One-Step Targeted Minimum Loss-based Estimation Based on Universal Least Favorable One-Dimensional Submodels

Statistical methods to control for confounders in rare disease settings that use external control

In the drug development for rare disease, the number of treated subjects in the clinical trial is often very small, whereas the number of external controls can be relatively large. There is no clear guidance on choosing an appropriate statistical method to control baseline confounding in this situation. To fill this gap, we conduct extensive simulations to evaluate the performance of commonly used matching and weighting methods as well as the more recently developed targeted maximum likelihood estimation (TMLE) and cardinality matching in small sample settings, mimicking the motivating data from a pediatric rare disease. Among the methods examined, the performance of coarsened exact matching (CEM) and TMLE are relatively robust under various model specifications. CEM is only feasible when the number of controls far exceeds the number of treated, whereas TMLE has better performance with less extreme treatment allocation ratios. Our simulations suggest bootstrap is useful for variance estimation in small samples after matching.

Targeted maximum likelihood estimation for causal inference in survival and competing risks analysis

Targeted maximum likelihood estimation (TMLE) provides a general methodology for estimation of causal parameters in presence of high-dimensional nuisance parameters. Generally, TMLE consists of a two-step procedure that combines data-adaptive nuisance parameter estimation with semiparametric efficiency and rigorous statistical inference obtained via a targeted update step. In this paper, we demonstrate the practical applicability of TMLE based causal inference in survival and competing risks settings where event times are not confined to take place on a discrete and finite grid. We focus on estimation of causal effects of time-fixed treatment decisions on survival and absolute risk probabilities, considering different univariate and multidimensional parameters. Besides providing a general guidance to using TMLE for survival and competing risks analysis, we further describe how the previous work can be extended with the use of loss-based cross-validated estimation, also known as super learning, of the conditional hazards. We illustrate the usage of the considered methods using publicly available data from a trial on adjuvant chemotherapy for colon cancer. R software code to implement all considered algorithms and to reproduce all analyses is available in an accompanying online appendix on Github.

Application of targeted maximum likelihood estimation in public health and epidemiological studies: a systematic review

PURPOSE

The targeted maximum likelihood estimation (TMLE) statistical data analysis framework integrates machine learning, statistical theory, and statistical inference to provide a least biased, efficient, and robust strategy for estimation and inference of a variety of statistical and causal parameters. We describe and evaluate the epidemiological applications that have benefited from recent methodological developments.

METHODS

We conducted a systematic literature review in PubMed for articles that applied any form of TMLE in observational studies. We summarized the epidemiological discipline, geographical location, expertize of the authors, and TMLE methods over time. We used the Roadmap of Targeted Learning and Causal Inference to extract key methodological aspects of the publications. We showcase the contributions to the literature of these TMLE results.

RESULTS

Of the 89 publications included, 33% originated from the University of California at Berkeley, where the framework was first developed by Professor Mark van der Laan. By 2022, 59% of the publications originated from outside the United States and explored up to seven different epidemiological disciplines in 2021-2022. Double-robustness, bias reduction, and model misspecification were the main motivations that drew researchers toward the TMLE framework. Through time, a wide variety of methodological, tutorial, and software-specific articles were cited, owing to the constant growth of methodological developments around TMLE.

CONCLUSIONS

There is a clear dissemination trend of the TMLE framework to various epidemiological disciplines and to increasing numbers of geographical areas. The availability of R packages, publication of tutorial papers, and involvement of methodological experts in applied publications have contributed to an exponential increase in the number of studies that understood the benefits and adoption of TMLE.

Efficient targeted learning of heterogeneous treatment effects for multiple subgroups

In biomedical science, analyzing treatment effect heterogeneity plays an essential role in assisting personalized medicine. The main goals of analyzing treatment effect heterogeneity include estimating treatment effects in clinically relevant subgroups and predicting whether a patient subpopulation might benefit from a particular treatment. Conventional approaches often evaluate the subgroup treatment effects via parametric modeling and can thus be susceptible to model mis-specifications. In this paper, we take a model-free semiparametric perspective and aim to efficiently evaluate the heterogeneous treatment effects of multiple subgroups simultaneously under the one-step targeted maximum-likelihood estimation (TMLE) framework. When the number of subgroups is large, we further expand this path of research by looking at a variation of the one-step TMLE that is robust to the presence of small estimated propensity scores in finite samples. From our simulations, our method demonstrates substantial finite sample improvements compared to conventional methods. In a case study, our method unveils the potential treatment effect heterogeneity of rs12916-T allele (a proxy for statin usage) in decreasing Alzheimer's disease risk.

Efficient estimation of pathwise differentiable target parameters with the undersmoothed highly adaptive lasso

We consider estimation of a functional parameter of a realistically modeled data distribution based on observing independent and identically distributed observations. The highly adaptive lasso estimator of the functional parameter is defined as the minimizer of the empirical risk over a class of cadlag functions with finite sectional variation norm, where the functional parameter is parametrized in terms of such a class of functions. In this article we establish that this HAL estimator yields an asymptotically efficient estimator of any smooth feature of the functional parameter under a global undersmoothing condition. It is formally shown that the L 1-restriction in HAL does not obstruct it from solving the score equations along paths that do not enforce this condition. Therefore, from an asymptotic point of view, the only reason for undersmoothing is that the true target function might not be complex so that the HAL-fit leaves out key basis functions that are needed to span the desired efficient influence curve of the smooth target parameter. Nonetheless, in practice undersmoothing appears to be beneficial and a simple targeted method is proposed and practically verified to perform well. We demonstrate our general result HAL-estimator of a treatment-specific mean and of the integrated square density. We also present simulations for these two examples confirming the theory.

Nonparametric bootstrap inference for the targeted highly adaptive least absolute shrinkage and selection operator (LASSO) estimator

The Highly-Adaptive least absolute shrinkage and selection operator (LASSO) Targeted Minimum Loss Estimator (HAL-TMLE) is an efficient plug-in estimator of a pathwise differentiable parameter in a statistical model that at minimal (and possibly only) assumes that the sectional variation norm of the true nuisance functions (i.e., relevant part of data distribution) are finite. It relies on an initial estimator (HAL-MLE) of the nuisance functions by minimizing the empirical risk over the parameter space under the constraint that the sectional variation norm of the candidate functions are bounded by a constant, where this constant can be selected with cross-validation. In this article we establish that the nonparametric bootstrap for the HAL-TMLE, fixing the value of the sectional variation norm at a value larger or equal than the cross-validation selector, provides a consistent method for estimating the normal limit distribution of the HAL-TMLE. In order to optimize the finite sample coverage of the nonparametric bootstrap confidence intervals, we propose a selection method for this sectional variation norm that is based on running the nonparametric bootstrap for all values of the sectional variation norm larger than the one selected by cross-validation, and subsequently determining a value at which the width of the resulting confidence intervals reaches a plateau. We demonstrate our method for 1) nonparametric estimation of the average treatment effect when observing a covariate vector, binary treatment, and outcome, and for 2) nonparametric estimation of the integral of the square of the multivariate density of the data distribution. In addition, we also present simulation results for these two examples demonstrating the excellent finite sample coverage of bootstrap-based confidence intervals.

The obesity paradox in critically ill patients: a causal learning approach to a casual finding

Background

While obesity confers an increased risk of death in the general population, numerous studies have reported an association between obesity and improved survival among critically ill patients. This contrary finding has been referred to as the obesity paradox. In this retrospective study, two causal inference approaches were used to address whether the survival of non-obese critically ill patients would have been improved if they had been obese.

Methods

The study cohort comprised 6557 adult critically ill patients hospitalized at the Intensive Care Unit of the Ghent University Hospital between 2015 and 2017. Obesity was defined as a body mass index of ≥ 30 kg/m². Two causal inference approaches were used to estimate the average effect of obesity in the non-obese (AON): a traditional approach that used regression adjustment for confounding and that assumed missingness completely at random and a robust approach that used machine learning within the targeted maximum likelihood estimation framework along with multiple imputation of missing values under the assumption of missingness at random. 1754 (26.8%) patients were discarded in the traditional approach because of at least one missing value for obesity status or confounders.

Results

Obesity was present in 18.9% of patients. The in-hospital mortality was 14.6% in non-obese patients and 13.5% in obese patients. The raw marginal risk difference for in-hospital mortality between obese and non-obese patients was − 1.06% (95% confidence interval (CI) − 3.23 to 1.11%, P = 0.337). The traditional approach resulted in an AON of − 2.48% (95% CI − 4.80 to − 0.15%, P = 0.037), whereas the robust approach yielded an AON of − 0.59% (95% CI − 2.77 to 1.60%, P = 0.599).

Conclusions

A causal inference approach that is robust to residual confounding bias due to model misspecification and selection bias due to missing (at random) data mitigates the obesity paradox observed in critically ill patients, whereas a traditional approach results in even more paradoxical findings. The robust approach does not provide evidence that the survival of non-obese critically ill patients would have been improved if they had been obese.

Causal models adjusting for time-varying confounding-a systematic review of the literature

BACKGROUND

Obtaining unbiased causal estimates from longitudinal observational data can be difficult due to exposure-affected time-varying confounding. The past decade has seen considerable development in methods for analysing such complex longitudinal data. However, the extent to which those methods have been implemented is unclear. This study describes and characterizes the state of the field in methods adjusting for exposure-affected time-varying confounding, and examines their use in the literature.

METHODS

We systematically reviewed the literature from 2000 to 2016 for studies adjusting for time-dependent confounding, including use of specific methods like inverse probability of treatment weighting (IPTW). Articles were coded based on the methods used and, for applied articles, the topic areas covered.

RESULTS

We screened 4239 abstracts, and subsequently reviewed 1100 articles, leaving 542 relevant articles in the analyses. The number of published articles increased from two in 2000, to 112 in 2016. This increase was primarily in applied articles using IPTW, which increased from one study in 2000, to 90 in 2016. Of the 432 studies with applications to observed data, 60.9% were on at least one of: HIV (30.6%), cardiopulmonary health (13.2%), kidney disease (11.8%) or mental health (10.0%).

CONCLUSIONS

There has been marked growth in reports addressing exposure-affected time-varying confounding. This was driven by work in a small number of topic areas, with other areas showing relatively little uptake. In addition, despite developments in more advanced methods such doubly robust techniques and estimation via machine learning, implementation has been largely concentrated on the simpler, yet potentially less robust, IPTW.

Improved precision in the analysis of randomized trials with survival outcomes, without assuming proportional hazards

We present a new estimator of the restricted mean survival time in randomized trials where there is right censoring that may depend on treatment and baseline variables. The proposed estimator leverages prognostic baseline variables to obtain equal or better asymptotic precision compared to traditional estimators. Under regularity conditions and random censoring within strata of treatment and baseline variables, the proposed estimator has the following features: (i) it is interpretable under violations of the proportional hazards assumption; (ii) it is consistent and at least as precise as the Kaplan-Meier and inverse probability weighted estimators, under identifiability conditions; (iii) it remains consistent under violations of independent censoring (unlike the Kaplan-Meier estimator) when either the censoring or survival distributions, conditional on covariates, are estimated consistently; and (iv) it achieves the nonparametric efficiency bound when both of these distributions are consistently estimated. We illustrate the performance of our method using simulations based on resampling data from a completed, phase 3 randomized clinical trial of a new surgical treatment for stroke; the proposed estimator achieves a 12% gain in relative efficiency compared to the Kaplan-Meier estimator. The proposed estimator has potential advantages over existing approaches for randomized trials with time-to-event outcomes, since existing methods either rely on model assumptions that are untenable in many applications, or lack some of the efficiency and consistency properties (i)-(iv). We focus on estimation of the restricted mean survival time, but our methods may be adapted to estimate any treatment effect measure defined as a smooth contrast between the survival curves for each study arm. We provide R code to implement the estimator.

Doubly robust nonparametric inference on the average treatment effect

Doubly robust estimators are widely used to draw inference about the average effect of a treatment. Such estimators are consistent for the effect of interest if either one of two nuisance parameters is consistently estimated. However, if flexible, data-adaptive estimators of these nuisance parameters are used, double robustness does not readily extend to inference. We present a general theoretical study of the behaviour of doubly robust estimators of an average treatment effect when one of the nuisance parameters is inconsistently estimated. We contrast different methods for constructing such estimators and investigate the extent to which they may be modified to also allow doubly robust inference. We find that while targeted minimum loss-based estimation can be used to solve this problem very naturally, common alternative frameworks appear to be inappropriate for this purpose. We provide a theoretical study and a numerical evaluation of the alternatives considered. Our simulations highlight the need for and usefulness of these approaches in practice, while our theoretical developments have broad implications for the construction of estimators that permit doubly robust inference in other problems.

A Generally Efficient Targeted Minimum Loss Based Estimator based on the Highly Adaptive Lasso

Suppose we observe n $n$ independent and identically distributed observations of a finite dimensional bounded random variable. This article is concerned with the construction of an efficient targeted minimum loss-based estimator (TMLE) of a pathwise differentiable target parameter of the data distribution based on a realistic statistical model. The only smoothness condition we will enforce on the statistical model is that the nuisance parameters of the data distribution that are needed to evaluate the canonical gradient of the pathwise derivative of the target parameter are multivariate real valued cadlag functions (right-continuous and left-hand limits, (G. Neuhaus. On weak convergence of stochastic processes with multidimensional time parameter. Ann Stat 1971;42:1285-1295.) and have a finite supremum and (sectional) variation norm. Each nuisance parameter is defined as a minimizer of the expectation of a loss function over over all functions it its parameter space. For each nuisance parameter, we propose a new minimum loss based estimator that minimizes the loss-specific empirical risk over the functions in its parameter space under the additional constraint that the variation norm of the function is bounded by a set constant. The constant is selected with cross-validation. We show such an MLE can be represented as the minimizer of the empirical risk over linear combinations of indicator basis functions under the constraint that the sum of the absolute value of the coefficients is bounded by the constant: i.e., the variation norm corresponds with this L1 $L_1$-norm of the vector of coefficients. We will refer to this estimator as the highly adaptive Lasso (HAL)-estimator. We prove that for all models the HAL-estimator converges to the true nuisance parameter value at a rate that is faster than n-1/4 $n^{-1/4}$ w.r.t. square-root of the loss-based dissimilarity. We also show that if this HAL-estimator is included in the library of an ensemble super-learner, then the super-learner will at minimal achieve the rate of convergence of the HAL, but, by previous results, it will actually be asymptotically equivalent with the oracle (i.e., in some sense best) estimator in the library. Subsequently, we establish that a one-step TMLE using such a super-learner as initial estimator for each of the nuisance parameters is asymptotically efficient at any data generating distribution in the model, under weak structural conditions on the target parameter mapping and model and a strong positivity assumption (e.g., the canonical gradient is uniformly bounded). We demonstrate our general theorem by constructing such a one-step TMLE of the average causal effect in a nonparametric model, and establishing that it is asymptotically efficient.