Targeted maximum likelihood estimation

Targeted Maximum Likelihood Estimation (TMLE) (also, more accurately referenced as Targeted Minimum Loss-Based Estimation) is a general statistical estimation framework for causal inference and semiparametric models. TMLE combines ideas from maximum likelihood estimation, semiparametric efficiency theory, and machine learning. It was introduced by Mark J. van der Laan and colleagues in the mid-2000s as a method that yields asymptotically efficient plug-in estimators while allowing the use of flexible, data-adaptive algorithms such as ensemble machine learning for nuisance parameter estimation. ^{[1] [2]}

TMLE is used in epidemiology, biostatistics, and the social sciences to estimate causal effects in observational and experimental studies. Applications of TMLE include Longitudinal TMLE (LTMLE) for time-varying treatments and confounders. Variations in how the targeting step in TMLE is carried out have resulted in various versions of TMLE such as Collaborative TMLE (CTMLE) and Adaptive TMLE for improved finite-sample performance and automated variable selection.

History

The TMLE framework was first described by van der Laan and Rubin (2006) as a general approach for the construction of efficient plug-in estimators of smooth features of the data density. It was demonstrated in the context of causal inference and missing data problems. ^[1] It was developed to address limitations of traditional doubly robust methods, such as Augmented Inverse Probability Weighting (AIPW), by respecting the plug-in principle in the sense that it respects that the target parameter is a function of the data density that is an element of the statistical model. TMLE estimates the data density or relevant parts of it with machine learning and targets these machine learning fits before it is plugged in the target parameter mapping. In this manner, a TMLE always respects global knowledge and satisfies known bounds such as that the target parameter is a probability . ^[3]

Since its introduction, TMLE has been developed in a series of theoretical and applied papers, culminating in book-length treatments of the method and its applications to survival analysis, adaptive designs, and longitudinal data. ^{[2] [4]}

Methodology

At its core, TMLE is a two-step estimation procedure:

1. Initial estimation: Machine learning methods (such as the Super Learner ensemble) are used to obtain flexible estimates of nuisance parameters, such as outcome regressions and propensity scores. ^[5]
2. Targeting step: The initial estimate is updated by solving a score equation (the efficient influence function) so that the final estimator is consistent, asymptotically normal, and efficient under mild regularity conditions. The targeted machine learning fit is then mapped into the corresponding estimator of the target parameter by simply plugging it in the target parameter mapping.

This approach balances the bias–variance trade-off by combining data-adaptive estimation with semiparametric efficiency theory. TMLE is doubly robust, meaning it remains consistent if either the outcome model or the treatment model is consistently estimated. ^[6]

Formula

Here we explain the TMLE of the average treatment effect of a binary treatment on an outcome adjusting for baseline covariates. Consider i.i.d. observations ${\displaystyle O_{i}=(W_{i},A_{i},Y_{i})}$ from a distribution ${\displaystyle P_{0}}$, where ${\displaystyle W}$ are baseline covariates, ${\displaystyle A}$ is a binary treatment, and ${\displaystyle Y}$ is an outcome. Let ${\displaystyle Q_{0}(a,w)=\mathbb {E} [Y\mid A=a,W=w]}$ represent the outcome model and ${\displaystyle g_{0}(a\mid w)=P(A=a\mid W=w)}$ represent the propensity score.

The average treatment effect (ATE) is given by ${\displaystyle \psi _{0}=\mathbb {E} \{Q_{0}(1,W)-Q_{0}(0,W)\}.}$

A basic TMLE for the ATE proceeds:

1. Estimate ${\displaystyle {\hat {Q}}^{0}(a,w)}$ and ${\displaystyle {\hat {g}}^{0}(a\mid w)}$ (often via Super Learner).
2. Define a (logistic) fluctuation submodel through ${\displaystyle {\hat {Q}}^{0}}$:

${\displaystyle \operatorname {logit} {\big (}{\hat {Q}}^{\varepsilon }(A,W){\big )}=\operatorname {logit} {\big (}{\hat {Q}}^{0}(A,W){\big )}+\varepsilon \,H(A,W),}$ where the clever covariate is ${\displaystyle H(A,W)={\frac {\mathbb {1} \{A=1\}}{{\hat {g}}^{0}(1\mid W)}}-{\frac {\mathbb {1} \{A=0\}}{{\hat {g}}^{0}(0\mid W)}}}$.

1. Choose ${\displaystyle {\hat {\varepsilon }}}$ to solve the score equation

${\displaystyle {\frac {1}{n}}\sum _{i=1}^{n}H(A_{i},W_{i})\{Y_{i}-{\hat {Q}}^{\varepsilon }(A_{i},W_{i})\}=0.}$

1. Update ${\displaystyle {\hat {Q}}^{*}={\hat {Q}}^{\hat {\varepsilon }}}$ and compute

${\displaystyle {\hat {\psi }}_{\text{TMLE}}={\frac {1}{n}}\sum _{i=1}^{n}{\big [}{\hat {Q}}^{*}(1,W_{i})-{\hat {Q}}^{*}(0,W_{i}){\big ]}.}$

For inference, the efficient influence function (EIF) is ${\displaystyle D^{*}(O_{i})=H(A_{i},W_{i})\{Y_{i}-{\hat {Q}}^{*}(A_{i},W_{i})\}+{\hat {Q}}^{*}(1,W_{i})-{\hat {Q}}^{*}(0,W_{i})-{\hat {\psi }}_{\text{TMLE}}.}$ The variance is estimated by ${\displaystyle {\hat {\sigma }}^{2}=n^{-1}\sum _{i=1}^{n}{\big (}D^{*}(O_{i}){\big )}^{2}}$, yielding Wald intervals ${\displaystyle {\hat {\psi }}_{\text{TMLE}}\pm z_{1-\alpha /2}\,{\hat {\sigma }}/{\sqrt {n}}}$. ^[2]

Applications

TMLE has been applied in:

• Epidemiology: Estimating causal effects of exposures and interventions in observational cohort studies. ^[7]
• Clinical trials and real-world evidence: The Targeted Learning roadmap provides a structured framework for generating and validating real-world evidence (RWE), bridging randomized trials and observational data using TMLE and related estimation techniques. This approach enables transparency, sensitivity analysis, and stronger causal inference for regulatory and clinical trial contexts. ^[8]
• High-dimensional settings: Integration with ensemble methods for causal effect estimation. ^[9] TMLE has been successfully applied in pharmacoepidemiology where a large number of covariates are automatically selected to adjust for confounding. In a study of post–myocardial infarction statin use and 1-year mortality, TMLE demonstrated robust performance relative to inverse probability weighting in scenarios with hundreds of potential confounders. ^[10]

Derivatives and extensions

• Longitudinal TMLE (LTMLE): A methodological extension of TMLE for longitudinal data with time-varying treatments, confounders, and censoring. It allows the estimation of dynamic treatment regimes and intervention-specific causal effects over time. This framework was originally introduced by van der Laan & Gruber (2012). ^[11]
• Collaborative TMLE (CTMLE): Enhances finite-sample performance and variable selection by collaboratively fitting the treatment mechanism in conjunction with the target parameter. ^{[12] [13]}

Software

Several R packages implement TMLE and related methods:

• tmle: Functions for binary, categorical, and continuous outcomes. ^[14]
• ltmle: Implementation for longitudinal data with time-varying treatments and outcomes. ^[15]
• ctmle: Algorithms for collaborative TMLE and adaptive variable selection. ^[16]
• SuperLearner: A theoretically grounded, cross-validated ensemble learning method that combines predictions from multiple algorithms to minimize predictive risk. Widely used in TMLE for estimating nuisance parameters. The original implementation is available as the R package SuperLearner. ^[17] Recent machine learning platforms like H2O AutoML implement similar ensemble strategies, combining diverse learners in parallel and leveraging stacking and blending techniques, effectively functioning as a large-scale Super Learner. ^[18]

See also

• Causal inference

References

1. van der Laan, Mark J.; Rubin, Daniel (2006). "Targeted Maximum Likelihood Learning". International Journal of Biostatistics. 2 (1): Article 11. doi:10.2202/1557-4679.1043.
2. van der Laan, Mark J.; Rose, Sherri (2011). Targeted Learning: Causal Inference for Observational and Experimental Data. Springer. ISBN   978-1-4419-9781-4.
3. Bang, Heejung; Robins, James M. (2005). "Doubly Robust Estimation in Missing Data and Causal Inference Models". Biometrics. 61 (4): 962–973. doi:10.1111/j.1541-0420.2005.00377.x.
4. van der Laan, Mark J.; Rose, Sherri (2018). Targeted Learning in Data Science: Causal Inference for Complex Longitudinal Studies. Springer. ISBN   978-3-319-65303-7.
5. van der Laan, Mark J.; Polley, Eric C.; Hubbard, Alan E. (2007). "Super Learner". Statistical Applications in Genetics and Molecular Biology. 6 (1): Article 25. doi:10.2202/1544-6115.1309. PMID   17910531.
6. Bang, Heejung; Robins, James M. (2005). "Doubly Robust Estimation in Missing Data and Causal Inference Models". Biometrics. 61 (4): 962–973. doi:10.1111/j.1541-0420.2005.00377.x.
7. Petersen, Maya L.; Porter, Kristin E.; Gruber, Susan; Wang, Yue; van der Laan, Mark J. (2012). "Diagnosing and responding to violations in the positivity assumption". Statistical Methods in Medical Research. 21 (1): 31–54. doi:10.1177/0962280210386207. PMC   4107929 . PMID   21030422.
8. Gruber, Susan; Phillips, Rachael V.; Lee, Hana; Concato, John; van der Laan, Mark J. (2023). "Evaluating and improving real-world evidence with Targeted Learning". BMC Medical Research Methodology. 23 (1): 178. doi: 10.1186/s12874-023-01998-2 . PMC   10394864 . PMID   37533017.
9. Schuler, Megan; Rose, Sherri (2017). "Targeted maximum likelihood estimation for causal inference in observational studies". American Journal of Epidemiology. 185 (1): 65–73. doi:10.1093/aje/kww165. PMID   27941068.
10. Pang, Menglan; Schuster, Tibor; Filion, Kristian; Eberg, Maria; Platt, Robert W. (2016). "Targeted Maximum Likelihood Estimation for Pharmacoepidemiologic Research". Epidemiology. 27 (4): 570–577. doi:10.1097/EDE.0000000000000487. PMC   4890840 .
11. van der Laan, Mark J.; Gruber, Susan (2012). "Targeted minimum loss based estimation of causal effects of multiple time point interventions". International Journal of Biostatistics. 8 (1): Article ?. doi:10.1515/1557-4679.1370.
12. Ju, Cheng; Gruber, Susan; Lendle, Samuel D.; van der Laan, Mark J. (2019). "Scalable collaborative targeted learning for high-dimensional data". Statistical Methods in Medical Research. 28 (2): 532–554. arXiv: 1703.02237 . doi:10.1177/0962280217729845.
13. Gruber, Susan; van der Laan, Mark J. (2010). "An application of collaborative targeted maximum likelihood estimation in causal inference and genomics". The International Journal of Biostatistics. 6 (1): Article 18. doi:10.2202/1557-4679.1182. PMC   3126668 . PMID   21731530.
14. Gruber, Susan; van der Laan, Mark J. (2012). "tmle: An R Package for Targeted Maximum Likelihood Estimation". Journal of Statistical Software. 51 (13): 1–35. doi: 10.18637/jss.v051.i13 .
15. Lendle, Samuel D.; Schwab, Jenny; Petersen, Maya L.; van der Laan, Mark J. (2017). "ltmle: An R Package Implementing Targeted Minimum Loss-Based Estimation for Longitudinal Data". Journal of Statistical Software. 81 (1): 1–21. doi: 10.18637/jss.v081.i01 .
16. Ju, Cheng; Gruber, Susan; van der Laan, Mark J. (2017). "ctmle: Collaborative Targeted Maximum Likelihood Estimation in R". Journal of Statistical Software. 80 (3): 1–30. doi: 10.18637/jss.v080.i03 .
17. van der Laan, Mark J.; Polley, Eric C.; Hubbard, Alan E. (2007). "Super Learner". Statistical Applications in Genetics and Molecular Biology. 6 (1): Article 25. doi:10.2202/1544-6115.1309.
18. "H2O AutoML: Automatic Machine Learning". H2O.ai. 2025-03-27.