Causal Inference Opening Workshop - Bipartite Causal Inference with Interference for Evaluating Air Pollution Regulations - Corwin Zigler, December 10, 2019

cory.zigler@austin.utexas.edu 1 /32
Bipartite Causal Inference with Interference
for Evaluating Air Pollution Regulations
Corwin M. Zigler
Department of Statistics and Data Sciences
Department of Women’s Health
University of Texas at Austin and Dell Medical School
December 10, 2019

The Motivating Problem
Evaluating Pollution Interventions on
Power Plants

Major Pollution Source: Power Plants
⇒ Many regulations to reduce emissions
• Some install “scrubbers” in
response to regulations
→ Reduce SO2 emissions
→ Reduce ambient
particulate matter (PM2.5 )
→ Improve health
∗Note: Due to regulations in the Clean Air
Act estimated to be the the most costly and
most beneﬁcial of all monetized federal
regulations.

Major Pollution Source: Power Plants
⇒ Many regulations to reduce emissions
• Some install “scrubbers” in
response to regulations
→ Reduce SO2 emissions
→ Reduce ambient
particulate matter (PM2.5 )
→ Improve health
∗Note: Due to regulations in the Clean Air
Act estimated to be the the most costly and
most beneficial of all monetized federal
regulations.
Question: Do scrubbers on coal-fired power plants causally
affect hospitalizations for Ischemic Heart Disease (IHD) among
Medicare beneficiaries?

Long-range Pollution Transport
Movement of pollution from sources → populations
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• Emissions originating at a power plant
(yellow triangle) move
• Long distances towards conversion to
harmful pollution
• Intervening at a power plant impacts
health at many zip codes (black dots)
• ⇒ inference on a “network” of
interconnected power plants/population
locations (interference)

Scrubber Effects Propagate Across a
Network
Network of 100s of power plants and 10Ks of population
locations
Interventions at Power Plants:
“Scrubber” Installation
No Scrubber
Scrubbed Facility
Affect
−→
Health at Population Locations
(Zip Codes)
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Interference Due to Treatment Diffusion
or Complex Exposure Dependencies
(compare with unit-to-unit outcome dependencies in a social network)
• Here, interference arises due to treatment diffusion
• Resulting in complex exposure dependencies
• Governed by atmospheric transport of pollution
• Scrubber impacts diffuse across a network
• Knowledge of the complex “interference process” is
available
• Deterministic physical-chemical models of air pollution
• Need some repurposing for this purpose

Bipartite Causal Inference with Interference
Potential Outcomes Formalization

Defining Characteristics of Bipartite
Causal Inference with Interference
Evaluating the causal effects of a (binary) intervention where:
1. “Bipartite” Structure
• The intervention is defined on one type of observational
unit
• E.g., whether a coal-fired power plant installs a “scrubber”
• Outcomes of interest are defined and measured on a
second, distinct, type of unit
• E.g., hospitalizations measured at residential zip codes

Defining Characteristics of Bipartite
Causal Inference with Interference
Evaluating the causal effects of a (binary) intervention where:
1. “Bipartite” Structure
• The intervention is defined on one type of observational
unit
• E.g., whether a coal-fired power plant installs a “scrubber”
• Outcomes of interest are defined and measured on a
second, distinct, type of unit
• E.g., hospitalizations measured at residential zip codes
2. Interference
Interconnectedness between units ⇒ outcomes for a particular
unit depend upon treatments assigned to (multiple) other units

Formalization of Bipartite Structure
Two types of observational units:
1 Interventional Units
• P = {p1, p2, . . . , pi , . . . , pP}: a set of P power plants
• Where interventions occur (or not)
• Ai = 1/0 denotes presence/absence of intervention
• Treatment allocation: A = (A1, A2, . . . , AP)
• Covariates W = (W1, W2, . . . , WP)

Formalization of Bipartite Structure
Two types of observational units:
1 Interventional Units
• P = {p1, p2, . . . , pi , . . . , pP}: a set of P power plants
• Where interventions occur (or not)
• Ai = 1/0 denotes presence/absence of intervention
• Treatment allocation: A = (A1, A2, . . . , AP)
• Covariates W = (W1, W2, . . . , WP)
2 Outcome Units
• M = {m1, m2, . . . , mj , . . . , mM }: a set of M locations
• Where outcomes are measured/of interest
• Yj denotes outcome if interest
• Covariates: X = (X1, X2, . . . , XM )

Potential Outcomes with Bipartite
Interference
Yj (A) = Yj (A1 = a1, A2 = a2, . . . , AP = aP )
Ai : Treatment assignment of the the ith power plant
A(−i) Treatment assignments of all power plants except i
Most primitive individual-level causal effect:
Yj(Ai = a, A(−i) = a(−i)) − Yj(Ai = a , A(−i) = a(−i))
Effect on outcome unit mj of treatment allocation a with ai = a
versus treatment allocation a with ai = a
Key bipartite feature: P such effects deﬁned for every
j = 1, 2, . . . , M
Complicates typical notions of which treatments “directly” apply
to a unit

Direct and Indirect (or Spillover) Effects
Direct Effect (at individual M-level)
DE(i,j) = Yj(Ai = 1, A(−i) = a) − Yj(Ai = 0, A(−i) = a)
• Effect on location j of intervening (vs. not) at power plant i,
under a given allocation to all other plants

Direct and Indirect (or Spillover) Effects
Direct Effect (at individual M-level)
DE(i,j) = Yj(Ai = 1, A(−i) = a) − Yj(Ai = 0, A(−i) = a)
• Effect on location j of intervening (vs. not) at power plant i,
under a given allocation to all other plants
Indirect Effect (at individual M-level)
IE(i,j) = Yj(Ai = 0, A(−i) = a) − Yj(Ai = 0, A(−i) = a )
• Effect of changing intervention allocation to other upwind
plants

Many Types of Average Effects
owing to the (i, j) indexing of individual-level estimands
1 P-indexed effects
• Effects of treatments on pi ∈ P, averaged over locations
mj ∈ M that contain pi in their interference set
• E.g., the “typical” effect across all connected locations of
treating a single power plant

1 P-indexed effects
• Effects of treatments on pi ∈ P, averaged over locations
mj ∈ M that contain pi in their interference set
• E.g., the “typical” effect across all connected locations of
treating a single power plant
2 M-indexed effects
• Effects on mj ∈ M, averaged over treatments at pi ∈ P in
the interference set
• E.g., the “typical” effect on a single location of treating a
single power plant

2 M-indexed effects
• Effects on mj ∈ M, averaged over treatments at pi ∈ P in
the interference set
• E.g., the “typical” effect on a single location of treating a
single power plant
→ Subset of M∗-indexed effects
• Do not average over all (i, j)
• Effects on mj ∈ M, deﬁned according to a single i∗
in the
interference set
• E.g., the “typical” effect of treating a key associated i∗

“Key-Associated” Causal Effects
In the bipartite setting
In this talk, estimands will be developed based on the following
rationale (although more general development available):
• Consider for each mj, a particular power plant of interest
• E.g., the closest (or most inﬂuential) plant to location j
• Denote with p∗
i(j)
• Simplify notation: p∗
i(j) ≡ i∗
• For each j, only consider one
Yj(Ai∗ = a, A(−i∗) = a(−i∗)), DE(i∗,j), IE(i∗,j)

Joint Propensity Score Estimation of Causal
Effects of Scrubbers
Modeling the Mechanism of Interference

HYSPLIT Average Dispersions
(HyADS)
Computationally scalable approximation of air pollution transport
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• Model movement of air
parcel through time/space

(HyADS)
• Model movement of air
parcel through time/space
• Repeat multiple times per
power plant per hour per
day per year, . . .
• ≈ 108 parcels per year
tracked in time and space

(HyADS)
• ⇒ tij to denote “inﬂuence”
of pi on mj
• tj = (tji, tj2, . . . , tjP) ≡
“interference map” for the
jth location

Estimands and GPS Estimation

Bivariate Treatment
Forastiere et al. (2019+)
1 “Individual” Treatment: Scrubber status of the key
associated power plant: Ai∗
• i∗
denotes the power plant with the most inﬂuence on zip
code j
• i∗
≡ i; tji = maxi {tj }
• Ai∗ = 0, 1 according to whether the key power plant has a
scrubber

Bivariate Treatment
1 “Individual” Treatment: Scrubber status of the key
associated power plant: Ai∗
• i∗
denotes the power plant with the most inﬂuence on zip
code j
• i∗
≡ i; tji = maxi {tj }
• Ai∗ = 0, 1 according to whether the key power plant has a
scrubber
2 “Upwind” Treatment: Function of the scrubber statuses of
all other “linked” power plants
• “Exposure Mapping:” gj (·; T) : {0, 1}P−1
→ Gi
• gj (A, T) = Gj = i=i∗ tji Ai
• HyADS-weighted upwind treatment rate

“Upwind Interference” Assumption
(Adaptation of SUTVA)
For any A, A such that Ai∗ = Ai∗ and Gj = Gj :
Yj(A) = Yj(Ai∗ , Gj) = Yj(Ai∗ , Gj ) = Yj(A )
• Reduces interference to depend only on i∗ and Gj (not on
the entire A)
• Hospitalizations are the same under different A if the
treatment of the key power plant and the “upwind”
treatment rate are the same
• The key assumption about interference

“Direct” Effects in the Bipartite Setting
The “direct” effect of treating the key plant while holding ﬁxed
the “upwind” treatments:
τ(g) = E Yj(Ai∗ = 1, Gj = g) − Yi(Ai∗ = 0, Gj = g)
“Overall” average over the distribution of “upwind” treatments:
τ =
g∈G
τ(g)P(Gj = g)
“The average effect on IHD hospitalizations of installing a
scrubber on the key associated power plant”

“Indirect” or “Spillover” Effects in the
Bipartite Setting
The “indirect” or “upwind” effect of installing scrubbers on
upwind power plants without changing the key plant:
δ(g; a) = E Yj(Ai∗ = a, Gj = g) − Yj(Ai∗ = a, Gj = 0)
“Overall” average over the distribution of “upwind” treatments:
∆(a) =
g∈G
δ(g; a)P(Gj = g)
“The average effect on IHD hospitalizations of installing more
scrubbers on upwind plants”

Key-Associated and Neighborhood
Propensity Scores
View “key-associated” and “upwind” treatments as bivariate
treatment and estimate a joint propensity score:
ψ(a; g; x) = P(Ai∗ = a, Gj = g|Xj = x)
Further decompose into:
ψ(a; g; x) = P(Ai∗ = a, Gj = g|Xj = x)
= P(Ai∗ = a|Xi∗ = x) (1)
× P(Gj = g|Ai∗ = a, X = x) (2)
(1) ≡ φ(a, x) ≡ “key-associated propensity sore”
(2) ≡ λ(g, z, x) ≡ “upwind propensity score”

Estimation Strategy (1/3)
Extension of Forastiere et al. (2019+), other variations on this theme
1. Estimate the key-associated propensity score
φ(a, x) = P(Ai∗ = a|Xj) = fA∗
(z, Xj; γ)
• Logistic regression
• Key-associated PS estimates: ˆφj for j = 1, 2, . . . ,27,009
2. Stratify zip codes
• Into K = 5 strata based on ˆφj

Within each k = 1, 2, . . . , K strata:
3. Estimate the “upwind” propensity score
λ(g, z, x) = P(Gj = g|Ai∗ = a, Xj) = fG
(g, a, Xj; δk
)
• Normal regression model
• Upwind PS estimates: ˆλj
4. Estimate parametric dose-response function
Yj(a, g)|ˆλj ∼ fy
(a, g, ˆλ; θk
)
• Poisson regression model
• Predicted potential outcomes: ˆYj(a, g)
• For every (a, g) across a grid of g

5. Calculate average causal estimates
• Straightforward with ˆYj(a, g) in hand
• “Direct Effect”: τ
• “Upwind Effects”: ∆(a) for a = 0, 1
6. Bootstrap for standard errors and intervals
• “Egocentric” bootstrap, independent samples of zip codes,
preserving network structure
• Not exactly right for this sampling mechanism, but shown
to be reasonable

Effects of Scrubbers on Medicare IHD
Hospitalizations

Power Plant and Zip Code Data
Integrated with HyADS atmospheric model
472 Coal-ﬁred power plants
operating in 2005
• 106 with scrubbers
installed
• Information on emissions,
plant size, operating
capacity, other controls,
etc.

Causal Inference Opening Workshop - Bipartite Causal Inference with Interference for Evaluating Air Pollution Regulations - Corwin Zigler, December 10, 2019

Causal Inference Opening Workshop - Bipartite Causal Inference with Interference for Evaluating Air Pollution Regulations - Corwin Zigler, December 10, 2019

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