Propensity Score in the Face of Interference

Abstract

Rosenbaum and Rubin’s (1983) propensity score revolutionized the field of causal inference and has emerged as a standard tool when researchers reason about cause-and-effect relationship across many disciplines. This discussion centers around the key “no interference” assumption in Rosenbaum and Rubin’s original development of the propensity score and reviews some recent advances in extending the propensity score to studies involving dependent happenings.

Four decades ago, Paul Rosenbaum and Donald Rubin wrote one of the most consequential articles about drawing causal inference from non-randomized observational data. Rosenbaum and Rubin examined the key role of “assignment mechanism” in non-randomized studies and developed a suite of propensity-score-based tools in Rosenbaum and Rubin (1983) and subsequent works, including propensity score matching (Rosenbaum and Rubin, 1983, 1985), propensity score subclassification (Rosenbaum and Rubin, 1984), and weighting/direct adjustment based on the propensity score (Rosenbaum, 1987). Propensity score is also a key element in doubly-robust estimators, such as augmented inverse probability weighted estimators (Robins and Rotnitzky, 1995) and targeted maximum likelihood estimation (Van der Laan and Rose, 2011). Propensity-score-based methods have now become standard tools in many branches of medical research, epidemiology, and social science, and have contributed to many important scientific endeavors. For instance, a propensity score based analysis was recently used to assess the effectiveness of the mRNA vaccines against SARS-CoV-2 infection requiring hospitalization, ICU admission, or an emergency department or urgent care clinic visit using electronic health records data (Thompson et al., 2021).

A key assumption made in Rosenbaum and Rubin (1983) is the no interference assumption. According to Cox (1958), there is no interference if “the observation on one unit [is] unaffected by the particular assignment of treatments to the other units.” This no interference assumption is a key element in Rubin’s “stable unit-treatment value assumption” or SUTVA (Rubin, 1980). The no interference assumption is often not plausible in the infectious disease context; for instance, Rubin (1990, Page 475) wrote that “interference between units can be a major issue when studying medical treatments for infectious disease.” In a series of articles, Halloran and Struchiner (1995), Sobel (2006) and Hong and Raudenbush (2006) discussed how to relax the no interference assumption under the potential outcomes framework. Rosenbaum (2007) distinguished between two sharp null hypotheses under interference, the null hypothesis of no primary effect and that of no effect. Rosenbaum (2007) also developed randomization-based inferential procedures for them. Hudgens and Halloran (2008) formally defined different kinds of causal effects of interest, including direct, indirect (spillover), total, and overall effects, under a partial interference context (Sobel, 2006) where individuals are partitioned into $N$ clusters, each with size $n_i$, such that there is no interference between individuals in different clusters. Hudgens and Halloran (2008) also proposed unbiased estimators for these estimands under an experimental design using two-stage randomization.

A natural question then emerged: Can researchers still identify these scientifically meaningful causal effects (e.g., direct and indirect effects) in a non-experimental setting using observational data, and if so, under what assumptions? In the absence of two-stage randomization, researchers, not surprisingly, turned their attention to the propensity score. Tchetgen Tchetgen and VanderWeele (2012) first discussed how to estimate the direct, indirect, total, and overall effects using data from observational studies in a partial interference setting. Their key identification assumption is a cluster-level generalization of Rosenbaum and Rubin’s (1983) strong ignorability assumption:

$$P\left({\mathbf{A}}_i={\mathbf{a}}_i\mid {\mathbf{X}}_i,{\mathbf{Y}}_i(\cdot )\right)=P\left({\mathbf{A}}_i={\mathbf{a}}_i\mid {\mathbf{X}}_i\right),\text{for}i=1,\dots ,N,$$ (1)

where ${\mathbf{A}}_i$ is the treatment assignment of all $n_i$ individuals in cluster $i,{\mathbf{a}}_i$ is one realization in the set of all possible length-$n_i$ treatment assignments $𝒜\left(n_i\right),{\mathbf{X}}_i$ is the observed pre-treatment covariates including both cluster- and individual-level covariates, and ${\mathbf{Y}}_i(\cdot )$ is the collection of all possible potential outcomes of $n_i$ individuals under all possible treatment allocations in $𝒜\left(n_i\right)$. The cluster-level propensity score $P\left({\mathbf{A}}_i={\mathbf{a}}_i\mid {\mathbf{X}}_i\right)$ describes the relationship between observed treatment assignments and covariates, and is the probability of the joint treatment statuses of all individuals within the cluster. The cluster-level propensity score collapses to Rosenbaum and Rubin’s (1983) original propensity score when $n_i=1$; it also collapses to the original propensity score when $𝒜\left(n_i\right)=\left\{\right(1,\dots ,1),(0,\dots ,0\left)\right\}$, i.e., when there are only two possible treatment allocations for each cluster: assigning each individual to 1 or each to 0 (Zubizarreta and Keele, 2017).

The cluster-level propensity score $P\left({\mathbf{A}}_i={\mathbf{a}}_i\mid {\mathbf{X}}_i\right)$ may be modeled using a mixed effects model that takes into account non-null associations among individuals in the same cluster as follows (Tchetgen Tchetgen and VanderWeele, 2012; Perez-Heydrich et al., 2014):

$$P\left({\mathbf{A}}_i={\mathbf{a}}_i\mid {\mathbf{X}}_i;\mathbf{\psi }\right)=\int \prod _{j=1}^{n_i}{h}_{ij}{\left(b_i;{\psi }_a\right)}^{A_{ij}}\times {\left\{1-h_{ij}\left(b_i;{\psi }_a\right)\right\}}^{A_{ij}}{f}_b\left(b_i;{\psi }_b\right)d{b}_i,$$ (2)

where $b_i\sim N\left(0,{\psi }_b\right)$ is a cluster-level random effect with density function $f_b\left(b_i;{\psi }_b\right)$, and $h_{ij}\left(b_i;{\psi }_a\right)=P\left(A_{ij}=1\mid {\mathbf{X}}_{ij};b_i,{\psi }_a\right)={\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{t}}^{-1}\left({\mathbf{X}}_{ij}{\psi }_a+b_i\right)$ is an individual-level propensity score, i.e., probability that individual $j$ in cluster $i$ receives the treatment conditional on her covariates ${\mathbf{X}}_{ij}$ (possibly including cluster-level covariates) and the cluster-level random effect $b_i$. The cluster-level propensity score is parameterized by $\mathbf{\psi }=\left({\psi }_a,{\psi }_b\right)$, and $\mathbf{\psi }$ is estimated by $\widehat{\mathbf{\psi }}$ using maximum likelihood.

Under Assumption (1), a cluster-level analogue of the positivity assumption $P\left({\mathbf{A}}_i={\mathbf{a}}_i\mid \right.\left.{\mathbf{L}}_i\right)>0$ for any ${\mathbf{a}}_i\in 𝒜\left(n_i\right)$, and assuming model (2) is correctly specified, Tchetgen Tchetgen and VanderWeele (2012) showed that

$${\widehat{Y}}_{i,ipw}(a,\alpha )=\frac{\sum _{i=1}^{n_{ij}} \mathbf{\pi }\left(A_{i,-j};\alpha \right)\mathbb{1}\left\{A_{ij}=a\right\}{Y}_{ij}}{P\left({\mathbf{A}}_i\mid {\mathbf{X}}_i;\mathbf{\psi }\right)n_i}\text{and}{\widehat{Y}}_{i,ipw}(\alpha )=\frac{\sum _{i=1}^{n_{ij}} \mathbf{\pi }\left(A_i;\alpha \right)Y_{ij}}{P\left({\mathbf{A}}_i\mid {\mathbf{X}}_i;\mathbf{\psi }\right)n_i}$$ (3)

are unbiased estimators of the cluster-level average potential outcome for $a\in \left\{\mathrm{0,1}\right\}$ and the cluster-level marginal average potential outcome, respectively. In expression (3), $\mathbf{\pi }\left(A_{i,-j};\alpha \right)$ denotes the conditional probability $\mathbf{\pi }\left(A_{i,-j}={\mathbf{a}}_{i,-j}\mid A_{ij}=a\right)$ and $\mathbf{\pi }\left(A_i;\alpha \right)$ denotes the probability $\mathbf{\pi }\left(A_i={\mathbf{a}}_i\right)$, both under a “stochastic policy” parameterized by $\alpha$. The conditional probability $\mathbf{\pi }\left(A_{i,-j}={\mathbf{a}}_{i,-j}\mid A_{ij}=a\right)$ equals the unconditional probability $\mathbf{\pi }\left(A_{i,-j}={\mathbf{a}}_{i,-j}\right)$ when the stochastic policy treats participants independently but not otherwise (VanderWeele and Tchetgen Tchetgen, 2011). For instance, if a policy under evaluation treats individuals in a cluster independently from each other and with the same probability $\alpha =0.5$, then $\pi \left(A_{i,-j};\alpha \right)={\prod }_{k=1,k\ne j}^{n_{ij}} {0.5}^{{\alpha }_{ik}}\times (1-0.5{)}^{1-a_{ik}}$. Averaging ${\widehat{Y}}_{i,ipw}(a,\alpha )$ and ${\widehat{Y}}_{i,ipw}(\alpha )$ over $i=1,\dots ,N$ clusters, one obtains ${\widehat{Y}}_{ipw}(a,\alpha )=N^{-1}{\sum }_{i=1}^N {\widehat{Y}}_{i,ipw}(a,\alpha )$ and ${\widehat{Y}}_{ipw}(\alpha )=N^{-1}{\sum }_{i=1}^N {\widehat{Y}}_{i,ipw}(\alpha )$, from which one may readily construct estimators for the population average direct effect of the treatment policy parameterized by $\alpha :\widehat{DE}(\alpha )={\widehat{Y}}_{ipw}(1,\alpha )-{\widehat{Y}}_{ipw}(0,\alpha )$, indirect effect that compares two policies parameterized by $\alpha$ and ${\alpha }^{\prime }$ holding an individual’s treatment status fixed at $0:\widehat{IE}\left(\alpha ,{\alpha }^{\prime }\right)={\widehat{Y}}_{ipw}(0,\alpha )-{\widehat{Y}}_{ipw}\left(0,{\alpha }^{\prime }\right)$, total effect $\widehat{TE}\left(\alpha ,{\alpha }^{\prime }\right)={\widehat{Y}}_{ipw}(1,\alpha )-{\widehat{Y}}_{ipw}\left(0,{\alpha }^{\prime }\right)$, and overall effect $\widehat{OE}\left(\alpha ,{\alpha }^{\prime }\right)={\widehat{Y}}_{ipw}(\alpha )-{\widehat{Y}}_{ipw}\left({\alpha }^{\prime }\right)$.

Perez-Heydrich et al. (2014) applied these estimators to a study of the effectiveness of cholera vaccine in Matlab, Bangladesh. While vaccine was individually randomly assigned in the cholera vaccine trial, individuals self-selected into participation in the vaccine trial, making the actual vaccination status observational and subject to confounding bias. Moreover, the formulation of clusters was post hoc, so the vaccine coverage within clusters was non-randomized. Given the observational nature of the data, Perez-Heydrich et al. (2014) assumed individuals’ vaccination statuses in each neighborhood were “ignorable” conditional on age and distance to the nearest river. Perez-Heydrich et al. (2014) found that the estimated direct effect in general decreased as vaccine coverage increased, and that the indirect effect comparing $\alpha$ and ${\alpha }^{\prime }>\alpha$ seemed to increase as ${\alpha }^{\prime }-\alpha$ increased.

There are many possible directions to extend the framework of Tchetgen Tchetgen and VanderWeele (2012) and Perez-Heydrich et al. (2014) using inverse probability weighted estimators under interference. Liu et al. (2016) extended IPW estimators from a partial interference regime to a general interference regime where each individual is allowed to have their own interference set. Liu et al. (2016) proposed IPW and stabilized IPW estimators, and established their asymptotic normality in a partial interference setting. In a subsequent article, Liu et al. (2019) further proposed doubly-robust estimators for various causal estimands under interference. Liu et al. (2019)’s doubly-robust estimators combine IPW and outcome-regression-based estimators and are consistent and asymptotically normal if either model, but not necessarily both, is correctly specified. Park and Kang (2020) showed one of Liu et al.’s doubly robust estimators is locally efficient. Park and Kang (2020) also considered non-parametric data-adaptive estimators for the partial interference setting which utilize machine learning methods to estimate the propensity score and outcome regression models. Papadogeorgou et al. (2019) and Barkley et al. (2020) considered causal estimands that allow for within-cluster dependence in the individual treatment selection and developed corresponding IPW estimators. Chakladar et al. (2022) developed IPW estimators that accommodate partial interference and right censoring. Zigler and Papadogeorgou (2021) considered IPW estimators that allow for interference in the bipartite causal inference setting where treatments are assigned to units that are distinct from those on which outcomes are measured.

Causal inference with interference is a rapidly growing research area. Above we have summarized some extensions of the propensity score to the interference setting, primarily focusing on inverse probability weighted estimators. Extensions of propensity score-based subclassification estimators which allow for interference have also been developed (e.g., Hong and Raudenbush, 2006, Forastiere et al., 2021). Likewise, estimators that utilize propensity score matching have also been proposed for the interference setting (e.g., Forastiere et al., 2018, Zirkle et al., 2021). While much progress has been made in relaxing the no interference assumption, there is still much room for future research that extends the use of the propensity score proposed by Rosenbaum and Rubin (1983) to the setting of interference.