Matching Estimators of Causal Eﬀects in Clustered Observational Studies with Application to Quantifying the Impact of

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Matching Estimators of Causal Eﬀects in

Clustered Observational Studies with

Application to Quantifying the Impact of

Marine Protected Areas on Biodiversity

Can Cui, Shu Yang, Brian J Reich

Department of Statistics, North Carolina State University

and

David A Gill

Nicholas School of the Environment, Duke University

Abstract

Marine conservation preserves ﬁsh biodiversity, protects marine and coastal ecosys-

tems, and supports climate resilience and adaptation. Despite the importance of

establishing marine protected areas (MPAs), research on the eﬀectiveness of MPAs

with diﬀerent conservation policies is limited due to the lack of quantitative MPA

information. In this paper, leveraging a global MPA database, we investigate the

causal impact of MPA policies on ﬁsh biodiversity. To address challenges posed by

this clustered and confounded observational study, we construct a matching estimator

of the average treatment eﬀect and a cluster-weighted bootstrap method for variance

estimation. We establish the theoretical guarantees of the matching estimator and

its variance estimator. Under our proposed matching framework, we recommend

matching on both cluster-level and unit-level covariates to achieve eﬃciency. The

simulation results demonstrate that our matching strategy minimizes the bias and

achieves the nominal conﬁdence interval coverage. Applying our proposed matching

method to compare diﬀerent MPA policies reveals that the no-take policy is more

eﬀective than the multi-use policy in preserving ﬁsh biodiversity.

Key words: Causal inference, conservation, potential outcomes, weighted bootstrap

arXiv:2210.03890v1 [stat.ME] 8 Oct 2022

1 Introduction

1.1 Causal Impact of Marine Protected Areas on Biodiversity

Preserving marine biological diversity is an important objective of governments, scien-

tists, local communities, and conservationists. Marine protected areas (MPAs) have been

established worldwide to keep sustainable and resilient marine ecosystems by restricting

destructive and extractive activities within their boundaries (Grorud-Colvert et al., 2021;

UNEP-WCMC et al., 2021). Despite widespread use, the eﬀectiveness of many MPAs and

diﬀerent types of MPA policies in conserving marine biodiversity remain unclear (Grorud-

Colvert et al., 2021). Very few studies employ rigorous causal inference methods to assess

MPA impacts, and even less so to investigate the relative eﬀects of diﬀerent conservation

policies (Ferraro et al., 2019). Such studies, however, are important and have signiﬁcant

policy implications, as prohibiting ﬁshing activities that are potentially important for local

food and livelihood security can result in signiﬁcant social costs and harm (e.g., Kamat

(2014); Bennett and Dearden (2014)).

Gill et al. (2017) investigated the eﬀectiveness of MPA management and its impacts

on ﬁsh populations. They developed a database of ecological, management, social, and

environmental conditions in and around hundreds of MPAs globally. In their study, man-

agement attributes such as available capacity were strongly associated with increases in

ﬁsh biomass observed in MPAs. Nonetheless, the relative eﬀects of diﬀerent types of MPAs

(referred to as policies or treatments), such as those that restrict ﬁshing (hereafter called

multi-use or MU MPAs) and those that prohibit all ﬁshing (hereafter called no-take or NT

MPAs) require further investigation.

While the Gill et al. (2017) database represents one of the largest global datasets of MPA

conditions and ecological outcomes to date, its properties present signiﬁcant challenges for

applying traditional causal inference methods. First, given the intractability of conducting

randomized experiments in many conservation settings, the global MPA dataset is observa-

tional, and thus subject to confounding biases not present when treatment is randomized

(Pynegar et al., 2021). MU and NT MPAs are likely to be located in areas with diﬀerent

social, environmental and regulatory conditions. Direct comparisons of the biodiversity

between MU and NT MPAs are fallible. Second, the MPA data are spatially clustered as

nearby sites are usually under the same conservation policy, whether it be because they lie

within the same MPA, speciﬁc management zone within an MPA (e.g., no diving area),

or larger-scale management policy area (e.g., regional or national level ﬁshing policies).

Individual sites also share similar geographical, environmental, and social features that are

possibly dependent on each other. Therefore, estimating the causal impacts of policies such

as MPAs requires appropriate methods for clustered and confounded data.

1.2 Previous Work: Causal Inference in Observational Studies

Although randomized experiments serve as the gold standard, observational studies can

estimate causal eﬀects when all confounding variables are well balanced between treatment

groups. To adjust for the imbalance in observed confounding covariates, matching (Stuart,

2010) is often applied to isolate causal eﬀects due to its transparency and intuitive appeal.

While statistical methods to estimate causal eﬀect in observational studies are growing,

most methods apply to unstructured data (i.e., without clustering). However, clustering

often exists because subjects may be grouped by experimental design, geography, or by

sharing higher-level features. Examples include health and educational studies, where

patients are nested in hospitals and students are clustered in classrooms or schools. Such

clustered data structure poses additional challenges when inferring the causal eﬀect. In

our motivating example, the MPA database is naturally clustered, where several sites are

nested in the MPA. Capturing MPA-level as well as site-level features (e.g., local social or

environmental conditions) is important to remove confounding biases when evaluating the

eﬀectiveness of environmental policies.

To estimate causal eﬀects in clustered data, Cafri et al. (2019) showed that treatment

eﬀect estimation is more accurate when accounting for cluster-level confounding variables.

Even if suﬃcient individual-level covariates are included, ignoring cluster-level confounding

covariates would leave a bias on estimation. Within the matching framework, several

propensity score methods are developed for the clustered data (Hong and Raudenbush,

2006; Arpino and Mealli, 2011; Li et al., 2013; Yang, 2018). However, King and Nielsen

(2019) discussed the ineﬃciency and failure of balancing covariate distributions between

treatment groups using the propensity score. They attribute the ineﬃciency of matching

on propensity scores to its goal of mimicking a completely randomized trial rather than a

block-randomized trial as well as error in estimating the propensity score.

1.3 Our Contribution: A Matching Strategy in Clustered Obser-

vational Studies

This article focuses on matching as a nonparametric approach and intuitively mimics a

cluster-randomized experiment. We aim to estimate the causal eﬀect by matching estima-

tors under the framework in Abadie and Imbens (2006). Following the characteristics in

the MPA database, we analyze the clustered data where the treatment is clustered within

the MPA. Nearby sites tend to be assigned the same MPA policy, and one MPA usually

contains a single policy only. Cluster-level and unit-level covariates are available, and the

outcome is collected at the unit level. To account for the conditional bias when matching on

multiple covariates, we adopt the bias-corrected matching estimator (Abadie and Imbens,

2011) in clustered data for two common estimands, the average treatment eﬀect and aver-

age treatment eﬀect on the treated, and establish the large sample properties. Under this

data structure, matching on cluster-level covariates is suﬃcient to remove the confound-

ing biases. However, we recommend including relevant unit-level covariates in matching

to achieve higher eﬃciency and lower variance. We show reduced variance in theory and

simulation to demonstrate the advantages of matching on both cluster-level and unit-level

covariates.

To account for clustered dependence, we propose a cluster-weighted bootstrap method

for variance estimation, which combines the idea of cluster bootstrap (Davison and Hink-

ley, 1997) and weighted bootstrap (Otsu and Rai, 2017). Based on a linearization of the

matching estimator, the weighted bootstrap method creates residuals so that matching

estimators can be viewed as the sample averages of residuals. The variance of the match-

ing estimator can then be approximated by bootstrapping the residuals with appropriate

weights. This method preserves the distribution of the number of times that each unit is

matched in the resampling procedure. Thus, it avoids the failure of the standard bootstrap

in this setting, as discussed in Abadie and Imbens (2008).

The rest of this paper is organized as follows. In Section 2, we introduce the motivating

data and describe challenges in establishing causal eﬀects due to the nature of the data

structure. Section 3 introduces the notation, assumptions, and estimands of interests. Sec-

tion 4 explores the large sample properties of matching estimators in clustered data. Section

5 presents the cluster-weighted bootstrap procedure for variance estimation. An extension

to unit-level treatment assignments for matching estimators is described in Section 6. In

Section 7, we apply the proposed matching estimator in the MPA data to investigate the

causal eﬀect of diﬀerent marine protection policies on ﬁsh biodiversity. In Section 8, a sim-

ulation study is reported to evaluate the performance of the proposed matching estimator

in clustered data. Finally, we conclude our ﬁndings in Section 9.

2 MPA Data and Exploratory Analysis

The MPA dataset created by Gill et al. (2017) includes social, environmental, and ecological

information in 9987 sites within 215 MPAs worldwide (Figure 1). The number of sites in

each MPA ranges from 1 to 1619, with a mean of 46 and a median of 8. The multi-use

(MU) and the no-take (NT) policy represent two broad categories of types of MPAs. The

MU policy regulates ﬁshing activities to reduce negative impacts, while the NT policy is

more rigorous and prohibits all ﬁshing within the MPA boundaries. Among 9987 sites,

3988 sites receive the NT policy, whereas 5999 are under the MU policy. The outcome

variable is total ﬁsh biomass at each site, recorded in underwater visual surveys. There

are 13 continuous covariates and 4 categorical covariates that describe the MPA-level and

site-level features (Table 1). MPA-level covariates include MPA size and country. The

other covariates include site-level social and environmental conditions, as well as sampling

protocol, location, and date.

Figure 1: Map showing MPA location, size and policy type (MU = multi-use, NT = no-

take); MPA policies are present by the majority within each MPA.

Table 1: Feature list in the MPA database with units in parentheses for continuous variables

and number of levels in parentheses for categorical variables. A detailed summary of the

covariates is given in Gill et al. (2017) Supplementary Table 5.

Site-level Covariates MPA-level Covariates

Continuous (13)

Latitude (degree)

Longitude (degree)

Depth (m)

Wave exposure (kW/m)

Distance to shoreline (km)

Distance to population center (“market”; km)

Coastal population (million/100km2)

Sample date (year)

Minimum sea surface temperature (◦C)

Chlorophyll-A (mg/m3)

Reef area within 15km (km2)

MPA age (years)

MPA size (km2)

Categorical (4)

Habitat type (16)

Marine ecoregion (56)

Sampling protocol (6)

Country (43)

Sites within the same MPA usually receive the same policy (i.e., same ﬁshing regula-

tions), and each site belongs only to one MPA. As a result, the dataset is naturally clustered

where observed sites are nested within the MPA, and conservation policies are geographi-

cally clustered. The cluster structure brings diﬃculty in causal inference due to potential

confounding. Both cluster-level and site-level covariates could contribute to the confound-

ing bias. Sites in the same MPA share common environmental, MPA-level and geographical

characteristics, aﬀecting both the ﬁsh population and MPA policy assignment (Ahmadia

et al., 2015; Gill et al., 2017; Ferraro et al., 2019). Site-speciﬁc covariates, including depth,

distance to population centers (also called “markets”), size of neighboring human popu-

lation, and chlorophyll concentration, are also relevant to the ecological outcome (Brewer

et al., 2013; Edgar et al., 2014; Gill et al., 2017; Campbell et al., 2020). Within the same

MPA, implementing either the MU or NT policy could be heavily inﬂuenced by pre-existing

ecological conditions, local tourism, ﬁshing, or politics (Toth et al., 2014; Karr et al., 2015),

which are ideally captured as site-speciﬁc covariates.

Confounding and clustering present two major challenges. We compare the covariate

distributions under the two MPA policies for both unadjusted and adjusted samples. The

unadjusted sample refers to the raw observation, while the adjusted sample results from

multiple matching (one-to-three) using the Mahalanobis distance and with replacement.

Figure 2 is a hypothetical example to illustrate the applied multiple matching. The letters

A, G, and K represent sites under the multi-use policy, while the rest of the letters represent

sites under the no-take policy. With 1:3 matching, one multi-use site is matched with three

no-take sites. Meanwhile, the matched no-take sites can be paired with other multi-use

sites. For example,the matched no-take site E is used twice to match site A and G.

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MatchingEstimatorsofCausalEectsinClusteredObservationalStudieswithApplicationtoQuantifyingtheImpactofMarineProtectedAreasonBiodiversityCanCui,ShuYang,BrianJReichDepartmentofStatistics,NorthCarolinaStateUniversityandDavidAGillNicholasSchooloftheEnvironment,DukeUniversityAbstractMarineconservationpre...

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