Nested-Fixable Estimation and Missing Data

This vignette shows the two pieces that go beyond standard backdoor/front-door workflows: nested-fixable effects and missing-data weighting. All datasets are simulated.

The common theme is identification before estimation. The graph determines whether the target law can be represented using observed data. Only after that step does the package construct weights and nuisance regressions.

Nested-Fixable Effects

A treatment is nested-fixable when the target effect is identified by a One-Line ID style fixing sequence, but the treatment is neither a-fixable nor p-fixable. This happens in ADMGs where hidden-confounding structure is too complex for ordinary backdoor adjustment or a front-door/NPS argument.

The word "nested" comes from nested Markov models for ADMGs. Identification is described by repeatedly "fixing" variables: moving a variable from random to fixed status and updating the kernel for the remaining random variables. If a valid fixing sequence isolates the target counterfactual distribution, the effect is identified.

using CausalGraphs, DataFrames, Random

graph = make_graph(
    vertices = [:ViralLoad, :Income, :Exercise, :T, :Toxicity, :Adherence, :CD4],
    di_edges = [
        (:ViralLoad, :T), (:ViralLoad, :CD4),
        (:Income, :T), (:Exercise, :CD4),
        (:T, :Toxicity), (:T, :CD4),
        (:Toxicity, :Adherence), (:Adherence, :CD4),
    ],
    bi_edges = [
        (:Income, :Toxicity),
        (:Exercise, :Income),
        (:Exercise, :T),
        (:ViralLoad, :Adherence),
        (:ViralLoad, :CD4),
    ],
)

identify(graph, :T, :CD4).strategy

:nested_fixable

(district_T = district(graph, :T),
 children_T = children(graph, :T),
 p_fixable = is_p_fix(graph, :T))

(district_T = [:T, :Exercise, :Income, :Toxicity], children_T = [:Toxicity, :CD4], p_fixable = false)

draw_graph(graph; direction="TB")

For nested-fixable effects, estimate_causal() returns augmented nested IPW (ANIPW) and nested IPW (NIPW) estimates.

Nested rebalancing weights are analogous to ordinary IPW, but they balance the pieces of the observed law that appear in the nested identification formula. ANIPW augments those weights with outcome regression terms, in the same spirit that AIPW augments ordinary IPW.

In this example, Toxicity is both a child of T and in the same district as T, so p-fixability fails. The nested route is needed because the effect is identified by a fixing sequence rather than by one adjustment set or one front-door mediator formula.

Random.seed!(42)
n = 500
U_EI = randn(n); U_ET = randn(n); U_ITox = randn(n)
U_VA = randn(n); U_VCD4 = randn(n)
ViralLoad = randn(n)
Exercise  = randn(n) .+ 0.5 .* U_EI .+ 0.5 .* U_ET
Income    = randn(n) .+ 0.4 .* ViralLoad .+ 0.4 .* U_EI .+ 0.4 .* U_ITox
T         = Float64.(rand(n) .< 1 ./(1 .+ exp.(-(0.5 .* ViralLoad .+ 0.3 .* Income .+ 0.4 .* U_ET))))
Toxicity  = randn(n) .+ 0.6 .* T .+ 0.3 .* U_ITox
Adherence = randn(n) .+ 0.5 .* Toxicity .+ 0.3 .* U_VA
CD4       = randn(n) .+ 0.4 .* T .+ 0.3 .* Adherence .+
            0.2 .* Exercise .+ 0.3 .* ViralLoad .+ 0.3 .* U_VCD4 .+ 0.2 .* U_VA
df = DataFrame(ViralLoad=ViralLoad, Income=Income, Exercise=Exercise,
               T=T, Toxicity=Toxicity, Adherence=Adherence, CD4=CD4)

res = estimate_causal(a=[1, 0], data=df, graph=graph, treatment=:T, outcome=:CD4)
r = x -> round(x, sigdigits=4)
(ACE=r(res[:ANIPW].ACE), lower_ci=r(res[:ANIPW].lower_ci), upper_ci=r(res[:ANIPW].upper_ci))

(ACE = 0.4492, lower_ci = 0.2097, upper_ci = 0.6887)

Missing-Data Weighting

An mDAG extends a causal graph with missingness indicators. A variable such as X has an indicator Rx showing whether X is observed. Edges into Rx encode the missingness mechanism.

The key identification question is whether the target full-data law can be recovered from the observed-data law. If it can, the package estimates observation probabilities and forms inverse-probability weights. Those weights can be passed to estimate_causal(...; sample_weights=wts) so complete cases are weighted back toward the target population.

This separates two assumptions: the causal ADMG describes treatment, outcome, and covariates; the mDAG describes how values become missing.

For a simple MAR graph, if the probability of observing X depends only on fully observed variables, complete cases can be weighted by the inverse observation probability. If missingness depends directly on the unobserved value of X, the target law may not be recoverable without additional graph structure.

mdag = make_mdag(
    obs_variables = ["A", "Y"],
    missing_variables = ["X"],
    missing_indicators = ["Rx"],
    di_edges = [("X", "A"), ("X", "Y"), ("X", "Rx"), ("A", "Y")],
)

ID = ID_algorithm(mdag)
ID.target.id_status

false

draw_graph(mdag)

compute_missing_weights() estimates the identified missingness propensities and returns inverse-probability weights that can be passed to estimate_causal(...; sample_weights=wts).