Confounding, DAGs and what to adjust for

What this page answers

The cohort study chapter set out the central difficulty of observational research: the two groups differ before any treatment is given, so an observed association is not a treatment effect. Its answer was to put the measured confounders into a Cox model, with one honest qualifier attached — statistical adjustment can only handle variables you actually measured.

This page goes one step earlier, to the more basic question: among the variables you did measure, which ones belong in the model?

The intuitive answer is “all of them — the more you adjust for, the cleaner it gets”. That intuition is wrong in three situations, and it is wrong in a specific direction: the problem is not that you adjusted too little, it is that the adjustment itself created the bias. Telling those three situations apart cannot be done with p-values, stepwise regression, or AIC. All of those look only at associations in the data, and whether a variable should be adjusted for is a question about causal structure, which the data alone will never answer.

A directed acyclic graph (DAG) is the tool for writing that structure down. It will not find the truth for you. It does exactly one thing: it puts your assumptions out in the open, so that “what should I adjust for” becomes a question you can check line by line instead of a judgement call made by feel.

The three conditions for a confounder

For a variable $C$ to be a confounder of the exposure $A$ and the outcome $Y$, the traditional definition asks for three conditions at once:

1. $C$ is associated with the exposure $A$ — the treated and untreated groups differ on $C$
2. $C$ is an independent risk factor for $Y$ — even setting $A$ aside, $C$ affects the outcome
3. $C$ does not lie on the causal path from $A$ to $Y$ — it is not caused by the exposure, and it is not a way station through which the exposure acts

The third is the one people skip, and it is exactly the line separating a confounder from a mediator. The first two are visible in the data (you can compute a correlation, you can run a regression). The third is not. It is knowledge about timing and mechanism, and it can only come from the clinic, from the literature, and from your understanding of the disease.

Drawing the problem as a DAG

The rules of a DAG fit in one sentence: draw a node for each variable, draw an arrow for each “I believe X directly causes Y”, point every arrow at the consequence, and never let a path loop back on itself (acyclic). An absent arrow is a substantive claim — “I believe there is no direct causal link between these two” — so the places where you cannot draw an arrow matter as much as the places where you can.

Once the graph is drawn, association flows along three kinds of path:

The third row is the least intuitive and the most useful thing in the whole DAG language. A collider path is closed by default, and adjustment opens it. It is the single most important counterexample to “more adjustment is better”.

Four structures, four answers

Each of the four structures was simulated 500 times with 2000 participants per run, and each run fits two models: exposure only (Y ~ A), and exposure plus one more variable (Y ~ A + Z). Every arrow in all four structures was given the same strength, so the only thing that differs between them is the shape.

set.seed(20260822)
n <- 2000; g <- 0.8      # g = strength of every non-causal arrow, shared by all four

# 1. Confounder: C causes both A and Y; true effect = 1
gen_confounder <- function(n) {
  C <- rnorm(n); A <- g * C + rnorm(n); Y <- 1 * A + g * C + rnorm(n)
  data.frame(A, Y, Z = C)
}
# 2. Collider: A and Y are independent, both cause S; true effect = 0
gen_collider <- function(n) {
  A <- rnorm(n); Y <- 0 * A + rnorm(n); S <- g * A + g * Y + rnorm(n)
  data.frame(A, Y, Z = S)
}
# 3. Mediator: A acts through M; total effect = 0.2 + 0.8 * 0.8
gen_mediator <- function(n) {
  A <- rnorm(n); M <- g * A + rnorm(n); Y <- 0.2 * A + g * M + rnorm(n)
  data.frame(A, Y, Z = M)
}
# 4. M-bias: Z precedes the exposure, but is a common effect of two
#    unmeasured variables; true effect = 0
gen_mbias <- function(n) {
  U1 <- rnorm(n); U2 <- rnorm(n)
  Z <- g * U1 + g * U2 + rnorm(n)
  A <- g * U1 + rnorm(n)
  Y <- 0 * A + g * U2 + rnorm(n)
  data.frame(A, Y, Z)
}

d <- gen_collider(n)
coef(lm(Y ~ A, data = d))["A"]        # unadjusted
coef(lm(Y ~ A + Z, data = d))["A"]    # after adjusting for the collider

import numpy as np, statsmodels.api as sm

rng = np.random.default_rng(20260822)
n, g = 2000, 0.8

def gen_collider(n):
    A = rng.normal(size=n)
    Y = 0 * A + rng.normal(size=n)
    S = g * A + g * Y + rng.normal(size=n)
    return A, Y, S

A, Y, S = gen_collider(n)
crude = sm.OLS(Y, sm.add_constant(A)).fit()
adj   = sm.OLS(Y, sm.add_constant(np.column_stack([A, S]))).fit()
print(crude.params[1], adj.params[1])   # near zero vs clearly away from zero

Only in the first row does the extra variable improve anything. In the other three, the variable you added took an unbiased estimate and broke it.

Colliders: adjustment manufactures an association

The second row is worth stopping on. The exposure and the outcome are completely independent in the simulation — the true effect is set to zero — and the unadjusted estimate is +0.001. After adjustment it becomes −0.388. In a single run of the simulation the confidence interval for that association runs from -0.455 to -0.372, nowhere near zero, with a p-value too small for the output to print.

Two variables with nothing whatsoever to do with each other look strongly related, purely because one common effect was added to the model.

Here is why, intuitively. Suppose there are only two reasons to be admitted to hospital ($S$): the exposure, and the disease itself. Now look only at admitted patients — among them, anyone who was not exposed must have been admitted because their disease was more severe. So within admitted patients, “unexposed” and “severely ill” travel together, and that association does not exist in the general population at all. You created it by choosing whom to look at.

M-bias: it happens before the exposure, and you still must not adjust for it

That test invites a convenient shortcut: “adjust only for things that happened before the exposure, and you can never hit a collider.” The shortcut is right most of the time. M-bias is the rest of the time.

The fourth DAG has this structure. Two things go unmeasured: $U_1$ (say, a tendency toward health-seeking behaviour) and $U_2$ (say, undiagnosed latent disease). $U_1$ influences whether a patient receives the exposure, $U_2$ influences the outcome, and both of them influence the same variable $Z$ (say, a lab value measured years earlier).

$Z$ clearly happened before the exposure, it is associated with the exposure, and it is associated with the outcome — run it through the traditional three-condition checklist and it looks like a textbook confounder. In fact it is a collider for $U_1$ and $U_2$. Adjusting for it opens the path $A \leftarrow U_1 \to Z \leftarrow U_2 \to Y$, which was closed: the true effect is zero, and the adjusted estimate is −0.124.

Mediators: you adjust away the thing you wanted to estimate

The third row is a different kind of problem. It is not bias — it is answering a different question.

In the simulation the drug acts along two paths: a direct one, and one through a mediator (blood pressure). The total effect is 0.84; the unadjusted model estimates 0.841, which is accurate. Put the mediator in and the estimate falls to 0.200 — which is precisely the direct effect.

Nobody miscalculated. The two numbers answer different questions:

• Total effect: “if I give this drug, how much does the outcome change?” — the number a clinical decision needs
• Direct effect: “with the blood-pressure pathway taken out, how much of the drug’s action is left?” — the number a mechanistic study needs

The backdoor criterion in plain words

With those three path shapes in hand, “what should I adjust for” becomes a criterion you can execute mechanically. A backdoor path is any path that starts at the exposure with an arrow pointing into the exposure (the arrow leaves “backwards”) and ends at the outcome. The association such a path carries is not causal; it is noise.

The plain-language backdoor criterion is three sentences:

1. Find every backdoor path
2. Choose a set of variables such that every backdoor path is blocked — either a non-collider on the path has been adjusted for, or the path contains a collider that has not been adjusted for
3. That set must contain no descendant of the exposure (anything the exposure causes, mediators included)

A set of variables satisfying all three is a valid adjustment set. Two things about it run against intuition:

• There can be more than one valid adjustment set, and the valid one is not necessarily the largest
• Adding a variable can turn a valid set into an invalid one — which is exactly what M-bias is

You do not have to do this by hand: the dagitty package (there is also a web version) takes the graph and returns every minimal valid adjustment set, and the next section shows exactly how. What still needs a human is getting the arrows right, and no tool can do that part for you.

Handing the DAG to a machine: from picture to adjustment set

The previous section said you do not have to work this out by hand. This section is that promise paid off — write the same picture as code and let it produce the answer.

The dagitty syntax is close to just typing the arrows out. What follows is the confounder structure from the top left of this page’s first figure, character for character:

library(dagitty)

g <- dagitty('dag {
  A [exposure,pos="0,0"]
  Y [outcome,pos="1,0"]
  C [pos="0.5,-0.85"]
  A -> Y
  C -> A
  C -> Y
}')

adjustmentSets(g)                      # every minimal sufficient adjustment set
impliedConditionalIndependencies(g)    # what this graph claims you will not see

# There is no equivalent Python package. networkx can draw the graph, but solving
# the backdoor criterion would have to be written from scratch, which is not worth it.
# Do the DAG step in R or on dagitty.net, then bring the variable list back to Python.

The pos= coordinates affect layout only, never the computation. They are there so that pasting the string into dagitty.net gives you the same picture as the figure above. (The y values are negative because the web version points its y axis downwards.)

What the four structures answer

Running all four structures from this page through adjustmentSets():

Only the confounder structure has instinct and tool agreeing. For the other three the tool returns { } — adjust for nothing.

That is the same conclusion the simulations earlier on this page reached, arrived at by a different route. The simulations answer “how far does the estimate move if you adjust”; this answers “the graph never asked you to adjust in the first place”. The second kind of evidence needs no data at all.

A DAG can be refuted by data

impliedConditionalIndependencies() lists the places where the graph claims you will find no association. For both the collider and the M-bias structures it returns the same line, A _||_ Y — exposure and outcome should be unrelated, because neither of those graphs contains a causal arrow from A to Y, and the one path that does join them in the M-bias graph runs through the collider Z, which stays closed for as long as you do not adjust for it.

That is useful because you can check it against data. If A and Y are plainly associated in the data, the data is not the problem: your graph is wrong — a missing arrow, or one pointing the wrong way.

The confounder and mediator graphs return an empty list. On their three observed nodes those graphs are complete — every pair is joined by a direct arrow — and a complete graph forbids nothing, so there is nothing left for data to contradict. The asymmetry is worth remembering: a DAG that forbids nothing is a DAG that cannot be shown to be wrong.

So what should you adjust for

Everything above, compressed into a checklist you can take into a meeting:

From the DAG to an actual analysis

A DAG tells you which paths need blocking. It does not tell you how to block them. There are several ways, each with its own price, and the next few pages take them one at a time:

• Adjust inside a regression model — the most direct route; see the Cox model and linear regression
• Propensity score matching — compress “who gets the exposure” into a single score, then match two comparable groups
• Inverse probability weighting — use the same score as a weight to construct a hypothetical population
• Instrumental variables — the only route left when the crucial confounder was never measured at all
• Self-controlled designs — let each person serve as their own control, removing every time-invariant personal characteristic at once

The first three share one prerequisite: the confounders were measured. Matching and weighting can do nothing at all about the grey, unmeasured nodes on a DAG — they are simply different ways of using the variables you already have. The cohort study chapter said this once; it is worth saying again.

Common misuses

Reproducing every number on this page

/opt/homebrew/bin/Rscript figures/scripts/B6-01-dag.R

Watch next

▶Clearing Up Confounding

▶因果圖 DAG 是什麼？一張圖看穿因果的陷阱

▶Introduction to Causal Graphs

▶Directed Acyclic Graphs (DAGs)

▶Principles of Epidemiology 05. Causal Inferences, Bias, Confounding, and Interaction

Report a problem on this page