Causal Inference for Intervention Spillover in a Stepped Wedge Cluster-randomized Trial: Lessons from a Physician Network

Abstract

Often motivated by concerns about contamination of control subjects, stepped wedge cluster-randomized trial designs assign interventions to distinct clusters (e.g., hospitals) and protect against confounding by randomizing the timing of intervention delivery. However, when trial units are embedded in professional networks that span clusters, such designs cannot prevent spillover. In this study, we use longitudinal cluster-randomized trial data and contemporaneous physician networks to estimate the direct, indirect and direct-indirect interaction effects of an advance care planning (ACP) intervention. The direct effect reflects the intervention’s impact in a counterfactual world absent contamination, the indirect effect captures spillover to control subjects, and the interaction effect assesses whether spillover modifies the intervention’s effect, a phenomenon known as contamination. In addition, we exploit the staggered nature of the stepped-wedge design to test whether intervened peers impact the sustainment of the intervention over follow-up and, specifically, of whether spillover reinforces or attenuates the intervention. On a national provider organization’s physician network, we find that accounting for spillover and contamination had a profound effect on the direct effect of the intervention. We also found evidence that physician spillover effects reinforced the intervention over follow-up. These findings suggest that the diffusion of the intervention’s effect through the network was primarily driven by a large physician spillover effect that modified the direct effect at the time of intervention and subsequently over follow-up.

1. Introduction

The stepped wedge cluster-randomized trial (SW-CRT) has emerged as a powerful design for evaluating complex service delivery and policy interventions (Cook and Campbell, 1979; Brown and Lilford, 2006; Ellenberg, 2018; Hemming and Taljaard, 2020; Li et al, 2025). Unlike conventional parallel cluster randomized trials where clusters are fixed in either the control or intervention condition, the SW-CRT introduces the intervention sequentially. At each prespecified interval, a randomly chosen subset of clusters transitions from control to intervention until every cluster has crossed over. Because each cluster provides data from both phases, the design naturally accommodates within-cluster comparisons while addressing practical or ethical barriers that often preclude full individual randomization (Cook and Campbell, 1979; Brown and Lilford, 2006; Ellenberg, 2018; Hemming and Taljaard, 2020; Li et al, 2025).

A key feature and advantage of this staggered rollout is that it facilitates examination of both immediate and longer-term effects. Once clusters have completed the crossover, follow-up data can reveal whether intervention effects are sustained beyond the active implementation period. In contrast, outcome data are typically collected from the units of analysis (e.g., patients) at a single point in time, under either the control or intervention condition applicable to their cluster at that moment.

The SW-CRT framework has been successfully applied in a variety of large-scale public health and healthcare system interventions. One of the earliest and most well-known examples is the Gambia hepatitis intervention study of the 1980s, which evaluated the effectiveness of hepatitis B vaccination using a phased regional rollout (The Gambia Hepatitis Study Group, 1987; Hemming et al, 2015). Since then, stepped wedge designs have been used for initiatives ranging from national vaccination programs to complex health system interventions, such as collaborative care for trauma-related PTSD (TSOS) and efforts to reduce unnecessary lumbar imaging (LIRE) (Jarvik et al, 2015; Zatzick et al, 2021; Hemming et al, 2017; Hemming and Taljaard, 2020). The use of SW-CRTs has expanded rapidly, with more than 375 registered clinical trials on ClinicalTrials.gov since 2020.

Despite their appeal, SW-CRTs pose unique analytical and practical challenges. Because intervention rollout is time-ordered, data collected under control conditions often precede those collected under intervention conditions. This temporal asymmetry raises the risk that clusters yet to receive the intervention may nevertheless adopt some of its elements prematurely through information sharing or professional interaction. This is of particular concern in settings like healthcare where communication among practitioners is frequent. Such early adoption can dilute or obscure intervention effects, biasing estimates toward the null (Hemming et al, 2017). Moreover, exposure to intervention components before formal rollout may alter later engagement once the intervention is officially implemented. Ironically, this potential for “contamination” is often cited as the very reason for adopting a stepped wedge design rather than randomizing individual providers or patients.

We use methods from social network analysis to detect, model, and account for contamination, yielding a more accurate and informative compilation of effects for a SW-CRT. In addition, we further extend our model to allow direct and indirect intervention effects to change over follow-up and test whether sustainment is bolstered by increased exposure to intervened peer physicians (a phenomenon we refer to as reinforcement). As will be demonstrated in this paper, measurement of a network linking the intervened subjects allows for many embellishments of the standard model for analyzing SW-CRTs.

The motivation for this research stems from a SW-CRT conducted in collaboration with a National Provider Organization (NPO) to evaluate the effectiveness of a video game designed to impact physicians subconsciousness and conscious cognition to make them more likely to have Advance Care Planning (ACP) conversations with their patients. The intervention responded to a recognized gap in patient-centered care for serious illness and the belief that having more discussions on end-of-life care preferences that are reflective of patient values would improve the experience of patients, their caregivers and their family members towards and at the end of a patient’s life. In this trial, concerns about contamination within and between physicians led to hospitals being the clusters and thus the units of randomization, as is commonly practiced with educational-type interventions in medicine. The participating physicians in each trial hospital received iPads containing a video game educational intervention at the start of the step to which their hospital was randomized (Mohan et al, 2021, 2023). Medical insurance billing data made available by the NPO facilitated implementation of the extended analyses developed herein as this allowed a physician network to be constructed at baseline and specifically for each step of the trial (Bobak et al, 2023, 2026).

In the next section we describe how the medical insurance billing data is used to construct professional social networks of the trial participating physicians. This is followed in Section 3 by the development of causal models for extended SW-CRT designs involving potential outcomes based definitions of the targeted causal estimands and in Section 4 by the statistical models that seek to estimate these causal effects. The motivating SW-CRT is described in greater detail in Section 5 with the results of our analyses of it reported in Section 6. The paper concludes with a discussion of the findings, limitations and future research possibilities in Section 7.

2. Physician network and exposure to potential physician peer-effects

A social network consists of nodes or actors and the relationships between them. In a SW-CRT with hospitals as units of randomization but physicians as the delivery agents of the intervention to patients, we seek a network in which nodes represent physicians and the edges between them capture professional ties that quantify the potential transmission of peer influence. In our study, the availability of the NPO’s medical insurance billing data allows physician networks to be constructed from the co-occurrence of physicians on the same insurance claims. This requirement is a stricter condition than typically used in the construction of shared-patient physician networks of only requiring a patient to have encountered a pair of physicians at any point within a specified time-period, not necessarily on the same bill, to contribute to the existence or weight of a network edge (Korsberg et al, 2024; O’Malley et al, 2023; Onnela et al, 2018).

From the insurance claims, we first construct a bipartite network with edges linking a patient bill to a physician if the physician’s NPI appeared on the bill. Two physicians appearing on the same bill suggests that they encountered the patient during the same hospital stay, likely coordinating care or collaborating (within the same hospital, these might include handoffs from one physician to another such as between shifts). Denoting the mixing matrix representing this bipartite network by $\mathit{B}$ with $B_{ik}$ indicating whether physician $k$ cared for patient $i$ during their hospital stay, we compute $\mathit{A}={\mathit{B}}^T\mathit{B}$ to obtain a weighted physician network adjacency matrix in which $A_{kl}$ quantifies the number of patient bills involving both physicians $k$ and $l$.

We may compute $\mathit{A}$ at baseline or using patient admissions over a specified period of time. If the network evolves with time, we denote the network at the beginning of step $t$ by ${\mathit{A}}_t$ and its $k{l}^{\prime }th$ element by $A_{klt}$ (throughout the paper we will use the subscripts $k$ and $l$ for physicians), whereas if the network is time-invariant, $\mathit{A}$ is used throughout the study. Under any network construction, the network given by $\mathit{A}=\left[A_{kl}\right]$ or ${\mathit{A}}_t=\left[A_{klt}\right]$ may be left as a weighted network or transformed about a threshold believed to define a true relationship to form a binary network. If the influence of one physician on another is believed to be proportional to the corresponding edge weight, the rows of $\mathit{A}$ or ${\mathit{A}}_t$ may be scaled to sum to 1, yielding weight matrices $\mathit{W}$ or ${\mathit{W}}_t$ whose $kl$th element is

$$W_{kl}=\frac{A_{kl}}{\sum _{l\ne k}{A}_{kl}}\text{or}{W}_{klt}=\frac{A_{klt}}{\sum _{l\ne k}{A}_{klt}}$$

Let ${\mathit{\text{PhysInt}}}_t$ denote a vector whose $k$th element contains the intervention status (1 = intervened on by the start of the current step or earlier, 0 otherwise) of physician $k$. Then define:

$${\mathit{\text{PhysPeer}}}_t={\mathit{\text{WPhysInt}}}_t\text{or}{\mathit{\text{PhysPeer}}}_t={\mathit{W}}_t{\mathit{\text{PhysInt}}}_t$$ (1)

with

$${\mathit{\text{PhysPeer}}}_{kt}=\frac{\sum _{l\ne k}{A}_{kl}{\mathit{\text{PhysInt}}}_{lt}}{\sum _{l\ne k}{A}_{kl}}\text{or}{\mathit{\text{PhysPeer}}}_{kt}=\frac{\sum _{l\ne k}{A}_{klt}{\mathit{\text{PhysInt}}}_{lt}}{\sum _{l\ne k}{A}_{klt}},$$

as the average intervention status of physician $k$’s peers at the start of step $t$. A concern with using ${\mathit{W}}_t$ as opposed to $\mathit{W}$ in defining $\mathit{P}\mathit{h}\mathit{y}\mathit{s}\mathit{P}\mathit{e}\mathit{e}{\mathit{r}}_t$ occurs if the intervention in earlier steps impacts the network in subsequent steps as the network becomes an endogenous variable with respect to the intervention – a phenomenon known as reverse causality – with the consequence that adjusting for $\mathit{P}\mathit{h}\mathit{y}\mathit{s}\mathit{P}\mathit{e}\mathit{e}{\mathit{r}}_t$ may absorb some of the intervention effect. Reverse causality often leads to bias towards the null, which tends to be of less concern than bias away from the null. However, contamination is likely to be more precisely accounted for by using ${\mathit{W}}_t$ to define $\mathit{P}\mathit{h}\mathit{y}\mathit{s}\mathit{P}\mathit{e}\mathit{e}{\mathit{r}}_t$. Due to these competing reasons, in our motivating application we perform the analysis both with $\mathit{P}\mathit{h}\mathit{y}\mathit{s}\mathit{P}\mathit{e}\mathit{e}{\mathit{r}}_t$ defined using $\mathit{W}$ and with $\mathit{P}\mathit{h}\mathit{y}\mathit{s}\mathit{P}\mathit{e}\mathit{e}{\mathit{r}}_t$ defined using ${\mathit{W}}_t$.

3. Extended Causal Models for SW-CRT

In this section we first introduce a theoretical sociology model that describes possible physician peer-influence mechanisms that lead to contamination in a SW-CRT, define notation for specifying causal effects (targeted estimands on the scale of the outcome) and causal models, and specify statistical models that seek to estimate the targeted causal effects.

3.1. Sociological model leading to spillover

An unanticipated phenomenon during the ACP intervention trial was the high degree of physician mobility and collaboration across hospitals. According to NPO employment records, many physicians practiced at multiple sites or worked alongside colleagues based at other hospitals. We believe that physicians exposed to the intervention may have independently increased their facilitation of ACP conversations, leading to spillover effects and contamination risks as they moved between clinical sites. The reverse may also have occurred, with physicians transferring into hospitals that received the intervention earlier, or locum physicians working concurrently at multiple hospitals. Team-based care structures could have further contributed to these effects, as intervention-exposed physicians encouraged colleagues to initiate ACP discussions during handoffs or influenced peers through observation and modeling. Informal peer discussions about the intervention, the video game, or ACP practices may also have reinforced these behaviors—consistent with evidence that social connections often exert stronger influence on clinical practice than formal institutional policies.

3.2. Causal direct and indirect intervention effects

For simplicity in defining causal effects, models and methodological results, we temporarily suppose each patient is treated by a single physician and each patient only enters the study once. (In Section 3.5 we describe how the causal effect definitions and models may be generalized to account for patients encountering multiple physicians.) Let ${\mathit{P}\mathit{a}\mathit{t}\mathit{I}\mathit{n}\mathit{t}}_t$ and ${\mathit{P}\mathit{a}\mathit{t}\mathit{P}\mathit{e}\mathit{e}\mathit{r}}_t$ be the vectors obtained by expanding ${\mathit{P}\mathit{a}\mathit{t}\mathit{I}\mathit{n}\mathit{t}}_t$ and $\mathit{P}\mathit{h}\mathit{y}\mathit{s}\mathit{P}\mathit{e}\mathit{e}{\mathit{r}}_t$ to the patient-level, respectively. Denoting the physician who treated patient $i$ by the shorthand notation $k(i)$, it follows that ${\mathit{\text{PatInt}}}_{it}={\mathit{\text{PhysInt}}}_{k(i)t}$ and ${\mathit{\text{PatPeer}}}_{it}={\mathit{\text{PhysPeer}}}_{k(i)t}$, where ${\mathit{\text{PatInt}}}_{it}$ and ${\mathit{\text{PatPeer}}}_{it}$ are the $i$th elements of ${\mathit{P}\mathit{a}\mathit{t}\mathit{I}\mathit{n}\mathit{t}}_t$ and $\mathit{P}\mathit{h}\mathit{y}\mathit{s}\mathit{P}\mathit{e}\mathit{e}{\mathit{r}}_t$, respectively. The set of physicians other than $k(i)$ is denoted $-k(i)$.

To define the direct and indirect effects of the intervention in our motivating study we partition ${\mathit{P}\mathit{a}\mathit{t}\mathit{I}\mathit{n}\mathit{t}}_t$ into ${\mathit{\text{PhysInt}}}_{k(i)t}$, the intervention status of the physician that patient $i$ encountered, and ${\mathit{P}\mathit{h}\mathit{y}\mathit{s}\mathit{I}\mathit{n}\mathit{t}}_{-k(i)t}$, the intervention status of the other study physicians in time-period $t$. Let $Y_{it}\left({\mathit{\text{PhysInt}}}_{k(i)t}\right.=z,{\mathit{P}\mathit{h}\mathit{y}\mathit{s}\mathit{I}\mathit{n}\mathit{t}}_{-k(i)t})$ denote the potential outcome for patient $i$ under the counterfactual that physician $k(i)$ is assigned intervention status $z\in \{0,1\}$ at $t$ with the intervention status of all other physicians unchanged. We define the average directed effect (ADE) of the intervention across all patients in the trial as

$$ADE=N^{-1}\sum _{t=1}^T\sum _{i=1}^{n_t}\left(Y_{it}\left({\mathit{\text{PhysInt}}}_{k(i)t}=1,{\mathit{P}\mathit{h}\mathit{y}\mathit{s}\mathit{I}\mathit{n}\mathit{t}}_{-k(i)t}\right)-Y_{it}\left({\mathit{\text{PhysInt}}}_{k(i)t}=0,{\mathit{P}\mathit{h}\mathit{y}\mathit{s}\mathit{I}\mathit{n}\mathit{t}}_{-k(i)t}\right)\right),$$

where $N=\sum _{t=1}^T{n}_t$ and $n_t$ is the number of patients admitted to any hospital during step $t$. This definition adapts the ADE in Hu et al (2022) to the case when the outcomes are measured on a sub-unit (patients) of the nodes (physicians) defining the network and distinct panels of patients enter the study in each time-period. In Appendix A we show that the ADE is a weighted counterpart of the physician-level ADE in which the weights are the number of patients treated by each physician.

We define the average indirect effect (AIE) as the difference in the potential outcomes of patient $j$ across the counterfactual intervention states of patient $i$’s physician averaged over all patient dyads for which $k(j)\ne k(i)$ and trial-steps $t=1,\dots ,T$:

$$\begin{gathered} AIE=(N\bar{d}{)}^{-1}\sum _{t=1}^T\sum _{i=1}^{n_t}\sum _{j:k(j)\ne k(i)}^{n_t}\left(Y_{jt}\right.\left({\mathit{\text{PhysInt}}}_{k(i)t}=1,{\mathit{P}\mathit{h}\mathit{y}\mathit{s}\mathit{I}\mathit{n}\mathit{t}}_{-k(i)t}\right) \\ \left.-Y_{jt}\left({\mathit{\text{PhysInt}}}_{k(i)t}=0,{\mathit{P}\mathit{h}\mathit{y}\mathit{s}\mathit{I}\mathit{n}\mathit{t}}_{-k(i)t}\right)\right), \end{gathered}$$

where $\bar{d}$ is the average number of patients a physician co-treated with other physicians averaged across physicians and steps (the AIE’s summation over all pairs of impacts of one patient’s treatment on another’s outcome resembles the Shapley Value’s summation over the value of one player to another in a game). The rationale for requiring that patient $j$ see a different physician from patient $i$ is to avoid the scenario that they were treated by the same physician and so had the same direct treatment, which would result in the ADE and the AIE overlapping. In our primary motivating analyses, this condition will automatically be enforced by restricting peer physicians to those at other hospitals. As shown in Appendix A, the resulting AIE is a co-treated patient count (or physician edge) weighted generalization of the AIE in Hu et al (2022). Because the counterfactual comparison is of the intervention statuses of the physicians, not the exchange of patients between steps, we restrict the inner average of the AIE to patient dyads within $t$.

3.2.1. Characterization of ADE and AIE as predictor coefficients in a linear model

Because the ADE and AIE depend on the counterfactual intervention status of all physicians, we seek a more parsimonious representation. Suppose that

$$Y_{it}\left({\mathit{\text{PhysInt}}}_{\mathit{t}}\right)={\beta }_{0t}+{\beta }_{\text{dir}}{\mathit{\text{PhysInt}}}_{k(i)t}+{\beta }_{\text{indir}}{\mathit{\text{PhysPeer}}}_{k(i)t}+{ϵ}_{it},$$ (2)

where $E\left[{ϵ}_{it}\mid \right.\left.{\mathit{\text{PhysInt}}}_t\right]=0$, generated potential outcomes $Y_{it}(\mathit{v})$ given the vector of counterfactual intervention assignments ${\mathit{\text{PhysInt}}}_t=\mathit{v}$. Under the conditions assumed in Hu et al (2022), $ADE={\beta }_{\text{dir}}$ and $AIE={\beta }_{\text{indir}}$ (see Appendix A for proofs). Therefore, if one assumes (2) holds, it is sufficient to represent counterfactual states in terms of the scalars ${\mathit{\text{PatInt}}}_{it}={\mathit{\text{PhysInt}}}_{k(i)t}=z$ and ${\mathit{\text{PatPeer}}}_{it}={\mathit{\text{PhysPeer}}}_{k(i)t}=w$, avoiding specifying each element of ${\mathit{P}\mathit{h}\mathit{y}\mathit{s}\mathit{I}\mathit{n}\mathit{t}}_{-k(i)t}$ to represent peer-physician intervention exposure for each patient.

The above characterization of the non-model-based ADE and AIE in terms of ${\beta }_{\text{dir}}$ and ${\beta }_{\text{indir}}$ leads us to define the parameterized or level-specific causal effects:

$$\begin{gathered} ADE(w)=N^{-1}\sum _{t=1}^T\sum _{i=1}^{n_t}\left(Y_{it}\left({\mathit{\text{PhysInt}}}_{k(i)t}=1,{\mathit{\text{PhysPeer}}}_{k(i)t}=w\right) \\ \left.-Y_{it}\left({\mathit{\text{PhysInt}}}_{k(i)t}=0,{\mathit{\text{PhysPeer}}}_{k(i)t}=w\right)\right)\right. \end{gathered}$$

$$\begin{gathered} AIE(z,w)=N^{-1}\sum _{t=1}^T\sum _{i=1}^{n_t}\left(Y_{it}\left({\mathit{\text{PhysInt}}}_{k(i)t}=z,{\mathit{\text{PhysPeer}}}_{k(i)t}=w\right) \\ \left.-Y_{it}\left({\mathit{\text{PhysInt}}}_{k(i)t}=z,{\mathit{\text{PhysPeer}}}_{k(i)t}=0\right)\right)\right. \end{gathered}$$

where $ADE(w)$ now depends on $w$, the counterfactual proportion of peer physicians who have received the intervention, and $AIE(z,w)$ also depends on $z$, the direct intervention status. Next define the average contamination effect as $ACE(w)=ADE(w)-ADE(0)$, the difference in the ADE at peer-physician exposure level $w$ and at 0 (no peer-exposure). The $ACE(w)$ is an interaction effect as it quantifies how the effect of the ADE depends on the level of the AIE and vice-versa. Because $w\in [0,1]$, there are infinitely many realizations of $ADE(w)$, $AIE(z,w)$ and $ACE(w)$; herein we present results for the counterfactual change $w_0\to w_0+w$ with $w_0=0$ and $w=0.1$ unless otherwise stated.

3.3. Stepped wedge cluster-randomized trial (SW-CRT) causal model extended to direct and indirect effects

In our motivating application, $Y_{it}(z,w)=Y_{it}({\mathit{\text{PhysInt}}}_{k(i)t}=z,\left.{\mathit{\text{PhysPeer}}}_{k(i)t}=w\right)$ represents the existence of an insurance claim for ACP and so is binary. Therefore, it is natural to assume that the causal model for our SW-CRT has the form

$$g\left(E\left[Y_{it}(z,w)\right]\right)=g\left(E\left[Y_{it}(0,0)\right]\right)+{\beta }_{\text{dir}}z+{\beta }_{\text{indir}}w+{\beta }_{\text{contam}}zw$$ (3)

where $g\left(E\left[Y_{it}(0,0)\right]\right)={\beta }_{0t}+{\mathit{\beta }}_X^T{\mathit{X}}_{it}+{\mathit{\beta }}_Z^T{\mathit{Z}}_{h(i)}+{\theta }_{h(i)},g(.)$ is a link function (e.g., the logit function) that maps values in the domain of $E\left[Y_{it}(z,w)\right]$ (the (0, 1) interval in the case of binary outcomes) to the real number line, ${\mathit{X}}_{it}$ and ${\mathit{Z}}_{h(i)}$ denote patient and baseline hospital-level covariates (e.g., representing any variables used for blocking), respectively, and $h(i)$ is the hospital that patient $i$ attended. Because the covariates impact $Y_{it}(z,w)$ across all counterfactual values for $(z,w)$, the causal model for $Y_{it}(z,w)$ may be expressed in terms of $Y_{it}(0,0)$ without explicitly displaying covariates in (3).

3.4. SW-CRT causal model extended to sustainment and reinforcement

The causal model in (3) assumes that the effect of the intervention is homogeneous over follow-up. We are also interested in whether the intervention has a heterogeneous effect over follow-up and if exposure to a higher proportion of intervened peer physicians sustains, enhances or diminishes the intervention effect over follow-up. Augmenting the ADE and ACE with $u$ denoting counterfactual values of time since intervention (defining $u=0$ when $z=0$ as the intervention is yet to be rolled-out), we define

$$\begin{gathered} ASE(w,u)=ADE(w,u)-ADE(w,0) \\ ARE(w,u)=ASE(w,u)-ASE(0,u) \\ =ADE(w,u)-ADE(w,0)-(ADE(0,u)-ADE(0,0)) \end{gathered}$$

as the average sustainment effect (ASE) and the average reinforcement effect (ARE) at $u$ time units following intervention rollout, respectively. Augmenting (3) with the interaction terms corresponding to $ASE(w,u)$ and $ARE(w,u)$, the causal model for our SW-CRT becomes:

$$g\left(E\left[Y_{it}(z,w,u)\right]\right)=g\left(E\left[Y_{it}(0,0,0)\right]\right)+{\beta }_{\text{dir}}z+{\beta }_{\text{indir}}w+{\beta }_{\text{contam}}zw+{\beta }_{\text{sustain}}zu+{\beta }_{\text{reinforce}}zwu$$ (4)

where $g\left(E\left[Y_{it}(0,0,0)\right]\right)={\beta }_{0t}+{\mathit{\beta }}_X^T{\mathit{X}}_{it}+{\mathit{\beta }}_Z^T{\mathit{Z}}_{h(i)}+{\theta }_{h(i)}$. When $z$ is binary, $zu=u$ as by construction $u=0$ if $z=0$, allowing (4) to be simplified. However, when $z$ is more general (e.g., a count as in Section 3.5) this simplification does not occur and the more general representation in (4) is needed. Over follow-up, the intervention is enhanced if $ASE(w,u)>0$ or ${\beta }_{\text{sustain}}>0$, diminished if $ASE(w,u)<0$ or ${\beta }_{\text{sustain}}<0$ and reinforced at spillover level $w$ if $ARE(w,u)>0$ or ${\beta }_{\text{reinforce}}>0$. These dynamics are visualized in Figure 1.

Fig. 1:

Conceptual diagram of the spillover-extended SW-CRT model. Five nodes represent the hospital (direct assignment), peer physicians (spillover), time since rollout, the patient at time $t$, and the outcome $Y_{it}$. Arrows indicate (i) the direct path (hospital → patient → outcome, ${\beta }_1$), (ii) spillover (peers → patient → outcome, ${\beta }_2$), (iii) contamination interaction of direct and spillover effects $\left({\beta }_3\right)$, (iv) sustainment of direct-effect of intervention over follow-up time $\left({\beta }_4\right)$, and (v) peer reinforcement of the direct effect over follow-up $\left({\beta }_5\right)$.

3.5. Accounting for patients encountering multiple physicians

When the direct intervention is represented by the number of intervened physicians, $z$ generalizes from a binary variable to a non-negative count $z\in \{0,1,2,\dots \}$. When the direct intervention undergoes the unit change $z\leftarrow z+1$, we denote the causal estimands by $ADE(z,w)$, $AIE(z,w)$, $ACE(z,w)$, $ASE(z,w,u)$ and $ARE(z,w,u)$ with $Y_{it}\left({\mathit{\text{PhysInt}}}_{k(i)t}\right.=z+1,{\mathit{\text{PhysPeer}}}_{k(i)t}\left.=w\right)$ and $Y_{it}\left({\mathit{\text{PhysInt}}}_{k(i)t}\right.=z,\left.{\mathit{\text{PhysPeer}}}_{k(i)t}=w\right)$ replacing $Y_{it}\left({\mathit{\text{PhysInt}}}_{k(i)t}\right.=1,\left.{\mathit{\text{PhysPeer}}}_{k(i)t}=w\right)$ and $Y_{it}\left({\mathit{\text{PhysInt}}}_{k(i)t}\right.=0,\left.{\mathit{\text{PhysPeer}}}_{k(i)t}=w\right)$, respectively, in the earlier causal definitions due to the dependence of all estimands on $z$. Analogous effects can also be defined for other counterfactual differences; for example, $z+k$ versus $z$. The parametric structural models in (3) and (4) already accommodate integer $z$ (see justification following Equation 4) and so do not require further specification.

Because the number of study physicians a patient encounters is a post-randomization variable, it may be impacted by the intervention (a phenomena known as reverse causality) even after adjusting for the total number of (study and non-study) physicians encountered. Therefore, including it as a predictor risks adjusting away part of the effect of the intervention and compromising the estimation of the causal effects in (3) and (4). However, if one is primarily interested in testing whether the intervention has any effect and less about the exact mechanism, letting $z$ be the number of intervened physicians may provide more scope to detect a departure from the null hypothesis of 0 intervention effect; recall that a model only needs to hold under the null hypothesis for a statistical test to be valid (VanderWeele et al, 2012). Because the focus in this paper is on causal inference, in the main text we only present results for the standard and causally-protective intention-to-treat (ITT) case when the direct intervention is the binary indicator of whether the hospital at which the patient is admitted received the intervention during the current step or earlier.

4. Estimation of causal effects

We now describe the general statistical analysis used to estimate the causal effects defined in (3) and (4). Recall that in the motivating application, hospitals are randomized to study steps that are inherited by the physicians working at them at baseline. These physicians receive the intervention during the rollout period corresponding to their randomized study step. In addition, each patient may encounter one or more physicians who participated in the study during their stay.

4.1. Notation for statistical analyses

In presenting the statistical model, we explicitly denote the hospital the patient attended and map the direct and indirect intervention status vectors to the patient level. Let $i=1,\dots ,n_h$ denote patient $i$ within hospital $h=1,\dots ,H$ and chronological study time-period $t=1,\dots ,T$. Often $t$ will correspond to the step of the study with $t=1$ and $t=T$ denoting the baseline and final time-periods when no and all hospitals have received the intervention, respectively. However, finer-grained time-units may be used. The outcome (billed for ACP conversation), direct intervention, indirect or peer intervention, a vector of patient-level covariates and baseline hospital-level covariates for the $i$th patient admitted to hospital $h$ during time-period $t$ are denoted $Y_{iht}$, ${\mathit{\text{PatInt}}}_{ht}$, ${\mathit{\text{PatPeer}}}_{iht}$, ${\mathit{X}}_{iht}$ and ${\mathit{Z}}_h$, respectively.

The subscript $i$ is omitted from ${\mathit{\text{PatInt}}}_{ht}$, the binary indicator of whether the intervention has been received at hospital $h$ by time-interval $t$, as all study physicians at a hospital have the same intervention status and no distinction is made between the healthcare experience (e.g., number of physicians encountered) from one patient to the next. Because there is no exposure to the intervention prior to the onset of the study, ${\mathit{\text{PatInt}}}_{ih0}={\mathit{\text{PatPeer}}}_{ih0}=0$. The amount of follow-up time elapsed at the time when patient $i$ enters hospital $h$ equals ${\mathit{\text{FuTime}}}_{iht}=\text{max}\left\{{\mathit{\text{AdmitTime}}}_{iht}\right.-{\mathit{\text{StepTime}}}_h\left.,0\right\}$, where ${\mathit{\text{AdmitTime}}}_{iht}$ and ${\mathit{\text{StepTime}}}_h$ represent the corresponding patient admission date and the starting date of intervention rollout at hospital $h$, respectively.

4.2. Statistical models for a SW-CRT with physician network spillover effects

Because the occurrence of an Advance Care Planning (ACP) conversation is a binary random variable, it is natural to assume a logistic or other form of binary regression model. To adjust for differences across steps and account for statistical dependencies due to patients being clustered within hospitals, it is standard to account for step (or more fine-grained time-period) effects and hospital random effects. Because a SW-CRT contains relatively little randomization, chance imbalances with respect to intervention status in patient and hospital characteristics may occur. Adjusting for ${\mathit{X}}_{iht}$ may help defend the assumption that the expected value of the counterfactual outcomes absent the intervention are parallel across the study groups (i.e., that determine the step when intervention rollout occurs). Adjusting for baseline hospital covariates, ${\mathit{Z}}_h$, may help account for differences in hospital characteristics across the trial steps and reduce the amount of unexplained variation between hospitals in the outcome, possibly increasing statistical precision, but due to randomization the benefit of doing is likely less than that of adjusting for ${\mathit{X}}_{iht}$. Therefore, our statistical model companion to the causal model in (3) has the form $Y_{iht}\mid {\mathit{X}}_{iht},{\mathit{Z}}_h,{\theta }_h~\text{Bern}\left({\mathit{X}}_{iht},{\mathit{Z}}_h;{\theta }_h\right)$ with

$$g\left(E\left[Y_{iht}\mid {\theta }_h\right]\right)={\beta }_{0t}+{\beta }_1{\mathit{\text{PatInt}}}_{ht}+{\beta }_2{\mathit{\text{PatPeer}}}_{iht}+{\beta }_3{\mathit{\text{PatInt}}}_{ht}{\mathit{\text{PatPeer}}}_{iht}+{\mathit{\beta }}_4^T{\mathit{X}}_{iht}+{\mathit{\beta }}_5^T{\mathit{Z}}_h+{\theta }_h,$$ (5)

where Bern(.) denotes the Bernoulli distribution; ${\left\{{\theta }_h\right\}}_{1:H}$ denotes the effects of the $H$ hospitals in the study; ${\beta }_{0t}$ denotes the effect of time-period $t=1,\dots ,T$; ${\beta }_1$ is the direct effect of the intervention; ${\beta }_2$ is the spillover or indirect effect of the intervention, ${\beta }_3$ is the contamination effect; and ${\mathit{\beta }}_4$ and ${\mathit{\beta }}_5$ are the effects of the patient- and hospital-level covariates, respectively. It is commonly assumed that ${\theta }_h~\text{Normal}\left(0,{\tau }^2\right)$.

The extension of (5) to accommodate intervention sustainment and reinforcement, the statistical model counterpart to the causal model in (4), is:

$$\begin{gathered} g\left(E\left[Y_{iht}\mid {\theta }_h\right]\right)={\beta }_{0t}+{\beta }_1{\mathit{\text{PatInt}}}_{ht}+{\beta }_2{\mathit{\text{PatPeer}}}_{iht}+{\beta }_3{\mathit{\text{PatInt}}}_{ht}{\mathit{\text{PatPeer}}}_{iht} \\ +{\beta }_4{\mathit{\text{PatInt}}}_{ht}{\mathit{\text{FuTime}}}_{iht}+{\beta }_5{\mathit{\text{PatInt}}}_{ht}{\mathit{\text{PatPeer}}}_{iht}{\mathit{\text{FuTime}}}_{iht} \\ +{\mathit{\beta }}_6^T{\mathit{X}}_{iht}+{\mathit{\beta }}_7^T{\mathit{Z}}_h+{\theta }_h, \end{gathered}$$ (6)

Exact sustainment, enhancement, or diminishment corresponds to ${\beta }_4=0$, ${\beta }_4>0$ and ${\beta }_4<0$, respectively. Likewise, if spillover increasingly reinforces the intervention over follow-up, then ${\beta }_5>0$.

4.3. Alternative model specifications

The specifications in (5) and (6) are indifferent to whether $\mathit{W}$ or ${\mathit{W}}_t$ is used in the construction of ${\mathit{\text{PatPeer}}}_{iht}$. The former emulates an intention-to-treat (ITT) analysis but may imprecisely capture spillover and thus contamination from intervened peer physicians whereas the latter is more susceptible to reverse-causality acting through the network. As a robustness check of the hospital random effects specification in (5) and (6), we also estimate these models with ${\left\{{\theta }_h\right\}}_{1:H}$ as fixed effects (as for a study with a staggered roll-out intervention such as in Xiong et al (2024)) and compare the results to those of the mixed-effect models. In our ACP application, the results differed minimally; see Section 6.2 for summarized results and Section 3 of the Supplementary Material for detailed results.

Another alternative to the mixed-effect model with hospital random-effects is the Generalized Estimating Equations (GEE) procedure with hospital as the clustering variable. GEE avoids modeling the hospital-effects thereby providing the model parameters with marginal interpretations and yielding standard errors that are robust to misspecification of the within-cluster covariance structure (Lange and Ryan, 1989; Huang, 2022). The GEE results for the key predictors in our models were similar to those for the mixed-effect (and thus fixed-effect) models (see Section 3.1 of the Supplementary Material for more detailed comments).

The models in (5) and (6) may be generalized to allow multiple lagged values of the peer-physician spillover variable (PhysPeer) or their cumulative counterparts as predictors. Furthermore, the model in (6) may be made more flexible by modeling ${\mathit{\text{FuTime}}}_{iht}$ non-parametrically, which as noted in the Discussion is an important direction for future research.

4.4. Evaluation of statistical inferences

Estimation of (6) is aided in a SW-CRT by the variation in the length of follow-up post intervention across hospitals randomized to different trial steps and by the cross-sectional heterogeneity of intervention status across the clusters during intervention rollout. These design features allow the effects of sustainment and reinforcement to be identified with less reliance on the specified model and estimated with more precision. Because the network is either fixed at baseline or is evaluated at the prior time-period, ${\mathit{\text{PatPeer}}}_{iht}$ may be conditioned on and treated like ${\mathit{\text{PatInt}}}_{ht}$ and the other predictors in the model when evaluating standard errors and associated statistical inferences without inducing simultaneity (the phenomena whereby outcomes of different actors have simultaneous direct effects on each other); this is true even in models that include multiple lagged physician spillover variables. A chi-square or $z$-test under the fitted model in (6) may be performed to evaluate the statistical significance of the estimated values $\left({\hat{\beta }}_4,{\hat{\beta }}_5\right)$ and thus test if there is sufficient evidence to conclude that intervention enhancement, diminishment or reinforcement is present.

All of the variance estimators used herein assume statistical independence of observations in different clusters (hospitals in our ACP application) conditional on the observed predictors. Clearly, this assumption is threatened by the presence of physician connections between hospitals. Therefore, a key consideration to the validity of the variance calculations and the appropriateness of assuming asymptotic normality in computing statistical inferences is the between-hospital density of physician network connections (i.e., the proportion of possible edges between physicians at different hospitals with non-zero shared patient values). Prior research suggests that the lower the density of between network edges (i.e., the more that the between cluster component of the network is sparse), the more accurate that methods only accounting for clustering by hospital are likely to be (Aronow and Samii, 2017; Lu et al, 2025; Papadogeorgou et al, 2025). Because the network of 163 physicians in our ACP study has a between-hospital density of 0.007 or less at baseline and through the study period (the overall network density ranges between 0.033 and 0.036), the level of sparsity is very high suggesting that our statistical inferences should be sufficiently accurate. Section 2.2 of the Supplementary Material includes further discussion on the impact of network induced statistical dependence on the accuracy of both asymptotic and resampling (e.g., bootstrap) based variance estimation while the Discussion describes future research possibilities involving alternative methods for variance estimation that may overcome network induced between cluster statistical dependence.

4.5. Counterfactual outcome evaluation of causal estimands on the outcome scale

To estimate the causal effects in Sections 3.3 and 3.4, the estimated model parameters may be mapped to the counterfactual causal estimands, which have the benefit for interpretation of being on the scale of the outcome and thus interpreted as probability differences. For example, if ${\theta }_h$ is modeled as a fixed-effect, the average direct effect in the absence of spillover, $ADE(0)$, may be estimated by fitting (5) and then evaluating:

$$\widehat{ADE}(0)=N^{-1}\sum _{iht}\left({\hat{g}}^{-1}\left(1,0,{\mathit{X}}_{iht},{\mathit{Z}}_h\right)-{\hat{g}}^{-1}\left(0,0,{\mathit{X}}_{iht},{\mathit{Z}}_h\right)\right)$$

where

$$\widehat{g}\left({\mathit{\text{PatInt}}}_{ht}=z,{\mathit{\text{PatPeer}}}_{iht}=w,{\mathit{X}}_{iht},{\mathit{Z}}_h\right)={\hat{\beta }}_{0t}+{\hat{\beta }}_{1}z+{\hat{\beta }}_{2}w+{\hat{\beta }}_{3}zw+{\stackrel{\mathit{ˆ}}{\mathit{\beta }}}_4^T{\mathit{X}}_{iht}+{\stackrel{\mathit{ˆ}}{\mathit{\beta }}}_5^T{\mathit{Z}}_h+{\hat{\theta }}_h.$$

If ${\theta }_h$ is modeled as a random-effect, we instead estimate $ADE(0)$ by

$$\widehat{ADE}\left(0\right)=N^{-1}\sum _{iht}\int \left({\hat{g}}^{-1}\left(1,0,{\mathit{X}}_{iht},{\mathit{Z}}_h,{\theta }_h\right)-{\hat{g}}^{-1}\left(0,0,{\mathit{X}}_{iht},{\mathit{Z}}_h,{\theta }_h\right)\right)\hat{\pi }\left({\theta }_h\mid {\mathit{X}}_{iht},{\mathit{Z}}_h\right)d{\theta }_h,$$

where

$$\widehat{g}\left({\mathit{\text{PatInt}}}_{ht}=z,{\mathit{\text{PatPeer}}}_{iht}=w,{\mathit{X}}_{iht},{\mathit{Z}}_h,{\theta }_h\right)={\hat{\beta }}_{0t}+{\hat{\beta }}_{1}z+{\hat{\beta }}_{2}w+{\hat{\beta }}_{3}zw+{\stackrel{\mathit{ˆ}}{\mathit{\beta }}}_4^T{\mathit{X}}_{iht}+{\stackrel{\mathit{ˆ}}{\mathit{\beta }}}_5^T{\mathit{Z}}_h+{\theta }_h$$

and $\hat{\pi }\left({\theta }_h\mid {\mathit{X}}_{iht},{\mathit{Z}}_h\right)$ is the estimated distribution of ${\theta }_h$ given the observed data. If we alternatively use Generalized Estimating Equations (GEE), integration over ${\theta }_h$ is avoided. In Appendix B.1, based on the fitted models in (3) and (4), we derive analogous expressions for the estimators of $ADE(w)$, $AIE(z,w)$, $ACE(z,w)$, $ASE(w,u)$ and $ARE(w,u)$, where $z$, $w$ and $u$ are counterfactual values for the status of the direct intervention, the level of the indirect intervention or spillover and the time since intervention receipt.

To compute standard errors for the estimated vales of the counterfactual causal estimands (e.g., $ADE(w),AIE(z,w)$), we use the Delta method (Oehlert, 1992); see Appendix B.2 for an overview of the calculation and Section 2.1 of the Supplementary Material for detailed calculations. In reporting the results of our analyses, we report both the regression-based ORs and the counterfactual causal estimands (the marginal effects) as both are informative. Regression coefficients are convenient for testing specific hypotheses of interest without needing to specify the values of other predictors in the model while the counterfactual estimands target specific causal quantities and by virtue of being defined on the scale of the outcome are readily interpretable.

5. Motivating Case Study: Data Wrangling and Analyses

Because patients may interact with multiple physicians during a single hospital stay, we define two types of direct effects: the intention-to-treat (ITT) effect corresponding to the hospital’s random assignment of a study step and the as-treated (AT) effect reflecting the extent to which the patient was directly exposed to the intervention during their hospital stay. Two variants of the indirect effect that pair with the ITT and the AT can be similarly defined. For each patient, we define:

• Direct intervention (ITT): Indicates whether, at the time of the patient’s admission, the admitting hospital (and thereby its study physicians) had already received the intervention.

• Indirect intervention (paired with ITT): The sum of the exposures of the study physicians encountered by the patient to external peer physicians (only those outside the admitting hospital) who had already received the intervention by the time of the patient’s admission to hospital.

• Direct intervention (AT): The number of study physicians encountered by the patient during their stay who had received the intervention at the time of their admission to hospital.

• Indirect intervention (paired with AT): The sum of the exposures of the study physicians encountered by the admitted patient to peer physicians both within and external to the admitting hospital who had already received the intervention by the time of the patient’s admission to hospital.

As discussed in Section 3.5, AT analyses risk adjusting away part of the effect of the intervention (e.g., if it impacts the number of physicians a patient encounters) or being exposed to confounding, even when adjusting for the total number of physicians encountered during patient hospital stays. Therefore, the ITT direct intervention and its paired indirect exposure measures are the basis of the key predictors in the analyses reported herein; the parallel AT analyses are in Section 4 of the Supplemental Materials.

5.1. ACP billing data

We re-analyze the SW-CRT motivating this work (Mohan et al, 2021, 2023). That study sought to evaluate the impact of a video game intervention, designed to motivate hospitalists toward having more Advance Care Planning (ACP) conversations with older patients, on ACP billing frequency. The trial involved 163 hospitalists from 35 hospitals across the U.S., with intervention rollout occurring from October 2020 to February 2021. Participating physicians were provided iPads preloaded with the game and asked to engage with it for at least two hours. The primary outcome, ACP billing, served as a proxy for ACP conversation initiation, with data abstracters blinded to the intervention status of physicians. The baseline patient characteristics are: trial step, calendar month and day of the trial, patient age category, COVID-19 status, and key comorbid conditions (e.g., coronary artery disease, dementia, heart failure, diabetes, lung disease), as well as a composite Charlson score and prior ACP engagement. The above patient covariates were all included in ${\mathit{X}}_{iht}$ while two baseline hospital characteristics were included in ${\mathit{Z}}_h$: hospital region and baseline COVID-19 hospitalization rates. Patient-level intervention and exposure measures are shown by assigned study step in Table 1; see Mohan et al (2023) for a summary of patient and hospital baseline characteristics.

Table 1:

Unadjusted patient-level outcome, context, and intervention exposure variables by assigned intervention cluster. Reported values are the mean (and its associated 95% CI).

	Assigned intervention cluster
	1 N=11,279	2 N=7,368	3 N=7,009	4 N=8,925	5 N=9,654
Outcome and care context
ACP occurrence	0.21 (0.20–0.22)	0.20 (0.19–0.21)	0.21 (0.20–0.22)	0.25 (0.24–0.26)	0.27 (0.26–0.28)
Number of physicians associated with stay	2.71 (2.68–2.73)	2.63 (2.61–2.66)	2.84 (2.81–2.86)	2.48 (2.45–2.50)	2.62 (2.60–2.64)
Time since intervention rollout	81.83 (80.40–83.25)	64.14 (62.59–65.70)	44.05 (42.75–45.34)	31.18 (30.25–32.10)	18.07 (17.42–18.71)
Intervention exposure measures
Number of intervened Physicians associated with stay	0.93 (0.92–0.95)	0.79 (0.77–0.81)	0.67 (0.65–0.69)	0.64 (0.62–0.66)	0.42 (0.40–0.43)
Direct intervention	0.72 (0.71–0.72)	0.64 (0.63–0.65)	0.53 (0.52–0.54)	0.44 (0.43–0.45)	0.31 (0.31–0.32)
External-hospital indirect intervention	1.66 (1.55–1.77)	1.33 (1.23–1.43)	0.74 (0.69–0.78)	0.89 (0.82–0.97)	0.85 (0.79–0.91)
Total indirect intervention	4.45 (4.36–4.54)	3.14 (3.04–3.24)	2.24 (2.16–2.32)	2.89 (2.79–2.98)	0.43 (0.41–0.45)
Contamination	1.66 (1.55–1.77)	1.33 (1.23–1.43)	0.74 (0.69–0.78)	0.89 (0.82–0.97)	0.80 (0.74–0.86)
Sustainment	0.91 (0.89–0.92)	0.71 (0.70–0.73)	0.49 (0.48–0.50)	0.35 (0.34–0.36)	0.20 (0.19–0.21)
Reinforcement	2.36 (2.19–2.53)	2.07 (1.91–2.24)	0.77 (0.72–0.82)	0.84 (0.76–0.91)	0.51 (0.47–0.56)

As expected, the mean of the variables involving the intervention (direct and indirect intervention, contamination, sustainment and reinforcement) linearly decline from the first to the last step (Table 1). Notably, across the duration of the study, the utilization of ACP was much higher among patients who were admitted to hospitals in steps 4 and 5 than for those admitted in other steps while the number of physicians encountered by patients varied less systematically.

5.2. Shared-Patient Network Construction

To construct the physician networks described in Section 2, we use health insurance billing records maintained by the NPO. The dataset comprises patient encounters from multiple hospitals and was filtered to include encounters occurring between January 1, 2019 and June 1, 2021 (predating and overlapping with the trial). As described in Section 2, we sought to directly measure spillover by evaluating a network of physicians based on their involvement in the same patient encounters. This rule is stricter than that commonly used to construct shared-patient networks as it requires that the physicians cared for patients during the same patient encounter as opposed to only having to have cared for the same patient within a specified time-interval. Only the 163 physicians directly involved in the intervention were included in the network. We recapitulate from Section 2 that we perform the analysis both with the undirected baseline network $\mathit{A}$ to guard against reverse causality and with the time-varying undirected network ${\mathit{A}}_t$ evaluated over the prior study step to reflect evolving professional interactions in computing ${\mathit{P}\mathit{h}\mathit{y}\mathit{s}\mathit{P}\mathit{e}\mathit{e}\mathit{r}}_t$ and other network-based predictors involving the intervention. Because $\mathit{A}$ and ${\mathit{A}}_t$ are sparse, we leveraged the Matrix package (Bates et al, 2024) to efficiently store and manipulate them with data manipulation and cleaning carried out using dplyr (Wickham et al, 2019). Networks were plotted using the Graphopt layout algorithm (Trivedi et al, 2020) while data wrangling and descriptive network analysis was performed using the Matrix (Bates et al, 2024) and igraph (Csardi, 2014) packages, respectively. All of these analyses were conducted using R version 4.3.3; a link to an online repository containing our complete code is available in the Availability of data and materials section.

6. Findings from Extended ACP SW-CRT Analyses

The progression of the intervention’s influence across the physician network is summarized in Figure 2. Panel A depicts the baseline network structure, in which each node represents a physician and edges denote shared patients between physicians. Nodes are colored by the hospital’s assigned intervention step, with white nodes indicating physicians not directly assigned to the intervention. Panels B–F illustrate the sequential diffusion of spillover exposure across trial steps. Within each step, grayscale shading represents the degree of indirect exposure experienced by physicians who were not directly intervened, capturing the accumulation of spillover effects through professional ties to intervened peers. The visualization demonstrates how exposure extended beyond directly treated physicians, with increasing portions of the network indirectly exposed as rollout progressed.

Fig. 2:

Panel A displays the baseline physician collaboration network across the entire network, where nodes correspond to individual physicians. Edges represent shared patients between physicians. Nodes are colored by hospital intervention step; white nodes indicate physicians not directly assigned to the intervention (their presence here helps allow spillover to be illustrated). Panels B–F depict the progression of spillover exposure from external hospitals across steps 1–5 where node shading in grayscale reflects the degree of indirect exposure to the intervention through network peers (scaled 0–1).

Temporal patterns of exposure are summarized in Figures 3 and 4. The overall exposure trajectories (Figure 3) show a monotonic increase in direct interventions over time, consistent with the SW-CRT design, accompanied by a lagged but pronounced rise in indirect exposure across hospitals. When stratified by randomization step (Figure 4), direct exposure follows the expected staggered adoption pattern, while indirect exposure exhibits substantial variability across hospitals. Notably, hospitals randomized to earlier steps show higher levels of between-hospital spillover, reflecting denser inter-hospital connectivity and greater potential for cross-hospital diffusion. Collectively, these figures illustrate how both direct and indirect pathways contributed to the cumulative reach of the intervention within and across institutional boundaries.

Fig. 3:

Diffusion across the study of direct and indirect patient exposure to the intervention through their physicians and their physicians peers, respectively

Fig. 4:

Diffusion across the study of the direct and indirect patient exposure to the intervention through their physicians and their physicians peers, respectively, by randomization group

6.1. Adjusting for Physician Spillover

As described in Section 4.2, we used generalized linear mixed models (GLMM) with a logit link function and a random effect for hospital to model the binary event that ACP was billed. We report the key ITT intervention estimates from reduced and full versions of the model in Equation (5) and for the extended model in Equation (6) when the network is fixed at its baseline value (Table 2) and when the network is evaluated at the prior time-period (Table 3). For each model, the estimated values and associated inferences for the counterfactual causal estimands (ATE, AIE, ACE, ASE and ADE) are presented along-side the most relevant regression parameters. The analogous results for the as-treated (AT) versions of the four model specifications based on (5) and (6) when the physician network is fixed at baseline and when it is evaluated at the prior time-period are shown in Tables 6 and 7 of the Supplementary Material, respectively. The parameter estimates of all the predictors in the extended model in (6) under these specifications of the physician network for the ITT and AT analyses are in Tables 8 and 9 of the Supplementary Material, respectively.

Table 2:

Estimated ITT intervention and physician spillover effects on ACP billing under a mixed-effect logistic regression model with a time-invariant baseline physician network and adjusting for patient and baseline hospital characteristics

Regression Model Parameters		Counterfactual Estimands
Model/Variable	OR (95% CI)	p-value	Term	Est (95% CI)
Traditional SW-CRT
Direct Intervention	0.935 (0.852, 1.027)	0.161	$ADE(0)$	−0.011 (−0.026, 0.004)
Network-Adjusted
Direct Intervention	0.915 (0.833, 1.004)	0.062	$ADE(0)$	−0.014 (−0.029, 0.001)
Physician Spillover	1.039 (1.032, 1.047)	< 0.001	$AIE(0,w)$	0.006 (0.005, 0.008)
Network-Adjusted with Interaction
Direct Intervention	0.925 (0.842, 1.016)	0.102	$ADE(0)$	−0.013 (−0.028, 0.003)
Physician Spillover	1.176 (1.064, 1.300)	0.002	$AIE(0,w)$	0.027 (0.010, 0.045)
Intervention × Spillover	0.883 (0.799, 0.976)	0.015	$ACE(w)$	−0.021 (−0.039, −0.004)
Network-adjusted sustainment-retention
Direct Intervention	0.950 (0.863, 1.045)	0.291	$ADE(0)$	−0.008 (−0.023, 0.007)
Physician Spillover	1.182 (1.069, 1.306)	0.001	$AIE(0,w)$	0.027 (0.010, 0.044)
Intervention × Spillover	0.863 (0.781, 0.955)	0.004	$ACE(w)$	−0.024 (−0.041, −0.007)
Intervention × FUTime	1.162 (1.045, 1.292)	0.006	$ASE(w,u)$	0.024 (0.007, 0.040)
Intervention × Spillover × FUTime	1.014 (1.005, 1.024)	0.004	$ARE(w,u)$	0.003 (0.001, 0.004)

Note: A unit-change of the physician spillover variable $({\mathit{\text{PatPeer}}}_{iht})$ corresponds to the equivalent of a $w=0.1$ increase in the proportion of intervened peer physicians at that moment (a shift approximately equal to the average value of ${\mathit{\text{PatPeer}}}_{iht}$ in the dataset), with ${\mathit{\text{PatPeer}}}_{iht}$ summing over the physicians a patient encountered during their hospital stay, while ${\mathit{\text{FUTime}}}_{ijt}$ is scaled to be in terms of quarters. ADE, AIE, ACE, ASE and ARE denote average (direct, indirect, contamination, sustainment and reinforcement) effect. We set $u=1$ (one quarter of a year) following intervention rollout when computing $ASE(w,u)$ and $ARE(w,u)$. Because the approximate value of the p-values for the counterfactual estimands are almost identical to those for the corresponding odds-ratios, they are excluded.

Table 3:

Estimated ITT intervention and physician spillover effects on ACP billing under a mixed-effect logistic regression model with a time-varying physician network evaluated in the prior step and adjusting for patient and baseline hospital characteristics

Regression Model Parameters		Counterfactual Estimands
Model/Variable	OR (95% CI)	p-value	Term	Est (95% CI)
Traditional SW-CRT
Direct Intervention	0.935 (0.852, 1.027)	0.161	$ADE(0)$	−0.011 (−0.026, 0.004)
Network-Adjusted
Direct Intervention	0.910 (0.828, 0.999)	0.048	$ADE(0)$	−0.015 (−0.030, 0.000)
Physician Spillover	1.047 (1.038, 1.055)	< 0.001	$AIE(0,w)$	0.008 (0.006, 0.009)
Network-Adjusted with Interaction
Direct Intervention	0.919 (0.836, 1.011)	0.082	$ADE(0)$	−0.013 (−0.029, 0.002)
Physician Spillover	1.108 (1.015, 1.209)	0.021	$AIE(0,w)$	0.017 (0.002, 0.032)
Intervention × Spillover	0.944 (0.865, 1.031)	0.200	$ACE(w)$	−0.010 (−0.025, 0.005)
Network-adjusted sustainment-retention
Direct Intervention	0.948 (0.861, 1.044)	0.277	$ADE\left(0\right)$	−0.008 (−0.023, 0.007)
Physician Spillover	1.114 (1.021, 1.216)	0.016	$AIE(0,w)$	0.017 (0.003, 0.003)
Intervention × Spillover	0.922 (0.844, 1.007)	0.070	$ACE(w)$	−0.013 (−0.027, 0.001)
Intervention × FUTime	1.155 (1.038, 1.284)	0.008	$ASE(w,u)$	0.023 (0.006, 0.039)
Intervention × Spillover × FUTime	1.015 (1.005, 1.025)	0.002	$ARE(w,u)$	0.003 (0.001, 0.004)

Note: A unit-change of the physician spillover variable $({\mathit{\text{PatPeer}}}_{iht})$ corresponds to the equivalent of a $w=0.1$ increase in the proportion of intervened peer physicians (a shift approximately equal to the average value of ${\mathit{\text{PatPeer}}}_{iht}$ in the dataset), with ${\mathit{\text{PatPeer}}}_{iht}$ summing over the physicians a patient encountered during their hospital stay, while ${\mathit{\text{FUTime}}}_{ijt}$ is scaled to be in terms of quarters. ADE, AIE, ACE, ASE and ARE denote average (direct, indirect, contamination, sustainment and reinforcement) effect. We set $u=1$ (one quarter of a year) following intervention rollout when computing $ASE(w,u)$ and $ARE(w,u)$. Because the approximate value of the p-values for the counterfactual estimands are almost identical to those for the corresponding odds-ratios, they are excluded.

When the network was fixed at its value at the end of the baseline period, the ITT direct effect of the intervention showed no significant effect on ACP billing ($\text{OR}=0.935,p=0.161$), emulating the findings in (Mohan et al, 2023, 2021). After adding the spillover term the estimated direct effect decreased to OR = 0.915 (95% CI (0.833, 1.004)) whereas the physician spillover effect was positive and highly statistically significant (OR = 1.039, 95% CI (1.032, 1.047)). The corresponding estimates of $ADE(w=0)$ (Est = −0.014, 95% CI (−0.029,−0.001)) and $AIE(z=0,w=0.1)$ (Est = 0.006, 95% CI (0.005, 0.008)), which are probability-differences, recapitulate the finding observed for the OR estimates that accounting for spillover offsets the direct effect. Together these results suggest that the direct and the indirect effects of the intervention capture different mechanisms with the direct effect suggesting that the experience of playing the game was unhelpful towards increasing ACP conversations with patients whereas having a higher proportion of peers play the game might have generated more ACP discussions among physicians and their external peers. This finding also suggests that the diffusion of intervention behaviors through the network may have outpaced the formal delivery of the intervention itself – a pattern consistent with theories of social reinforcement and contagion.

When the contamination term was also added to the statistical model, the indirect intervention (OR = 1.176, p < 0.001) and interaction $(\text{OR}=0.883,p=0.015)$ effects were both highly significant but in opposite directions. Because of the inclusion of the interaction, the main effect of the intervention (OR = 0.925, 95% CI (0.842, 1.016)) may be directly interpreted as the counterfactual effect in a world absent contamination and the spillover effect as the effect of external peer physicians when the status of the direct intervention is 0. While both effects are estimated under the statistical model in (5) by exploiting simultaneous variation in the predictors across the observations, the former corresponds to the impact of the intervention on the likelihood of ACP prior to a physician being directly intervened on (the period of time when the direct intervention equals 0) while the latter estimates the effect of intervention spillover on the non-intervened. The negative interaction reveals that the spillover effect modified (hence, contaminated) the direct ITT effect with higher levels of spillover occurring in unison with more negative direct effects. The corresponding estimates for the ADE, AIE and ACE reinforce these findings by revealing that $ADE(0)$ is only marginally significant (Est = −0.013, 95% CI (−0.028, 0.003)) whereas the impact of a $w=0.1$ increase in the proportion of intervened peer-physicians (a modest increase) translates to a highly significant $AIE(0,0.1)$ (Est 0.027, 95% CI (0.010, 0.045)) and a significant $ACE(w=0.1)$ (Est −0.021, 95% CI (−0.039, −0.004)). These effects suggest that patients may have benefited from a sizeable indirect effect throughout the study and especially so if admitted to a hospital during that hospital’s pre-intervention period.

When the sustainment and reinforcement effects were incorporated via the model in (6), the direct effect further increased towards its null value of OR = 1 while the spillover and contamination effects remained positive and negative, respectively, along with being statistically significant. Notably, both the sustainment (OR = 1.162, 95% CI (1.041, 1.295)) and reinforcement (OR = 1.014, 95% CI (1.005, 1.024)) effects were statistically significant ($p=0.006$ and $p=0.004$, respectively). These results are consistent with the intervention having a learning effect that benefited from having more intervened external peer physicians by the time of the patient’s admission, offsetting the initial negative effect of the interaction. In other words, physicians in the most densely connected groups of intervened peers at a given time were in the most advantageous position to becoming the highest users of ACP (insofar as billing reflects use of ACP). Because $ASE(w=0.1,u=1)$ and in particular $ARE(w=0.1,u=1)$ are highly statistically significant, the fourth model in the sequence presented in Table 2 is the models whose results should be used for drawing conclusions. This is also the case for the results reported in Table 3.

Allowing the network adjacency matrix $\mathit{A}$ to update at the end of each trial step (i.e., using $\left.{\mathit{A}}_t\right)$ in constructing $\mathit{P}\mathit{h}\mathit{y}\mathit{s}\mathit{P}\mathit{e}\mathit{e}{\mathit{r}}_t$ and thus $\mathit{P}\mathit{a}\mathit{t}\mathit{P}\mathit{e}\mathit{e}{\mathit{r}}_t$, led to an estimated spillover coefficient that was highly positively associated with ACP billing (OR = 1.047, $p<0.001$) and an estimated direct effect that was more negative (OR = 0.91, p=0.048) than in the standard SW-CRT analysis (OR = 0.935). Adding the Intervention × Spillover term to the model, the estimated direct effect moved towards 1 (ADE estimated as −0.013), losing its statistical significance, whereas the estimated effect of spillover increased but due to colinearity with the interaction term was less significant (OR = 1.108, $p=0.021;$ AIE = 0.017, 95% CI (0.002, 0.032)). The interaction was negative but non-significant (OR = 0.944, 95% CI (0.865, 1.031)); ACE = −0.010, 95% CI (−0.025, 0.005)). Although the interaction term and thus the ACE were non-significant, the pattern of the results emulates those for the earlier case when the network is fixed at baseline.

Extending the model to assess whether post-intervention changes in spillover exposure support intervention reinforcement, both sustainment (OR = 1.155, $p=0.008$) and reinforcement (OR = 1.015, $p=0.002$) had significant positive effects with the other terms involving the effect of the intervention not substantially deviating from what they were under the time-invariant $\mathit{A}$. The corresponding estimate of $ASE(\mathrm{0.1,1})$ of 0.023 (95% CI (0.006, 0.039)) reveals that the magnitude of the effect on the probability scale is quite large. Due to the modest level of spillover used for $w$ and length of time for $u$, the estimate of $ARE(\mathrm{0.1,1})$ of 0.003 is close to 0 despite being highly statistically significant. However, under a $w=0.5$ counterfactual increase in spillover, the estimate of $ARE(\mathrm{0.5,1})$ is 5 times greater (Est = 0.014, 95% CI (0.007, 0.021)). These results suggest that individual and group learning might have led the physicians in the most densely connected groups of intervened peers to become the highest at billing for (and presumably users of) ACP.

6.2. Results of hospital random-effects robustness check

The results of the robustness check to the specification of the hospital effects as random found that there was almost no difference between the estimated regression coefficients when the hospital effects were specified as random versus fixed (see Section 4 and Tables 10 and 11 of the Supplementary Material for the fitted values and a discussion of the findings). The minimal differences can be explained by the fact that the level of exposure across physicians at the same hospital to intervened peers at other hospitals varies extensively across the physicians. Therefore, ${\mathit{\text{PatInt}}}_{ht}$ and ${\mathit{\text{PatPeer}}}_{iht}$ act as within-hospital predictors, making them largely resilient to how the blocking variable (hospital) is modeled.

7. Discussion

We developed a causal model for direct and indirect effects and interactions involving them in a SW-CRT in which a network is measured on the study units. The causal model informed statistical analyses that can be performed to test for the existence of spillover effects and most critically of contamination, which we represented as the modification of the direct effect of the intervention by the indirect (spillover) effect, which if not accounted for risks obtaining misleading results from SW-CRTs. We also allowed the direct and indirect effect of the intervention to change over follow-up by defining sustainment as the phenomenon that the direct effect of the intervention is enhanced or maintained, and similarly of its interaction with the indirect effect, defining reinforcement as the phenomenon that exposure to physicians with a higher proportion of intervened peers helps to sustain the direct effect of the intervention over follow-up. Five intervention effects were defined: the direct effect in a counterfactual world absent spillover, the indirect or spillover effect, the interaction of the direct and indirect effects termed contamination, sustainment and reinforcement. Spillover, contamination and reinforcement are network dependent. We also extended the traditional SW-CRT model to allow the effect of the intervention to be represented by a count variable reflecting the extent to which a patient received the intervention leading to an as-treated (AT) formulation of the model; due to the non-causal nature of the AT, results for it are consigned to Section 3 of the Supplementary Material.

We applied our methodology to a case study assessing the effect of an intervention aimed at increasing Advance Care Planning (ACP). We restricted exposure to peer physicians to those at external hospitals so that the targeted direct effect of the intervention corresponded to the intervention effect in a standard SW-CRT. We found that the spillover and direct-indirect intervention interaction effect (contamination) dominated the results, suggesting that the intervention primarily impacted physicians through a sociability effect (e.g., from the chatter it created among them). However, when we allowed the direct intervention effect and the contamination effect to be modified over follow-up, there was evidence of both individual and group (reinforcement) learning effects.

Our primary goal has been to bring attention to the network relevant causal assumptions inherent in traditional SW-CRT frameworks (the lack of interference known as the Stable Unit Treatment Value Assumption) and to introduce social network analysis methodology that can account for the presence of contamination from collaborative physician networks (Angrist et al, 1996). Instead of assuming the absence of inter-hospital spillover effects, investigators conducting trials like ours involving intervention recipients like physicians who may influence one another should consider measuring a network with the recipients as nodes and whose edges reflect the dyadic strength of the relationships between them. If the measurement of such a network is feasible, a study-design that accounts for spillover embodies numerous benefits; for example, bias in the estimated direct intervention effect could be averted, the spillover and contamination effects may be estimated as additional outputs, further refinements may be used to allow direct and indirect effects to vary over follow-up in the form of sustainment and reinforcement. Furthermore, if the network can be measured in advance, designs that purposefully select physicians or hospitals in strategically important positions of the network (e.g., those that will lead to the most efficient diffusion of the intervention through the network) in order to maximize the impact of the intervention and the statistical precision of the estimation of the intervention effects could be used.

This initial contribution can be extended in multiple ways. One is to allow direct intervention and spillover effect heterogeneity with respect to one or more observed predictors modeled parametrically, semi-parametrically or non-parametrically. The extension to non-parametric modeling of effect-heterogeneity would allow estimation of when the intervention and spillover effects are strongest. As noted in Section 4.3, another non-parametric extension is to flexibly model the physicians’ full exposure history by including multiple PatPeer lagged predictors or otherwise. Future research should also aim to explore the consequences of misspecification of such models.

While time-invariant networks offer a cleaner causal interpretation consistent with trial design, time-varying networks reflect an “as-treated” reality - introducing causal ambiguity but enabling detection of dynamic spillover processes, including potential reverse causality. Another topic for future research is to explore methods to account for reverse causality by simultaneously modeling the network over time and accounting for the impact of the network on patient outcomes in a SW-CRT. One approach to testing for reverse causality is to use a longitudinal model for the network with lagged intervention statuses of the physicians as attributes (e.g., as node covariates and nodematch or homophily terms) to test if the intervention is associated with subsequent change in the physician network (O’Malley, 2013). For example, the network at step $t$ may be regressed on ${\mathit{\text{PhysInt}}}_{t-1}$ (and functions thereof) for $t=2,\dots ,T$ using Temporal Exponential Random Graph Models (Leifeld et al, 2018), adjusting for other network statistics and configurations involving actor attributes. The joint network and attribute evolution models in SIENA could also be used to test for reverse causality (e.g., Steglich et al (2010)). Furthermore, Granger causality type tests could be adapted to network data to form more general tests involving multiple lags of the network and intervention status.

The impact of network sparsity on the accuracy of traditional standard error and statistical inference calculations (e.g., confidence intervals and p-values) together with the development of methods that account for statistical dependence due to physician connections between hospitals (and generally the clusters in a SW-CRT) are important topics for future research. Such research could develop a specialized GEE procedure (e.g., as in VanderWeele et al (2012)) that relaxes the assumption of statistical independence between clusters made by the “sandwich estimator,” the standard robust GEE variance estimator, to account for network-induced statistical dependence between hospitals. The resulting procedure would be robust to misspecification of the covariance structure of the observations within and between the clusters (i.e., to the network structure). Alternatively, the correlation structure could be directly accounted for by jointly modeling the network (see preceding paragraph) and the patient outcomes using the extended SW-CRT models developed herein. To account for the complication of patients who encounter multiple physicians during their hospital stay, physician random-effects with multiple membership structures (Zhao et al, 2025) may be added to the SW-CRT models with spillover effects developed herein.

We focused on regression modeling approaches, in part because we wanted to remain closely connected to standard SW-CRT analyses. However, in analyses of observational studies, there has been a surge of interest in matching or weighting based propensity score methods. An advantage of propensity score methods is that they avoid relying on correct specification of the outcome model for statistical inferences to be valid (Li et al, 2018). While propensity score methods for the ADE are well developed, those for the AIE and the other causal estimands derived in this paper have received much less attention; although there are exceptions (Aronow and Samii, 2017). Hence, another important topic for future research is to derive propensity score based counterparts for each of the counterfactual causal estimands.

A limitation of our motivating example is that the use of billing data to infer social connections among physicians is relatively unproven. Prior studies suggest that patient-sharing networks derived from claims data reasonably approximate real-world professional relationships, yet these networks remain an imperfect representation of the underlying social structure (Barnett et al, 2011; O’Malley et al, 2025). Without gold-standard validation data, such as direct surveys or observational studies of physician interactions, we cannot rule out the possibility of missing or misclassified connections. Physicians may engage in collaborative decision-making through informal or unrecorded interactions that are not captured in claims data, potentially leading to an underestimation of spillover effects. However, even if the sensitivity and specificity of the network may be suboptimal, imperfections should bias the estimates of the network-based effects towards null findings. The fact that we obtained statistically significant network effects both when the network was fixed at baseline and when it was evaluated in the prior step suggests that the existence of spillover and contamination is real.

Our findings underscore the importance of accounting for spillover effects in SW-CRTs, particularly in settings where intervention recipients are embedded in localized dense professional networks. The substantial influence of physician collaboration on Advance Care Planning (ACP) billing highlights how interventions can extend beyond their intended targets to intervene via diffusion of the intervention to untreated physicians and hospitals, leading to potential underestimation or misattribution of effects if not properly accounted for using statistical modeling. Further, the estimates of sustainment and reinforcement illustrate the added insights able to be obtained from allowing for dynamic effects of the intervention over follow-up. Given the increasing reliance on SW-CRTs to evaluate healthcare interventions, we encourage researchers to assess the feasibility of measuring connections among intervention recipients and integrating spillover controls into analytical plans. Where possible, pre-trial network assessments should be conducted to estimate the potential for contamination, allowing for the use of appropriate study designs and development of statistical models that explicitly adjust for these effects. By incorporating spillover-aware methodologies, trial designs can more accurately estimate both direct and indirect intervention effects, leading to stronger, more generalizable conclusions in clinical and health services research. We hope that future work will apply and refine these methods in trials involving other healthcare interventions, particularly those where provider collaboration is likely to influence treatment diffusion.

Abbreviations

ACP

Advance care planning

ADE

Average Direct Effect

AIE

Average Indirect Effect

ACE

Average Contamination Effect

ASE

Average Sustainment Effect

ARE

Average Reinforcement Effect

ITT

Intention-to-treat

AT

As-treated

CI

Confidence interval

DID

Difference-in-difference

GLMM

Generalized linear mixed model

OR

Odds ratio

NPI

National provider identifier

SW-CRT

Stepped wedge cluster-randomized trial

LRT

Likelihood-ratio test

Appendix A. Mapping of ADE and AIE to linear model coefficients

The linear potential outcomes model in Equation (2) of Section 3.2.1 can be expressed as

$$Y_{it}\left({\mathit{\text{PhysInt}}}_{k(i)t},{\mathit{\text{PhysInt}}}_{-k(i)t}\right)={\beta }_{0t}+{\beta }_{\text{dir}}{\mathit{\text{PhysInt}}}_{k(i)t}+{\beta }_{\text{indir}}{\mathit{\text{PhysPeer}}}_{k(i)t}+{ϵ}_{it},$$

where ${\mathit{\text{PhysPeer}}}_{k(i)t}=\sum _{j:k(j)\ne k(i)}{A}_{k(i)k(j)t}{\mathit{\text{PhysInt}}}_{k(i)t}/\sum _{k(j)\ne k(i)}{A}_{k(i)k(j)t}$. Therefore, substituting the right-hand-side of the linear causal model for $Y_{it}\left({\mathit{\text{PhysInt}}}_{k(i)t}\right.=z,\left.{\mathit{\text{PhysInt}}}_{-k(i)t}\right)$ for $z\in \{0,1\}$ into the expressions for the ADE and the AIE in Section 3.2, it follows that

$$\begin{gathered} ADE=N^{-1}\sum _{t=1}^T\sum _{i=1}^{n_t}\left(Y_{it}\left({\mathit{\text{PhysInt}}}_{k(i)t}=1,{\mathit{\text{PhysInt}}}_{-k(i)t}\right)-Y_{it}\left({\mathit{\text{PhysInt}}}_{k(i)t}=0,{\mathit{\text{PhysInt}}}_{-k(i)t}\right)\right) \\ =N^{-1}\sum _{t=1}^T\sum _{i=1}^{n_t}{\beta }_{\text{dir}} \\ ={\beta }_{\text{dir}}. \end{gathered}$$

Let $d_{klt}$ be the number of patients who encountered both physicians $k$ and $l$ and ${\bar{d}}_{lt}$ be the average number of patients physician $l$ co-treated with other physicians in step $t$. Then

$$\begin{gathered} AIE=(N\bar{d}{)}^{-1}\sum _{t=1}^T\sum _{i=1}^{n_t}\sum _{j:k(j)\ne k(i)}^{n_t}\left(Y_{jt}\left({\mathit{\text{PhysInt}}}_{k(i)t}=1,{\mathit{\text{PhysInt}}}_{-k(i)t}\right)\left.-Y_{jt}\left(\mathit{\text{PhysIn}}{t}_{k(i)t}=0,{\mathit{\text{PhysInt}}}_{-k(i)t}\right)\right)\right. \\ =(N\bar{d}{)}^{-1}\sum _{t=1}^T\sum _{i=1}^{n_t}\sum _{j:k(j)\ne k(i)}^{n_t}{\beta }_{\text{indir}}\frac{A_{k(j)k(i)t}}{\sum _{k(i)\ne k(j)}{A}_{k(j)k(i)t}} \\ ={\beta }_{\text{indir}}(N\bar{d}{)}^{-1}\sum _{t=1}^T\sum _{k=1}\sum _{l\ne k}\frac{d_{lkt}{A}_{lkt}}{\sum _{k\ne l}{A}_{lkt}} \\ ={\beta }_{\text{indir}}(N\bar{d}{)}^{-1}\sum _{t=1}^T\sum _{l=1}\sum _{k\ne l}\frac{d_{lkt}{A}_{lkt}}{\sum _{k\ne l}{A}_{lkt}}\text{as}{A}_{klt}=A_{lkt} \\ ={\beta }_{\text{indir}}(N\bar{d}{)}^{-1}\sum _{t=1}^T\sum _{l=1}{\bar{d}}_{lt} \\ ={\beta }_{\text{indir}}(N\bar{d}{)}^{-1}N\bar{d} \\ ={\beta }_{\text{indir}} \end{gathered}$$

where $\bar{d}=N^{-1}\sum _{t=1}^T\sum _{l=1}{\bar{d}}_{lt}$ is the average number of patients a physician co-treated with other physicians averaged across physicians and steps.

Therefore, under (2) the ADE and AIE correspond to ${\beta }_{\mathit{\text{dir}}}$ and ${\beta }_{\mathit{\text{indir}}}$, respectively, avoiding needing to specify each element of ${\mathit{P}\mathit{h}\mathit{y}\mathit{s}\mathit{I}\mathit{n}\mathit{t}}_{-k(i)t}$ to quantify the extent to which physician $i$’s peers have received the intervention by time $t$.

Appendix B. Estimators of Causal Estimands on the scale of the outcome

In Section 4.5, we mapped an estimator of ${\beta }_1$, denoted ${\hat{\beta }}_1$, in (3) to the scale of the outcome to define an estimator of the $ADE(0)$, firstly under the assumption that ${\theta }_h$ is a fixed-effect and then assuming ${\theta }_h~\mathit{\text{Normal}}\left(0,{\tau }^2\right)$. Continuing with the nomenclature of denoting the counterfactual state of the direct intervention and level of the indirect intervention by $z$ and $w$, respectively, we first derive estimators of $ADE(w)$, $AIE(0,w)$ and $ACE(w)$ with respect to the model in (3). We then derive the estimators of the $ASE(w,u)$ and $ARE(w,u)$, where $u$ is a counterfactual value of ${\mathit{\text{FuTime}}}_{iht}$, the time since intervention receipt, with respect to the model in (4). Because the same adaption of the estimator from the ${\theta }_h$ fixed-effect to the ${\theta }_h$ random-effect case applies in each case, in the following we only present the simpler fixed-effect case.

B.1 Point estimation

To make the dependence of the fitted model on the estimated model parameters explicit and emphasize that the fitted values are probabilities, let $\left(z,w,{\mathit{X}}_{iht},{\mathit{Z}}_h,\stackrel{\mathit{ˆ}}{\mathit{\beta }},{\hat{\theta }}_h\right)=g^{-1}\left({\mathit{\text{PatInt}}}_{ht}\right.=z$, ${\mathit{\text{PatPeer}}}_{iht}\left.=w,{\mathit{X}}_{iht},{\mathit{Z}}_h\right)$, where ${\mathit{\text{PatInt}}}_{ht}=z$ and ${\mathit{\text{PatPeer}}}_{iht}=w$ are counterfactuals. Then the estimator of $ADE(w)$, the ADE when the level of indirect intervention or spillover is $w$, is:

$$\widehat{ADE}(w)=N^{-1}\sum _{iht}\left(\pi \left(1,w,{\mathit{X}}_{iht},{\mathit{Z}}_h,\stackrel{\mathit{ˆ}}{\mathit{\beta }},{\hat{\theta }}_h\right)-\pi \left(0,w,{\mathit{X}}_{iht},{\mathit{Z}}_h,\stackrel{\mathit{ˆ}}{\mathit{\beta }},{\hat{\theta }}_h\right)\right)$$ (B1)

where $\pi \left(z,w,{\mathit{X}}_{iht},{\mathit{Z}}_h,\stackrel{\mathit{ˆ}}{\mathit{\beta }},{\hat{\theta }}_h\right)$ is a fitted probability having the inverse-logit form:

$${\pi }^{-1}\left(z,w,{\mathit{X}}_{iht},{\mathit{Z}}_h,\stackrel{\mathit{ˆ}}{\mathit{\beta }},{\hat{\theta }}_h\right)={\hat{\beta }}_{0t}+{\hat{\beta }}_{1}z+{\hat{\beta }}_{2}w+{\hat{\beta }}_{3}zw+{\stackrel{\mathit{ˆ}}{\mathit{\beta }}}_4^T{\mathit{X}}_{iht}+{\stackrel{\mathit{ˆ}}{\mathit{\beta }}}_5^T{\mathit{Z}}_h+{\hat{\theta }}_h,$$

$\mathit{X}$ is the matrix of observed values of the covariates having ${\mathit{X}}_{\text{iht}}^T$ as its $iht$’th row and $\stackrel{\mathit{ˆ}}{\mathit{\beta }}={\left({\hat{\beta }}_{0t},{\hat{\beta }}_1,{\hat{\beta }}_2,{\hat{\beta }}_3,{\stackrel{\mathit{ˆ}}{\mathit{\beta }}}_4^T,{\stackrel{\mathit{ˆ}}{\mathit{\beta }}}_5^T\right)}^T$ is the vector of the estimated model parameters. Similarly, the estimator of the AIE when the direct intervention status is $z$, $AIE(z,w)$, is:

$$\widehat{AIE}(z,w)=N^{-1}\sum _{iht}\left(\pi \left(z,w,{\mathit{X}}_{iht},{\mathit{Z}}_h,\stackrel{\mathit{ˆ}}{\mathit{\beta }},{\hat{\theta }}_h\right)-\pi \left(z,0,{\mathit{X}}_{iht},{\mathit{Z}}_h,\stackrel{\mathit{ˆ}}{\mathit{\beta }},{\hat{\theta }}_h\right)\right).$$ (B2)

One may substitute 0 for $z$ to obtain the special case $\widehat{AIE}(0,w)$ (the indirect effect prior to intervention known as spillover).

The ACE is evaluated as a contrast of the ADE at two different levels of spillover. We are particularly interested in the values $w$ and 0 (the absence of spillover), in which case $\widehat{ACE}(w)=\widehat{ADE}(w)-\widehat{ADE}(0)$.

The two remaining quantities, $ASE(w,u)$ and $ARE(w,u)$, are only defined under the causal model in Equation (4). These depend on the generalized ADE described in Section 3.5, $ADE(w,u)$, where ${\mathit{\text{FuTime}}}_{iht}=u$ and spillover-level $w$ are counterfactual variables. After evaluating $\widehat{ADE}(w,u)$ by emulating the calculation in (B1) but with $u$ added as the counterfactual value for ${\mathit{\text{FuTime}}}_{iht}$ and using the model in (4), it follows that $\widehat{ASE}(w,u)=\widehat{ADE}(w,u)-\widehat{ADE}(w,0)$ and $\widehat{ARE}(w,u)=\widehat{ASE}(w,u)-\widehat{ASE}(0,u)$ may then in turn be evaluated.

B.2 Standard error calculation and statistical inferences

The above estimators allow evaluation of population average counterparts of the individual causal effects specified in Sections 3.3 and 3.4. In Section 2.1 of the Supplementary Material we derive the variance of $\widehat{ADE}(w),\widehat{AIE}(z,w),\widehat{ACE},\widehat{ASE}$ and $\widehat{ARE}$ using the Delta Method (Oehlert, 1992). The square root of the computed variance yields the standard errors of a given estimator, which is then used to form asymptotic normal 95% confidence intervals and p-values. All calculations are implemented in an R script available at https://github.com/CarlyBobak/ANSWERS.

Availability of data and materials

Access to a synthetic dataset that emulates the confidential actual dataset will be made available upon written request to the first author. Code for network construction, spillover variable generation, statistical modeling and counterfactual causal estimand evaluation is available at https://github.com/CarlyBobak/ANSWERS.