Targeted Data Adaptive Estimation of the Causal Dose–Response Curve

Abstract

Estimation of the causal dose-response curve is an old problem in statistics. In a non-parametric model, if the treatment is continuous, the dose-response curve is not a pathwise differentiable parameter, and no ffiffiffi n p -consistent estimator is available. However, the risk of a candidate algorithm for estimation of the dose-response curve is a pathwise differentiable parameter, whose consistent and efficient estimation is possible. In this work, we review the cross-validated augmented inverse probability of treatment weighted estimator (CV A-IPTW) of the risk and present a cross-validated targeted minimum loss-based estimator (CV-TMLE) counterpart. These estimators are proven consistent and efficient under certain consistency and regularity conditions on the initial estimators of the outcome and treatment mechanism. We also present a methodology that uses these estimated risks to select among a library of candidate algorithms. These selectors are proven optimal in the sense that they are asymptotically equivalent to the oracle selector under certain consistency conditions on the estimators of the treatment and outcome mechanisms. Because the CV-TMLE is a substitution estimator, it is more robust than the CV-AIPTW against empirical violations of the positivity assumption. This and other small sample size differences between the CV-TMLE and the CV-A-IPTW are explored in a simulation study.

Targeted Learning of the Mean Outcome under an Optimal Dynamic Treatment Rule

“…The key property that we need from the ε n and the corresponding update $Q_{nj}^{d_{nj}\ast }$ is that it (approximately) solves the cross-validated empirical mean of the efficient influence curve: $$E_{B_n}{P}_{n,B_n}^1{D}^{\ast }true(d_{nj},Q_{nj}^{d_{n{j}^{\ast }}},g_{nj}true)=O_{P_0}(1∕\sqrt{n}).$$ The CV-TMLE implementation presented in the appendix satisfies this equation with $o_{P_0}(1∕\sqrt{n})$ replaced by the 0. The proposed estimator of ${\tilde{\psi }}_{0n}$ is given by $${\tilde{\psi }}_n^{\ast }\equiv E_{B_n}{\Psi }_{d_{nj}}true(Q_{nj}^{{d_{nj}}^{\ast }}true).$$ In the current literature we have referred to this estimator as the CV-TMLE [53–56]. We give a concrete CV-TMLE algorithm for ${\tilde{\psi }}_n^{\ast }$ in “CV-TMLE of the mean outcome under data adaptive V -optimal rule” in Appendix B, but note that other CV-TMLE algorithms can be derived using the approach in this section for different choices of loss function ϕ and submodels.…”

“…With this minor twist, the (same) CV-TMLE is now used to target the average of training sample-specific target parameters averaged across the J training samples. This utilization of CV-TMLE was already used to estimate the average (across training samples) of the true risk of an estimator based on a training sample in van der Laan and Petersen [53] and Díaz and van der Laan [54], so that this just represents a generalization of that application of CV-TMLE to estimate general data adaptive target parameters as proposed in van der Laan et al [46]. …”

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

“…Due to this generality, its statistical applications are diverse and widespread, going beyond the construction of an efficient estimator of a pathwise differentiable target parameter for arbitrary semi-parametric models and pathwise differentiable target parameter mappings: collaborative targeted maximum likelihood estimation (CTMLE) for targeted estimation of the nuisance parameter in the canonical gradient (van der Laan and Rose, 2011; van der Laan and Gruber, 2010; Gruber and van der Laan, 2012; Stitelman and van der Laan, 2010; Gruber and van der Laan, 2010); cross-validated TMLE (CV-TMLE) to robustify the bias-reduction of the TMLE-step (Zheng and van der Laan, 2011; van der Laan and Rose, 2011); guaranteed improvement w.r.t. a user supplied asymptotically linear estimator (Gruber and van der Laan, 2012; Lendle et al, 2013); targeted initial estimator through empirical efficiency maximization (Rubin and van der Laan, 2008; van der Laan and Rose, 2011); double robust inference by targeting censoring/treatment mechanism (van der Laan, 2012); CV-TMLE to estimate data adaptive target parameters such as the risk of a candidate estimator and thereby develop a super-learner that uses CV-TMLE instead of the normal cross-validated empirical risk (van der Laan and Petersen, 2012; Díaz and van der Laan, 2013, In press); higher-order TMLE in order to replace in the above proof R 2 () by a higher order term (Carone et al, 2014; Diaz et al, 2015). …”

Super-Learning of an Optimal Dynamic Treatment Rule

“…The CV- TMLE for the second time point is identical to the CV-TMLE presented in Appendix B.2 of van der Laan and Luedtke [34], with the exception that the covariate for ε 2 is replaced by $$\frac{A_{2}false(0false)Ifalse(Afalse(1false)=I\left({\displaystyle {\sum }_j{\alpha }_{A(1),j}{f}_{2,u,j}(A(0),V(1))>0)}\right)}{g_{0}false(Ofalse)},$$and the covariate for ε 1 is replaced by A 2 (0)/ g 0, A (0) ( O ). The conditions for the validity of the resulting risk estimate are not presented here, but are analogous to those presented in Diaz et al [39]. The CV-TMLE has the same double robustness and asymptotic efficiency properties as the cross-validated empirical mean of the double robust loss.…”

“…The CV-TMLE was originally proposed in Zheng and van der Laan [38]. Diaz et al [39] use a CV-TMLE to estimate the risk of the causal dose response curve. In van der Laan and Luedtke [34] we presented a CV-TMLE for the cross-validated mean outcome under a fitted rule.…”