Analysis of the Binary IV Model or The case of the missing 9 - - PowerPoint PPT Presentation

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Analysis of the Binary IV Model or The case of the missing 9 - - PowerPoint PPT Presentation

Mar 29, 2023 6 likes •160 views

Analysis of the Binary IV Model or The case of the missing 9 dimensions Thomas Richardson Department of Statistics University of Washington This is joint work with James Robins (Harvard). Outline 1. The binary IV potential outcomes model

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SLIDE 1

Analysis of the Binary IV Model

  • r

The case of the missing 9 dimensions Thomas Richardson

Department of Statistics University of Washington This is joint work with James Robins (Harvard).

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SLIDE 2

Outline

  • 1. The binary IV potential outcomes model
  • ‘partial’ identification
  • 2. Even simpler: X → Y
  • 3. Analysis of the binary IV model
  • pictures
  • separating the identified from the unidentified
  • 4. ?Implications for Bayesian Inference?

Thomas Richardson Page 2

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SLIDE 3

Binary IV model

tX, tY Z X Y Z is assigned treatment; X is received treatment; Y is the final outcome tX ∈ {Always Taker, Never Taker, Complier, Defier } ≡ {AT, NT, CO, DE} tY ∈ {Always Recover, Never Recover, Helped, Hurt} ≡ {AR, NR, HE, HU} p(tX, tY ) lives in 15-dim. simplex. p(x, y|z) lives in a 6-dim. space (product of two 3-dim. simplices).

Thomas Richardson Page 3

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SLIDE 4

Even Simpler model: X → Y

tY X Y p(y|x) is 2-dimensional p(tY ) lives in 3-dim. simplex

Set of possible distributions p(tY ) compatible with p(y|x) is of dimension 1. Cannot determine ‘causes of effects’

Thomas Richardson Page 4

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SLIDE 5

p(AR | tX) p(HE | tX) p(HU | tX) p(NR | tX) γ1 tX γ0 tX 1

Here γ0

tX ≡ p(y = 1 | tX, do(X = 0)), γ1 tX ≡ p(y = 1 | tX, do(X = 1)).

This accounts for 4 missing dimensions:

  • ne dimension for ‘causes of effects’ for each compliance ‘type’

This leaves just 5 to go!

Thomas Richardson Page 5

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SLIDE 6

γ0

AT

γ0

NT

p(tX) γ1

NT

γ1

AT

p(y|x=0, z =1) p(x|z =1) p(y|x=1, z =1) p(y|x=0, z =0) p(x|z =0) p(y|x=1, z =0) γ0

CO

γ1

DE

γ0

DE

γ1

CO

Here γ1

NT and γ0 AT are completely unrestricted. (2 dimensions)

One dimension is associate with each of:

(γ0

CO, γ0 DE, γ0 NT)

(γ1

CO, γ1 DE, γ1 AT)

p(tX) ≡ distribution over compliance types

This accounts for the remaining 5 dimensions.

Thomas Richardson Page 6

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SLIDE 7

Thomas Richardson Page 7

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SLIDE 8

Pr[Y=1|do(X=0),NT]

Possible values for risk for unexposed never takers

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.11 0.29 0.47

AT

0.53 0.35 0.17

CO

0.36 0.18

DE

0.18 0.36

NT Compliance Type Proportions: Pr[Y=1|do(X=0),CO]

Possible values for risk for unexposed compliers

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.11 0.29 0.47

AT

0.53 0.35 0.17

CO

0.36 0.18

DE

0.18 0.36

NT Compliance Type Proportions: Pr[Y=1|X=0,DE]

Possible values for risk for unexposed defiers

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.11 0.29 0.47

AT

0.53 0.35 0.17

CO

0.36 0.18

DE

0.18 0.36

NT Compliance Type Proportions: Pr[Y=1|do(X=1),AT]

Possible values for risk for exposed always takers

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.11 0.29 0.47

AT

0.53 0.35 0.17

CO

0.36 0.18

DE

0.18 0.36

NT Compliance Type Proportions: Pr[Y=1|do(X=1),CO]

Possible values for risk for exposed compliers

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.11 0.29 0.47

AT

0.53 0.35 0.17

CO

0.36 0.18

DE

0.18 0.36

NT Compliance Type Proportions: Pr[Y=1|X=1,DE]

Possible values for risk for exposed defiers

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.11 0.29 0.47

AT

0.53 0.35 0.17

CO

0.36 0.18

DE

0.18 0.36

NT Compliance Type Proportions:

Depiction of the set of values for πX vs. γ0

CO, γ0 DE, γ0 NT (upper row), and πX vs. γ1 CO, γ1 DE, γ1 AT.

Thomas Richardson Page 8

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SLIDE 9

Pr[Y=1|do(X=0),NT]

Possible values for risk for unexposed never takers

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.11 0.29 0.47

AT

0.53 0.35 0.17

CO

0.36 0.18

DE

0.18 0.36

NT Compliance Type Proportions: Pr[Y=1|do(X=0),CO]

Possible values for risk for unexposed compliers

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.11 0.29 0.47

AT

0.53 0.35 0.17

CO

0.36 0.18

DE

0.18 0.36

NT Compliance Type Proportions: Pr[Y=1|X=0,DE]

Possible values for risk for unexposed defiers

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.11 0.29 0.47

AT

0.53 0.35 0.17

CO

0.36 0.18

DE

0.18 0.36

NT Compliance Type Proportions: Pr[Y=1|do(X=1),AT]

Possible values for risk for exposed always takers

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.11 0.29 0.47

AT

0.53 0.35 0.17

CO

0.36 0.18

DE

0.18 0.36

NT Compliance Type Proportions: Pr[Y=1|do(X=1),CO]

Possible values for risk for exposed compliers

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.11 0.29 0.47

AT

0.53 0.35 0.17

CO

0.36 0.18

DE

0.18 0.36

NT Compliance Type Proportions: Pr[Y=1|X=1,DE]

Possible values for risk for exposed defiers

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.11 0.29 0.47

AT

0.53 0.35 0.17

CO

0.36 0.18

DE

0.18 0.36

NT Compliance Type Proportions:

Depiction of the set of values for πX vs. γ0

CO, γ0 DE, γ0 NT (upper row), and πX vs. γ1 CO, γ1 DE, γ1 AT.

Thomas Richardson Page 9

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SLIDE 10

Possible risks for unexposed COs and unexposed DEs

Pr[Y=1|X=0,CO] Pr[Y=1|X=0,DE] 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Possible risks for exposed COs and exposed DEs

Pr[Y=1|X=1,CO] Pr[Y=1|X=1,DE] 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Thomas Richardson Page 10

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SLIDE 11

Separating identified from non-identified

θ ℵ p(tX, tY ) Z X Y θ is a 6 dim. parameter, (completely!) identifiable from p(x, y|z). ℵ is a 9 dim. parameter, (completely!) non-identifiable. p(tX, tY ) = f(θ, ℵ).

Thomas Richardson Page 11

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SLIDE 12

Partial identification: an analogy

θ δ X Y

  • Randomized study to find average effect (δ) of X on Y in a target

population containing (a known proportion of) males and females.

  • Researcher reports priors on model params (θ), and posterior for δ

– Aware of partial identification but posterior is clearly different from prior ⇒ ‘informed’ by data

Thomas Richardson Page 12

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SLIDE 13

Partial identification: an analogy

θ δ θM θF δ X Y X Y

  • Randomized study of the effect of a drug X on outcome Y in a target

population of males and females.

  • Later you learn that sample contained no females!

partial identification + proper priors = opaque inference

Thomas Richardson Page 13

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SLIDE 14

Future work: incorporating covariates

p(tX, tY ) Z C X Y

An identified parameterization makes it possible to parameterize the IV model

p(x, y|z, c) conditional on baseline covariates (C).

Thomas Richardson Page 14

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SLIDE 15

Parameters For Plots

P(X = 1|Z = 0) = 0.47 P(X = 1|Z = 1) = 0.64 P(Y = 1|X = 0, Z = 0) = 0.655 P(Y = 1|X = 0, Z = 1) = 0.52 P(Y = 1|X = 1, Z = 0) = 0.49 P(Y = 1|X = 1, Z = 1) = 0.53

Thomas Richardson Page 15