11

ON BIAS AMPLIFIERS

Judea Pearl University of California

Los Angeles(www.cs.ucla.edu/~judea/)

22

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:

zxdoyPzPzxyP ))(|()(),|(

ON BIAS AMPLIFIERSJudea Pearl

University of CaliforniaLos Angeles

(www.cs.ucla.edu/~judea/)

33

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?

Z6

Z3

Z2

Z5

Z1

XY

Z10

Z7 Z8Z9

Z4

U

e.g., Z = {U, Z4, Z5}

44

U

c1

X Y

Z

c2c3

c0

SURPRISING RESULT:Instrumental variables are Bias-Amplifiers in linear models (Bhattarcharya & Vogt 2007; Wooldridge 2009)

“Naive” bias

Adjusted bias

2123

2102

3

210

11))(|(),|( cc

c

ccc

c

cccxdoYE

xzxYE

xBz

2102100 ))(|()|( ccccccxdoYEx

xYEx

B

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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)

U

c1

X Y

Z

c2c3

c0

U

c1

X Y

Z

c2c3

c0

66

c0

c2

Z

c3

U

YX

c4

T1

c1

WHAT’S BETWEEN AN INSTRUMENT AND A CONFOUNDER?Should we adjust for Z?

T2

ANSWER:

CONCLUSION:

23

12

3

4

1 c

cccc

Yes, if

No, otherwise

Adjusting for a parent of Y is safer than a parent of X

77

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.

88

CAN AN IV AMPLIFY SELECTION BIAS?

ANSWER: NoExercise: which selection bias will be amplifiedby Z? S1? S2? or S3?

1

X Y

Zc3 c0

2

S

UY

S= s0

U1

S1

X Y

Z

S2 S3

UY

U2

99

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