TMLE for the ATE with a missing-at-random outcome

Missingness is a second intervention node. Writing Delta = 1 for an observed outcome, the joint intervention is "set A = a and Delta = 1", so the clever covariate carries both probabilities, H = Delta/pi(A, X) * [A/g(X) - (1 - A)/(1 - g(X))], with pi(a, X) evaluated at the counterfactual arm. The initial outcome regression is fitted on complete cases only, and the fluctuation Q*(a, X) = Q(a, X) + eps/(g_a(X) pi(a, X)) uses the arm-specific denominator. With no missingness pi == 1 and the estimator collapses onto the standard point-treatment TMLE.

Usage

Tmlmda(y, D, X, missing)

Arguments

y

Outcome; entries with missing = 1 are ignored.

D

Binary treatment.

X

Covariates.

missing

1 if the outcome is missing, 0 if observed.

Value

List with estimate, se, eps, n_obs, n.

References

Rotnitzky, A., Robins, J. M. & Scharfstein, D. O. (1998). JASA 93(444):1321-1339; Bang, H. & Robins, J. M. (2005). Biometrics 61(4):962-973.

Examples

set.seed(1)
r <- Tmlmda(y = rnorm(10), D = rbinom(10, 1, 0.5), X = rnorm(10), missing = rnorm(10)); TRUE
#> [1] TRUE