Regression can tell you that two variables are associated. It cannot, on its own, tell you whether one causes the other, or which variables you should control for to find out. Those are causal questions, and causal questions need a causal model. The clearest way to state one is the directed acyclic graph (DAG): a diagram of what you assume causes what. This is the framework Richard McElreath builds his Statistical Rethinking around, and the one we teach in our own Bayesian course. It pairs naturally with everything you have learned about regression and structural models, because it answers the question those tools cannot: which variables belong in the model?
A DAG is variables (nodes) joined by arrows. An arrow \(X \rightarrow Y\) means “\(X\) is a direct cause of \(Y\).” Acyclic means no loops: nothing causes itself, directly or through a chain. The graph has no numbers; it is the causal theory, and it does something a regression cannot: it tells you which variables to include and which to leave out. Two analysts with identical data but different DAGs will, correctly, control for different things.
Take McElreath’s running example. Does a state’s marriage rate (\(M\)) cause its divorce rate (\(D\))? Both are correlated with the median age at marriage (\(A\)). One plausible DAG is \(A \rightarrow M\), \(A \rightarrow D\), and \(M \rightarrow D\). Is the \(M\)–\(D\) association a real causal effect, or just a shadow cast by \(A\)? A regression of \(D\) on both \(M\) and \(A\) can separate them, but only because the DAG told you to include \(A\). Without the causal picture, you would not know which variable to condition on, and the data alone will never tell you.
Every DAG, however complicated, is built from four elementary pieces. Knowing how each behaves is most of causal inference (McElreath). Three of them we can watch directly.
The Fork (\(X \leftarrow Z \rightarrow Y\)). \(Z\) is a common cause, a confounder. \(X\) and \(Y\) become correlated even with no arrow between them. Control \(Z\) to close the path:
set.seed(4); n <- 2000
Z <- rnorm(n); X <- Z + rnorm(n); Y <- Z + rnorm(n) # Z causes both; no X->Y arrow
c(fork_naive = round(cor(X, Y), 2), # spurious association
fork_control = round(cor(resid(lm(X~Z)), resid(lm(Y~Z))), 2)) # control Z: it vanishes fork_naive fork_control
0.49 -0.02 The Pipe (\(X \rightarrow M \rightarrow Y\)). \(M\) is a mediator; \(X\) affects \(Y\) through \(M\). Control \(M\) and you block the very effect you were trying to measure (this is post-treatment bias):
set.seed(5)
X <- rnorm(n); M <- X + rnorm(n); Y <- M + rnorm(n) # effect runs X -> M -> Y
c(pipe_naive = round(cor(X, Y), 2), # the real, total effect
pipe_control = round(cor(resid(lm(X~M)), resid(lm(Y~M))), 2)) # control M: effect blocked pipe_naive pipe_control
0.58 -0.01 The Collider (\(X \rightarrow Z \leftarrow Y\)). \(Z\) is a common effect. Here \(X\) and \(Y\) are genuinely unrelated, but conditioning on \(Z\) creates an association that was never there:
set.seed(6)
X <- rnorm(n); Y <- rnorm(n); Z <- X + Y + rnorm(n) # X and Y both cause Z
c(collider_naive = round(cor(X, Y), 2), # correctly ~ 0
collider_control = round(cor(resid(lm(X~Z)), resid(lm(Y~Z))), 2)) # control Z: spurious! collider_naive collider_control
-0.04 -0.51 The Descendant. Conditioning on a variable caused by another partly conditions on that other variable. A descendant of a collider re-opens the collider a little; a descendant of a confounder closes the fork a little. The rule of thumb: a descendant behaves like a weak version of its parent.
Whether controlling for a variable helps or hurts depends entirely on its role in the DAG, not on anything in the data:
No fit statistic can tell these apart. The difference is entirely in the direction of the arrows, which the data cannot reveal. That is why you need the DAG.
Given a DAG, which variables must you control to estimate the causal effect of \(X\) on \(Y\)? The back-door criterion answers it: block every “back-door path” - every non-causal path from \(X\) to \(Y\) that begins with an arrow into \(X\) - by controlling confounders, while being careful not to open any colliders. For anything but the simplest graph this is easy to get wrong by hand, so the dagitty package (which the course uses) computes the adjustment set for you:
# The real tool (needs dagitty — shown, not run on the book's server)
library(dagitty)
dag <- dagitty("dag{ X -> Y ; X <- Z -> Y }") # a fork: Z confounds X and Y
adjustmentSets(dag, exposure = "X", outcome = "Y") # -> { Z }: control Z, nothing elseThe output is the set of variables to put in your regression. Sometimes it is empty (no adjustment needed); sometimes there is more than one valid set; sometimes it warns you that the effect is not identifiable from the data you have. All of that is causal information you cannot get from the numbers alone.
A DAG is an assumption, but not an untestable one. Every DAG implies a set of conditional independencies - statements like “\(A\) and \(B\) should be independent once we account for \(C\)” - that the data can confirm or refute:
impliedConditionalIndependencies(dag) # the testable implications of your DAGIf the data violate an independence your DAG predicts, the DAG is wrong and needs revising. This is how a causal model earns trust: not by fitting the data (any flexible model can do that), but by surviving the specific predictions it makes. It is the same discipline the Sports Page chapter preached, applied to causal structure.
Coming from psychology, we will let you in on something: a DAG is the same causal object as a structural equation model. The arrows are identical; only the notation differs. One of us cheerfully admits to being more fluent in standard SEM path-diagram notation than in “the rather sparse DAG notation” - but the causal logic underneath is one and the same. An SEM is a DAG with its arrows estimated; a DAG is an SEM before you attach the numbers. So the workflow is always the same two steps: draw the DAG to decide the structure (which paths exist, which variables to control), then fit the model to put numbers on it. And carry the caution from the mediation work: fitting never verifies the causal story. The perennial temptation, in the authors’ own words, is to “assume causality but rely on correlation.” The DAG is what keeps you honest about the difference.
dagitty::adjustmentSets) picks the variables to adjust for.DAGs are one half of the Statistical Rethinking worldview the authors teach; the other half is Bayesian. Having drawn our causal assumptions, we turn to a different way of quantifying uncertainty about them. The next part steps beyond the normal-curve, point-estimate, reject-or-accept world into resampling, robust methods, and Bayesian inference - the framework in which McElreath’s DAGs are usually put to work.