1 CONFOUNDING EQUIVALENCE Judea Pearl – UCLA, USA Azaria Paz – Technion, Israel (

2 CONFOUNDING EQUIVALENCE W1W1 W4W4 XY V2V2 V1V1 W2W2 W3W3 Z T L Z  T ? PROBLEM: When would two measurements be equally effective for reducing confounding bias? Bias =

3 CONFOUNDING EQUIVALENCE DEFINITION AND SOLUTION DEFINITION: T and Z are c -equivalent if DEFINITION (Markov boundary): Markov boundary S m of S (relative to X ) is the minimal subset of S that d -separates X from all other members of S. THEOREM: Z and T are c -equivalent iff 1. Z m =T m, or 2. Z and T are admissible (i.e., satisfy the back-door condition)

4 ARE Z AND T CONFOUNDING EQUIVALET? ANSWER: Yes! Because both are admissible W1W1 W4W4 XY V2V2 V1V1 W2W2 W3W3 Z T Z  T

5 ARE Z AND T CONFOUNDING EQUIVALET? W1W1 W4W4 XY V2V2 V1V1 W2W2 W3W3 Z T L ANSWER: No! None is admissible and Z  T

6 The definition of c -equivalence is purely statistical. The graphical criterion we found invokes “admissibility,” which is causal – why? Corollary 1: c -equivalence is an invariant property of M -equivalent graphs Corollary 2: There ought to be a graphical criterion that does not invoke “admissibility,” based on d -separation only. Indeed, Theorem: Z ~ T iff A PUZZLE: and

7 EXAMPLE: TESTING IF Z ~ T WITHOUT INVOKING ADMISSIBILITY W1W1 W4W4 XY V2V2 V1V1 W2W2 W3W3 ZT Z  T L W1W1 W4W4 V2V2 V1V1 W2W2 W3W3 ZT XY

8 W1W1 W2W2 Z1Z1 Z2Z2 XY V Z1Z1 Z2Z2 W1W1 W2W2 XY V (b)(a) APPLICATIONS TO MODEL TESTING Answer: No,  a CI that holds in (b) but not in (a) Answer: No, because { Z 1,W 1,W 2 } ~ { W 1,W 2,Z 2 } holds in (b) but not in (a) Which is easier to detect? Which is easier to test? Are these two models Markov-equivalent?

9 W1W1 W2W2 Z1Z1 Z2Z2 XY V Z1Z1 Z2Z2 W1W1 W2W2 XY V (b)(a) WHICH IS EASIER TO TEST WHEN Z IS HIGH DIMENSIONAL? Conditional independence? or c -equivalence? c -equivalence can invoke propensity scores (Scalars): And we can ignore the outcome process 

10 CONCLUSIONS c -equivalence provides: A simple, polynomial time test for deciding whether one set of measurements has the same bias-reducing potential as another. A powerful tool for: 1.Testing from sampled data, the validity of a given graphical model and, eventually, and 2.systematic search for graph structures.

11 CONCLUSIONS c -equivalence provides: A simple, polynomial time test for deciding whether one set of measurements has the same bias-reducing potential as another. A powerful tool for: 1.Testing from sampled data, the validity of a given graphical model and, eventually, and 2.systematic search for graph structures.