Causal Backdoor Adjustment
Separate observational association from an intervention under an explicit causal model.
By Riano · Published · Updated · Report a correction
Causal Backdoor Adjustment
When estimating the causal effect of an action (X) on a response (Y), simple observational association may be biased due to confounding variables (Z) that influence both the action and the response.
Worked Example
Suppose we want to know the effect of a treatment X on recovery Y. A confounder Z (e.g., age or severity) affects both whether treatment is taken and whether recovery occurs.
If P(Z=0) = 0.5 and P(Z=1) = 0.5, and the treated group mostly consists of people with a high baseline risk (Z=1), then a simple comparison of treated vs. untreated (observational difference) will falsely suggest the treatment is harmful.
Using backdoor adjustment, we compute the intervention probability (do-calculus) by weighting the conditional probabilities by the marginal probability of Z, removing the spurious path X ← Z → Y. For instance, if P(Y=1|X=1, Z=0) = 0.1 and P(Y=1|X=1, Z=1) = 0.8, with P(Z=0) = 0.5 and P(Z=1) = 0.5, the intervention response P(Y_(X=1)=1) is calculated as follows:
- Take the product for Z=0: 0.1 × 0.5 = 0.05.
- Take the product for Z=1: 0.8 × 0.5 = 0.40.
- Sum them: 0.05 + 0.40 = 0.45, so the expected result is 0.45.
Experiment
Explore the interactive module to see how observational and interventional responses differ. Open the Causal Backdoor Adjustment module.
- Adjust the "Propensity at Z0" to 0.1 and "Propensity at Z1" to 0.9. Notice how the "Observed difference" becomes 0.56 due to confounding.
- Adjust the propensities both to 0.5. Notice how the "Observed difference" drops to 0, matching the true "SCM causal difference" of 0.
Limitations
Backdoor adjustment requires all confounders Z to be observed. If there are unobserved confounders, this adjustment cannot fully remove bias.
Furthermore, the positivity assumption must hold. If P(X=x|Z=z) is 0 or 1 for any stratum z (e.g. if one group never receives the treatment), then the conditional probability P(Y|X,Z) cannot be estimated from data, rendering the intervention effect unidentifiable without extrapolation.
References
The following resources provide a comprehensive background on causal inference and the do-calculus framework used in this module.
Experience it interactively
Adjust parameters, observe in real time, and build deep intuition with Riano’s interactive Causal Backdoor Adjustment module.
Try Causal Backdoor Adjustment on Riano →