where H(t)=1 denotes the bad health outcome (later the analysis will look at food insecurity, obesity, anemia and reported subjective health) in treatment t.
Without any data, one can already see that the ATE will lie between -1 and 1. Due to the selection problem neither of the two probabilities can be directly inferred from the data.
However, applying the law of total probability with respect to the treatment status FS*, one can decompose each of the two probabilities: