Collaborative Targeted Maximum Likelihood for Time to Event Data

Targeted maximum likelihood estimation of a parameter of a data generating distribution, known to be an element of a semi-parametric model, involves constructing a parametric model through an initial density estimator with parameter ɛ representing an amount of fluctuation of the initial density estimator, where the score of this fluctuation model at ɛ = 0 equals the efficient influence curve/canonical gradient. The latter constraint can be satisfied by many parametric fluctuation models since it represents only a local constraint of its behavior at zero fluctuation. However, it is very important that the fluctuations stay within the semi-parametric model for the observed data distribution, even if the parameter can be defined on fluctuations that fall outside the assumed observed data model. In particular, in the context of sparse data, by which we mean situations where the Fisher information is low, a violation of this property can heavily affect the performance of the estimator. This paper presents a fluctuation approach that guarantees the fluctuated density estimator remains inside the bounds of the data model. We demonstrate this in the context of estimation of a causal effect of a binary treatment on a continuous outcome that is bounded. It results in a targeted maximum likelihood estimator that inherently respects known bounds, and consequently is more robust in sparse data situations than the targeted MLE using a naive fluctuation model. When an estimation procedure incorporates weights, observations having large weights relative to the rest heavily influence the point estimate and inflate the variance. Truncating these weights is a common approach to reducing the variance, but it can also introduce bias into the estimate. We present an alternative targeted maximum likelihood estimation (TMLE) approach that dampens the effect of these heavily weighted observations. As a substitution estimator, TMLE respects the global constraints of the observed data model. For example, when outcomes are binary, a fluctuation of an initial density estimate on the logit scale constrains predicted probabilities to be between 0 and 1. This inherent enforcement of bounds has been extended to continuous outcomes. Simulation study results indicate that this approach is on a par with, and many times superior to, fluctuating on the linear scale, and in particular is more robust when there is sparsity in the data.

Objectives Highly flexible nonparametric estimators have gained popularity in causal inference and epidemiology. Popular examples of such estimators include targeted maximum likelihood estimators (TMLE) and double machine learning (DML). TMLE is often argued or suggested to be better than DML estimators and several other estimators in small to moderate samples – even if they share the same large-sample properties – because TMLE is a plug-in estimator and respects the known bounds on the parameter, while other estimators might fall outside the known bounds and yield absurd estimates. However, this argument is not a rigorously proven result and may fail in certain cases. Methods In a carefully chosen simulation setting, I compare the performance of several versions of TMLE and DML estimators of the average treatment effect among treated in small to moderate samples. Results In this simulation setting, DML estimators outperforms some versions of TMLE in small samples. TMLE fluctuations are unstable, and hence empirically checking the magnitude of the TMLE fluctuation might alert cases where TMLE might perform poorly. Conclusions As a plug-in estimator, TMLE is not guaranteed to outperform non-plug-in counterparts such as DML estimators in small samples. Checking the fluctuation magnitude might be a useful diagnosis for TMLE. More rigorous theoretical justification is needed to understand and compare the finite-sample performance of these highly flexible estimators in general.

Targeted maximum likelihood estimation is a versatile tool for estimating parameters in semiparametric and nonparametric models. We work through an example applying targeted maximum likelihood methodology to estimate the parameter of a marginal structural model. In the case we consider, we show how this can be easily done by clever use of standard statistical software. We point out differences between targeted maximum likelihood estimation and other approaches (including estimating function based methods). The application we consider is to estimate the effect of adherence to antiretroviral medications on virologic failure in HIV positive individuals.

BackgroundLong-term effectiveness of asthma control medication has been shown in clinical trials but results from observational studies with children and adolescents are lacking. Marginal structural models estimated using targeted maximum likelihood methods are a novel statistiscal approach for such studies as it allows to account for time-varying confounders and time-varying treatment. Therefore, we aimed to calculate the long-term risk of reporting asthma symptoms in relation to control medication use in a real-life setting from childhood to adulthood applying targeted maximum likelihood estimation.MethodsIn the prospective cohort study SOLAR (Study on Occupational Allergy Risks) we followed a German subsample of 121 asthmatic children (9–11 years old) of the ISAAC II cohort (International Study of Asthma and Allergies in Childhood) until the age of 19 to 24. We obtained self-reported questionnaire data on asthma control medication use at baseline (1995–1996) and first follow-up (2002–2003) as well as self-reported asthma symptoms at baseline, first and second follow-up (2007–2009). Three hypothetical treatment scenarios were defined: early sustained intervention, early unsustained intervention and no treatment at all. We performed longitudinal targeted maximum likelihood estimation combined with Super Learner algorithm to estimate the relative risk (RR) to report asthma symptoms at SOLAR I and SOLAR II in relation to the different hypothetical scenarios.ResultsA hypothetical intervention of early sustained treatment was associated with a statistically significant risk increment of asthma symptoms at second follow-up when compared to no treatment at all (RR: 1.51, 95% CI: 1.19–1.83) or early unsustained intervention (RR:1.38, 95% CI: 1.11–1.65).ConclusionsWhile we could confirm the tagerted maximum likelihood estimation to be a usable and robust statistical tool, we did not observe a beneficial effect of asthma control medication on asthma symptoms. Because of potential due to the small sample size, lack of data on disease severity and reverse causation our results should, however, be interpreted with caution.

Efficient local IV estimation of an empirical auction model

Consider a study in which one observes n independent and identically distributed random variables whose probability distribution is known to be an element of a particular statistical model, and one is concerned with estimation of a particular real valued pathwise differentiable target parameter of this data probability distribution. The targeted maximum likelihood estimator (TMLE) is an asymptotically efficient substitution estimator obtained by constructing a so called least favorable parametric submodel through an initial estimator with score, at zero fluctuation of the initial estimator, that spans the efficient influence curve, and iteratively maximizing the corresponding parametric likelihood till no more updates occur, at which point the updated initial estimator solves the so called efficient influence curve equation. In this article we construct a one-dimensional universal least favorable submodel for which the TMLE only takes one step, and thereby requires minimal extra data fitting to achieve its goal of solving the efficient influence curve equation. We generalize these to universal least favorable submodels through the relevant part of the data distribution as required for targeted minimum loss-based estimation. Finally, remarkably, given a multidimensional target parameter, we develop a universal canonical one-dimensional submodel such that the one-step TMLE, only maximizing the log-likelihood over a univariate parameter, solves the multivariate efficient influence curve equation. This allows us to construct a one-step TMLE based on a one-dimensional parametric submodel through the initial estimator, that solves any multivariate desired set of estimating equations.

Covariate adjustment using linear models for continuous outcomes in randomized trials has been shown to increase efficiency and power over the unadjusted method in estimating the marginal effect of treatment. However, for binary outcomes, investigators generally rely on the unadjusted estimate as the literature indicates that covariate-adjusted estimates based on the logistic regression models are less efficient. The crucial step that has been missing when adjusting for covariates is that one must integrate/average the adjusted estimate over those covariates in order to obtain the marginal effect. We apply the method of targeted maximum likelihood estimation (tMLE) to obtain estimators for the marginal effect using covariate adjustment for binary outcomes. We show that the covariate adjustment in randomized trials using the logistic regression models can be mapped, by averaging over the covariate(s), to obtain a fully robust and efficient estimator of the marginal effect, which equals a targeted maximum likelihood estimator. This tMLE is obtained by simply adding a clever covariate to a fixed initial regression. We present simulation studies that demonstrate that this tMLE increases efficiency and power over the unadjusted method, particularly for smaller sample sizes, even when the regression model is mis-specified.

Causal attribution, which seeks to explain the reasons behind events or behaviors, plays a critical role in causal inference and deepens our understanding of cause-and-effect relationships in scientific research. The probabilities of necessary causation (PN) and sufficient causation (PS) are two of the most common quantities for attribution in causal inference. While several works have explored the identification or bounds of PN and PS, efficient estimation remains unaddressed. To fill this gap, this paper focuses on obtaining semiparametric efficient estimators of PN and PS under two sets of identifiability assumptions: strong ignorability and monotonicity, and strong ignorability and conditional independence. We derive efficient influence functions and semiparametric efficiency bounds for PN and PS under the two sets of identifiability assumptions, respectively. Based on this, we propose efficient estimators for PN and PS and show their large sample properties. Extensive simulations validate the superiority of our estimators compared to competing methods. We apply our methods to a real-world dataset to assess various risk factors affecting stroke.

Several cluster-randomized trials are underway to investigate the implementation and effectiveness of a universal test-and-treat strategy on the HIV epidemic in sub-Saharan Africa. We consider nesting studies of pre-exposure prophylaxis within these trials. Pre-exposure prophylaxis is a general strategy where high-risk HIV- persons take antiretrovirals daily to reduce their risk of infection from exposure to HIV. We address how to target pre-exposure prophylaxis to high-risk groups and how to maximize power to detect the individual and combined effects of universal test-and-treat and pre-exposure prophylaxis strategies. We simulated 1000 trials, each consisting of 32 villages with 200 individuals per village. At baseline, we randomized the universal test-and-treat strategy. Then, after 3 years of follow-up, we considered four strategies for targeting pre-exposure prophylaxis: (1) all HIV- individuals who self-identify as high risk, (2) all HIV- individuals who are identified by their HIV+ partner (serodiscordant couples), (3) highly connected HIV- individuals, and (4) the HIV- contacts of a newly diagnosed HIV+ individual (a ring-based strategy). We explored two possible trial designs, and all villages were followed for a total of 7 years. For each village in a trial, we used a stochastic block model to generate bipartite (male-female) networks and simulated an agent-based epidemic process on these networks. We estimated the individual and combined intervention effects with a novel targeted maximum likelihood estimator, which used cross-validation to data-adaptively select from a pre-specified library the candidate estimator that maximized the efficiency of the analysis. The universal test-and-treat strategy reduced the 3-year cumulative HIV incidence by 4.0% on average. The impact of each pre-exposure prophylaxis strategy on the 4-year cumulative HIV incidence varied by the coverage of the universal test-and-treat strategy with lower coverage resulting in a larger impact of pre-exposure prophylaxis. Offering pre-exposure prophylaxis to serodiscordant couples resulted in the largest reductions in HIV incidence (2% reduction), and the ring-based strategy had little impact (0% reduction). The joint effect was larger than either individual effect with reductions in the 7-year incidence ranging from 4.5% to 8.8%. Targeted maximum likelihood estimation, data-adaptively adjusting for baseline covariates, substantially improved power over the unadjusted analysis, while maintaining nominal confidence interval coverage. Our simulation study suggests that nesting a pre-exposure prophylaxis study within an ongoing trial can lead to combined intervention effects greater than those of universal test-and-treat alone and can provide information about the efficacy of pre-exposure prophylaxis in the presence of high coverage of treatment for HIV+ persons.

Abstract: Real-world evidence (RWE) studies face bias from confounding, missingness, and informative censoring, limiting credible effect estimation for policy and practice. This study proposes an AI-augmented targeted learning workflow in Stata for average and heterogeneous treatment effects. The approach integrates programmatic cohort construction, principled handling of missing data and censoring, and automated model selection via a Super Learner ensemble that screens candidate learners using cross-validation and meta-learning heuristics. Methods include estimation of nuisance functions for propensity and outcome, targeted maximum likelihood estimation (TMLE) for average treatment effects, and doubly robust DR-learners for conditional effects; robustness checks cover overlap diagnostics, sensitivity to unmeasured confounding, and transportability assessments across sites. In synthetic yet realistic RWE benchmarks, the pipeline attains lower absolute bias and shorter confidence intervals while retaining nominal coverage, with stable runtimes suitable for routine analyses. Implications include an auditable, reproducible pathway to deploy targeted learning within standard Stata workflows, enabling decision-grade RWE under common data irregularities; limitations involve reliance on positivity, data quality, and the external validity of simulated scenarios. This matters because health and policy stakeholders require defensible causal estimates from imperfect observational data. What is provided is an end-to-end template, with diagnostics and reporting artifacts aligned to regulatory expectations. Keywords Real-world evidence, Stata, Targeted maximum likelihood estimation, Super Learner, Doubly robust estimation, Propensity score, Causal inference, Treatment effect heterogeneity, DR-learner, Sensitivity analysis, Positivity and overlap, Reproducible analytics

ABSTRACTBackgroundSurvey data are essential in paediatric epidemiology, providing valuable insights into child health outcomes. The potential outcomes framework has advanced causal inference using observational data. However, traditional design‐based adjustments, especially sample weights, are often overlooked. This omission limits the ability to generalise findings to the broader population.ObjectiveThis study demonstrates three approaches for estimating the population average treatment effect (PATE) in a practical example, examining the impact of household second‐hand smoke (SHS) exposure on blood pressure in school‐aged children.MethodsUsing data from the National Health and Nutrition Examination Survey (NHANES) 2017–2020, we assessed the effect of household SHS exposure, a non‐randomised treatment, on blood pressure in school‐aged children. We applied estimators based on Inverse Probability of Treatment Weighting (IPTW), G‐computation, Targeted Maximum Likelihood Estimation (TMLE), and regression adjustment. Models without adjustments were run for comparison. We examined point estimates and the efficiency of the estimates obtained from these methods.ResultsThe largest differences were observed between the unadjusted regression models and the fully adjusted methods (IPTW, G‐computation, and TMLE), which account for both confounding and survey weights. While the inclusion of the sample weights leads to wider confidence intervals for all methods, G‐computation and TMLE showed comparatively narrower confidence intervals. Confidence intervals for the models not adjusted for sample weights were likely underestimated.ConclusionsThis study highlights the important role of sample weights in causal inference. Generalisability of the average treatment effect as estimated on data sampled using common survey designs to a defined population requires the use of sample weights. The estimators described provide a framework for incorporating sample weights, and their use in health research is recommended.

Efficient estimation for partially linear varying coefficient models when coefficient functions have different smoothing variables

The authors consider semiparametric efficient estimation of parameters in the conditional mean model for a simple incomplete data structure in which the outcome of interest is observed only for a random subset of subjects but covariates and surrogate (auxiliary) outcomes are observed for all. They use optimal estimating function theory to derive the semiparametric efficient score in closed form. They show that when covariates and auxiliary outcomes are discrete, a Horvitz‐Thompson type estimator with empirically estimated weights is semiparametric efficient. The authors give simulation studies validating the finite‐sample behaviour of the semiparametric efficient estimator and its asymptotic variance; they demonstrate the efficiency of the estimator in realistic settings.

We provide an overview of classical and newer methods for the control of confounding of time-invariant interventions to permit causal inference in public health evaluations. We estimated the causal effect of gender on all-cause mortality in a large HIV care and treatment program supported by the President's Emergency Program for AIDS Relief in Dar es Salaam, Tanzania, between 2004 and 2012. We compared results from multivariable modeling, three propensity score methods, inverse-probability weighting, doubly robust methods, and targeted maximum likelihood estimation. Considerable confounding was evident, and, as expected by theory, all methods considered gave the same result, a statistically significant approximately 20% increased mortality rate in men. In general, there is no clear advantage of any of these methods for causal inference over classical multivariable modeling, from the point of view of either bias reduction or efficiency. Rather, given sufficient data to adequately fit the multivariable model to the data, multivariable modeling will yield causal estimates with the greatest statistical efficiency. All methods can adjust only for well-measured confounders-if there are unmeasured or poorly measured confounders, none of these methods will yield causal estimates.

The misspecification of gravity spatial interaction models has recently been described by the author. The bias in parameter estimates that results from such misspecification appears to produce the ‘map pattern effect’ or ‘spatial structure bias’ in estimated distance-decay parameters. A further aspect of the misspecification bias in gravity parameter estimates is explored here. The severity of the bias is shown to vary in a predictable manner with variations in spatial structure. In particular, the bias is shown to be dependent upon the pattern of accessibility that exists within a spatial system. The relevant aspects of this pattern are discussed for intraurban and interurban (or interregional) flow matrices. It is shown that from an examination of the spatial structure of centres in a spatial system it is possible, a priori, to identify whether significant bias will arise in the calibration of a gravity model. Certain configurations of centres are shown theoretically to produce maximal bias. The author thus answers the question, “why do gravity parameter estimates appear to be biased in some systems but not in others?”