Targeted maximum likelihood estimator of the ATE

Clever covariates (Gruber & van der Laan 2012, eqs 2-3) \(H_1^*=I(A=1)/g(1|W)\), \(H_0^*=I(A=0)/g(0|W)\), and the fluctuation \(logit(Q^*) = logit(Q) + \epsilon_0 H_0^* + \epsilon_1 H_1^*\) fitted by logistic regression of Y on the clever covariates with offset \(logit(Q)\). y must be bounded in [0, 1] so the fluctuation stays inside the model space. When the initial fit already solves the score, epsilon is 0 and the estimate equals plugin.

Usage

Caustmle(y, T, ps, Q1, Q0, n_iter = 100L)

Arguments

y

Outcomes in [0, 1].

T

Treatment indicator, 0/1.

ps

Propensity \(g(1|W)\), strictly in (0, 1).

Q1, Q0

Initial outcome fits under treatment and control.

n_iter

Fixed Newton steps for the fluctuation.

Value

Named list with estimate, ATE_TMLE, IF, se, ci_lower, ci_upper, epsilon, EY1, EY0, plugin, n, method.

References

van der Laan & Rubin (2006) Int J Biostat 2(1):11.

Examples

Caustmle(c(0.8, 0.3, 0.8, 0.3), c(1, 0, 1, 0), rep(0.5, 4),
         rep(0.8, 4), rep(0.3, 4))
#> $estimate
#> [1] 0.5
#> 
#> $ATE_TMLE
#> [1] 0.5
#> 
#> $IF
#> [1] 0 0 0 0
#> 
#> $se
#> [1] 0
#> 
#> $ci_lower
#> [1] 0.5
#> 
#> $ci_upper
#> [1] 0.5
#> 
#> $epsilon
#> [1] 0 0
#> 
#> $EY1
#> [1] 0.8
#> 
#> $EY0
#> [1] 0.3
#> 
#> $plugin
#> [1] 0.5
#> 
#> $n
#> [1] 4
#> 
#> $method
#> [1] "TMLE of the ATE (van der Laan & Rubin 2006)"
#>