Targeted Maximum Likelihood for ATE, why not just GLM for ATE?

I have been attempting to learn targeted maximum likelihood (TMLE), which is a struggle. The estimand of interest is often the average treatment effect. If we have an outcome $$Y$$, intervention $$A$$ and a set of covariates $$W$$ this is defined as,

$$ATE =$$ $$E_W( E(Y|A = 1, W) - E(Y | A = 0, W)) =$$ $$E_W( E(Y|A = 1 ,W)) - E_W( E(Y|A = 0,W))$$

where $$E_W$$ denotes the expectation with respect to the distribution of the covariates. Why do we need targeted maximum likelihood here? For any GLM I can estimate $$E_W( E(Y|A = 1,W))$$ by first just finding $$E(Y|A = 1,W)$$ explicitly from the GLM (similarly for $$E(Y|A = 0,W)$$) and marginalizing out over the distribution of my covariates. What is TMLE getting me here that traditional MLE of a GLM is not?

• TMLE allows you to estimate $E[Y \mid A, W]$ nonparametrically using machine-learning. Unlike glm, TMLE makes no parametric assumptions on the form of $E[Y \mid A, W]$ (i.e. does not assume it is a linear model). Thus, TMLE gives consistent and efficient estimates and inference under much weaker assumptions than glm.
– Lars
Oct 30 '21 at 2:06
• Also, if $E[Y \mid A, W]$ is estimated using machine-learning then the marginalized estimator is usually not root-n consistent nor asymptotically normally distributed due to bias. Thus, inference is unavailable. TMLE allows you to correct/target/adjust an initial estimator of $E[Y \mid A, W]$ in a way such that the resulting marginalized estimator is root-n consistent, asymptotically normal and statistically efficient.
– Lars
Oct 30 '21 at 3:08

The answer is double robustness. Like all doubly-robust methods, with TMLE you get two chances to get the model correct and the treatment effect estimate is consistent if either is correct (or approach at a given rate). With g-computation using a GLM, you only get one chance to get the model right. If that model is wrong (and it almost certainly is), then your treatment effect estimate is not consistent. You can use flexible machine learning methods with TMLE that are more likely to approximate the true form of either the treatment and outcome model. Indeed, given the inability of an outcome model GLM to even plausibly capture the true outcome model, one should almost never use it to estimate treatment effect, whereas TMLE should be among your first choices.

• To build on Noah's point regarding flexible ML methods with TMLE, the form of the TML estimator means that the approximation errors of the both models (the treatment and outcome models) are multiplied together. This slower convergence is not something that happens with other singly-robust estimators. See the following for further details on doubly robust estimators and convergence stats.stackexchange.com/questions/482445/… Nov 3 '21 at 19:45
• @pzivich Your comment is referring to the need for cross-splitting when using doubly robust methods right? (so-called "bias from overfitting?") Dec 19 '21 at 0:15
• @aranglol not exactly. Doubly robust methods don't require cross-fitting (or splitting) when used with parametric models. Cross-fitting is needed when using machine learning algorithms that are non-Donsker, which is separate from the convergence issue. The link above gives more details on the distinction Dec 20 '21 at 1:06

If the dimension of $$W$$ is large, computing the MLE of the whole GLM and then marginalising need not give the efficient estimator of the ATE. From an asymptotic viewpoint, suppose the dimension of $$W$$ grows fast enough that the whole parameter vector of the GLM is not estimable at $$\sqrt{n}$$ rate. It's still possible that the ATE is estimable at $$\sqrt{n}$$ rate, and that the plug-in estimator you describe isn't efficient.

Heuristically, efficient estimation of the GLM (by maximum likelihood or any other way) involves getting the bias:variance tradeoff right for the whole parameter vector. But for causal inference what you care about is a particular one-dimensional functional of the parameter vector. It turns out that the optimal tradeoffs can be different. Targeted MLE focuses on estimating the one-dimensional parameter of interest, and accounts for the need to estimate the other parameters by considering the one-dimensional submodel of the GLM in which estimation of the ATE is hardest.

If the dimension of $$W$$ is small enough that the whole GLM can be estimated well, the simple plug-in estimator will be efficient. This is what you get with an asymptotic argumen that treats the dimension of $$W$$ as fixed as the sample size increases.