Backdoor criterion

As explained in Causal Inference for Data Science

The back-door criterion is a graphical criterion for determining whether conditioning on a set of variables identifies a causal effect in a directed acyclic graph.

When choosing a shoe price from historical sales rather than an A/B test, the goal is not to predict sales but to estimate what sales would be under each price. The back-door criterion uses a causal DAG to decide which observed variables belong in an adjustment set for estimating the average treatment effect. section 7.0 section 7.2.1

For treatment X and outcome Y, a set Z satisfies the criterion when it blocks every back-door path from X to Y and contains no descendant of X. A back-door path begins with an arrow entering X; such paths can create spurious association through confounding. Excluding descendants prevents conditioning from blocking part of X's causal effect on Y. section 7.2

Use the criterion after drawing a DAG from domain knowledge. List the paths between treatment and outcome, identify the back-door paths that remain open, and select variables that close all of them without conditioning on descendants of the treatment. Colliders require care: conditioning can open paths rather than close them. section 7.2 section 7.8

Warm-up exercises and injury

For a DAG relating warm-up exercises to injury, several distinct sets can satisfy the criterion, including Coach plus Fitness Level, or Fitness Level plus Genetics. Once a valid set is found, it can be used in the adjustment formula; software such as Dagitty can compute valid sets from a drawn graph. section 7.2.1

A valid adjustment set supports the adjustment formula regardless of whether estimation uses linear models, propensity scores, machine learning, or another variation. Statistical modeling does not replace the causal work of deciding what to adjust for. In a linear regression for a total effect, controlling for a set that closes every back-door path and excludes descendants of X yields an unbiased coefficient for X under the stated DAG conditions. section 7.2.1 section 7.4.5

Table 7.7 Different stages in searching for back-door closing variables

General pricing We cannot estimate the causal effect.

Stock is the key. We can estimate the causal effect, adjusting for Remaining Stock.

Overcoming the lack of stock information We can estimate the causal effect if we have Purchase Orders and # Shoes Sold Last Week.

Follow the pricing progression from an unidentifiable effect to adjustment using remaining stock, then to proxy information when stock is unavailable. section 7.2.1

The criterion depends on the DAG being credible. Data alone generally cannot establish that graph; domain knowledge is needed. If an important confounder is unobserved and no alternative observed variables close the relevant paths, the causal effect cannot be estimated by this adjustment route. section 7.2 section 7.8

How Causal Inference for Data Science covers it

How deep it goes

Defines the graphical conditions, explains back-door paths and descendants, and connects valid sets to adjustment and linear regression.

Examples

Uses shoe pricing and sales, plus warm-up exercises and sports injury, to compare possible adjustment sets.

What it assumes

Assumes a credible DAG informed by domain knowledge and builds adjustment decisions around d-separation.

What you can do after

Apply it via Dagitty or path analysis to choose adjustment variables before estimating a causal effect.

How it connects

Connects DAGs, d-separation, confounders, colliders, mediators, adjustment formulas, propensity scores, and linear models.

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Also explained in

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  Causal AIThe backdoor criterion is a graphical condition requiring a covariate set to block every backdoor path from treatment to outcome while excluding descendants of the treatment.

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Related concepts

• adjustment setused by · An adjustment set is a set of variables conditioned on to block noncausal paths between a treatment and outcome while preserving the causal paths.
• causal adjustmentconfigures · Causal adjustment estimates a causal effect by conditioning on variables that block noncausal pathways and averaging over their target-population distribution.
• backdoor pathuses · A backdoor path is a path between a putative cause and an outcome that begins with an arrow into the cause and can create a noncausal association.
• descendantuses · A descendant is a node or element reachable from another by following one or more downward links in a hierarchy or directed structure.
• d-separationhas part · D-separation is a graphical criterion for determining whether a conditioning set blocks all relevant paths between variables in a graph, implying conditional independence under the graph’s assumptions.
• directed acyclic graphused by · A directed acyclic graph is a graph with directed edges and no directed cycles.

This explainer was generated by Manning's knowledge graph from the text of Causal Inference for Data Science. The section links point to the parts of the book it draws on.