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

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 • ‘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

Binary IV model t X , t Y Z X Y Z is assigned treatment; X is received treatment; Y is the final outcome t X ∈ { Always Taker, Never Taker, Complier, Defier } ≡ { AT, NT, CO, DE } t Y ∈ { Always Recover, Never Recover, Helped, Hurt } ≡ { AR, NR, HE, HU } p ( t X , t Y ) 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

Even Simpler model: X → Y t Y X Y p ( y | x ) is 2-dimensional p ( t Y ) lives in 3-dim. simplex Set of possible distributions p ( t Y ) compatible with p ( y | x ) is of dimension 1 . Cannot determine ‘causes of effects’ Thomas Richardson Page 4

p ( NR | t X ) p ( HE | t X ) p ( HU | t X ) 1 p ( AR | t X ) γ 1 γ 0 t X t X Here γ 0 t X ≡ p ( y = 1 | t X , do ( X = 0)) , γ 1 t X ≡ p ( y = 1 | t X , do ( X = 1)) . This accounts for 4 missing dimensions: one dimension for ‘causes of effects’ for each compliance ‘type’ This leaves just 5 to go! Thomas Richardson Page 5

p ( y | x =0 , z =0) p ( x | z =0) p ( y | x =1 , z =0) γ 0 γ 1 CO DE γ 0 γ 0 γ 1 γ 1 p ( t X ) NT AT NT AT γ 0 γ 1 DE CO p ( y | x =0 , z =1) p ( x | z =1) p ( y | x =1 , z =1) Here γ 1 NT and γ 0 AT are completely unrestricted. (2 dimensions) One dimension is associate with each of: ( γ 0 CO , γ 0 DE , γ 0 NT ) ( γ 1 CO , γ 1 DE , γ 1 AT ) p ( t X ) ≡ distribution over compliance types This accounts for the remaining 5 dimensions. Thomas Richardson Page 6

Thomas Richardson Page 7

Possible values for risk for unexposed never takers Possible values for risk for unexposed compliers Possible values for risk for unexposed defiers 1.0 1.0 1.0 0.9 0.9 0.9 0.8 0.8 0.8 0.7 0.7 0.7 Pr[Y=1|do(X=0),NT] Pr[Y=1|do(X=0),CO] Pr[Y=1|X=0,DE] 0.6 0.6 0.6 0.5 0.5 0.5 0.4 0.4 0.4 0.3 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0.1 0.0 0.0 0.0 AT AT AT Compliance Compliance Compliance 0.11 0.29 0.47 0.11 0.29 0.47 0.11 0.29 0.47 CO CO CO Type Type Type 0.53 0.35 0.17 0.53 0.35 0.17 0.53 0.35 0.17 DE DE DE Proportions: Proportions: Proportions: 0.36 0.18 0 0.36 0.18 0 0.36 0.18 0 NT NT NT 0 0.18 0.36 0 0.18 0.36 0 0.18 0.36 Possible values for risk for exposed always takers Possible values for risk for exposed compliers Possible values for risk for exposed defiers 1.0 1.0 1.0 0.9 0.9 0.9 0.8 0.8 0.8 0.7 0.7 0.7 Pr[Y=1|do(X=1),CO] Pr[Y=1|do(X=1),AT] Pr[Y=1|X=1,DE] 0.6 0.6 0.6 0.5 0.5 0.5 0.4 0.4 0.4 0.3 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0.1 0.0 0.0 0.0 AT AT AT Compliance Compliance Compliance 0.11 0.29 0.47 0.11 0.29 0.47 0.11 0.29 0.47 CO CO CO Type Type Type 0.53 0.35 0.17 0.53 0.35 0.17 0.53 0.35 0.17 DE DE DE Proportions: Proportions: Proportions: 0.36 0.18 0 0.36 0.18 0 0.36 0.18 0 NT NT NT 0 0.18 0.36 0 0.18 0.36 0 0.18 0.36 Depiction of the set of values for π X vs. � γ 0 CO , γ 0 DE , γ 0 NT � (upper row), and π X vs. � γ 1 CO , γ 1 DE , γ 1 AT � . Thomas Richardson Page 8

Possible values for risk for unexposed never takers Possible values for risk for unexposed compliers Possible values for risk for unexposed defiers 1.0 1.0 1.0 0.9 0.9 0.9 0.8 0.8 0.8 0.7 0.7 0.7 Pr[Y=1|do(X=0),CO] Pr[Y=1|do(X=0),NT] Pr[Y=1|X=0,DE] 0.6 0.6 0.6 0.5 0.5 0.5 0.4 0.4 0.4 0.3 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0.1 0.0 0.0 0.0 AT AT AT Compliance Compliance Compliance 0.11 0.29 0.47 0.11 0.29 0.47 0.11 0.29 0.47 CO CO CO Type Type Type 0.53 0.35 0.17 0.53 0.35 0.17 0.53 0.35 0.17 DE DE DE Proportions: Proportions: Proportions: 0.36 0.18 0 0.36 0.18 0 0.36 0.18 0 NT NT NT 0 0.18 0.36 0 0.18 0.36 0 0.18 0.36 Possible values for risk for exposed always takers Possible values for risk for exposed compliers Possible values for risk for exposed defiers 1.0 1.0 1.0 0.9 0.9 0.9 0.8 0.8 0.8 0.7 0.7 0.7 Pr[Y=1|do(X=1),AT] Pr[Y=1|do(X=1),CO] Pr[Y=1|X=1,DE] 0.6 0.6 0.6 0.5 0.5 0.5 0.4 0.4 0.4 0.3 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0.1 0.0 0.0 0.0 AT AT AT Compliance Compliance Compliance 0.11 0.29 0.47 0.11 0.29 0.47 0.11 0.29 0.47 CO CO CO Type Type Type 0.53 0.35 0.17 0.53 0.35 0.17 0.53 0.35 0.17 DE DE DE Proportions: Proportions: Proportions: 0.36 0.18 0 0.36 0.18 0 0.36 0.18 0 NT NT NT 0 0.18 0.36 0 0.18 0.36 0 0.18 0.36 Depiction of the set of values for π X vs. � γ 0 CO , γ 0 DE , γ 0 NT � (upper row), and π X vs. � γ 1 CO , γ 1 DE , γ 1 AT � . Thomas Richardson Page 9

Possible risks for unexposed COs and unexposed DEs Possible risks for exposed COs and exposed DEs 1.0 1.0 0.8 0.8 Pr[Y=1|X=0,DE] Pr[Y=1|X=1,DE] 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0.0 Pr[Y=1|X=0,CO] Pr[Y=1|X=1,CO] 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

Separating identified from non-identified ℵ θ p ( t X , t Y ) 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 ( t X , t Y ) = f ( θ, ℵ ) . Thomas Richardson Page 11

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

Partial identification: an analogy θ M θ F θ δ 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

Future work: incorporating covariates p ( t X , t Y ) Z X Y C 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

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