The Back-Door Criterion
By the end of this lesson, you should be able to: find the back-door paths in a causal graph, choose a set of variables that blocks them, explain why more controls aren't always better, and recognise controls that add bias.
Back-door paths
In a causal graph, the effect of a treatment T on an outcome Y travels along the arrows out of T: T → Y, or T → M → Y. Those are front-door paths, and they're the effect you want.
A back-door path is any path from T to Y that starts with an arrow pointing into T, like T ← Z → Y. It connects treatment and outcome through a common cause, so it produces association without causation. That's confounding.
The criterion
A set of variables is enough to adjust for if:
- it blocks every back-door path from T to Y, and
- it contains nothing caused by T, such as mediators, colliders, or anything downstream of the treatment.
If such a set exists and you measure it, comparing treated and untreated units with the same values of those variables gives the causal effect. If even one back-door path stays open, the estimate is biased, however many other variables you add.