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L4-05 · Frontdoor identification

Calculate a frontdoor effect and state all three graphical conditions that justify it.

Level 4 icon Level 4 · Undergraduate
Objective. Calculate a frontdoor effect and state all three graphical conditions that justify it.
Before this lesson. L4-04
Hands-on lab. Confounding and intervention

Learn

Sometimes treatment and outcome have an unmeasured common cause, so ordinary backdoor adjustment is unavailable. A measured mediator can still permit identification in a special graph. The frontdoor criterion requires that the mediator intercept every directed path from treatment to outcome.

Two further path conditions are required. There must be no open backdoor path from treatment to mediator. Treatment must block every backdoor path from mediator to outcome. A measured mediator alone is insufficient. A direct treatment-to-outcome path or an unblocked mediator-outcome common cause can invalidate the ordinary formula.

First estimate the mediator distribution for each treatment. Then calculate a standardized mediator-outcome response by averaging over the observed treatment distribution. Combine these two quantities. This derivation separates the treatment-to-mediator effect from the mediator-to-outcome effect using different path conditions, rather than pretending the hidden cause was observed.

Practice

Solution

At X=1, the risk is 0.75(0.6)+0.25(0.2)=0.50. At X=0, it is 0.25(0.6)+0.75(0.2)=0.30. Subtract the second intervention risk from the first to obtain the risk difference of 0.20.

Check your understanding

Which condition is required by the ordinary frontdoor criterion?

Show the answer

B. The mediator must intercept every directed X-to-Y path

The mediator must intercept every directed treatment-to-outcome path. Hidden treatment-outcome confounding can remain when the other frontdoor conditions hold.

Sources

This lesson draws on Pearl, 1995Pearl, 2009.

Next: L4-06 · Potential outcomes and randomization

References
  1. Pearl, J. (1995). Causal diagrams for empirical research. https://ftp.cs.ucla.edu/pub/stat_ser/R218-B.pdf
  2. Pearl, J. (2009). Causal inference in statistics: An overview. Statistics Surveys, 3(none), 96–146. 10.1214/09-ss057
  3. Bareinboim, E. (2025). Causal Artificial Intelligence: A Roadmap for Building Causally Intelligent Systems. https://causalai-book.net/