Graphs and Identification

This vignette introduces the graph layer and the identify() function.

A graph is a compact statement of causal assumptions. It tells us which variables can directly affect other variables and which observed variables may share hidden common causes. Identification asks whether an interventional quantity, such as E[Y(a)] or E[Y(1)-Y(0)], can be written in terms of the observed data distribution under those assumptions.

Constructing an ADMG

An acyclic directed mixed graph has directed edges for observed causal relationships and bidirected edges for latent common causes.

In an ADMG:

• A -> B means A may directly cause B;
• A <-> B means A and B share an unmeasured common cause;
• directed arrows cannot form cycles.

ADMGs are useful because they represent latent confounding without drawing each unobserved variable. Instead of adding a hidden U with U -> A and U -> Y, we draw A <-> Y.

using CausalGraphs

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

draw_graph(g_fd; direction="LR")

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

draw_graph(g_bd; direction="LR")

Graph Properties

Graph properties are the vocabulary used by identification algorithms. Parents and children describe direct arrows. Ancestors and descendants describe all directed upstream or downstream variables. Districts are bidirected-connected components and encode latent confounding. Markov pillows are local adjustment sets used by the a-fixable/backdoor estimator.

parents(g_fd, :M)

1-element Vector{Symbol}:
 :A

children(g_fd, :A)

1-element Vector{Symbol}:
 :M

district(g_fd, :A)

2-element Vector{Symbol}:
 :A
 :Y

markov_pillow(g_bd, :A; treatment=:A)

1-element Vector{Symbol}:
 :X

Identification

identify() routes the graph to the implemented estimation strategy.

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

:a_fixable

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

:p_fixable

The implemented strategies are:

These strategies correspond to different identification arguments:

• :a_fixable covers backdoor-style effects. The treatment can be adjusted using its Markov pillow.
• :p_fixable covers front-door/primal-fixable/NPS effects. The treatment may be confounded with the outcome, but its children are not in the treatment's hidden-confounding district.
• :nested_fixable covers ADMGs where neither simple adjustment nor p-fixability is enough, but a One-Line ID style fixing sequence still identifies the effect.
• :id_algorithm covers more general ADMGs where the recursive ID algorithm finds a symbolic functional. The package can estimate such functionals with a finite-support plug-in estimator with EIF confidence intervals when the variables are discrete.

The returned strategy is an estimation instruction for estimate_causal() when it is :a_fixable, :p_fixable, :nested_fixable, or :id_algorithm. For :id_algorithm, inspect identify(...).id_expression, call ID_algorithm() directly, or use estimate_id() for the finite-support plug-in estimator.

Optional NPCausal.jl Estimation

CausalGraphs.jl can also act as the identification layer for NPCausal.jl. Install and load both packages, then call either estimate_causal_npcausal(...) or NPCausal.admg_estimate_causal(...):

using Pkg
Pkg.add(url="https://github.com/xiangao/NPCausal.jl")

using CausalGraphs, NPCausal

res = estimate_causal_npcausal(
    a = [1, 0],
    data = df,
    graph = g_bd,
    treatment = :A,
    outcome = :Y,
    nsplits = 5,
)

r = x -> round(x, sigdigits=4)
(ACE = r(res[:NPCausalOnestep].ACE),
 SE = r(res[:NPCausalOnestep].standard_error))

The route covers :a_fixable/backdoor effects through NPCausal.ate, :p_fixable effects through NPS TMLE, :nested_fixable effects through ANIPW, and finite-support discrete :id_algorithm functionals through :IDPlugin.

For the backdoor graph, the a-fixability condition holds because A is not in the same hidden-confounding district as any descendant other than itself. The Markov pillow is X, which becomes the adjustment set.

(descendants_A = descendants(g_bd, [:A]),
 district_A = district(g_bd, :A),
 markov_pillow_A = markov_pillow(g_bd, :A; treatment=:A))

(descendants_A = [:A, :Y], district_A = [:A], markov_pillow_A = [:X])

If a graph has both A -> Y and A <-> Y, the directed causal effect and the hidden common cause are entangled. That bow graph is the simplest non-identified case.

General ID Algorithm

ID_algorithm() runs the Pearl-Shpitser recursive ID algorithm for ADMG queries. It is useful for separating mathematical identification from estimator routing. On the front-door graph, it recovers the familiar front-door functional:

id_general = ID_algorithm(g_fd, :A, :Y)
(identified = id_general.identified,
 expression = string(id_general.expression))

(identified = true, expression = "sum_{M} P(M | A) * sum_{A} P(A) * P(Y | A, M)")

On the bow graph, the general ID algorithm fails and returns a hedge witness.

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

id_bow = ID_algorithm(g_bow, :A, :Y)
(identified = id_bow.identified, hedge = id_bow.hedge)

(identified = false, hedge = (S = [:Y], F = [:A, :Y], treatment = [:A], outcome = [:Y]))