Assessing spillover effects: Handling missing outcomes in network-based studies

Abstract

Estimating causal effects in the presence of spillover among individuals within a social network poses challenges due to missing information. Spillover effects refer to the impact of an intervention on individuals not directly exposed themselves but connected to intervention recipients within the network. In network-based studies, outcomes may be missing due to study termination or participant dropout, termed censoring. We introduce an inverse probability censoring weighted estimator which extends the inverse probability weighted estimator for network-based observational studies to handle possible outcome censoring. We prove the consistency and asymptotic normality of the proposed estimator and derive a closed-form estimator for its asymptotic variance. Applying the inverse probability censoring weighted estimator, we assess spillover effects in a network-based study of a nonrandomized intervention with outcome censoring. A simulation study evaluates the finite-sample performance of the inverse probability censoring weighted estimator, demonstrating its effectiveness with sufficiently large sample sizes and number of connected subnetworks. We then employ the method to assess spillover effects of community alerts on self-reported human immunodeficiency virus risk behavior among people who inject drugs and their contacts in the Transmission Reduction Intervention Project (TRIP), from 2013 to 2015, Athens, Greece. Results suggest that community alerts may help reduce human immunodeficiency virus risk behavior for both the individuals who receive them and others in their network, possibly through shared information. In this study, we found that the risk of human immunodeficiency virus behavior was reduced by increasing the proportion of a participant’s immediate contacts exposed to community alerts.

Keywords

Causal inference interference dissemination/spillover network studies people who inject drugs HIV/AIDS inverse probability weights

1. Introduction

Estimating causal effects in the presence of spillover (dissemination) among individuals in a network is often complicated by missing information. We need to consider missing outcome data when evaluating spillover within networks due to the dependencies between individuals. Ignoring missing data could not only introduce selection bias but also distort the network structure, resulting in possible unmeasured confounding or mismeasured exposures and covariates. There are various sources of missingness in the data, such as missing outcomes, baseline covariates, exposure status, and network connections. However, we focus on missing outcomes due to potential censoring, for example, the differential loss to follow-up due to study dropout. This censoring can occur when participants who drop out differ from those who do not with respect to their exposure or outcome (or both).

One common approach for estimating causal effects is to exclude all observations with missing outcomes, known as complete case analysis. The complete case estimator is consistent if the observations are missing completely at random (MCAR), indicating that missingness is unrelated to any measured and unmeasured variables. When the MCAR assumption does not hold, and the missing at random (MAR) assumption is appropriate for the observed data, several approaches such as maximum likelihood, multiple imputation, fully Bayesian, and inverse probability weighting^1,2 can be applied in this case.

Social networks can be represented as graphs illustrating connections among individuals, such as those delineated by engagement in human immunodeficiency virus (HIV) risk behaviors (e.g. sexual or injection behaviors) among people who inject drugs (PWID). The exposure of one individual could affect the outcome of another, a phenomenon known as interference.³ In this context, a key interest lies in the spillover (or indirect) effect, often conceptualized as a contrast in average potential outcomes if an individual is unexposed, comparing different exposure vectors for their spillover set (e.g. neighbors). In network-based studies, the spillover (or interference) set can be defined as an individual’s neighbors, namely those who share a connection (edge or link) with that individual. Therefore, the assumption of partial interference, which posits that spillover is possible within a set or group of individuals but not between groups,^4,5 may not hold. That is, while no interference occurs between individuals in different groups, interference is possible among individuals within the same group defined by the interference set. Additionally, the collection of interference sets forms a partition of the study population.

Inverse probability weighted (IPW) estimators for causal effects in observational studies in the presence of interference are often formulated under the assumption of partial interference.³ Alternatively, some approaches define interference based on spatial proximity or network ties,^6,7 allowing for possibly overlapping interference sets. Liu et al.⁷ proposed an IPW estimator for a generalized interference set that permits overlaps between interference sets; however, the asymptotic variance was estimated under the assumption of partial interference defined by larger groupings or clusters. Forastiere et al.⁶ quantified effects using a subclassification estimator with a generalized propensity score, employing a bootstrapping procedure with resampling either at the individual-level or the cluster-level to quantify variance. Nevertheless, these methods either hinge on partial interference defined by larger clusters or resort to bootstrapping for variance estimation. In practice, overlooking overlapping interference sets during variance estimation can yield inaccurate inference, and resampling approaches may be computationally intensive.

To address these challenges, the study by Lee et al.⁸ utilized the IPW estimators proposed by Forastiere et al.⁶ and Liu et al.,⁷ and introduced closed-form variance estimators capable of incorporating overlapping interference sets. Through statistical simulations, their proposed variance estimators demonstrated greater efficiency in network-based studies by leveraging additional information on connections between individuals. McNealis et al.⁹ proposed doubly robust estimators that include models for both treatment and outcome, ensuring consistency and asymptotic normality as long as at least one of the models is correctly specified. Additionally, alongside the IPW approach, van der Laan¹⁰ proposed a targeted maximum likelihood estimator for estimating causal effects in time invariant and longitudinal network studies. Their focus was on specific types of causal quantities, particularly the counterfactual mean under stochastic intervention on the unit-specific treatment nodes. When the outcome of interest is subject to possible right censoring (i.e. events occurring after a certain period of follow-up are missing), Chakladar et al.¹¹ and Loh et al.¹² extended IPW estimators under partial interference in observational studies to accommodate censoring using inverse probability of censoring weighted (IPCW). In these methods, survival models, such as proportional hazards frailty models and accelerated failure time models, were employed to estimate censoring weights. Moreover, a grouping of observations was utilized to define the interference set (e.g. study clusters and geographic location), allowing for spillover within but not between clusters. However, these approaches did not account for spillover between individuals with connections (i.e., edges) within a cluster or component, such as those observed connections with HIV risk transmission risk in network-based studies.¹³ Liu et al.⁷ proposed generalized weighted type estimators with neighbor-level exposure weights, relaxing the partial interference assumption. However, this method did not incorporate censoring weights, and the variance was estimated assuming partial interference.

In this work, we developed methods to estimate causal effects in network-based observational studies in the presence of missing binary or continuous outcomes due to possible censoring. In this study, we employed a IPW estimator by Lee et al.,⁸ which allows for spillover to each participant from their immediate connections in the network (i.e., neighbors) to quantify causal effects in network-based observational studies with a nonrandomized intervention. More precisely, each participant has a unique interference set determined by the observed network and interference sets can now overlap and no longer partition the network. We extended their IPW estimator to a setting with missing outcomes due to censoring using IPCW, where we consider two different censoring mechanism: (1) censoring indicators are independent across participants conditional on baseline covariates and exposures or (2) censoring indicators are correlated between participants within a connected subnetwork or component in the network. Components are defined as subsets of connected participants, but not connected with participants in other components. We used the network structures including network connections between individuals and the independence of components to calculate a novel closed-form variance estimator by applying M-estimation. We evaluated finite-sample performance and the impact of the dependency assumption between censoring indicators of the proposed estimators in a simulation study.

We employed the proposed IPCW estimator to evaluate the direct and spillover effects of community alerts on self-reported HIV risk behavior at 6 months among PWID and their contacts in the Transmission Reduction Intervention Project (TRIP) from 2013 to 2015 in Athens, Greece.^14–16 Previous work found evidence of a possibly meaningful spillover effect of community alerts on HIV risk behavior in TRIP; however, the prior study used a complete case analysis and did not consider the impact of missing outcomes on the validity of the analysis.⁸ PWID often participate in behaviors with HIV transmission risk and are part of sexual, social, and drug use networks. In these networks, nodes represent individuals and edges between nodes represent the relationship between two people, such as sexual, social or drug use contacts, referred to as HIV risk networks. These individuals may or may not engage in behavior that increases HIV transmission risk in their partnership, including sharing equipment for injection drug use or condomless sex. In networks of PWIDs, educational and treatment-based interventions often have spillover causal effects, and intervention effects frequently depend on the network structure.^17,18

The proposed IPCW estimator is developed in Section 2. In Section 3, the results of a simulation study are reported that evaluates the performance of IPCW in finite-sample settings. In Section 4, we report the results that used the IPCW estimator to analyze the spillover effects of community alerts on self-reported HIV risk behavior among PWID in TRIP. We then discuss the performance of the IPCW estimators in the simulation study, limitation of the method, and future directions in Section 5.

2. Methods

In this section, we propose an IPCW estimator for network-based studies with binary or continuous outcomes subject to possible censoring due to differential loss to follow-up due to study drop out. If participants who drop out of the study differ from those who did not with respect to their outcome or exposure (or both), then this may result in selection bias, even under the null.¹⁹ There are several types of missing mechanisms including MCAR, MAR, and missing not at random (MNAR). In the case of MCAR, the missing data mechanism is unrelated to any measured or unmeasured study variable. That is, the participants with completely observed data are assumed to be a random sample of all the participants assigned a particular intervention, which is often an unrealistic assumption in practice. MAR occurs when the missingness can be accounted for by fully observed study variables.²⁰ In other words, censoring indicators are independent conditional on baseline variables (e.g. covariates and exposures). MNAR assumes the missingness depends on underlying variables that we do not observe, such as participants with more frequent injection drug use with others are more likely to drop out of the study. In this article, we focus on the MAR assumption, but the censoring models could be extended to some cases of MNAR.^21,22

The rest of this section is organized as follows: We first state the notation in Section 2.1 and the assumptions in Section 2.2. We then define the potential outcome framework and the population average causal effects of interest in Section 2.3. In Section 2.4, we define the inverse probability censoring weight estimator with censoring weights and neighbor-level exposure weights. The large sample variance estimators are derived using M-estimation theory²³ in Section 2.5.

2.1. Notation

A network is a structure consisting of a set of nodes, $V$ , and a set of edges between nodes, $E$ . Consider an HIV risk network, let $V = {1, \dots, n}$ denote a set of participants in the study and define $E$ a set of the edges, where $e_{i j} = 1$ if participants $i$ and $j$ engage in sexual or drug use behavior together and $e_{i j} = 0$ otherwise. Define the neighbors of participant $i$ by $N_{i} = {j; e_{i j} = 1}$ ; in words, for each participant $i$ , the neighbors are the immediate connections in the observed network excluding participant $i$ . We denote $N_{i}^{*} = N_{i} \cup {i}$ . Oftentimes, a network can be partitioned into connected subnetworks, $G_{1}, G_{2}, \dots, G_{m}$ with $ν = 1, \dots, m$ , each called components. There are no edges or connections between components, but possibly there are connections between participants within a component. Figure 1 is the TRIP network with one of the network components enlarged. This component includes individuals labeled with ${1, 2, 3, 4, 5, 6, 7, 8, 9, 10}$ , where, for example, the neighbors of node $6$ are ${2, 3, 5, 8}$ and the neighbors of node $2$ are ${1, 3, 6, 7}$ .

The degree of unit $i$ is denoted as $d_{i} = \sum_{j = 1}^{n} e_{i j} = | N_{i} |$ . Let $A_{i}$ be the binary baseline exposure of participant $i$ with $A_{i} = 1$ if exposed and 0 otherwise. Let $Z_{i}$ denote the vector of baseline pre-exposure covariates for participant $i$ . We denote the vector of intervention exposures for the neighbors for participant $i$ as $A_{N_{i}} = {A_{i j}; e_{i j} = 1}$ . We also denote the vector of baseline pre-exposure covariates for the neighbors for participant $i$ as $Z_{N_{i}} = {Z_{i j}; e_{i j} = 1}$ . Denote realizations of $A_{i}$ by $a_{i}$ and $Z_{i}$ by $z_{i}$ . A nearest neighbor-level random effect $b_{N_{i}^{*}}$ is used to account for possible correlation of exposures among individual $i$ and their neighbors $N_{i}$ . For example, $b_{N_{1}^{*}}$ is a random effect for participants 1 and 2, while $b_{N_{2}^{*}}$ is a random effect for participants 1, 2, 3, 6, and 7 (Figure 1).

2.2. Assumptions

The proposed approach requires identification assumptions in this observational network design.

Assumption 1
The neighbor interference assumption⁶ assumes that the outcome of participant $i$ only depend on their exposure and the exposures of their neighbors $N_{i}$ and not on the exposures of others in the network. Specifically,
$Y_{i} = y_{i} (A_{i}, A_{N_{i}})$

Let $y_{i} (a_{i}, a_{N_{i}})$ denote the potential outcome of individual $i$ if they received exposure $a_{i}$ and their neighbors received $a_{N_{i}}$ . Therefore, the potential outcome of participant $i$ depends not only on their own exposure, but also on the exposures of their neighbors. Let $Y_{i} = y_{i} (A_{i}, A_{N_{i}})$ denote the observed outcome, which holds by (causal) consistency. For example, in Figure 1, $Y_{7} = Y_{7} (a_{7}, (a_{2}, a_{9}))$ , which is that the outcome for individual 7 is affected by their own exposure and the exposures of individuals 2 and 9 only and no other individuals’ exposures in either the component or network.
Assumption 2
We assume that the intervention assignment mechanism does not affect the outcome beyond the assigned interventions themselves.⁶ That is, if multiple versions of the intervention exist, they are considered irrelevant to the causal contrasts of interest. We assume a single version of the intervention exposure and a single version of no exposure. For each $a_{i} = a_{i}^{'}$ and $a_{N_{i}} = a_{N_{i}}^{'}$ , we have
$y_{i} (a_{i}, a_{N_{i}}) = y_{i} (a_{i}^{'}, a_{N_{i}}^{'})$

There are no multiple versions of this intervention or exposure, and if there were multiple versions of exposure, those are assumed to be irrelevant for the causal effect of interest.²⁴
Assumption 3
The conditional exchangeability assumption,⁸
$y (a_{i}, a_{N_{i}}) ⊥ A_{i}, A_{N_{i}} | Z_{i}, Z_{N_{i}}$

Assumption 3 states that exposure is independent of potential outcomes, conditional on the pre-exposure covariates of participant $i$ and their neighbors. While component size is not included in the conditioning set, the assumption implicitly requires that component size either has no influence on exposure or outcomes, or that any such influence is fully captured by the observed covariates.
Assumption 4
Positivity assumption for the intervention exposure,⁷ $Pr (A_{i} = a_{i} | Z_{i} = z_{i}) > 0$ and $Pr (A_{i} = a_{i}, A_{N_{i}} = a_{N_{i}} | Z_{i} = z_{i}, Z_{N_{i}} = z_{N_{i}}) > 0$ for all $a_{i}$ , $a_{N_{i}}$ , $z_{i}$ , and $z_{N_{i}}$ .

Positivity assumes that each participant has a non-zero probability of being assigned every possible intervention exposure status given every combination for the levels of the observed covariates.
Assumption 5
Conditional independent censoring $C_{i} ⊥ y_{i} (a_{i}, a_{N_{i}}) | Z_{i}, Z_{N_{i}}, A_{i}, A_{N_{i}}$ .

Assumption 5 states that the censoring of participant $i$ is independent of their potential outcomes conditional on their own exposure and pre-exposure covariates and their neighbors’ exposures and pre-exposure covariates.
Proposition 1
(Identification) Under Assumptions 1–3, we have
$E [Y_{i} (a_{i}, a_{N_{i}})] = \int \int E [Y_{i} | A_{i} = a_{i}, A_{N_{i}} = a_{N_{i}}, Z_{i} = z_{i}, Z_{N_{i}} = z_{N_{i}}] \cdot Pr (Z_{i} = z_{i}, Z_{N_{i}} = z_{N_{i}}) .$

The proof of this proposition is in Supplemental Appendix A.
2.3. Causal estimands

We define individual average potential outcomes under counterfactual intervention assignment strategies specific to the individual and their neighbors, rather than network-wide estimands. We take the expectation with respect to the population of individuals from which our network components were sampled. In particular, we consider strategies that assign an individual $i$ to intervention $A_{i} = a$ and then assign the neighbors to of individual $i$ to the intervention under a hypothetical Bernoulli assignment with probability $α$ . We refer to this parameter as the intervention coverage among the neighbors. Let $π (a_{N_{i}}; α) = α^{\sum a_{N_{i}}} (1 - α)^{d_{i} - \sum a_{N_{i}}}$ denote the probability of the neighbors of individual $i$ receiving exposure $a_{N_{i}}$ under allocation strategy $α$ , where $d_{i} = | N_{i} |$ . Let $π (a_{i}; α) = α^{a_{i}} (1 - α)^{1 - a_{i}}$ denote the probability of individual $i$ receiving exposure $a_{i}$ and $π (a_{i}, a_{N_{i}}; α) = π (a_{N_{i}}; α) π (a_{i}; α)$ denote the probability of individual $i$ together with their neighbors receiving the set of exposures $(a_{i}, a_{N_{i}})$ .

Recall that $y_{i} (a_{i}, a_{N_{i}})$ denote the potential outcomes for participant $i$ under individual intervention set to $a$ and intervention coverage among their neighbors equal to $α$ by ${\bar{y}}_{i} (a, α) = \sum_{a_{N_{i}}} y_{i} (a, a_{N_{i}}) π (a_{N_{i}}; α) .$ Then, the average potential outcome is defined by the expectation of the individual average potential outcome:

\bar{y} (a, α) = E [{\bar{y}}_{i} (a, α)]

(1)

We also consider counterfactual treatment strategies that assign the intervention to both individual and their contacts using a Bernoulli assignment strategy with probability $α$ . Marginalizing over the individual-level exposure, the marginal individual average potential outcome is ${\bar{y}}_{i} (α) = \sum_{a_{i}, a_{N_{i}}} y_{i} (a_{i}, a_{N_{i}}) π (a_{i}, a_{N_{i}}; α) .$ Then, the average marginal potential outcome is defined by the expectation of the individual marginal potential outcome:

\bar{y} (α) = E [{\bar{y}}_{i} (α)]

(2)

Note that this definition is similar to that by Hudgens and Halloran,⁴ but here the Bernoulli assignment is applied to the neighbors of unit $i$ as opposed to cluster members. See also the definition by Liu et al.⁷

Different contrasts of these population average causal effects are often of interest in public health research. In particular, spillover effects can assess the public health impact of an intervention in the network among individuals who were not exposed to intervention but who are connected to recipients. An example of this type of intervention would be one delivered through contact tracing, where an individual and their contacts possibly received the intervention. We may be interested in the population-level impact of this intervention delivered via contact tracing. We defined these effects under allocation strategies $α, α_{0}, α_{1}$ as follows. The direct effect is $\bar{DE} (α) = \bar{y} (1, α) - \bar{y} (0, α)$ , which is a difference in the risk of the outcome when a participant is exposed to the intervention compared to when a participant is not exposed with exposure coverage level $α$ for the neighbors. The spillover or indirect effect is $\bar{IE} (α_{1}, α_{0}) = \bar{y} (0, α_{1}) - \bar{y} (0, α_{0})$ , which is a difference in the risk of the outcome if a participant is not exposed to the intervention under two different intervention coverage levels for the neighbors. The total effect is defined as $\bar{TE} (α_{1}, α_{0}) = \bar{y} (1, α_{1}) - \bar{y} (0, α_{0})$ , which is a measure of the maximal intervention effect, and is a difference in the risk if a participant is exposed under one intervention coverage level for the neighbors $α_{1}$ compared if a participant is unexposed under intervention coverage level for the neighbors $α_{0}$ . The overall effect is $\bar{OE} (α_{1}, α_{0}) = \bar{y} (α_{1}) - \bar{y} (α_{0})$ , which is the difference in average potential outcomes under one coverage for participant $i$ (index) and their neighbors compared to another coverage for the participant $i$ and the neighbors. The effects can also be considered on a ratio scale, such as a risk ratio $\bar{DE} (α) = \bar{y} (1, α) / \bar{y} (0, α) .$ In this study, we will focus on the estimation of risk differences.

According to Assumption 1, spillover is assumed to be possible to a participant from their neighbors in the network (i.e. a participant’s immediate or first-degree contacts). The spillover set or interference set of participant $i$ is defined as the set of their neighbors. Assumption 1 relaxes the partial interference assumption which implies no interference between individuals in different groups but allow for possible interference between individuals within the same group, requiring that individuals only belong to one interference set.^3,4 Assumption 1 allows for participants to be part of more than one interference set using information on connections in the network. For example, in Figure 1, the interference set of participant 3 is ${2, 4, 6}$ and the interference set of participant 5 is ${6, 8, 10}$ . Participants 3 and 5 are not neighbors but the interference sets overlap and both include participant $6$ .

2.4. Inverse probability of censoring weighted estimator

In a network-based study, data are ascertained from a network with $m$ distinct components, where components are connected subnetworks, $G_{1}, G_{2}, \dots, G_{m}$ , in the full network with $ν = 1, \dots, m$ . Suppose the outcomes are subject to possible censoring, for example, due to loss to follow-up. Let $C_{i}$ be the binary censoring indicator for participant $i$ where $C_{i} = 1$ if an individual is censored (i.e. missing outcome) and 0 otherwise (i.e. outcome observed). We assume that the $m$ components are random sampled from the infinite super-population of components and the size of each component is bounded.²³ Additionally, each sampled component contains all of its nodes and edges, meaning that each node and its corresponding edges are fully observed within the component.

For the causal effects of interest, we define two estimators. In this article, our estimands are not defined at the component level, but instead average individual-specific estimands, so we average over all individuals in the study (or take the expectation with respect to the population of individuals from which our network components was sampled). We are scientifically interested in the participant-average estimand, so we define our estimator similarly for this parameter. Under Assumptions 1–5, an extension of the nearest neighbor IPW estimator⁸ of the study population average outcome for exposure $a$ and intervention allocation strategy $α$ that now includes an inverse probability of censoring weight is defined as follows:

{\hat{Y}}^{IPCW} (a, α) = \frac{1}{n} \sum_{i = 1}^{n} \frac{y_{i} (A_{i}, A_{N_{i}}) I (C_{i} = 0) I (A_{i} = a) π (A_{N_{i}}; α)}{f (A_{i}, A_{N_{i}} | Z_{i}, Z_{N_{i}}) S_{C} (C_{i} | Z_{i}, A_{i})} .

Marginalizing over the individual-level exposure, an estimator of the population average marginal outcome for intervention allocation strategy

α

is defined as follows:

{\hat{Y}}^{IPCW} (α) = \frac{1}{n} \sum_{i = 1}^{n} \frac{y_{i} (A_{i}, A_{N_{i}}) I (C_{i} = 0) π (A_{i}, A_{N_{i}}; α)}{f (A_{i}, A_{N_{i}} | Z_{i}, Z_{N_{i}}) S_{C} (C_{i} | Z_{i}, A_{i})}

Under allocation strategies $α$ , $α_{0}$ , and $α_{1}$ , we consider the following risk difference estimators of the direct, spillover (indirect), composite (total), and overall effects:

\begin{aligned} \hat{DE} (α) & = {\hat{Y}}^{IPCW} (1, α) - {\hat{Y}}^{IPCW} (0, α) \\ \hat{IE} (α_{1}, α_{0}) & = {\hat{Y}}^{IPCW} (0, α_{1}) - {\hat{Y}}^{IPCW} (0, α_{0}) \\ \hat{TE} (α_{1}, α_{0}) & = {\hat{Y}}^{IPCW} (1, α_{1}) - {\hat{Y}}^{IPCW} (0, α_{0}) \\ \hat{OE} (α_{1}, α_{0}) & = {\hat{Y}}^{IPCW} (α_{1}) - {\hat{Y}}^{IPCW} (α_{0}) \end{aligned}

Proposition 2
If the exposure propensity scores and censoring weights are known, then the IPCW estimator is unbiased: $E [{\hat{Y}}^{IPCW} (a, α)] = \bar{y} (a, α)$ and $E [{\hat{Y}}^{IPCW} (α)] = \bar{y} (α)$ .

The proof of this proposition is in Supplemental Appendix A. We can easily get unbiased estimators for the causal effects because the causal effects are different contrasts of the average potential outcomes.

Following the approach of Lee et al.,⁸ we used a mixed-effects model to estimate the exposure propensity score for each participant and their interference set $N_{i}$ . We use a nearest neighbor-level random effect $b_{N_{i}^{}}$ accounts for possible correlation of exposures among individual $i$ and their neighbors $N_{i}$ . Conditional on the exposures and covariates for an individual’s neighbors and the neighbor-level random effect, we assume that the exposure $A_{i}$ for individual $i$ and the exposure $A_{j}$ for individual $j$ are independent. That is, given the neighbors’ exposures $A_{N_{i}}$ and $A_{N_{j}}$ and neighborhood-level random effect $b_{N_{i}^{}}$ and $b_{N_{j}^{}}$ ,
$A_{i} ⊥ A_{j} | A_{N_{i}}, b_{N_{i}^{}}, A_{N_{j}}, b_{N_{j}^{}}$
Under this modeling assumption, we used the following mixed-effects model to estimate the exposure propensity score^a :
$f (A_{i}, A_{N_{i}} | Z_{i}, Z_{N_{i}}) = \int \prod_{j \in N_{i}^{}} p_{j}^{A_{j}} (1 - p_{j})^{1 - A_{j}} f (b_{N_{i}^{}}; 0, τ) d b_{N_{i}^{}} .$
where $N_{i}^{} = N_{i} \cup {i}$ and $i \in G_{ν}$ , $p_{j} = Pr (A_{j} = 1 | Z_{j}, b_{N_{i}^{}}) = {logit}^{- 1} (Z_{j} \cdot θ_{z} + b_{N_{i}^{}})$ , and $f (b_{N_{i}^{}}; 0, τ) \sim N (0, τ^{2})$ .

Different censoring mechanisms require distinct model specifications to appropriately account for underlying dependencies and variability. When censoring indicators exhibit multiple sources of correlation among participants such as shared characteristics within a network, spatial clustering, or unobserved heterogeneity at different levels, models incorporating multiple random effects may be necessary to effectively capture these dependencies.

In this article, we consider a censoring mechanism in which the censoring indicators of participants within a component may be correlated, allowing us to demonstrate the estimators and estimate the model parameters. That is, if $i, j \in G_{ν}$ , then
$C_{i} ⊥ C_{j} | Z_{i}, Z_{N_{i}}, A_{i}, A_{N_{i}}, Z_{j}, Z_{N_{j}}, A_{j}, A_{N_{j}}, ρ_{ν}$
where $ρ_{ν} \sim N (0, ρ^{2})$ , for some $ρ$ , is the component-level random effect. Otherwise if $i, j \notin G_{ν}$ ,
$C_{i} ⊥ C_{j} | Z_{i}, Z_{N_{i}}, A_{i}, A_{N_{i}}, Z_{j}, Z_{N_{j}}, A_{j}, A_{N_{j}}$

This mechanism states the censoring of participant $i$ conditional on their own and their neighbors’ covariates and exposures as well as the component-level random effect is independent of the censoring of participant $j$ conditional on their own and their neighbors’ covariates, exposure, and the component-level random effect.^b While $ρ_{ν}$ is shared across the entire component $ν$ and therefore encodes information about all participants within it, the assumed conditional independence structure does not require conditioning on the full covariate or exposure history of all individuals in the component only on the relevant local neighborhoods of $i$ and $j$ , together with $ρ_{ν}$ . These considerations may be relevant in network intervention studies where participants’ decisions to return for study visits are independent of other network members attendance, conditional on their own exposures, covariates, and an aggregation of their neighbors’ exposures ( ${Agg}_{A})$ and covariates ( ${Agg}_{Z}$ ). The aggregation of neighbors’ exposures may be represented by the proportion of exposed neighbors. For example, if individual $i$ had four neighbors and two of them were exposed, then ${Agg}_{A} (A_{N_{i}}) = 0.5$ . Similarly, the aggregation of neighbors’ covariates may involve computing the average of continuous covariates, for example, if age is a covariate, ${Agg}_{Z} ({age}_{N_{i}}) = 1 / 4 \sum_{j \in N_{i}} {age}_{j}$ , or the proportion of categorical covariates. For instance, if one of individual $i$ ’s four neighbors is female, then ${Agg}_{sex} ({sex}_{N_{i}}) = 0.25$ . In our approach, we assume that the outcomes are MAR; that is, adjustment for missingness can be accounted for using measured baseline covariates and the exposure status to estimate a censoring weight.

To account for potential correlations in the censoring mechanisms among participants within the same component, the conditional probability of censoring can be modeled using a mixed-effects model with a logit link function.
$\begin{aligned} S_{C} (C_{i} | Z_{i}, Z_{N_{i}}, A_{i}, A_{N_{i}}, ρ_{ν}) \\ = & 1 - P r (C_{i} = 1 | Z_{i}, Z_{N_{i}}, A_{i}, A_{N_{i}}, ρ_{ν}) \\ = & 1 - {logit}^{- 1} ((Z_{i}, {Agg}_{Z} (Z_{N_{i}})) \cdot ξ_{z} + (A_{i}, {Agg}_{A} (A_{N_{i}})) \cdot ξ_{a} + ρ_{ν}) \end{aligned}$
where individual $i$ belongs to component $G_{ν}$ and $ρ_{ν} \sim N (0, ρ^{2})$ for some $ρ$ .

The vectors $ξ_{z}$ and $ξ_{a}$ represent the fixed-effect coefficients in the regression model. Specifically, the length of $ξ_{z}$ depends on the number of covariates included, while the length of $ξ_{a}$ is two. In other words, the conditional probability of censoring is the probability that the outcome of individual $i$ will not be observed at the follow-up visit. Note, that we inverse weight the uncensored observations in the final estimator by the probability of not being censored, so we can obtain estimates of causal contrasts as if the entire study population had not been censored.
2.5. Large sample properties of the IPCW estimator

The large sample variance estimators can be derived using M-estimation theory.²³ We assume that an observed network can be expressed as the union of connected subnetworks, referred to as components. Given a network with $n$ participants and $m$ components ${G_{1}, G_{2}, \dots, G_{m}}$ , let $Y_{ν} = {Y_{i} | i \in V (G_{ν})}$ , $A_{ν} = {A_{i} | i \in V (G_{ν})}$ , $Z_{ν} = {Z_{i} | i \in V (G_{ν})}$ , and $C_{ν} = {C_{i} | i \in V (G_{ν})}$ , where $V (G_{ν})$ is the set of nodes in $G_{ν}$ with $ν = 1, \dots, m$ . The observable random variables $(Y_{ν}, A_{ν}, Z_{ν}, C_{ν})$ for $ν = 1, \dots, m$ are assumed to be independent but not necessarily identically distributed. Recall, we assume that the $m$ components are a random sample from the infinite super-population of components and the size of each component is bounded.²³ Let $k = E [| V (G_{ν}) |]$ be the average component size in the population, and $\hat{k}$ be the average component size of the observed network.^c We assume that the components in the study are randomly sampled from a population of components with $k$ being the average size of the component in the population. Due to the random sampling, we assume that the average component size $\hat{k}$ in the observed network will be a reasonable approximation to the true average component size. We use the component-average to redefine the inverse probability weighted estimators to clarify the relationship to our individual-level causal estimands.²⁵

We denote $Θ = {θ_{z}, τ}$ as the set of coefficients in the exposure propensity score, including the fixed effects and the random effect standard deviation, and by $Φ$ the set of coefficients in the censoring weight model including fixed effect coefficients and random effect standard deviation, respectively. Under the assumption of correlated censoring indicators, the censoring model is a mixed effects model, and let $Φ = {ξ_{z}, ξ_{a}, ρ}$ represent the parameters, including fixed effects coefficients and random effect standard deviations.

Let $Y_{G_{ν}} = (Y_{0, G_{ν}}, Y_{1, G_{ν}}, Y_{2, G_{ν}})$ represent the sum of the potential outcomes in component $G_{v}$ under allocation strategy $α$ ,

\begin{aligned} Y_{0, G_{ν}} & = \sum_{j \in V (G_{ν})} \sum_{a_{N_{j}}} y_{j} (a_{j} = 0, a_{N_{j}}) π (a_{N_{j}}; α) \\ Y_{1, G_{ν}} & = \sum_{j \in V (G_{ν})} \sum_{a_{N_{j}}} y_{j} (a_{j} = 1, a_{N_{j}}) π (a_{N_{j}}; α) \\ Y_{2, G_{ν}} & = \sum_{j \in V (G_{ν})} \sum_{a_{j}, a_{N_{j}}} y_{j} (a_{j}, a_{N_{j}}) π (a_{j}, a_{N_{j}}; α) \end{aligned}

We use $m$ network components as the independent units to derive the variance to construct the Wald-type 95% confidence intervals (CIs). The estimating equations in the M-estimation were defined using the nearest neighbors. That is, by extending results for the variance by Liu et al.,⁷ each individual is now assigned their own propensity score based on their exposure and the exposure of their neighbors. Whereas by Liu et al.,⁷ statistical inference was conducted by assuming partial interference in which the study population was partitioned into non-overlapping groups and all individuals in a group were assigned one group-level propensity score. Let $θ_{0, α} = \bar{y} (0, α) = \sum_{ν = 1}^{m} Y_{0, G_{ν}} / n$ , $θ_{1, α} = \bar{y} (1, α) = \sum_{ν = 1}^{m} Y_{1, G_{ν}} / n$ , and $θ_{α} = \bar{y} (α) = \sum_{ν = 1}^{m} Y_{2, G_{ν}} / n$ .

Given the set of observed variables $(Y_{ν}, A_{ν}, Z_{ν}, C_{ν})$ , the maximum likelihood estimators of the coefficients $Θ$ and $Φ$ and can be written as system of estimating equations $\sum_{ν = 1}^{m} ψ (Y_{ν}, A_{ν}, Z_{ν}, C_{ν}; Θ, Φ) = 0$ , where $ψ (Y_{ν}, A_{ν}, Z_{ν}, C_{ν}; Θ, Φ)$ is a vector of estimating equations $ψ = (ψ_{γ}, ψ_{η}, ψ_{1}, ψ_{2}, ψ_{3})^{T}$ as defined below:

\begin{aligned} ψ_{γ} (Y_{ν}, A_{ν}, Z_{ν}, C_{ν}; γ) = \frac{1}{\hat{k}} \sum_{j \in V (G_{ν})} \frac{\partial \log f (A_{j}, A_{N_{j}} | Z_{j}, Z_{N_{j}})}{\partial γ}, γ \in Θ \\ ψ_{η} (Y_{ν}, A_{ν}, Z_{ν}, C_{ν}; η) = \frac{1}{\hat{k}} \sum_{j \in V (G_{ν})} \frac{\partial \log S_{C} (C_{j} | Z_{j}, A_{j})}{\partial η}, η \in Φ \end{aligned}

\begin{aligned} ψ_{0} (Y_{ν}, A_{ν}, Z_{ν}, C_{ν}; α, θ_{0, α}) = \frac{1}{\hat{k}} \sum_{j \in V (G_{ν})} \frac{y_{j} (A_{j}, A_{N_{j}}) I (C_{j} = 0) I (A_{j} = 0) π (A_{N_{j}}; α)}{f (A_{j}, A_{N_{j}} | Z_{j}, Z_{N_{j}}) S_{C} (C_{j} | Z_{j}, A_{j})} - θ_{1, α} \\ ψ_{1} (Y_{ν}, A_{ν}, Z_{ν}, C_{ν}; α, θ_{1, α}) = \frac{1}{\hat{k}} \sum_{j \in V (G_{ν})} \frac{y_{j} (A_{j}, A_{N_{j}}) I (C_{j} = 0) I (A_{j} = 1) π (A_{N_{j}}; α)}{f (A_{j}, A_{N_{j}} | Z_{j}, Z_{N_{j}}) S_{C} (C_{j} | Z_{j}, A_{j})} - θ_{0, α} \\ ψ_{2} (Y_{ν}, A_{ν}, Z_{ν}, C_{ν}; α, θ_{α}) = \frac{1}{\hat{k}} \sum_{j \in V (G_{ν})} \frac{y_{j} (A_{j}, A_{N_{j}}) I (C_{j} = 0) π (A_{j}, A_{N_{j}}; α)}{f (A_{j}, A_{N_{j}} | Z_{j}, Z_{N_{j}}) S_{C} (C_{j} | Z_{j}, A_{j})} - θ_{α} \end{aligned}

Let $θ = (Θ, Φ, θ_{0, α}, θ_{1, α}, θ_{α})$ , and

\begin{aligned} ψ_{ν} (Y_{ν}, A_{ν}, Z_{ν}, C_{ν}; α, θ) = {(\begin{matrix} ψ (Y_{ν}, A_{ν}, Z_{ν}, C_{ν}; γ) \\ ψ (Y_{ν}, A_{ν}, Z_{ν}, C_{ν}; η) \\ ψ_{0} (Y_{ν}, A_{ν}, Z_{ν}, C_{ν}; α, θ_{0, α}) \\ ψ_{1} (Y_{ν}, A_{ν}, Z_{ν}, C_{ν}; α, θ_{1, α}) \\ ψ_{2} (Y_{ν}, A_{ν}, Z_{ν}, C_{ν}; α, θ_{α}) \end{matrix})}_{γ \in Θ, η \in Ψ} \end{aligned}

such that

\sum_{ν = 1}^{m} ψ_{ν} (Y_{ν}, A_{ν}, Z_{ν}, C_{ν}; α, \hat{θ}) = 0

. Note that

\hat{θ}

is the solution for

θ

for this vector of estimating equations.

Under suitable regularity conditions that as $m \to \infty$ , the closed-form sandwich type estimator of the variance is a consistent and asymptotically normal by Proposition 3 below. Let $A (θ) = E [- \dot{ψ} (Y_{ν}, A_{ν}, Z_{ν}, C_{ν}; α, \hat{θ})]$ , and $B (θ) = E [ψ (Y_{ν}, A_{ν}, Z_{ν}, C_{ν}; α, \hat{θ}) ψ (Y_{ν}, A_{ν}, Z_{ν}, C_{ν}; α, \hat{θ})^{T}]$ .

Proposition 3
Under suitable regularity conditions and due to the unbiased estimating equations, $\sqrt{m} (\hat{θ} - θ)$ converges in distribution to $N (0, Σ_{m})$ as $m \to \infty$ where the covariance matrix is given by $Σ_{m} = \frac{1}{m} A^{- 1} (θ) B (θ) (A^{- 1} (θ))^{T} .$

More details of Proposition 3 can be found in Supplemental Appendix A. A consistent estimator of the variance for the IPCW estimator is given in Supplemental Appendix A. This variance estimator can be used to construct Wald-type CIs for the estimated direct, spillover, total, and overall effects.
3. Simulation

A simulation study was conducted to evaluate the finite-sample performance of the IPCW estimators and their corresponding closed-form variance estimators. We focused on assessing finite-sample bias and the coverage of the associated 95% Wald-type CIs. The observed network was generated as follows: we created $m$ regular network components of degree four (i.e. each node has four neighbors), where the number of nodes per component was sampled from a Poisson distribution with a mean of 10. The union of these $m$ components formed a single observed network. Simulations were conducted under varying numbers of components, with $m \in {10, 50, 100, 200}$ , and the average component size was 10 across all runs.

Given a fixed, randomly generated network, we simulated a total of 1000 datasets following the steps described below. This design reflects the applied setting in which researchers typically work with a single observed network. Our aim was to evaluate the performance of the proposed estimators conditional on an observed network structure, isolating variability due to the exposure, outcome, and censoring processes. The network characteristics (number of components and number of nodes in each component), parameters in the outcome model, and censoring models were informed by estimates in the TRIP data. We assumed that the censoring indicators were conditionally independent given their own and their neighbors’ covariates and exposures. We also did not find significant associations between the censoring indicators and neighbors’ covariates, their own and their neighbors’ exposures in TRIP analysis. Therefore, we did not include them in our simulation study. The data was generated in the following steps:

Generate a baseline covariate $Z_{i} \sim Bern (0.5)$ .

Simulate the random effects for the censoring model, $ρ_{ν} \sim N (0, {0.3}^{2})$ , and random effects for the component-level^d exposure propensity score $b_{ν}$ and $b_{ν} \sim N (0, {0.5}^{2})$ , $ν = 1, 2, \dots, m$ .

The censoring indicators are generated as

C_{i} \sim Bern ({logit}^{- 1} (- 3 + 2 \cdot Z_{i} + ρ_{ν}))

Additionally, we consider a censoring mechanism in which censoring indicators are independent across participants, given their covariates

Z_{i}

. In this case, the censoring indicator follows

C_{i} \sim Bern ({logit}^{- 1} (- 3 + 2 \cdot Z_{i}))

To estimate the probability of censoring, we utilized a logistic regression model.

We then generate the potential outcomes

\begin{aligned} y_{i} (a_{i}, a_{N_{i}}) \sim & Bern ({logit}^{- 1} (- 1.75 + 0.5 \cdot a_{i} + \frac{\sum a_{N_{i}}}{d_{i}} - 1.5 a_{i} \cdot \frac{\sum a_{N_{i}}}{d_{i}} + 0.5 Z_{i})) \end{aligned}

The observed exposures are generated as

A_{i} \sim Bern ({logit}^{- 1} (0.7 - 1.4 \cdot Z_{i} + b_{ν}))

where

i \in V (G_{ν})

Based on the observed exposures generated in Step 5, the observed outcome for each individual $i$ is $Y_{i} = y_{i} (A_{i}, A_{N_{i}})$ .

For each simulated data set, the ${\hat{Y}}^{IPCW} (a, α)$ and ${\hat{Y}}^{IPCW} (α)$ are evaluated for $a = 0, 1$ and $α = {0.25, 0.5, 0.75}$ . The true parameters were calculated by averaging the generated potential outcomes as in equations (1) and (2). The estimated standard errors (ASEs) were derived from the estimated variance matrix ${\hat{Σ}}_{m}$ in Supplemental Appendix A and were averaged across all simulations. The ASE in each data set were used to obtain corresponding Wald 95% CIs for each simulation. Empirical standard errors (ESEs) were the standard deviations of estimated averaged across all simulated data sets. Empirical coverage probability (ECP) was defined as the proportion of simulated data sets that the true parameter was within the 95% CI among the total 1000 simulations.

We compared the performance of the estimators for networks with 10, 50, 100, and 200 components. Figure 2 illustrated the absolute value of average bias and ECPs of $\hat{Y} (1, 0.5)$ , $\hat{Y} (0, 0.5)$ , and $\hat{Y} (0.5)$ . Although the absolute value of bias for networks with 100 components is slightly smaller ( $< 0.002$ ) than for networks with 200 components when considering the mixed effects censoring models with 50% allocation strategy, the absolute value of bias for 200 components network under allocation strategies 25% and 75% were smaller than for 100 components network (Supplemental Tables A3 and A4). We conclude that, in general, the absolute value of average bias decreases and the empirical coverage probability approaches 95% as the number of components increases. In Supplemental Table A4, we report the simulation results of network with 200 components. The absolute values of average bias over different allocation strategies are < $0.003$ , and the empirical coverage probabilities are around the nominal value of 95%. In addition to the main simulation scenarios above, we also considered the following scenarios:

We considered a scenario with a 50 component network, taking into account component sizes of 20 and 30, which are larger than those in the main scenario, to evaluate the effect on performance when varying the component sizes. The results indicate that networks with larger components exhibited smaller bias, which may be attributed to the larger number of participants in the entire study.

We evaluated the impact on the performance of the estimators when there was a higher level of dependency between censoring indicators for the mixed-effects model approach. We used the same simulated regular network with degree 4 and 100 components to compare the performance of the average outcome estimators. The censoring indicators were generated by

C_{i} \sim Bern ({logit}^{- 1} (- 3 + 2 \cdot Z_{i} + ρ_{ν}))

We increased the complexity of dependency by increasing the uncertainty of censoring random effects

ρ_{ν} \sim N (0, ρ^{2})

, where

ρ = 0.1, 0.2, 0.3, 0.4, 0.5

. The results demonstrated that the performance of both IPCW estimators were comparable when the variance of censoring random effects increased. Summarized results can be found in Supplemental Appendix C.

We considered the network structure from our motivating study TRIP. The TRIP network consisted of 10 components with 277 nodes and 542 edges. Based on the main simulation scenarios results, a small number of components may result in poor finite-sample performance of variance estimators. In addition, if there is substantial variation in the observed component sizes, the average component size used to derive the estimator of the variance may not be appropriate unless the average potential outcomes in the component are independent of component size.⁷ To increase the number of components based on network structure and generate components that were more comparable in size for estimation of the variance of the estimated causal effects, we employed an efficient modularity-based, fast greedy, approach to detect communities to further divide some large connected components of the TRIP network into a total of 24 smaller and denser components (Tables 3 and 4). By ignoring sparser edges between components, we treated the obtained communities as independent units to possibly improve the estimation of the variance with more components similar in size. Furthermore, community detection reduces the magnitude of the degree distribution resulting in estimators with lower variance. Importantly, we still defined the interference sets using the neighbors for point estimation of the causal effects. The results illustrate that the variance estimators were conservative on original TRIP network (10 components) and closer to or below the nominal level for most scenarios with the 24 components network under both censoring mechanisms (Tables 3 and 4). In network data sets with few components or large variability in the size of components, sensitivity analyses with different community detection methods in the network may be appropriate to assess the impact on statistical inference (Puleo et al., 2023).

Table 1.

Simulation results of IPCW estimators using the logistic regression censoring model (left) and the mixed-effects censoring model (right) under allocation strategies of 25%, 50%, and 75% for a network with 50 components, each with a component size of 20.

	Logistic regression					Mixed effects
	True	Bias	ESE	ASE	ECP	True	Bias	ESE	ASE	ECP
Y(1, 0.25)	0.2484	$-$ 0.0027	0.0565	0.0529	0.91	0.2488	0.0062	0.0489	0.0477	0.90
Y(1, 0.5)	0.2261	0.0005	0.0306	0.0319	0.94	0.2265	0.0035	0.0291	0.0297	0.94
Y(1, 0.75)	0.2051	$-$ 0.0018	0.0396	0.0399	0.92	0.2052	$- 0.0008$	0.0400	0.0366	0.91
Y(0, 0.25)	0.2279	$-$ 0.0029	0.0378	0.0413	0.93	0.2278	$- 0.0022$	0.0369	0.0350	0.93
Y(0, 0.5)	0.2744	0.0012	0.0314	0.0317	0.94	0.2744	0.0024	0.0292	0.0299	0.94
Y(0, 0.75)	0.3247	0.0020	0.0579	0.0547	0.92	0.3250	0.0012	0.0527	0.0526	0.92
Y(0.25)	0.2330	$-$ 0.0029	0.0313	0.0342	0.94	0.2330	$- 0.0001$	0.0296	0.0292	0.94
Y(0.5)	0.2503	0.0008	0.0209	0.0231	0.97	0.2505	0.0030	0.0199	0.0217	0.97
Y(0.75)	0.2350	$-$ 0.0009	0.0315	0.0329	0.93	0.2352	$- 0.0003$	0.0322	0.0307	0.94

IPCW: inverse probability censoring weighted; ESE: empirical standard error; ASE: average estimated standard error; ECP: empirical coverage probability.

Table 2.

	Logistic regression					Mixed effects
	True	Bias	ESE	ASE	ECP	True	Bias	ESE	ASE	ECP
Y(1, 0.25)	0.2489	0.0030	0.0412	0.0414	0.93	0.2487	0.0022	0.0437	0.0416	0.91
Y(1, 0.5)	0.2268	0.0011	0.0249	0.0256	0.95	0.2266	0.0013	0.0258	0.0256	0.93
Y(1, 0.75)	0.2057	$- 0.0043$	0.0337	0.0325	0.94	0.2056	$- 0.0018$	0.0339	0.0316	0.91
Y(0, 0.25)	0.2277	$- 0.0032$	0.0311	0.0307	0.93	0.2277	$- 0.0012$	0.0312	0.0295	0.92
Y(0, 0.5)	0.2742	0.0019	0.0254	0.0254	0.94	0.2744	0.0026	0.0246	0.0251	0.95
Y(0, 0.75)	0.3250	0.0021	0.0452	0.0442	0.93	0.3254	0.0028	0.0454	0.0435	0.92
Y(0.25)	0.2330	$- 0.0016$	0.0252	0.0255	0.94	0.2330	$- 0.0004$	0.0258	0.0245	0.93
Y(0.5)	0.2505	0.0015	0.0175	0.0185	0.96	0.2505	0.0019	0.0176	0.0183	0.95
Y(0.75)	0.2355	$- 0.0027$	0.0264	0.0268	0.95	0.2355	$- 0.0006$	0.0275	0.0262	0.92

IPCW: inverse probability censoring weighted; ESE: empirical standard error; ASE: average estimated standard error; ECP: empirical coverage probability.

Table 3.

Simulation results from 1000 simulated datasets of IPCW estimators using logistic regression censoring model on original TRIP network (10 components) (left) and the network divided by community detection into 24 components (right).

				10 components		24 components
	True	Bias	ESE	ASE	ECP	ASE	ECP
Y(1, 0.25)	0.2488	0.0130	0.121	0.196	0.970	0.088	0.872
Y(1, 0.5)	0.2269	0.0037	0.078	0.182	0.998	0.070	0.934
Y(1, 0.75)	0.2051	$- 0.0070$	0.145	0.176	0.969	0.076	0.904
Y(0, 0.25)	0.2288	$- 0.0045$	0.121	0.203	0.988	0.078	0.922
Y(0, 0.5)	0.2761	0.0039	0.078	0.226	1.000	0.079	0.956
Y(0, 0.75)	0.3262	0.0050	0.241	0.265	0.983	0.109	0.886
Y(0.25)	0.2338	$- 0.0002$	0.096	0.199	0.998	0.070	0.937
Y(0.5)	0.2515	0.0038	0.053	0.203	1.000	0.064	0.982
Y(0.75)	0.2354	$- 0.0040$	0.121	0.196	0.996	0.073	0.945

IPCW: inverse probability censoring weighted; TRIP: Transmission Reduction Intervention Project; ESE: empirical standard error; ASE: asymptotic standard error; ECP: empirical coverage probability.

Table 4.

Simulation results from 1000 simulated datasets of IPCW estimators using mixed effects censoring model on original TRIP network (10 components) (left) and the network divided by community detection into 24 components (right).

				10 components		24 components
	True	Bias	ESE	ASE	ECP	ASE	ECP
Y(1, 0.25)	0.2492	0.0139	0.125	0.229	0.960	0.090	0.859
Y(1, 0.5)	0.2276	0.0056	0.074	0.203	0.996	0.069	0.928
Y(1, 0.75)	0.2066	-0.0032	0.128	0.188	0.966	0.073	0.883
Y(0, 0.25)	0.2296	-0.0022	0.113	0.327	0.973	0.076	0.891
Y(0, 0.5)	0.2796	0.0090	0.073	0.314	0.996	0.080	0.941
Y(0, 0.75)	0.3270	0.0224	0.116	0.328	0.968	0.109	0.870
Y(0.25)	0.2345	0.0018	0.091	0.301	0.994	0.069	0.913
Y(0.5)	0.2523	0.0073	0.050	0.257	0.999	0.064	0.975
Y(0.75)	0.2367	0.0032	0.098	0.221	0.992	0.068	0.925

IPCW: inverse probability censoring weighted; TRIP: Transmission Reduction Intervention Project; ESE: empirical standard error; ASE: asymptotic standard error; ECP: empirical coverage probability.

4. Motivating study: The transmission reduction intervention project (TRIP)

Participants in the TRIP study were PWID and their contacts who lived in Athens, Greece between 2013 and 2015. PWID who participated in the ARISTOTLE project at HIV testing centers in Athens were initially recruited into the TRIP study if they were found to be recently infected with HIV. ARISTOTLE was a community-based program aiming to mitigate HIV transmission among PWID by implementing a care system that involved reaching out to high-risk PWID, engaging them in HIV testing, and initiating HIV care, opioid use disorder treatment, and antiretroviral therapy.^26,27 In TRIP, each newly diagnosed individual was asked to identify their recent sexual and drug use partners in the last 6 months. These partners were then recruited and asked to identify their sexual and drug use partners, who were also recruited and contacts to other individuals already recruited in the study were also ascertained. If any of these partners were determined to be recently infected with HIV, then their contacts and the contacts of their contacts (i.e. two waves of contact tracing) were recruited as well, including their possible connections to the other participants in the study. Complete details about the TRIP study design can be found in previously published papers.^15,28

In addition to HIV testing, the study provided access to treatment as prevention (TasP), referrals for medical care, and distributed community alerts to inform community members about temporary increases in the risk for HIV acquisition. For example, a community alert would be distributed to those in close proximity in the observed network to a recently infected participant. These alerts included paper flyers given to participants and posted in a location frequented by members of the local PWID community. We considered those who received the alerts from the study staff or flyers to be exposed to the community alert, while the remaining participants were not exposed but could have possibly received this information from their exposed neighbors. All participants completed computer-assisted interviews and also had their HIV status ascertained. They provided demographic information, answered questions about engagement in risk behaviors, HIV status, substance use, access to care, HIV knowledge, stigma, injection norms, and their opinions on the project. Follow-up interviews were conducted with participants about six months after they completed their baseline interview.¹⁵

We applied the IPCW estimators to evaluate the causal effects of community alerts on HIV risk behavior ascertained by the report of risk behavior at the 6-month follow-up visit. The exposure community alerts was defined as receiving a study flyer, either from study staff or viewing one posted in the community. We determined the outcome of HIV risk behavior ascertained at the 6-month visit as a self-report of shared drug equipment (e.g. needles and syringes) in the last 6 months. We consider the report of any injection HIV risk behavior at a 6-month visit as a binary outcome. The following pre-exposure baseline covariates are included in the adjusted models for both the exposure and censoring based on expert knowledge and were known or suspected risk factors for the outcome: HIV status, shared drug equipment (i.e. self-report of sharing or being shared drug equipment (e.g. needles and syringes)) in the last 6 months prior to baseline, the calendar date of the first interview (binary: before or after ARISTOLE program ended), education (primary school, high school, and post-high school), and employment status (employed, unemployed/looking for a job, cannot work because of health reason, etc.) (Table 5). The causal effects were estimated separately using the logistic regression censoring model under the assumption that censoring mechanisms are independent across participants and the mixed-effects censoring model under the assumption that there is a correlation between censoring of participants within a component. The results showed that the participant’s, their neighbors’ exposure to community alerts, and neighbors’ baseline covariates were not statistically significant associated with censoring indicators.

The network structure in TRIP included 356 participants and 542 shared connections. One of the participant was recruited twice as a network member of a recent seed and as a network member of a control seeds with long-term HIV infection. In our analysis, we only used the information for this participant corresponding to their records as a network member of a recent seed. A total of 79 participants were isolates (i.e. not sharing connection with other network members) and removed for our analysis as spillover is not possible for isolates. If these individuals were included, the assumption is made that having no contacts is the same as having contacts with no exposure. In TRIP, these are two different scenarios with different HIV transmission risks, so we exclude isolates from the analysis. In addition, two participants were removed due to missing values on HIV risk behavior in the past 6 months reported at baseline. The final TRIP network had 10 unique components (component sizes were 2, 2, 2, 2, 2, 4, 5, 7, 10, and 239) with 275 participants and 540 shared connections among those individuals after excluding isolates. There were 56 participants (21%) who were lost to follow-up by 6 months. Among the 275 participants in TRIP, 29 participants (11%) received a community alert about an increased risk for HIV infection. The point estimates and corresponding 95% Wald-type CIs use both censoring models under allocation strategies 25%, 50%, and 75%, representing low, moderate and high coverage strategies. The mixed-effects censoring model did not detect random effects in each component on the original TRIP network. Therefore, the results using either censoring model are identical (Table 6).

Table 5.
Descriptive statistics of network characteristics and baseline variables for study participants after excluding isolates in the Transmission Reduction Intervention Project, Athens, Greece, from 2013 to 2015.

Network characteristic Nodes 275

Edges 540

Average degree (SD) 3.9 (3.5)

Density 0.014

Community alert Exposed 29 (11%)

Not exposed 246 (89%)

HIV status Positive 142 (52%)

Negative 133 (48%)

Date of first interview Before ARISTOTLE ended 130 (47%)

After ARISTOTLE ended 145 (53%)

Education Primary school or less 87 (32%)

High school (first 3 years) 82 (30%)

High school (last 3 years) 68 (25%)

Post-high school 38 (13%)

Employment Employed 44 (16%)

Unemployed; looking for work 64 (23%)

Can’t work; health reason 128 (47%)

Other 39 (14%)

Shared injection equipment Yes 207 (75%)

in last 6 months No 68 (25%)

Outcome: sharing injection Yes Exposed 11 (4%)

equipment at the 6-month visit Not exposed 83 (30%)

No Exposed 14 (5%)

Not exposed 111 (40%)

Missing Exposed 4 (1%)

Not exposed 52 (20%)

Network characteristic	Nodes	275
Community alert	Exposed	29 (11%)
	Not exposed	246 (89%)
HIV status	Positive	142 (52%)
	Negative	133 (48%)
Date of first interview	Before ARISTOTLE ended	130 (47%)
	After ARISTOTLE ended	145 (53%)
Education	Primary school or less	87 (32%)
	High school (first 3 years)	82 (30%)
	High school (last 3 years)	68 (25%)
	Post-high school	38 (13%)
Employment	Employed	44 (16%)
	Unemployed; looking for work	64 (23%)
	Can’t work; health reason	128 (47%)
	Other	39 (14%)
Shared injection equipment	Yes	207 (75%)
in last 6 months	No	68 (25%)
Outcome: sharing injection	Yes	Exposed	11 (4%)
equipment at the 6-month visit		Not exposed	83 (30%)
	No	Exposed	14 (5%)
		Not exposed	111 (40%)
	Missing	Exposed	4 (1%)
		Not exposed	52 (20%)

Table 6.

The estimated RD and 95% Wald-type CIs of the effects of community alerts at baseline on HIV risk behavior at 6 months on the original TRIP network (10 components) (left) and estimated using completed cases ( $n = 216$ ) (right) under allocation strategies 25%, 50%, and 75%.

	Coverage	Censoring model		Complete cases
Effects	( $α$ , $α^{'}$ )	RD	95% CI	RD	95% CI
Direct	(25%, 25%)	$-$ 0.1228	( $-$ 0.288, 0.042)	$-$ 0.0772	( $-$ 0.181, 0.027)
Direct	(50%, 50%)	$-$ 0.2299	( $-$ 0.637, 0.177)	$-$ 0.1761	( $-$ 0.489, 0.137)
Direct	(75%, 75%)	$-$ 0.1741	( $-$ 0.514, 0.166)	$-$ 0.2393	( $-$ 0.700, 0.222)
Indirect	(50%, 25%)	$-$ 0.0565	( $-$ 0.117, 0.004)	$-$ 0.0404	( $-$ 0.081, 0.001)
Indirect	(75%, 50%)	$-$ 0.1502	( $-$ 0.382, 0.081)	$-$ 0.0818	( $-$ 0.186, 0.023)
Indirect	(75%, 25%)	$-$ 0.2066	( $-$ 0.494, 0.081)	$-$ 0.1222	( $-$ 0.262, 0.018)
Total	(50%, 25%)	$-$ 0.2864	( $-$ 0.747, 0.174)	$-$ 0.2165	( $-$ 0.551, 0.118)
Total	(75%, 50%)	$-$ 0.3243	( $-$ 0.894, 0.246)	$-$ 0.3211	( $-$ 0.873, 0.231)
Total	(75%, 25%)	$-$ 0.3808	( $-$ 1.004, 0.242)	$-$ 0.3615	( $-$ 0.937, 0.214)
Overall	(50%, 25%)	$-$ 0.1407	( $-$ 0.358, 0.076)	$-$ 0.1091	( $-$ 0.266, 0.048)
Overall	(75%, 50%)	$-$ 0.1658	( $-$ 0.447, 0.116)	$-$ 0.1732	( $-$ 0.458, 0.111)
Overall	(75%, 25%)	$-$ 0.3065	( $-$ 0.805, 0.192)	$-$ 0.2824	( $-$ 0.723, 0.158)

RD: risk difference; CIs: confidence intervals; HIV: human immunodeficiency virus; TRIP: Transmission Reduction Intervention Project.

These results indicate that the risk of HIV behavior was reduced by increasing the proportion of a participant’s neighbors exposed to community alerts, in addition to a participant’s exposure. The estimated direct effect under allocation strategy 75% was $- 0.17$ (CI: $- 0.51, 0.17$ ); that is, we would expect 17 fewer reports of risk behavior per 100 participants if an individual receives alert compared to if an individual did not receive an alert with 75% intervention coverage (i.e. 75% of their neighbors receiving alerts). The spillover effect under allocation strategies 25% versus 75% was $- 0.21$ (CI: $- 0.50, 0.08$ ). Although not statistically significant, we would expect 21 fewer reports of risk behavior per 100 participants if a participant does not receive an alert with 75% intervention coverage compared to only 25% intervention coverage. The total effect under allocation strategies 75% versus 25% was $- 0.38$ (CI: $- 1, 0.24$ ); that is, we expect 38 fewer reports of risk behavior per 100 participants if an individual receives the alert and 75% of their neighbors also receive an alert compared to if an individual does not receive an alert and only 25% of their neighbors receive an alert. The overall effect was $- 0.31$ (CI: $- 0.81, 0.19$ ), indicating we expect 31 fewer reports of risk behavior per 100 participants if 75% of the neighbors and the index participant receives alert compared to only 25% of the neighbors and index participant receive alerts. Interestingly, the estimated spillover effect was larger in magnitude than the estimated direct effect, possibly due to the information about the community alert already being available in the spillover set due to the intervention coverage level of 75%. The estimated spillover effect suggests a notable decrease in reported risk behavior, with a 21% reduction from a baseline prevalence of 75%.

5. Conclusion

In this article, we extended the neighbor inverse probability weighted estimator⁸ to allow for possible censoring of outcomes in network-based studies. The two IPCW estimators were obtained by including a censoring weight derived from two different censoring models, namely logistic regression and mixed-effects model, and an inverse probability weighted estimator that assumes the interference set comprises the first-degree connections for each participants. The additional simulation scenario of varying variances of random effects in the censoring weighted model suggested that the performance in terms of empirical coverage probabilities of both censoring models, with and without correlations of censoring indicators, was comparable. The main difference between the logistic model with fixed effects only and logistic mixed model is the model-based standard error but not the point estimation, while IPCW point estimators and variance estimators only use the information from the point estimation of fixed and random effects of the censoring models. In terms of estimating the average outcomes and their corresponding closed-form variances, both censoring models provided similar results for the simulation scenario described in Section 3. When one anticipates correlation of the censoring indicators within a network component either due to prior knowledge or estimated in the study data, the mixed effects censoring model can provide the information on the correlations between censoring indicators, a measure of the association between the missingness of participant outcomes. Therefore, the selection of the censoring model is based on what information is needed and if studying the association of the censoring mechanism between participants is of substantive interest.

We demonstrated both IPCW estimators to be consistent and asymptotically normal. We also derived a consistent estimator of the asymptotic variance. The simulation study showed that the IPCW estimator performed well in finite samples given a large number ( $> 100$ ) of components in the network. With the proposed method, we developed an approach to quantify social and biological spillover effects on the risk of HIV transmission in HIV risk networks among PWID.^15,29,30 Furthermore, when fixing the number of components at 50, larger component sizes resulted in more accurate point estimates, with coverage closer to the nominal level (Tables 1 and 2). In the main simulation scenario (Figure 2), where we generated a regular network with an average component size of 10, the empirical coverage probability reached the nominal level when the number of components exceeded 100. However, the simulation based on the TRIP network (10 components) exhibited slight over-coverage, which may again be attributed to the substantial variability in component sizes and node degrees, with some nodes having up to 16 contacts while some others had only one. This method leverages information on all participants in the network study to address the selection bias from differential loss to follow-up due to study drop out and avoids a possible distortion of the network structure resulting from removing individuals who had missing outcomes when a complete case analysis is conducted.

Using two different assumptions about the correlation of censoring mechanisms, we applied the IPCW estimators to assess the causal effects of community alerts on HIV risk behavior at a 6-month follow-up in TRIP. The TRIP network contained 79 isolates, which could be relevant for both direct and overall effects. However, since we are focusing on evaluating spillover effects, which are not possible for isolates, we excluded these 79 isolates from our analysis. Further, based on the results from additional simulation scenario 3, we observed that ECPs were over-estimated for 10 components of the TRIP network and slightly under-estimated for 24 components. As a result, we focused on the original TRIP 10-component network for our analysis. The mixed effects censoring model did not detect any random effects for each component, resulting in identical outcomes when using both the logistic censoring model and the mixed effects model. This outcome may be due to discordant random effects within the connected components of the network, which likely canceled each other out. The estimated CIs were quite wide when considering the original TRIP network, possibly due to the variability in the sizes of the components. In this study, community alerts were found to be protective, with potential spillover effects from neighbors to unexposed participants. The spillover effects ranged between $- 0.06$ and $- 0.21$ under allocation strategies of 25%, 50%, and 75%.

There are several interesting directions for future research. One prominent area involves the choice and correct specification of the censoring model. Although the additional simulation scenario varying the variance of random effects in the censoring weight model produced comparable results in our study setting, results may differ in other contexts, for example, in studies involving participants grouped in families, where dependency in censoring indicators among closely connected individuals may need to be modeled to improve variance estimation.³¹ Another important direction is to explore alternative censoring mechanisms, such as MNAR. Graphical models known as ”missingness graphs”—causal directed acyclic graphs offer a useful framework for representing MNAR mechanisms and may support the development of adjusted estimators for causal parameters using partially observed data.^21,22,32 Additionally, extending the simulation framework to incorporate varying network structures across replicates would allow evaluation of the estimators under structural variation, further aligning with real-world sampling scenarios. Finally, since the consistency of IPCW estimators depends on the correct specification of both the censoring and exposure weight models, assessing model fit and conducting sensitivity analyses is crucial to ensuring the validity of the proposed method.³³ It is worth noting that, due to the large sample properties, we require the network has fairly large number of components. The other direction is related to evaluating the accuracy of the variance estimator when the sample size or number of components is small.³⁴ Based on the simulation study, the CI coverage levels can be below the nominal level in a network with a small number of network components or above the nominal level when the variability of the size of components is high. Future research could include developing a methodology that has reasonable finite-sample performance when the network has a small number of components or small number of participants. Additionally, the random effect in the same component was assumed homogeneous in this study. However, this may not hold when some of components in the network have large size. Future work could involve a revised M-estimation procedure for the variance to use individually weighted estimators that are consistent when there are varying component sizes, extending results from a two-stage randomized trial to this network setting.²⁵

Supplemental Material

sj-pdf-1-smm-10.1177_09622802251382586 - Supplemental material for Assessing spillover effects: Handling missing outcomes in network-based studies

Supplemental material, sj-pdf-1-smm-10.1177_09622802251382586 for Assessing spillover effects: Handling missing outcomes in network-based studies by TingFang Lee, Ashley L Buchanan, Natallia Katenka, Laura Forastiere, M Elizabeth Halloran and Georgios Nikolopoulos in Statistical Methods in Medical Research

Footnotes

Acknowledgements

These findings are presented on behalf of the Transmission Reduction Intervention Project (TRIP). We would like to thank all of the TRIP investigators, data management teams, and participants who contributed to this project. The project described was supported by award number DP2DA046856 by the Avenir Award Program for Research on Substance Abuse and HIV/AIDS (DP2) and award number R01DA058994 from the National Institute on Drug Abuse of the National Institutes of Health (NIH), the NIH National Institute of Mental Health award number R01MH134715, the NIH National Institute on Drug Abuse award number DP1DA034989, which funded Preventing HIV Transmission by Recently-Infected Drug Users, the NIH National Institute on Drug Abuse award number P30DA011041, which supported the Center for Drug Use and HIV Research, and the NIH National Institute of Allergy and Infectious Diseases award number R01AI085073, which supported the project Causal Inference in Infectious Disease Prevention Studies. Special thanks to Dr Samuel R Friedman (Department of Population Health, New York University Grossman School of Medicine) for his generous support and comments. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.

ORCID iDs

TingFang Lee

Georgios Nikolopoulos

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Institutes of Health (NIH) through the following awards: the Avenir Award Program for Research on Substance Abuse and HIV/AIDS, National Institute on Drug Abuse [DP2DA046856]; the National Institute on Drug Abuse [R01DA058994, DP1DA034989, P30DA011041]; the National Institute of Mental Health [R01MH134715]; and the National Institute of Allergy and Infectious Diseases [R01AI085073].

Declaration of conflicting interest

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Supplemental material

Supplemental material for this article is available online.

Notes

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Network characteristic	Nodes		275
	Edges		540
	Average degree (SD)		3.9 (3.5)
	Density		0.014
Community alert	Exposed		29 (11%)
	Not exposed		246 (89%)
HIV status	Positive		142 (52%)
	Negative		133 (48%)
Date of first interview	Before ARISTOTLE ended		130 (47%)
	After ARISTOTLE ended		145 (53%)
Education	Primary school or less		87 (32%)
	High school (first 3 years)		82 (30%)
	High school (last 3 years)		68 (25%)
	Post-high school		38 (13%)
Employment	Employed		44 (16%)
	Unemployed; looking for work		64 (23%)
	Can’t work; health reason		128 (47%)
	Other		39 (14%)
Shared injection equipment	Yes		207 (75%)
in last 6 months	No		68 (25%)
Outcome: sharing injection	Yes	Exposed	11 (4%)
equipment at the 6-month visit		Not exposed	83 (30%)
	No	Exposed	14 (5%)
		Not exposed	111 (40%)
	Missing	Exposed	4 (1%)
		Not exposed	52 (20%)