4 89 TRANSFERRING TARGETED MAXIMUM LIKELIHOOD ESTIMATION INTO SPORT SCIENCE 1. Estimate \;( , ) (e.g., using machine learning or a parametric model). That is build an estimator for [ ∣ , ]. 2. Generate predictions from the estimator for each observation, where we set A for each observation (i.e., create counterfactual worlds). That is, we timator for each observation, where we set \;( = 0, ) ). With this we make predictions in and \;( = 1, ) for each Oi ∈ O (discarding the original values of A). With this we make predictions in the two counterfactual worlds ‘what if everyone received a treatment?’ versus ‘what if no one received treatment?’. 3. Estimate ψn using the G-computation formula as defined in Equation (1) Note that to estimate Q0, W we use the empirical distribution of W, and give each a weight of " ! . . In our initial example we assume a simplistic parametric linear model. Following the steps, we first estimate In our initial example we assume \;( , ) ≡ [ ∣ , ]. Using a linear model, such as GLM, this can be estimated as ` , ( , ) ≡ [ ∣ , ] = + + (2) With the formula in Equation (4) we can estimate Ŷ1 and Ŷ0 We use the subscript 1 and 0 on Ŷ to indicate that this value of Ŷ was calculated by respectively setting A = 1 and A = 0. That is, Ŷx is the evaluation of Equation (4) for all On, resulting in a list of tuples { Ŷ1, Ŷ 0} ∀ Oi ∈ O , which can be used to calculate the ATE as = ∑ Q g [ ∣ = , ]− [ ∣ = , ]h (3) = ∑ Q Z − Z (4) 3.6.2. Super learning and TMLE based estimation While the linear model provides an initial estimate, the underlying estimator follows a strictly parametric and linear nature, and thus poses various assumptions on the model that we currently cannot assume. To prevent these assumptions, the alternative is to use flexible machine learning techniques in a super learner approach and applying Targeted Maximum Likelihood estimation to perform the estimation of ψn. Note that we describe some of the background and intuition behind Super Learner and TMLE. For more information and formal proofs we would like to refer to Van der Laan and Rose [2]10. 10 There are also several R packages available that automate the process discussed below. For this, see https://tlverse.org/
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