ON BIAS AMPLIFIERS Judea Pearl University of California
ON BIAS AMPLIFIERS Judea Pearl University of California Los Angeles (www. cs. ucla. edu/~judea/) 1
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: 2
THE SOLUTION: Z must be admissible, i. e. , satisfy the back-door criterion Z Z 1 Z 3 2 Z 5 Z 4 Z 10 U X Y Z 6 Z 9 e. g. , Z = {U, Z 4, Z 5} Z 7 Z 8 But what if some confounders remain unmeasured (e. g. , U)? Would it help if we adjust for Z 10? Z 3? Perhaps Z 5? Or would it increase bias? 3
SURPRISING RESULT: Instrumental variables are Bias-Amplifiers in linear models (Bhattarcharya & Vogt 2007; Wooldridge 2009) Z U c 3 “Naive” bias c 1 X c 2 c 0 Y Adjusted bias 4
INTUTION: When Z is allowed to vary, it absorbs (or explains) some of the changes in X. Z U c 3 c 1 X c 2 Y c 0 When Z is fixed the burden falls on U alone, and transmitted to Y (resulting in a higher bias) Z U c 3 c 1 X c 2 c 0 Y 5
WHAT’S BETWEEN AN INSTRUMENT AND A CONFOUNDER? Should we adjust for Z? U Z c 4 c 1 c 3 c 2 T 1 X ANSWER: T 2 c 0 Y Yes, if No, otherwise CONCLUSION: Adjusting for a parent of Y is safer than a parent of X 6
WHAT ABOUT NON-LINEAR MODELS? 1. Conditioning on IVs may reduce or amplify bias; mostly amplify 2. Conditioning on IVs may introduce its own bias where none existed. 7
CAN AN IV AMPLIFY SELECTION BIAS? Z c 3 UY c 0 X 1 Y 2 S S= s 0 ANSWER: No Exercise: which selection bias will be amplified by Z? S 1? S 2? or S 3? Z U 1 U 2 UY X Y S 2 S 3 S 1 8
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 9
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