## 2 Methods
### Data Generation and Preparation for Uplift  
This section summarizes the process used to generate data from the 1-M matched control case study fitted with URL ([PERSON] et al., 2015), as the original numerical and categorical data of the study was unavailable, and its consequent preparation for ATE-Uplift, including factual and counterfactuals. The data generation involves drawing samples from the existing ULR model by [PERSON] et al. (2015).  
Let \(y\) be the case-control study, where \(y=1\) for a case and \(y=0\) for the control. Let \(x_{m}=\{x_{ml},x_{m2}\}\) a vector of matching variables, \(x_{e}\) and exposure associated with the case-control status, and \(x_{0}\) a vector of unmatched dummy variables corresponding to the strata of the matched variables. The ULR model, assuming no interaction between the predictors, is given by,  
\[logit(\pi)=\beta_{0}+\beta_{e}x_{e}+\beta_{m}^{T}x_{m}+\beta_{0}^{T}x_{0} \tag{1}\]  
where \(\pi\) is the probability of developing the disease, and \(\beta\)'s are the regression coefficients. \(T\) denotes transpose.  
The input data for the ATE model must be generated as a table of factual and counterfactual cases. 'Factuals' are persons in the factual universe (where a 'zero' value is assigned to benzene, or benzene = 0), and 'counterfactuals' are persons in a counterfactual universe (where 'one' value is assigned to benzene, or benzene = 1). The table is generated from the expected propensity of AML conditional on the inputs, which are predicted and converted into a binary variable, as shown in Table 1.  
In Table 1, person 1 and person 61 are identically theoretical persons, except for having different binary exposures to benzene and potentially different binary AML outcomes. Similarly, and with the same proviso, person 2 and person 62 are identical, person 3 and person 63 are identical, and so on.  
We used simulated data from the ULR of the case study; the simulation assumes no interaction between the benzene exposure and the rest of the predictors (socioeconomic status, race, birthplace, and parity); hence, [PERSON]'s paradox is ruled out. This paradox emerges when groups of data show one trend, but this is reversed when the groups are combined. Our factual and counterfactuals table, in particular, the BINaml column (the presence or absence of AML), is generated by the projection of the target variable of Equation (1) or its substitute, Equation (2), that is \(logit(\pi(x))\) of AML and its associated binary AML variable.  
\[logit(\pi(x))=\beta_{0}+\beta_{e}x_{e}+\beta_{m}^{T}x_{m} \tag{2}\]  
This projection requires Equation (1) to be completely specified, including the intercepts, which was omitted by [PERSON] et al. (2015) as is usual in matched case-controlled studies. Therefore, we substitute Equation (1) with Equation (2) to estimate the intercept, where \(\beta_{0}\) is a single number rather than a vector, yet \(\beta_{e}\), \(\beta_{m}^{T}\) are the same \(\beta_{e}\), \(\beta_{m}^{T}\)([PERSON] et al., 2015). We estimate \(\beta_{0}\) using Equation (2) following [PERSON]'s et al. (2018) optimization approach by using an estimated overall prevalence of AML for the years 1990-2007 and the simulated benzene exposure data. As input to this projection, we use two levels of benzene exposure: the minimum and maximum values of the benzene distribution. Having projected the binary AML variable, we inserted it into the BINaml column of the factual and counterfactuals table, along with the corresponding benzene exposures and other predictor values. Finally, we generated the factual and counterfactuals table by converting the benzene exposures from numerical to categorical (i.e., from a maximum value to 1 and from a minimum value to 0). We carry out this last conversion following the approach by [PERSON] (1993) and the California of Environmental Health Hazzard Assessment's reference exposure level for benzene as 0.940 ppm (0.903 ppb) ([PERSON] et al., 2015).