Estimation: Backdoor and Front-Door

This vignette covers the common estimate_causal() workflows for backdoor and front-door style graphs. All datasets are simulated with known true effects.

The workflow is: encode assumptions in an ADMG, call identify(), then call estimate_causal() only when the graph identifies the target effect. The estimand here is the average causal effect E[Y(1)-Y(0)].

Backdoor

Backdoor identification applies when observed pre-treatment variables block all noncausal paths from treatment to outcome. In this graph, X affects both A and Y, so a raw treated-control comparison is confounded. The adjustment formula is:

E[Y(a)] = sum_x E[Y | A=a, X=x] P(X=x)

The a-fixable check is the ADMG version of this adjustment logic. The package uses the treatment's Markov pillow as the adjustment set.

In this example, the only noncausal path from A to Y is A <- X -> Y. Conditioning on X blocks that path. After conditioning on X, treatment variation is treated as if randomized within levels of X; averaging E[Y | A=a, X] over the observed distribution of X gives E[Y(a)].

using CausalGraphs, DataFrames, Random, Statistics

Random.seed!(1)
n = 500
X = randn(n)
A = Float64.(rand(n) .< 1 ./ (1 .+ exp.(-X)))
Y = 2 .* A .+ X .+ randn(n)
data = DataFrame(X=X, A=A, Y=Y)

graph = make_graph(
    vertices = [:X, :A, :Y],
    di_edges = [(:X, :A), (:X, :Y), (:A, :Y)],
)

identify(graph, :A, :Y).strategy

:a_fixable

draw_graph(graph)

res = estimate_causal(
    a = [1, 0],
    data = data,
    graph = graph,
    treatment = :A,
    outcome = :Y,
)

raw = mean(data.Y[data.A .== 1]) - mean(data.Y[data.A .== 0])
r = x -> round(x, sigdigits=4)
(unadjusted = r(raw),
 ACE = r(res[:TMLE].ACE),
 lower_ci = r(res[:TMLE].lower_ci),
 upper_ci = r(res[:TMLE].upper_ci))

(unadjusted = 2.891, ACE = 2.032, lower_ci = 1.829, upper_ci = 2.236)

The true ACE is 2.0. For an a-fixable treatment, estimate_causal() uses the Markov pillow of the treatment as the adjustment set and returns TMLE, one-step/AIPW, G-computation, and IPW estimates.

G-computation models the outcome regression. IPW models the propensity score. One-step/AIPW and TMLE combine both nuisance models; TMLE additionally updates the outcome regression to solve the efficient influence-function equation.

Front-Door / P-Fixable

Front-door identification is useful when treatment and outcome are confounded, but the causal effect flows through observed post-treatment variables. The bidirected edge A <-> Y means ordinary backdoor adjustment fails. The mediator M helps because the graph identifies how A changes M and how M changes Y.

The classical front-door functional is:

E[Y(a)] = sum_m P(M=m | A=a) sum_a' E[Y | M=m, A=a'] P(A=a')

The package uses the broader p-fixable/NPS route. A treatment is p-fixable when none of its children are in its bidirected-connected district. Intuitively, treatment may be confounded with the outcome, but its immediate causal children are not hidden-confounded with treatment.

Here A is in the same district as Y, so backdoor adjustment fails. But M, the child of A, is not in that district. The graph identifies the effect by combining the treatment-mediator relationship with the mediator-outcome relationship while averaging over treatment to remove the A <-> Y confounding.

Random.seed!(2)
n = 700
A = Float64.(rand(n) .< 0.5)
M = 0.8 .* A .+ randn(n)
Y = 1.5 .* M .+ randn(n)
fd_data = DataFrame(A=A, M=M, Y=Y)

fd_graph = make_graph(
    vertices = [:A, :M, :Y],
    di_edges = [(:A, :M), (:M, :Y)],
    bi_edges = [(:A, :Y)],
)

identify(fd_graph, :A, :Y).strategy

:p_fixable

(district_A = district(fd_graph, :A),
 children_A = children(fd_graph, :A),
 p_fixable = is_p_fix(fd_graph, :A))

(district_A = [:A, :Y], children_A = [:M], p_fixable = true)

draw_graph(fd_graph)

fd_res = estimate_causal(
    a = [1, 0],
    data = fd_data,
    graph = fd_graph,
    treatment = :A,
    outcome = :Y,
)

r = x -> round(x, sigdigits=4)
(ACE=r(fd_res[:TMLE].ACE), lower_ci=r(fd_res[:TMLE].lower_ci), upper_ci=r(fd_res[:TMLE].upper_ci))

(ACE = 1.211, lower_ci = 0.9805, upper_ci = 1.441)

The true ACE is 1.2 (= 0.8 × 1.5, by the front-door formula). The package routes to the NPS/front-door style TMLE for p-fixable graphs.