4 92 CHAPTER 4 mechanism gn ( A| W) and the clever covariate Hn ( Ai, Wi). The treatment mechanism gn ( A∣ W) ≡ P( A∣ W), can be estimated using, for example, super learning. The clever covariate can balance the distributions of observed data of the samples under treatment versus the samples under control [11]. The clever covariate is defined for each individual as ( , )=Ü ( Q ) ( Q ∣ ) − ( Q ) ( Q ∣ )á (9) This clever covariate does not need estimation, but is used for fluctuating the initial estimate of \.;( , ), by relying on information collected about the treatment and control groups (i.e., t by relying on information collected about the treatment and control groups (i.e., the ratio between treated vs. control) [11]. Based on these definitions, the steps that are needed in order to estimate the TMLE are as follows: 1. Estimate \.;( , ) (e.g., using machine learning or a parametric model) 2. Generate predictions from the estimator for each observation, where we set A for each observation. That is, we estimate \.;( = 0, ) and \.;( = 1, ) for each + ∈ (discarding the original values of A) 3. Estimate the treatment mechanism .( ∣ ). 4. Create the clever covariate .( +, +). 5. Update / fluctuate the initial estimate of \.;( , ) using the clever covariate. The last step in this procedure describes updating the initial estimate. This is performed by applying a logistic regression on Y on H, using our initial estimate as offset. The logistic regression is used to ensure that TMLE is bounded, as introduced by min-max normalizing the outcome variable Y. The fluctuation can then be performed on a logistic scale [11]. ( ( ∣ , )) = ] ` ( , )^ + ( , ) (10) ` ∗( , ) = ] ] ` ( , )^ + ( , )^ (11) Case study For the current simulation study and the case study we did not implement these steps ourselves, but instead relied on an existing R packages that perform most of the calculations. We used the R ‘tmle’ package, version 1.5.0-1 for doing the Targeted Maximum Likelihood Estimation, and the ‘superlearner’ R package, version 2.0-26, for both the simulation study and the case study.
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