive Targeted Maximum Likelihood Estimation of the Mean Potential Outcome under a Treatment Rule

Estimating the mean counterfactual outcome under a treatment rule is a central problem in causal inference and policy evaluation. Standard estimators, including inverse probability weighting (IPW), augmented IPW (AIPW), and targeted maximum likelihood estimation (TMLE), can become unstable under practical positivity violations because their targeting or weighting steps depend on inverse propensity scores. We propose an adaptive targeted maximum likelihood estimation (A-TMLE) framework that uses a data-adaptive working model for the conditional average treatment effect (CATE). This working model induces a projected policy-value parameter, which coincides with the nonparametric mean potential outcome when the CATE is well represented by the adaptive basis. We derive the efficient influence function for the projected parameter and characterize its second-order remainder. We also introduce a regularized TMLE that targets the nonparametric policy value using a stabilized targeting covariate obtained by projecting the standard TMLE clever covariate onto the score space induced by the CATE working model. We quantify the first-order plug-in bias of regularized TMLE relative to the nonparametric target. The resulting targeting steps avoid direct inverse propensity score weighting, improving stability under limited overlap. In simulations, A-TMLE and regularized TMLE achieve lower mean squared error and improved coverage compared with IPW, AIPW, and standard TMLE under practical positivity violations, while remaining competitive when treatment overlap is strong. A real-data application to the Right Heart Catheterization study illustrates that the adaptive estimators produce stable policy-value estimates with substantially shorter confidence intervals than IPW and AIPW.

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arXiv2026

Investigating Targeting Strategies and Truncation in TMLE for the Average Treatment Effect under Practical Positivity Violations

Estimating average treatment effects from observational data is challenging under practical violations of the positivity assumption. Targeted Maximum Likelihood Estimators (TMLEs) are widely used because of their double robustness and efficiency, but they can remain sensitive to such violations. We conduct extensive simulation studies to examine how targeting strategies and truncation levels affect TMLE performance under varying degrees of outcome regression misspecification and practical positivity stress. We show that loss-weighted targeting can induce substantial systematic bias relative to clever-covariate-scaled targeting, while insufficient truncation for clever-covariate-scaled targeting leads to inflated variance and unstable estimation. We further find that fixed truncation rules of the form c/(sqrt(n) log n), especially with c = 5 or c = 6, provide robust practical defaults in many settings, although the optimal choice varies with sample size. Motivated by the limitations of standard Lepski selection, we propose a Lepski-type adaptive truncation procedure with a brake mechanism that improves stability in data-adaptive tuning. We also compare variance estimators and find that targeted bootstrap variance estimation provides a stable alternative across truncation levels.

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arXiv2026

Identifying and Estimating Causal Direct Effects Under Unmeasured Confounding

Causal mediation analysis provides techniques for defining and estimating effects that may be endowed with mechanistic interpretations. With many scientific investigations seeking to address mechanistic questions, causal direct and indirect effects have garnered much attention. The natural direct and indirect effects, the most widely used among such causal mediation estimands, are limited in their practical utility due to stringent identification requirements. Accordingly, considerable effort has been invested in developing alternative direct and indirect effect decompositions with relaxed identification requirements. Such efforts often yield effect definitions with nuanced and challenging interpretations. By contrast, relatively limited attention has been paid to relaxing the identification assumptions of the natural direct and indirect effects. Motivated by a secondary aim of a recent non-randomized vaccine prospective cohort study (NCT05168813), we present a set of relaxed conditions under which the natural direct effect is identifiable in spite of unobserved baseline confounding of the exposure-mediator pathway; we use this result to investigate the effect mediated by putative immune correlates of protection. Relaxing the commonly used but restrictive cross-world counterfactual independence assumption, we discuss strategies for evaluating the natural direct effect in non-randomized settings that arise in the analysis of vaccine studies. We revisit prior studies of semi-parametric efficiency theory to demonstrate the construction of flexible, multiply robust estimators of the natural direct effect and discuss efficient estimation strategies that do not place restrictive modeling assumptions on nuisance functions.

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arXiv2026

Highly Adaptive Empirical Risk Minimization with Principal Components

The Highly Adaptive Lasso (HAL) delivers unprecedented guarantees in nonparametric minimum loss estimation under minimal smoothness assumptions, such as dimension-free minimax optimal rates. However, the practical use of HAL has been severely limited by its exponentially growing computationally prohibitive indicator basis expansion in moderate to high dimensions. Existing screening strategies drastically reduce this dimension but lack any theoretical justification. We introduce the Principal Component Highly Adaptive (PC-HA) family of estimators, which for the first time provide a principled and theoretically valid dimension reduction. We establish formal results on the score equations solved by these PC-HA estimators, allowing to transfer plug-in efficiency and pointwise asymptotic normality results from HAL to these PC-HA estimators, under comparable complexity control.

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arXiv2026

Efficient Targeted Maximum Likelihood Estimators for Two-Phase Design Problems

In a typical two-phase design, a random sample is drawn from the target population in phase 1, during which only a subset of variables is collected. In phase 2, a subsample of the phase-1 cohort is selected, and additional variables are measured. This setting induces a coarsened data structure on the data from the second phase. We assume coarsening at random, that is, the phase-2 sampling mechanism depends only on variables fully observed. We review existing estimators, including the generalized raking estimator and the inverse probability of censoring weighted targeted maximum likelihood estimation (IPCW-TMLE) along with its extensions that also target the phase-2 sampling mechanism to improve efficiency. We further introduce a new class of estimators constructed within the TMLE framework that are asymptotically equivalent.

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arXiv2026

Powering RCTs for marginal effects with GLMs using prognostic score adjustment

In randomized clinical trials (RCTs), the accurate estimation of marginal treatment effects is crucial for determining the efficacy of interventions. Enhancing the statistical power of these analyses is a key objective for statisticians. The increasing availability of historical data from registries, prior trials, and health records presents an opportunity to improve trial efficiency. However, many methods for historical borrowing compromise strict type-I error rate control. Building on the work by Schuler et al. [2022] on prognostic score adjustment for linear models, this paper extends the methodology to the plug-in analysis proposed by Rosenblum et al. [2010] using generalized linear models (GLMs) to further enhance the efficiency of RCT analyses without introducing bias. Specifically, we train a prognostic model on historical control data and incorporate the resulting prognostic scores as covariates in the plug-in GLM analysis of the trial data. This approach leverages the predictive power of historical data to improve the precision of marginal treatment effect estimates. We demonstrate that this method achieves local semi-parametric efficiency under the assumption of an additive treatment effect on the link scale. We expand the GLM plug-in method to include negative binomial regression. Additionally, we provide a straightforward formula for conservatively estimating the asymptotic variance, facilitating power calculations that reflect these efficiency gains. Our simulation study supports the theory. Even without an additive treatment effect, we observe increased power or reduced standard error. While population shifts from historical to trial data may dilute benefits, they do not introduce bias.

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arXiv2025

Machine learning to optimize precision in the analysis of randomized trials: A journey in pre-specified, yet data-adaptive learning

Covariate adjustment is an approach to improve the precision of trial analyses by adjusting for baseline variables that are prognostic of the primary endpoint. Motivated by the SEARCH Universal HIV Test-and-Treat Trial (2013-2017), we tell our story of developing, evaluating, and implementing a machine learning-based approach for covariate adjustment. We provide the rationale for as well as the practical concerns with such an approach for estimating marginal effects. Using schematics, we illustrate our procedure: targeted machine learning estimation (TMLE) with Adaptive Pre-specification. Briefly, sample-splitting is used to data-adaptively select the combination of estimators of the outcome regression (i.e., the conditional expectation of the outcome given the trial arm and covariates) and known propensity score (i.e., the conditional probability of being randomized to the intervention given the covariates) that minimizes the cross-validated variance estimate and, thereby, maximizes empirical efficiency. We discuss our approach for evaluating finite sample performance with parametric and plasmode simulations, pre-specifying the Statistical Analysis Plan, and unblinding in real-time on video conference with our colleagues from around the world. We present the results from applying our approach in the primary, pre-specified analysis of 8 recently published trials (2022-2024). We conclude with practical recommendations and an invitation to implement our approach in the primary analysis of your next trial.

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arXiv2025

Hazard-Based Targeted Maximum Likelihood Estimation for Survival in Resampling Designs

Survival is a key metric for evaluating standards of care for people living with HIV. In resource-limited settings, high rates of loss to follow-up (LTFU) often result in underestimation of mortality when only observed deaths are considered. Resampling, which tracks a subset of LTFU patients to ascertain their outcomes, mitigates bias and improves survival estimates. However, common estimators for survival in resampling designs, such as weighted Kaplan-Meier (KM), fail to leverage covariate information collected during repeated clinic visits, even though this information is highly predictive of survival. We propose a Targeted Maximum Likelihood Estimator (TMLE) for survival in resampling designs, which addresses these limitations by leveraging baseline and longitudinal covariates to achieve greater efficiency. Our TMLE is a plug-in estimator and is robust to misspecification of the initial model for the conditional hazard of death, guaranteeing consistency of our estimator due to known resampling probabilities. We present: (1) a fully efficient TMLE for data from resampling studies with fixed follow-up time for all participants and (2) an inverse probability of censoring weighted (IPCW) TMLE that accounts for varied follow-up times by stratifying on patients with sufficient follow-up to evaluate survival. This IPCW-TMLE can be made highly efficient through nonparametric or targeted estimation of the follow-up censoring mechanism. In simulations, our TMLE reduced variance by up to 55% compared with the commonly used weighted KM estimator while preserving nominal confidence interval coverage. These findings demonstrate the potential of our TMLE to improve survival estimation in resampling designs, offering a robust and resource-efficient framework for HIV research. Keywords: Resampling designs, Survival analysis, Targeted Maximum Likelihood Estimation, Inverse probability weighting

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arXiv2025

Assessing treatment effects in observational data with missing confounders: A comparative study of practical doubly-robust and traditional missing data methods

In pharmacoepidemiology, safety and effectiveness are frequently evaluated using readily available administrative and electronic health records data. In these settings, detailed confounder data are often not available in all data sources and therefore missing on a subset of individuals. Multiple imputation (MI) and inverse-probability weighting (IPW) are go-to analytical methods to handle missing data and are dominant in the biomedical literature. Doubly-robust methods, which are consistent under fewer assumptions, can be more efficient with respect to mean-squared error. We discuss two practical-to-implement doubly-robust estimators, generalized raking and inverse probability-weighted targeted maximum likelihood estimation (TMLE), which are both currently under-utilized in biomedical studies. We compare their performance to IPW and MI in a detailed numerical study for a variety of synthetic data-generating and missingness scenarios, including scenarios with rare outcomes and a high missingness proportion. Further, we consider plasmode simulation studies that emulate the complex data structure of a large electronic health records cohort in order to compare anti-depressant therapies in a rare-outcome setting where a key confounder is prone to more than 50\% missingness. We provide guidance on selecting a missing data analysis approach, based on which methods excelled with respect to the bias-variance trade-off across the different scenarios studied.

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arXiv2025

Statistical learning for constrained functional parameters in infinite-dimensional models

We develop a general framework for estimating function-valued parameters under equality or inequality constraints in infinite-dimensional statistical models. Such constrained learning problems are common across many areas of statistics and machine learning, where estimated parameters must satisfy structural requirements such as moment restrictions, policy benchmarks, calibration criteria, or fairness considerations. To address these problems, we characterize the solution as the minimizer of a penalized population risk using a Lagrange-type formulation, and analyze it through a statistical functional lens. Central to our approach is a constraint-specific path through the unconstrained parameter space that defines the constrained solutions. For a broad class of constraint-risk pairs, this path admits closed-form expressions and reveals how constraints shape optimal adjustments. When closed forms are unavailable, we derive recursive representations that support tractable estimation. Our results also suggest natural estimators of the constrained parameter, constructed by combining estimates of unconstrained components of the data-generating distribution. Thus, our procedure can be integrated with any statistical learning approach and implemented using standard software. We provide general conditions under which the resulting estimators achieve optimal risk and constraint satisfaction, and we demonstrate the flexibility and effectiveness of the proposed method through various examples, simulations, and real-data applications.

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arXiv2025

Semi-parametric efficient estimation of small genetic effects in large-scale population cohorts

Population genetics seeks to quantify DNA variant associations with traits or diseases, as well as interactions among variants and with environmental factors. Computing millions of estimates in large cohorts in which small effect sizes are expected, necessitates minimising model-misspecification bias to control false discoveries. We present TarGene, a unified statistical workflow for the semi-parametric efficient and double robust estimation of genetic effects including k-point interactions among categorical variables in the presence of confounding and weak population dependence. k-point interactions, or Average Interaction Effects (AIEs), are a direct generalisation of the usual average treatment effect (ATE). We estimate AIEs with cross-validated and/or weighted versions of Targeted Minimum Loss-based Estimators (TMLE) and One-Step Estimators (OSE). The effect of dependence among data units on variance estimates is corrected by using sieve plateau variance estimators based on genetic relatedness across the units. We present extensive realistic simulations to demonstrate power, coverage, and control of type I error. Our motivating application is the targeted estimation of genetic effects on trait, including two-point and higher-order gene-gene and gene-environment interactions, in large-scale genomic databases such as UK Biobank and All of Us. All cross-validated and/or weighted TMLE and OSE for the AIE k-point interaction, as well as ATEs, conditional ATEs and functions thereof, are implemented in the general purpose Julia package TMLE.jl. For high-throughput applications in population genomics, we provide the open-source Nextflow pipeline and software TarGene which integrates seamlessly with modern high-performance and cloud computing platforms.

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arXiv2025

A cautionary note for plasmode simulation studies in the setting of causal inference

Plasmode simulation has become an important tool for evaluating the operating characteristics of different statistical methods in complex settings, such as pharmacoepidemiological studies of treatment effectiveness using electronic health records (EHR) data. These studies provide insight into how estimator performance is impacted by challenges including rare events, small sample size, etc., that can indicate which among a set of methods performs best in a real-world dataset. Plasmode simulation combines data resampled from a real-world dataset with synthetic data to generate a known truth for an estimand in realistic data. There are different potential plasmode strategies currently in use. We compare two popular plasmode simulation frameworks. We provide numerical evidence and a theoretical result, which shows that one of these frameworks can cause certain estimators to incorrectly appear overly biased with lower than nominal confidence interval coverage. Detailed simulation studies using both synthetic and real-world EHR data demonstrate that these pitfalls remain at large sample sizes and when analyzing data from a randomized controlled trial. We conclude with guidance for the choice of a plasmode simulation approach that maintains good theoretical properties to allow a fair evaluation of statistical methods while also maintaining the desired similarity to real data.

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arXiv2025

Nonparametric efficient causal estimation of the intervention-specific expected number of recurrent events with continuous-time targeted maximum likelihood and highly adaptive lasso estimation

Longitudinal settings involving outcome, competing risks and censoring events occurring and recurring in continuous time are common in medical research, but are often analyzed with methods that do not allow for taking post-baseline information into account. In this work, we define statistical and causal target parameters via the g-computation formula by carrying out interventions directly on the product integral representing the observed data distribution in a continuous-time counting process model framework. In recurrent events settings our target parameter identifies the expected number of recurrent events also in settings where the censoring mechanism or post-baseline treatment decisions depend on past information of post-baseline covariates such as the recurrent event process. We propose a flexible estimation procedure based on targeted maximum likelihood estimation coupled with highly adaptive lasso estimation to provide a novel approach for double robust and nonparametric inference for the considered target parameter. We illustrate the methods in a simulation study.

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