1. University of California
ON MEASUREMENT BIAS
IN CAUSAL INFERENCE
Judea Pearl
University of California
Los Angeles (

2. THE MEASUREMENT BIAS PROBLEM
Given causal diagram
Z-unobserved (latent)
Find P(y | do (x))
We know that
But
Minimize the bias Z W X Y

3. OUTLINE 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

4. MEASUREMENT BIAS AND EFFECT RESTORATION
Unobserved Z
P(w|z)
P(y | do(x)) is identifiable from measurement of W, if P(w | z) is given (Selen, 1986; Greenland & Lash, 2008) W X Y
Assume:
(local independence)
Solution:

5. EFFECT RESTORATION IN BINARY MODELS Z 1 W X Y 1 Weight distribution
from cell (x,y)
To cell
(x,y,z0) W X Y
To cell
(x,y,z1)
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6. WHAT IF Z IS MULTI-VARIATE?G Z1 Z2 W1 W2 Z3 W3 Z4 Z5 W4 W5 X Z6 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

7. PROPENSITY SCORE ESTIMATOR
(Rosenbaum & Rubin, 1983) Z1 Z2
P(y | do(x)) = ? L Z4 Z3 Z5 X Z6 Y
Adjustment for L replaces Adjustment for Z
Theorem:

8. PROPENSITY SCORE RESTORATIONG Z1 Z2 W1 W2 Z3 W3 Z4 Z5 W4 W5 X Z6 Y
From observed samples (x,w,y) to synthetic samples (x,z,y), to L(z), to

9. EFFECT RESTORATION IN LINEAR MODELS Z c1 c2 c3 W X c0 Y
The pivotal parameter needed is

10. EFFECT RESTORATION FROM A SECOND PROXY W Z X Y c1 c2 c3 c0 V Z X Y c1
(b)

11. THE THREE-PROXIES PRINCIPLEZ Y c2 c0 c1 c3 c4 W V X
Cai and Kuroki (2008)
c0 is identifiable

12. MODEL TESTING WITH MEASUREMENT ERRORS Z unobserved Problem: Test if c1
Solution: Test if c0 = 0
Theorem 1: If a latent variable Z d-separates two measured variables, X and Y, and Z has a proxy W, W = cZ + , then cov(XY) must satisfy: cov(XY)=cov(XW) cov(WY) / c2 var(Z)
Corollary: is testable if k=c2 var(Z) is estimable
Example: Given W and V, k is estimable, and X c0 Y V Z X Y c1 c2 c4 c0 W c3

13. CONCLUSIONS Effect restoration is feasible Rests on two principles
2.1 Matrix inversion in discrete models Requires synthetic population    and propensity score estimation
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 2SLS) for model testing