ON MEASUREMENT BIAS IN CAUSAL INFERENCE Judea Pearl
ON MEASUREMENT BIAS IN CAUSAL INFERENCE Judea Pearl University of California Los Angeles (www. cs. ucla. edu/~judea/) 1
THE MEASUREMENT BIAS PROBLEM Given causal diagram Z Z-unobserved (latent) Find P(y | do (x)) W X Y We know that But Minimize the bias 2
OUTLINE 1. 2. 3. 4. 5. 6. Effect Restoration using Matrix Inversion Example: Restoration in binary models Extension to multivariate confounders Effect Restoration in linear models The 3 -proxy principle and its variations Model testing with measurement errors 3
MEASUREMENT BIAS AND EFFECT RESTORATION Unobserved Z P(w|z) W X Assume: Y P(y | do(x)) is identifiable from measurement of W, if P(w | z) is given (Selen, 1986; Greenland & Lash, 2008) (local independence) Solution: 4
EFFECT RESTORATION IN BINARY MODELS Z 1 Weight distribution from cell (x, y) To cell (x, y, z 0) W X Y To cell (x, y, z 1) undefined 1 5
WHAT IF Z IS MULTI-VARIATE? G Z 1 Z 3 X Z 2 W 1 W 3 W 2 Z 4 Z 6 W 4 Z 5 W 5 Y If Z is high-dimensional, most cells will be empty of samples and, even if P(w | z) is known, P(x, w, y) cannot be estimated 6
PROPENSITY SCORE ESTIMATOR (Rosenbaum & Rubin, 1983) Z 1 Z 2 P(y | do(x)) = ? Z 4 Z 3 L X Z 6 Z 5 Y Theorem: Adjustment for L replaces Adjustment for Z 7
PROPENSITY SCORE RESTORATION G Z 1 Z 3 X Z 2 W 1 W 3 W 2 Z 4 Z 6 W 4 Z 5 W 5 Y From observed samples (x, w, y) to synthetic samples (x, z, y), to L(z), to 8
EFFECT RESTORATION IN LINEAR MODELS Z c 1 c 3 c 2 W X c 0 Y The pivotal parameter needed is 9
EFFECT RESTORATION FROM A SECOND PROXY Z c 1 c 3 Z c 2 c 1 W X c 0 (a) c 3 c 4 W Y X c 2 V c 0 Y (b) 10
THE THREE-PROXIES PRINCIPLE Z c 1 X c 3 c 4 W V c 0 c 2 Y Cai and Kuroki (2008) c 0 is identifiable 11
MODEL TESTING WITH MEASUREMENT ERRORS Z unobserved Problem: Test if Solution: Test if c 0 = 0 c 1 c 3 c 2 W X Y c 0 Theorem 1: If a latent variable Z d-separates two measured variables, X and Y, and Z has a proxy W, W = c. Z + , then cov(XY) must satisfy: cov(XY)=cov(XW) cov(WY) / c 2 var(Z) is testable if k=c 2 var(Z) is estimable Z Example: Given W and V, k is estimable, and c 1 c 2 c 3 c 4 Corollary: W X V c 0 Y 12
CONCLUSIONS 1. Effect restoration is feasible 2. Rests on two principles 2. 1 Matrix inversion in discrete models Requires synthetic population and propensity score estimation 2. 2 3 -proxies per latent in linear models proxies can be decoupled by clever conditioning 3. Conditional independence tests can be replaced by tetrad-like tests (using 2 SLS) for model testing 13
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