Using Balancing Weights to Target the Treatment Eﬀect on the Treated when Overlap is Poor

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Using Balancing Weights to Target the Treatment Eﬀect on the Treated when

Overlap is Poor

Eli Ben-Michael∗Luke Keele†

October 5, 2022

Abstract

Inverse probability weights are commonly used in epidemiology to estimate causal eﬀects in ob-

servational studies. Researchers can typically focus on either the average treatment eﬀect or the

average treatment eﬀect on the treated with inverse probability weighting estimators. However,

when overlap between the treated and control groups is poor, this can produce extreme weights that

can result in biased estimates and large variances. One alternative to inverse probability weights

are overlap weights, which target the population with the most overlap on observed characteristics.

While estimates based on overlap weights produce less bias in such contexts, the causal estimand

can be diﬃcult to interpret. One alternative to inverse probability weights are balancing weights,

which directly target imbalances during the estimation process. Here, we explore whether balancing

weights allow analysts to target the average treatment eﬀect on the treated in cases where inverse

probability weights are biased due to poor overlap. We conduct three simulation studies and an

empirical application. We ﬁnd that in many cases, balancing weights allow the analyst to still target

the average treatment eﬀect on the treated even when overlap is poor. We show that while overlap

weights remain a key tool for estimating causal eﬀects, more familiar estimands can be targeted by

using balancing weights instead of inverse probability weights.

causal inference; weighting, causal estimands, inverse probability weighting, balancing weights, over-

lap weights

∗Carnegie Mellon University, Email: ebenmichael@cmu.edu

†University of Pennsylvania, Philadelphia, PA, Email: luke.keele@gmail.com

arXiv:2210.01763v1 [stat.ME] 4 Oct 2022

1 Introduction

Evidence for the causal eﬀect of an intervention is often drawn from observational studies, especially in

settings where randomized trials are infeasible. The ﬁrst step of conducting an observational study is

deﬁning the causal eﬀect—i.e. estimand—of interest. Two common estimands targeted in observational

studies are the average treatment eﬀect (ATE) and the average treatment eﬀect on the treated (ATT).

The ATE measures the average diﬀerence in outcomes when all individuals in the study population

are assigned to treatment versus when all individuals are assigned to control. The ATT, on the other

hand, measures the average diﬀerence in outcomes among those individuals in the population that were

actually exposed to the treatment. These estimands answer diﬀerent scientiﬁc questions, so investigators

must select which to target based on substantive judgements.

Selection of the estimand may also depend on estimation considerations. Methods such as matching

and inverse probability weighting (IPW) can be used to target either the ATE or the ATT. However,

both methods depend on the assumption that each unit in the study has a non-zero probability of being

treated. Violation or near violation of this assumption can result in biased or imprecise estimates (Crump

et al., 2009). Critically, when overlap between the treated and control populations is limited, the ATT

may be identiﬁable when the ATE is not, for instance when only some units have a non-zero probability

of treatment. For some data conﬁgurations, overlap may be so limited that even the ATT may not

be identiﬁable. When this occurs, one strategy is to use an alternative estimand that only targets the

subset of treated units that overlap with the control units (Crump et al., 2009; Rosenbaum, 2012; Li

et al., 2018). One such estimand is the average treatment eﬀect for the overlap population (ATO) (Li

et al., 2018). Under the ATO, the estimand is focused on the marginal population that 0might or might

not receive the treatment of interest rather than a known, a priori well-deﬁned population such as the

treated group. The ATO can be estimated using methods that trim propensity scores, trim treated

units, or via overlap weights estimated from propensity scores (Crump et al., 2009; Rosenbaum, 2012;

Li et al., 2018). As such, use of the ATO is typically a response to overlap problems in the data. While

analysts might often be more interested in the ATT, choosing the ATO as the estimand allows them at

least to estimate a causal eﬀect from the data.

In this study, we explore whether a new form of weighting estimator, known as balancing weights,

may allow investigators to retain the ATT estimand in applications where overlap is limited. Balancing

weights are a generalization of inverse propensity score weights that solve a convex optimization problem

to ﬁnd a set of weights that directly target covariate balance (Hainmueller, 2011; Zubizarreta, 2015).

Recent theoretical work has demonstrated that balancing weights are implicitly estimates of the inverse

propensity score, ﬁt via a loss function that guarantees covariate balance (Zhao and Percival, 2016;

Zhao, 2019; Wang and Zubizarreta, 2019a; Chattopadhyay et al., 2020). Critically, balancing weights

can often outperform inverse propensity score weights estimated from logistic regression (Ben-Michael

et al., 2022; Chattopadhyay et al., 2020; Wang and Zubizarreta, 2019b). As such, balancing weights

may perform well in many scenarios where IPW fails due to limited overlap.

This paper is organized as follows. In the Methods section, we describe the IPW estimator for the ATT,

the overlap weighting (OW) estimator for the ATO, and balancing weights estimator for the ATT. We

then conduct three simulation studies to study if balancing weights can recover the ATT in scenarios

where IPW may fail. We include comparisons to the OW estimator to understand when balancing

weights can recover the same estimand as the ATO. We conclude with a practical application using

data on right heart catherization. We show that in this data, balancing weights and OW weights yield

the same estimate. Overall, we ﬁnd that balancing weights expand the set of applications where we

can reasonably expect to estimate the ATT, relative to IPW. However, as overlap degrades beyond a

certain point, neither can estimate the ATT well. In such settings, it is prudent to switch estimands to

the ATO using overlap weights.

2 Methods

First, we outline our notation. The population is indexed by units i= 1, . . . , n, and we denote a

binary treatment using Ziwhere Zi= 1 corresponds to the treated condition and Zi= 0 corresponds

to the control condition. We let Yidenote the observed outcome, and use the potential outcomes

framework, where each unit has two potential responses: (Yi(1), Yi(0)), corresponding to their response

under treatment and control, where the observed outcome is Yi=ZiYi(1) + (1 −Zi)Yi(0). Next, we

assume that the standard assumptions for an observational study based on weighting methods hold.

These assumptions are consistency, positivity (overlap), conditional exchangeability, and no interference

(Hernán and Robins, 2020). Finally, we denote Xias vector of baseline covariates suﬃciently rich to

ensure conditional exchangeability holds.

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UsingBalancingWeightstoTargettheTreatmentEectontheTreatedwhenOverlapisPoorEliBen-Michael*LukeKeeleOctober5,2022AbstractInverseprobabilityweightsarecommonlyusedinepidemiologytoestimatecausaleectsinob-servationalstudies.Researcherscantypicallyfocusoneithertheaveragetreatmenteectortheaveragetreatme...

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