21 From Graph to Estimate

library(dagitty)
library(igraph)
library(causaleffect)
library(ggplot2)

The previous chapters covered discovery: algorithms that learn a graph from data. But a graph is not an estimate. After we have a graph, we still need to ask two questions:

Identification: is the target effect a function of the observed-data distribution?
Estimation: once the identifying formula is known, estimate it from the data.

In R, dagitty handles identification via the backdoor and front-door criteria; causaleffect implements the full Pearl–Shpitser ID algorithm for acyclic directed mixed graphs (ADMGs) with hidden confounders. Estimation is done directly from the identifying formula — a step that is explicit in R, making the connection between the formula and the estimator transparent.

21.1 Backdoor: standard adjustment

Start with the simple case. $X$ is a common cause of treatment $A$ and outcome $Y$, and $X$ is observed. Backdoor adjustment applies.

# X is a common cause of A and Y
g_bd <- dagitty("dag { X -> A; X -> Y; A -> Y }")
exposures(g_bd) <- "A"
outcomes(g_bd)  <- "Y"

sets <- adjustmentSets(g_bd, exposure = "A", outcome = "Y")
cat("Adjustment sets:\n")

Adjustment sets:

print(sets)

{ X }

dagitty finds {X} as the minimal adjustment set. Adjusting on $X$ blocks the only open backdoor path $A \leftarrow X \rightarrow Y$.

set.seed(1)
n     <- 2000
X     <- rnorm(n)
A     <- as.numeric(runif(n) < 1 / (1 + exp(-X)))  # logistic: A depends on X
Y     <- 2 * A + X + 0.5 * rnorm(n)                 # true ACE = 2
data_bd <- data.frame(X = X, A = A, Y = Y)

# AIPW (doubly-robust) estimator for the backdoor-adjusted ACE
# Outcome model: E[Y | A, X]
mu1 <- predict(lm(Y ~ A + X, data = data_bd), newdata = transform(data_bd, A = 1))
mu0 <- predict(lm(Y ~ A + X, data = data_bd), newdata = transform(data_bd, A = 0))

# Propensity score: P(A = 1 | X)
ps  <- predict(glm(A ~ X, data = data_bd, family = binomial), type = "response")

# AIPW influence function
aipw <- mean(
  (data_bd$A / ps) * (Y - mu1) - ((1 - data_bd$A) / (1 - ps)) * (Y - mu0) +
  mu1 - mu0
)

cat(sprintf("Backdoor AIPW ACE (true = 2.0): %.3f\n", aipw))

Backdoor AIPW ACE (true = 2.0): 1.990

# Bootstrap 95% CI
set.seed(99)
boot_bd <- replicate(500, {
  idx  <- sample.int(n, replace = TRUE)
  bd   <- data_bd[idx, ]
  m1   <- predict(lm(Y ~ A + X, data = bd), newdata = transform(bd, A = 1))
  m0   <- predict(lm(Y ~ A + X, data = bd), newdata = transform(bd, A = 0))
  p    <- predict(glm(A ~ X, data = bd, family = binomial), type = "response")
  mean((bd$A / p) * (bd$Y - m1) - ((1 - bd$A) / (1 - p)) * (bd$Y - m0) + m1 - m0)
})
ci_bd <- quantile(boot_bd, c(0.025, 0.975))
cat(sprintf("95%% CI: [%.3f, %.3f]\n", ci_bd[1], ci_bd[2]))

95% CI: [1.942, 2.043]

The AIPW estimator is consistent if either the outcome model or the propensity score is correctly specified.

21.2 Front-door: p-fixable effects

Now suppose $A$ and $Y$ have an unmeasured common cause $U$, but there is a measured mediator $M$ on the path $A \rightarrow M \rightarrow Y$. Backdoor adjustment fails because $U$ is unobserved. The front-door formula can still identify the effect.

In dagitty, the bidirected edge $A \leftrightarrow Y$ (representing the latent $U$) appears as A <-> Y. For causaleffect, the same edge is encoded as two directed edges A -> Y and Y -> A both marked with description = "U" in the igraph object.

# dagitty for display and adjustment-set check
g_fd <- dagitty("dag { A -> M; M -> Y; A <-> Y }")
exposures(g_fd) <- "A"
outcomes(g_fd)  <- "Y"

cat("Backdoor adjustment sets (should be empty):\n")

Backdoor adjustment sets (should be empty):

print(adjustmentSets(g_fd, exposure = "A", outcome = "Y"))

# causaleffect for ID algorithm: bidirected A <-> Y = two directed edges + description "U"
g_fd_ig <- graph_from_literal(A -+ M, M -+ Y, A -+ Y, Y -+ A)
# Mark the A<->Y bidirected edge by endpoint name rather than a hardcoded
# edge index: igraph happens to sort edges by source vertex today (giving
# order A->M(1), A->Y(2), M->Y(3), Y->A(4)), but that ordering isn't part of
# igraph's documented contract, so indices like c(2,4) can silently break if
# igraph's internals or this graph's construction ever changes.
edge_ends <- ends(g_fd_ig, E(g_fd_ig))
confounded <- (edge_ends[, 1] == "A" & edge_ends[, 2] == "Y") |
              (edge_ends[, 1] == "Y" & edge_ends[, 2] == "A")
g_fd_ig <- set_edge_attr(g_fd_ig, "description", index = which(confounded), value = "U")

expr_fd <- causal.effect(y = "Y", x = "A", G = g_fd_ig, simp = TRUE)
cat("\nIdentifying expression:\n")


Identifying expression:

cat(expr_fd, "\n")

\sum_{M}P(M|A)\left(\sum_{A}P(Y|A,M)P(A)\right)

The expression returned is the front-door formula: $\sum_M P(M \mid A) \sum_{A'} P(Y \mid A', M)\, P(A')$. We now estimate it directly.

set.seed(2)
n     <- 3000
U     <- rnorm(n)                                     # unmeasured confounder
A_fd  <- as.numeric(runif(n) < 1 / (1 + exp(-U)))     # A depends on U
M_fd  <- as.numeric(runif(n) < 1 / (1 + exp(-(0.5 + 2 * A_fd))))  # M depends on A only
Y_fd  <- 1.5 * M_fd + 2 * U + 0.5 * rnorm(n)          # Y depends on M and U

# True ACE: E[Y | do(A=1)] - E[Y | do(A=0)]
# = 1.5 * (E[M | A=1] - E[M | A=0])
true_EMA1  <- 1 / (1 + exp(-(0.5 + 2)))   # scalar P(M=1 | A=1)
true_EMA0  <- 1 / (1 + exp(-0.5))         # scalar P(M=1 | A=0)
true_ace_fd <- 1.5 * (true_EMA1 - true_EMA0)
cat(sprintf("True front-door ACE = %.3f\n", true_ace_fd))

True front-door ACE = 0.453

data_fd <- data.frame(A = A_fd, M = M_fd, Y = Y_fd)

Estimate the front-door functional using the plug-in formula:

front_door_ace <- function(data) {
  pA <- mean(data$A)
  pM_given_A <- prop.table(table(data$A, data$M), margin = 1)
  pY_given_AM <- with(data, tapply(Y, list(A, M), mean))

  ey_do <- function(a) {
    sum(sapply(c(0, 1), function(m) {
      p_m_given_a <- pM_given_A[as.character(a), as.character(m)]
      inner <- sum(sapply(c(0, 1), function(ap)
        pY_given_AM[as.character(ap), as.character(m)] *
          ifelse(ap == 1, pA, 1 - pA)))
      p_m_given_a * inner
    }))
  }
  ey_do(1) - ey_do(0)
}

est_fd <- front_door_ace(data_fd)

set.seed(77)
boot_fd <- replicate(500, {
  idx <- sample.int(nrow(data_fd), replace = TRUE)
  front_door_ace(data_fd[idx, ])
})
ci_fd  <- quantile(boot_fd, c(0.025, 0.975))

cat(sprintf("Front-door ACE: %.3f  [95%% CI: %.3f, %.3f]\n",
            est_fd, ci_fd[1], ci_fd[2]))

Front-door ACE: 0.499  [95% CI: 0.432, 0.577]

# Naïve mean difference for comparison
naive_ace <- mean(Y_fd[A_fd == 1]) - mean(Y_fd[A_fd == 0])
cat(sprintf("Naïve mean difference (biased!): %.3f\n", naive_ace))

Naïve mean difference (biased!): 2.083

cat(sprintf("True ACE:                        %.3f\n", true_ace_fd))

True ACE:                        0.453

The front-door formula recovers the effect even though $U$ is unobserved. The naive mean difference is biased by the path through $U$.

21.3 When identification fails

Not every graph identifies the effect. If $A$ and $Y$ have a hidden common cause and there is no front-door variable, the effect is not identified.

# A → Y with unmeasured confounding (A <-> Y) and no mediator: the "bow"
# graph needs THREE igraph edges — the directed A → Y plus the
# bidirected pair (encoded as two opposite edges marked "U").
# graph_from_literal() silently collapses a duplicated A -+ Y, so we
# build the multigraph with make_graph().
g_unident_ig <- make_graph(c("A","Y",  "A","Y",  "Y","A"), directed = TRUE)
# make_graph() preserves the literal edge order given above (unlike
# graph_from_literal(), which re-sorts), so index 2:3 is reliable as long as
# the c(...) vector above isn't reordered -- mark by endpoint pair anyway so
# a future edit to that vector can't silently mismark the real A->Y edge.
edge_ends <- ends(g_unident_ig, E(g_unident_ig))
# The first A->Y edge is the real causal edge; the second (a duplicate,
# per `duplicated()`) together with Y->A forms the bidirected U pair.
u_edges <- which(((edge_ends[,1]=="A" & edge_ends[,2]=="Y") & duplicated(edge_ends)) |
                  (edge_ends[,1]=="Y" & edge_ends[,2]=="A"))
g_unident_ig <- set_edge_attr(g_unident_ig, "description",
                               index = u_edges, value = "U")

result <- tryCatch(
  causal.effect(y = "Y", x = "A", G = g_unident_ig, simp = TRUE),
  error = function(e) conditionMessage(e)
)
cat("Result from causal.effect:\n", result, "\n")

Result from causal.effect:
 Not identifiable.

causaleffect raises an error (“Not identifiable”), which the tryCatch captures, indicating the effect is not identifiable. The honest response in applied work is to report the non-identifiability, then either:

Find additional variables that close the unmeasured-confounding paths
Apply sensitivity analysis to bound the effect under maintained assumptions
Find an instrumental variable for a LATE-style identification (IV chapter)

Saying that the hidden confounding is probably small is sensitivity analysis, not identification.

21.4 End-to-end: discover, identify, estimate

Discovery algorithms return graphs that can be passed to the identification step:

library(pcalg)

# Step 1: discover the graph (chapters on causal discovery)
suffStat <- list(C = cor(data_obs), n = nrow(data_obs))
pc_fit   <- pc(suffStat, indepTest = gaussCItest,
               alpha = 0.05, p = ncol(data_obs))

# Step 2: check for backdoor adjustment set using dagitty
# (convert pcalg CPDAG → dagitty for convenience).
# NOTE the t(): as(graphNEL, "matrix") is from→to, but pcalg2dagitty
# expects the amat.cpdag coding, which is its transpose — without it
# every directed edge is silently reversed.
g_dag <- pcalg2dagitty(t(as(pc_fit@graph, "matrix")),
                        colnames(data_obs), type = "cpdag")
sets  <- adjustmentSets(g_dag, exposure = "A", outcome = "Y")

# Step 3: if backdoor set found, estimate with AIPW
# Step 4: if not, call causaleffect::causal.effect() for the ID expression
# Step 5: estimate the returned functional by plug-in or regression

In applied work, I would not trust the discovered graph blindly. A more reasonable workflow is:

Hypothesise a graph based on subject-matter knowledge
Discover a data-driven graph using PC/GES/FCI
Compare the two — disagreement points to substantive questions
Identify under both graphs separately
Estimate under both, and report the range as part of the result

If the estimate changes a lot across plausible graphs, the data alone is not settling the question.

21.5 Why this matters

The graph-identification-estimation sequence matters because:

Backdoor adjustment is not always available. Many real DAGs have hidden confounders. Front-door, nested, and general ID algorithms are the recourse, but they are too complex to derive by hand for non-trivial graphs.
The right estimator depends on the identification strategy. AIPW-for-backdoor uses a different formula than the front-door plug-in. Matching the estimator to the identification result avoids the common mistake of plugging a graph into the wrong estimator.
Reporting becomes more disciplined. Once the workflow forces you to specify a graph, it becomes harder to wave hands about “unconfoundedness” — you have to say exactly which variables are doing the unconfounding work.

21.6 Summary

A graph is not an estimate. Identification is the bridge.
dagitty handles backdoor and front-door checks; causaleffect handles general ADMG identification.
Backdoor effects can be estimated with AIPW.
Front-door effects require estimating the front-door functional.
If the effect is not identified, report that instead of forcing a biased regression estimate.

# From Graph to Estimate ```{r} #| include: false library(dagitty) library(igraph) library(causaleffect) library(ggplot2) library(dplyr) ``` ```{r} #| eval: false library(dagitty) library(igraph) library(causaleffect) library(ggplot2) ``` The previous chapters covered discovery: algorithms that learn a graph from data. But a graph is not an estimate. After we have a graph, we still need to ask two questions: 1. **Identification**: is the target effect a function of the observed-data distribution? 2. **Estimation**: once the identifying formula is known, estimate it from the data. In R, `dagitty` handles identification via the backdoor and front-door criteria; `causaleffect` implements the full Pearl–Shpitser ID algorithm for acyclic directed mixed graphs (ADMGs) with hidden confounders. Estimation is done directly from the identifying formula — a step that is explicit in R, making the connection between the formula and the estimator transparent. ## Backdoor: standard adjustment Start with the simple case. $X$ is a common cause of treatment $A$ and outcome $Y$, and $X$ is observed. Backdoor adjustment applies. ```{r} #| label: backdoor-graph #| cache: true # X is a common cause of A and Y g_bd <- dagitty("dag { X -> A; X -> Y; A -> Y }") exposures(g_bd) <- "A" outcomes(g_bd) <- "Y" sets <- adjustmentSets(g_bd, exposure = "A", outcome = "Y") cat("Adjustment sets:\n") print(sets) ``` `dagitty` finds `{X}` as the minimal adjustment set. Adjusting on $X$ blocks the only open backdoor path $A \leftarrow X \rightarrow Y$. ```{r} #| label: backdoor-sim #| cache: true set.seed(1) n <- 2000 X <- rnorm(n) A <- as.numeric(runif(n) < 1 / (1 + exp(-X))) # logistic: A depends on X Y <- 2 * A + X + 0.5 * rnorm(n) # true ACE = 2 data_bd <- data.frame(X = X, A = A, Y = Y) # AIPW (doubly-robust) estimator for the backdoor-adjusted ACE # Outcome model: E[Y | A, X] mu1 <- predict(lm(Y ~ A + X, data = data_bd), newdata = transform(data_bd, A = 1)) mu0 <- predict(lm(Y ~ A + X, data = data_bd), newdata = transform(data_bd, A = 0)) # Propensity score: P(A = 1 | X) ps <- predict(glm(A ~ X, data = data_bd, family = binomial), type = "response") # AIPW influence function aipw <- mean( (data_bd$A / ps) * (Y - mu1) - ((1 - data_bd$A) / (1 - ps)) * (Y - mu0) + mu1 - mu0 ) cat(sprintf("Backdoor AIPW ACE (true = 2.0): %.3f\n", aipw)) # Bootstrap 95% CI set.seed(99) boot_bd <- replicate(500, { idx <- sample.int(n, replace = TRUE) bd <- data_bd[idx, ] m1 <- predict(lm(Y ~ A + X, data = bd), newdata = transform(bd, A = 1)) m0 <- predict(lm(Y ~ A + X, data = bd), newdata = transform(bd, A = 0)) p <- predict(glm(A ~ X, data = bd, family = binomial), type = "response") mean((bd$A / p) * (bd$Y - m1) - ((1 - bd$A) / (1 - p)) * (bd$Y - m0) + m1 - m0) }) ci_bd <- quantile(boot_bd, c(0.025, 0.975)) cat(sprintf("95%% CI: [%.3f, %.3f]\n", ci_bd[1], ci_bd[2])) ``` The AIPW estimator is consistent if either the outcome model or the propensity score is correctly specified. ## Front-door: p-fixable effects Now suppose $A$ and $Y$ have an unmeasured common cause $U$, but there is a measured mediator $M$ on the path $A \rightarrow M \rightarrow Y$. Backdoor adjustment fails because $U$ is unobserved. The front-door formula can still identify the effect. In `dagitty`, the bidirected edge $A \leftrightarrow Y$ (representing the latent $U$) appears as `A <-> Y`. For `causaleffect`, the same edge is encoded as two directed edges `A -> Y` and `Y -> A` both marked with `description = "U"` in the igraph object. ```{r} #| label: frontdoor-graph #| cache: true # dagitty for display and adjustment-set check g_fd <- dagitty("dag { A -> M; M -> Y; A <-> Y }") exposures(g_fd) <- "A" outcomes(g_fd) <- "Y" cat("Backdoor adjustment sets (should be empty):\n") print(adjustmentSets(g_fd, exposure = "A", outcome = "Y")) # causaleffect for ID algorithm: bidirected A <-> Y = two directed edges + description "U" g_fd_ig <- graph_from_literal(A -+ M, M -+ Y, A -+ Y, Y -+ A) # Mark the A<->Y bidirected edge by endpoint name rather than a hardcoded # edge index: igraph happens to sort edges by source vertex today (giving # order A->M(1), A->Y(2), M->Y(3), Y->A(4)), but that ordering isn't part of # igraph's documented contract, so indices like c(2,4) can silently break if # igraph's internals or this graph's construction ever changes. edge_ends <- ends(g_fd_ig, E(g_fd_ig)) confounded <- (edge_ends[, 1] == "A" & edge_ends[, 2] == "Y") | (edge_ends[, 1] == "Y" & edge_ends[, 2] == "A") g_fd_ig <- set_edge_attr(g_fd_ig, "description", index = which(confounded), value = "U") expr_fd <- causal.effect(y = "Y", x = "A", G = g_fd_ig, simp = TRUE) cat("\nIdentifying expression:\n") cat(expr_fd, "\n") ``` The expression returned is the front-door formula: $\sum_M P(M \mid A) \sum_{A'} P(Y \mid A', M)\, P(A')$. We now estimate it directly. ```{r} #| label: frontdoor-sim #| cache: true set.seed(2) n <- 3000 U <- rnorm(n) # unmeasured confounder A_fd <- as.numeric(runif(n) < 1 / (1 + exp(-U))) # A depends on U M_fd <- as.numeric(runif(n) < 1 / (1 + exp(-(0.5 + 2 * A_fd)))) # M depends on A only Y_fd <- 1.5 * M_fd + 2 * U + 0.5 * rnorm(n) # Y depends on M and U # True ACE: E[Y | do(A=1)] - E[Y | do(A=0)] # = 1.5 * (E[M | A=1] - E[M | A=0]) true_EMA1 <- 1 / (1 + exp(-(0.5 + 2))) # scalar P(M=1 | A=1) true_EMA0 <- 1 / (1 + exp(-0.5)) # scalar P(M=1 | A=0) true_ace_fd <- 1.5 * (true_EMA1 - true_EMA0) cat(sprintf("True front-door ACE = %.3f\n", true_ace_fd)) data_fd <- data.frame(A = A_fd, M = M_fd, Y = Y_fd) ``` Estimate the front-door functional using the plug-in formula: ```{r} #| label: frontdoor-est #| cache: true front_door_ace <- function(data) { pA <- mean(data$A) pM_given_A <- prop.table(table(data$A, data$M), margin = 1) pY_given_AM <- with(data, tapply(Y, list(A, M), mean)) ey_do <- function(a) { sum(sapply(c(0, 1), function(m) { p_m_given_a <- pM_given_A[as.character(a), as.character(m)] inner <- sum(sapply(c(0, 1), function(ap) pY_given_AM[as.character(ap), as.character(m)] * ifelse(ap == 1, pA, 1 - pA))) p_m_given_a * inner })) } ey_do(1) - ey_do(0) } est_fd <- front_door_ace(data_fd) set.seed(77) boot_fd <- replicate(500, { idx <- sample.int(nrow(data_fd), replace = TRUE) front_door_ace(data_fd[idx, ]) }) ci_fd <- quantile(boot_fd, c(0.025, 0.975)) cat(sprintf("Front-door ACE: %.3f [95%% CI: %.3f, %.3f]\n", est_fd, ci_fd[1], ci_fd[2])) # Naïve mean difference for comparison naive_ace <- mean(Y_fd[A_fd == 1]) - mean(Y_fd[A_fd == 0]) cat(sprintf("Naïve mean difference (biased!): %.3f\n", naive_ace)) cat(sprintf("True ACE: %.3f\n", true_ace_fd)) ``` The front-door formula recovers the effect even though $U$ is unobserved. The naive mean difference is biased by the path through $U$. ## When identification fails Not every graph identifies the effect. If $A$ and $Y$ have a hidden common cause and there is no front-door variable, the effect is not identified. ```{r} #| label: not-identified #| cache: true # A → Y with unmeasured confounding (A <-> Y) and no mediator: the "bow" # graph needs THREE igraph edges — the directed A → Y plus the # bidirected pair (encoded as two opposite edges marked "U"). # graph_from_literal() silently collapses a duplicated A -+ Y, so we # build the multigraph with make_graph(). g_unident_ig <- make_graph(c("A","Y", "A","Y", "Y","A"), directed = TRUE) # make_graph() preserves the literal edge order given above (unlike # graph_from_literal(), which re-sorts), so index 2:3 is reliable as long as # the c(...) vector above isn't reordered -- mark by endpoint pair anyway so # a future edit to that vector can't silently mismark the real A->Y edge. edge_ends <- ends(g_unident_ig, E(g_unident_ig)) # The first A->Y edge is the real causal edge; the second (a duplicate, # per `duplicated()`) together with Y->A forms the bidirected U pair. u_edges <- which(((edge_ends[,1]=="A" & edge_ends[,2]=="Y") & duplicated(edge_ends)) | (edge_ends[,1]=="Y" & edge_ends[,2]=="A")) g_unident_ig <- set_edge_attr(g_unident_ig, "description", index = u_edges, value = "U") result <- tryCatch( causal.effect(y = "Y", x = "A", G = g_unident_ig, simp = TRUE), error = function(e) conditionMessage(e) ) cat("Result from causal.effect:\n", result, "\n") ``` `causaleffect` raises an error ("Not identifiable"), which the `tryCatch` captures, indicating the effect is not identifiable. The honest response in applied work is to report the non-identifiability, then either: 1. Find additional variables that close the unmeasured-confounding paths 2. Apply sensitivity analysis to bound the effect under maintained assumptions 3. Find an instrumental variable for a LATE-style identification (IV chapter) Saying that the hidden confounding is probably small is sensitivity analysis, not identification. ## End-to-end: discover, identify, estimate Discovery algorithms return graphs that can be passed to the identification step: ```{r} #| eval: false #| echo: true library(pcalg) # Step 1: discover the graph (chapters on causal discovery) suffStat <- list(C = cor(data_obs), n = nrow(data_obs)) pc_fit <- pc(suffStat, indepTest = gaussCItest, alpha = 0.05, p = ncol(data_obs)) # Step 2: check for backdoor adjustment set using dagitty # (convert pcalg CPDAG → dagitty for convenience). # NOTE the t(): as(graphNEL, "matrix") is from→to, but pcalg2dagitty # expects the amat.cpdag coding, which is its transpose — without it # every directed edge is silently reversed. g_dag <- pcalg2dagitty(t(as(pc_fit@graph, "matrix")), colnames(data_obs), type = "cpdag") sets <- adjustmentSets(g_dag, exposure = "A", outcome = "Y") # Step 3: if backdoor set found, estimate with AIPW # Step 4: if not, call causaleffect::causal.effect() for the ID expression # Step 5: estimate the returned functional by plug-in or regression ``` In applied work, I would not trust the discovered graph blindly. A more reasonable workflow is: 1. **Hypothesise** a graph based on subject-matter knowledge 2. **Discover** a data-driven graph using PC/GES/FCI 3. **Compare** the two — disagreement points to substantive questions 4. **Identify** under both graphs separately 5. **Estimate** under both, and report the range as part of the result If the estimate changes a lot across plausible graphs, the data alone is not settling the question. ## Why this matters The graph-identification-estimation sequence matters because: - **Backdoor adjustment is not always available.** Many real DAGs have hidden confounders. Front-door, nested, and general ID algorithms are the recourse, but they are too complex to derive by hand for non-trivial graphs. - **The right estimator depends on the identification strategy.** AIPW-for-backdoor uses a different formula than the front-door plug-in. Matching the estimator to the identification result avoids the common mistake of plugging a graph into the wrong estimator. - **Reporting becomes more disciplined.** Once the workflow forces you to specify a graph, it becomes harder to wave hands about "unconfoundedness" — you have to say exactly which variables are doing the unconfounding work. ## Summary - A graph is not an estimate. Identification is the bridge. - `dagitty` handles backdoor and front-door checks; `causaleffect` handles general ADMG identification. - Backdoor effects can be estimated with AIPW. - Front-door effects require estimating the front-door functional. - If the effect is not identified, report that instead of forcing a biased regression estimate.