Causal Effect Estimation Bias Amplifiers in Linear Models

1. ON BIAS AMPLIFIERS Judea Pearl University of California Los Angeles (www.cs.ucla.edu/~judea/)

2. ON BIAS AMPLIFIERS Judea Pearl University of California Los Angeles (www.cs.ucla.edu/~judea/) THE PROBLEM: We wish to estimate the causal effect P(y|do(x)) by adjusting for a set Z of variables. Given a graph, G, find Z so as to minimize the bias:

3. THE SOLUTION: Z must be admissible, i.e., satisfy the back-door criterion But what if some confounders remain unmeasured (e.g., U)? Would it help if we adjust for Z10? Z3? Perhaps Z5? Or would it increase bias? Z1 Z2 e.g., Z = {U, Z4, Z5} Z3 Z5 Z4 Z10 U Y X Z6 Z9 Z7 Z8

4. SURPRISING RESULT: Instrumental variables are Bias-Amplifiers in linear models (Bhattarcharya & Vogt 2007; Wooldridge 2009) Z U c3 c1 c2 X Y c0 “Naive” bias Adjusted bias

5. Z U c3 c1 c2 X Y c0 INTUTION: When Z is allowed to vary, it absorbs (or explains) some of the changes in X. When Z is fixed the burden falls on U alone, and transmitted to Y (resulting in a higher bias) Z U c3 c1 c2 X Y c0

6. WHAT’S BETWEEN AN INSTRUMENT AND A CONFOUNDER? Should we adjust for Z? U Z c4 c1 c3 c2 T1 T2 c0 Y X Yes, if No, otherwise Adjusting for a parent of Y is safer than a parent of X ANSWER: CONCLUSION:

7. WHAT ABOUT NON-LINEAR MODELS? • Conditioning on IVs may reduce or amplify bias; mostly amplify • Conditioning on IVs may introduce its own bias where none existed.

8. CAN AN IV AMPLIFY SELECTION BIAS? Z UY c3 c0 X Y 1 2 S S= s0 ANSWER: No Exercise: which selection bias will be amplified by Z? S1? S2? or S3? Z U1 U2 UY X Y S2 S3 S1

9. CONCLUSIONS • The prevailing practice of adjusting for all covariates, especially those that are good predictors of X(the “treatment assignment,” Rubin, 2009) is totally misguided. • The “outcome mechanism” is as important, and much safer • As X-rays are to the surgeon, graphs are for causation