TMLE for a multi-arm treatment: arm means and pairwise contrasts

With more than two arms a single clever covariate cannot solve every arm's score, so the fluctuation is saturated in the arm: one eps_a per arm with H_a = I(A = a)/g_a(X). That is what makes every arm mean, not just one contrast, a valid plug-in.

Usage

Tmlmlt(y, D, X, arm_set)

Arguments

y

Outcome.

D

Arm label of each subject.

X

Covariates.

arm_set

The distinct arm labels; the first is the reference.

Value

List with estimate (last arm minus first), se, psi, contrasts, n_arms, n.

Details

The multinomial is built from one-versus-rest logistics normalised to sum to one across the arm set – a working model, chosen because it is deterministic and mirrors exactly across the language arms; the targeting step repairs its bias as long as it or the outcome model is right. psi_a = mean_i [Q(a, X_i) + eps_a/g_a(X_i)].

References

Lendle, S. D. et al. (2017). Journal of Statistical Software 81(1).

Examples

Tmlmlt(y = c(1, 2, 3, 4, 5, 6, 7, 8), D = c(1, 2, 3, 4, 5, 6, 7, 8),
  X = c(1, 2, 3, 4, 5, 6, 7, 8), arm_set = c(1, 2, 3, 4, 5, 6, 7, 8))
#> $estimate
#> [1] -5.177984e+16
#> 
#> $se
#> [1] 9.998214
#> 
#> $psi
#> [1]  2.588992e+16  1.849280e+16  1.109568e+16  3.698560e+15 -3.698560e+15
#> [6] -1.109568e+16 -1.849280e+16 -2.588992e+16
#> 
#> $contrasts
#> [1]  0.000000e+00 -7.397120e+15 -1.479424e+16 -2.219136e+16 -2.958848e+16
#> [6] -3.698560e+16 -4.438272e+16 -5.177984e+16
#> 
#> $n_arms
#> [1] 8
#> 
#> $n
#> [1] 8
#> 
#> $method
#> [1] "TMLE for a multi-arm treatment with pairwise contrasts"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"