Targeted maximum likelihood estimation for a binary treatment: A tutorial

Abstract

When estimating the average effect of a binary treatment (or exposure) on an outcome, methods that incorporate propensity scores, the G-formula, or targeted maximum likelihood estimation (TMLE) are preferred over naïve regression approaches, which are biased under misspecification of a parametric outcome model. In contrast propensity score methods require the correct specification of an exposure model. Double-robust methods only require correct specification of either the outcome or the exposure model. Targeted maximum likelihood estimation is a semiparametric double-robust method that improves the chances of correct model specification by allowing for flexible estimation using (nonparametric) machine-learning methods. It therefore requires weaker assumptions than its competitors. We provide a step-by-step guided implementation of TMLE and illustrate it in a realistic scenario based on cancer epidemiology where assumptions about correct model specification and positivity (ie, when a study participant had 0 probability of receiving the treatment) are nearly violated. This article provides a concise and reproducible educational introduction to TMLE for a binary outcome and exposure. The reader should gain sufficient understanding of TMLE from this introductory tutorial to be able to apply the method in practice. Extensive R-code is provided in easy-to-read boxes throughout the article for replicability. Stata users will find a testing implementation of TMLE and additional material in the Appendix S1 and at the following GitHub repository: https://github.com/migariane/SIM-TMLE-tutorial.

Keywords: causal inference; ensemble Learning; machine learning; observational studies; targeted maximum likelihood estimation.

Figure 1

Direct acyclic graph. Legend: Conditional exchangeability of the treatment effect or exposure (A) on cancer mortality (Y) is obtained through conditioning on a set of available covariates (Y(1),Y(0) ⊥ A|W). The average treatment effect for the structural framework is estimated as the average risk difference between the expected effect of the treatment conditional on W among those treated (E(Y|A = 1; W)) and the expected effect of the treatment conditional on W among those untreated (E(Y|A = 0; W)). Y: mortality binary indicator (1 death, 0 alive), A: binary treatment for cancer with monotherapy versus dual therapy (1 Mono; 0 Dual); W: W ₁: sex; W ₂: age at diagnosis; W ₃: cancer stage, TNM classification; W ₄: comorbidities [Colour figure can be viewed at http://wileyonlinelibrary.com]

Figure 2

Probability density function of the propensity score by treatment status for one randomly selected sample from 1000 Monte Carlo simulations [Colour figure can be viewed at http://wileyonlinelibrary.com]