Tutorial for Causal Inference 375
way to obtain statistical inference for parameters, such as the G-computation estimand
Ψ(P
0
). Treating the final algorithm as if it were prespecified ignores the selection process.
Furthermore, the selected algorithm was tailored to maximize/minimize some criterion
with regard to the conditional expectation E
0
(Y |A, W ) and will, in general, not provide
the best bias–variance trade-off for estimating the statistical parameter Ψ(P
0
). Indeed,
estimating the conditional mean outcome Y in every stratum of (A, W )isamuch
more ambitious task than estimating one number (the difference in conditional means,
averaged with respect to the covariate distribution). Thus, without an additional step, the
resulting estimator will be overly biased relative to its standard error, preventing accurate
inference.
Targeted maximum likelihood estimation (TMLE) provides a way forward [3,40]. TMLE
is a general algorithm for the construction of double robust, semiparametric, efficient
substitution estimators. TMLE allows for data-adaptive estimation while obtaining valid
statistical inference. The algorithm is detailed in Chapter 22. Although TMLE is a
general algorithm for a wide range of parameters, we focus on its implementation for the
G-computation estimand. Briefly, the TMLE algorithm uses information in the estimated
exposure mechanism
ˆ
P (A|W ) to update the initial estimator of the conditional mean
E
0
(Y |A, W ). The targeted estimates are then substituted into the parameter mapping.
The updating step achieves a targeted bias reduction for the parameter of interest Ψ(P
0
)
and serves to solve the efficient score equation. As a result, TMLE is a double robust
estimator; it will be consistent for Ψ(P
0
) is either the conditional expectation E
0
(Y |A, W )
or the exposure mechanism P
0
(A|W ) is estimated consistently. When both functions are
consistently estimated at a fast enough rate, the TMLE will be efficient in that it achieves the
lowest asymptotic variance among a large class of estimators. These asymptotic properties
typically translate into lower bias and variance in finite samples. The advantages of TMLE
have been repeatedly demonstrated in both simulation studies and applied analyses [37,41–
43]. The procedure is available with standard software such as the tmle and ltmle packages
in R [44–46].
Thus far, we have discussed obtaining a point estimate from a simple or targeted substi-
tution estimator. To create confidence intervals and test hypotheses, we also need to quantify
uncertainty. A simple substitution estimator based on a correctly specified parametric model
is asymptotically linear, and its variance can be approximated by the variance of its influence
curve, divided by sample size n. It is worth emphasizing that our estimand Ψ(P
0
) often does
not correspond to a single coefficient, and therefore we usually cannot read off the reported
standard error from common software. Under reasonable conditions, the TMLE is also
asymptotically linear and inference can be based on an estimate of its influence curve.
Overall, this chapter focused on substitution estimators (simple and targeted) of the
G-computation identifiability result [27]. The simple substitution estimator only requires
an estimate of the marginal distribution of baseline covariates P
0
(W ) and the conditional
expectation of the outcome, given the exposure and covariates E
0
(Y |A, W ). TMLE also
requires an estimate of the exposure mechanism P
0
(A|W ). There are many other algorithms
available for estimation of Ψ(P
0
). A popular class of estimators relies only on estimation
of the exposure mechanism [47–49]. Inverse probability of treatment weighting (IPTW)
estimators, for example, control for measured confounders by up-weighting exposure–
covariate groups that are underrepresented and down-weighting exposure–covariate groups
that are overrepresented (relative to what would be seen were the exposure randomized).
Its double robust counterpart, augmented-IPTW, shares many of the same properties as
TMLE [50,51]. A key distinction is that IPTW and augmented-IPTW are solutions to
estimating equations and therefore respond differently in the face of challenges due to strong
confounding and rare outcomes [37,52]. Throughout, we maintain that estimators should