Function for targeting Qbarbar_M1_times_M2_star_a

There are two interesting features of this targeting problem. First, we see that the nuisance parameter Qbarbar_M1_times_M2_star_a can be viewed in two ways: (1) the conditional mean of Qbarbar_M1_a given C with respect to the marginal of M_2 given A = a_star, C; (2) the conditional mean of Qbarbar_M2_star_a given C with respect to the marginal of M_1 given A = a, C. The natural inclination then is to use a sum loss function. It seems that we're able to do that here. However, to generate the proper score, we would need to consider a submodel for the conditional mean of Qbarbar_M1_a/Qbarbar_M2_star_a given A and C; since, the inverse probability of treatment weight is a function of A. We also cannot include the IPTW as part of the loss function since we need one of the weights to be negative. So, we resort to an iterative approach, where we define a submodel and loss for the conditional mean of Qbarbar_M1_a and then a loss for the conditional mean of Qbarbar_M2_star_a. We iterate until the empirical mean of this portion of the canonical gradient is smaller than tol.

target_Qbarbar_M1_times_M2_star_a(
  Qbarbar,
  Y,
  A,
  a,
  a_star,
  gn,
  tol = 1/(sqrt(length(Y)) * log(length(Y))),
  max_iter = 25,
  iterative = FALSE,
  ...
)

Arguments

Qbarbar	Iterated mean estimates
Y	A vector of continuous or binary outcomes.
A	A vector of binary treatment assignment (assumed to be equal to 0 or 1).
a	The label for the treatment. The effects estimates returned pertain to estimation of interventional effects of a versus a_star.
a_star	The label for the treatment. The effects estimates returned pertain to estimation of interventional effects of a versus a_star.
gn	Power users may wish to pass in their own properly formatted list of the propensity score so that nuisance parameters can be fitted outside of intermed.
tol	The tolerance for stopping the iterative targeting procedure.
max_iter	The maximum number of iterations for the TMLE