Spillover-Adjusted NNT in Interference Settings

• Spillover-adjusted NNT is a causal efficacy measure that extends the classical reciprocal risk difference to settings with interference and spillover across units.
• It leverages exposure mappings and dynamic epidemic or mediation models to derive direct, indirect, and overall NNT measures under various interference structures.
• Applications include infectious disease transmission, network trials, and mediation analyses, highlighting diverse estimands and methodological challenges.

Searching arXiv for the cited papers and closely related interference/NNT work. Spillover-adjusted Number Needed to Treat (NNT) denotes an NNT-type efficacy measure defined in settings where treatment assigned to one unit can alter outcomes in other units, groups, or exposure states through interference, spillovers, or mediated pathways. In the simplest no-interference formulation, NNT is the reciprocal of an absolute risk reduction. Under interference, that reciprocal must instead be tied to a precisely specified causal contrast: a direct effect at fixed peer exposure, an indirect or spillover effect on untreated units, a total or overall effect under a saturation regime, a dynamic cross-group incidence reduction in a transmission model, or a conservative lower-bound construct that quantifies how many units were causally affected by others’ treatment without identifying events prevented (Piazzola et al., 11 Aug 2025). The term therefore refers less to a single canonical estimand than to a family of NNT-like measures whose interpretation depends on the interference structure, the exposure mapping, and the target of intervention.

1. Conceptual scope and basic definition

In the standard setting with no interference and binary outcomes, the NNT is the inverse of a causal risk difference. One formulation writes

$$\mathrm{NNT} = \frac{1}{p_1 - p_0} = \frac{1}{p_t},$$

with $p_t = \mathbb E[I_1-I_0]$, and a modified version is

$\mathrm{NNT} \equiv g(p_t) = \begin{cases} 1/p_t, & p_t>0,\[4pt] \infty, & p_t\le 0. \end{cases}$

This structure is retained in spillover settings, but the relevant risk difference is no longer determined solely by own treatment (Vancak et al., 23 Apr 2025).

A spillover-adjusted NNT is therefore an NNT defined on causal risk differences that are themselves defined in a framework allowing interference. One influential formulation makes this explicit through exposure mappings. Units $i=1,\ldots,N$ have treatment assignment vector $z=(z_1,\ldots,z_N)'$, and outcomes are written $y_i(z)=y_i(z_i;z_{-i})$. An exposure mapping $f:\Omega\times\Theta\to\Delta$ compresses the full assignment vector into exposure conditions such as direct plus indirect exposure, isolated direct exposure, indirect exposure, and no exposure. Average causal contrasts then take the form

$\tau(d_k,d_l)=\mu(d_k)-\mu(d_l),$

where $\mu(d_k)=N^{-1}\sum_i y_i(d_k)$. For binary adverse outcomes, an NNT is then defined as the reciprocal of the relevant exposure-specific risk difference (Aronow et al., 2020).

This general principle admits several non-equivalent operationalizations. In network experiments, it yields direct, spillover, total, and overall NNTs defined by contrasts of $Y(z,g)$ or $p_t = \mathbb E[I_1-I_0]$0. In mediation settings, it yields direct and indirect NNTs based on nested potential outcomes. In epidemic ODE models, it yields a time-resolved NNT derived from sensitivity of infections in one group to PrEP uptake in another. In assumption-lean finite-population analysis, it yields only lower-bound NNT-like quantities for “people affected” rather than “events prevented” (Piazzola et al., 11 Aug 2025).

2. Causal formulations under interference

A common framework for interference partitions the population into groups or networks and indexes potential outcomes by own treatment and peer exposure. In an egocentric network-based randomized design, only egos are randomized, alters are untreated, and observed exposure states reduce to $p_t = \mathbb E[I_1-I_0]$1. The principal estimands are the average individual effect

$p_t = \mathbb E[I_1-I_0]$2

the average spillover effect

$p_t = \mathbb E[I_1-I_0]$3

and the overall effect

$p_t = \mathbb E[I_1-I_0]$4

when each egonetwork has one ego and $p_t = \mathbb E[I_1-I_0]$5 alters (Fang et al., 2023).

These risk differences map directly into NNT-type quantities for binary adverse outcomes. The direct NNT is

$p_t = \mathbb E[I_1-I_0]$6

the spillover NNT per alter is

$p_t = \mathbb E[I_1-I_0]$7

and a natural spillover-adjusted NNT per treated ego is

$p_t = \mathbb E[I_1-I_0]$8

A per-person version uses the overall effect: $p_t = \mathbb E[I_1-I_0]$9 These are not alternative labels for the same quantity; they answer different questions about who is treated and where outcomes are counted (Fang et al., 2023).

A closely related but more general partial-interference framework defines group-average direct, indirect, total, and overall effects under treatment saturation regimes $\mathrm{NNT} \equiv g(p_t) = \begin{cases} 1/p_t, & p_t>0,\[4pt] \infty, & p_t\le 0. \end{cases}$0 and $\mathrm{NNT} \equiv g(p_t) = \begin{cases} 1/p_t, & p_t>0,\[4pt] \infty, & p_t\le 0. \end{cases}$1. The direct effect at saturation $\mathrm{NNT} \equiv g(p_t) = \begin{cases} 1/p_t, & p_t>0,\[4pt] \infty, & p_t\le 0. \end{cases}$2 is

$\mathrm{NNT} \equiv g(p_t) = \begin{cases} 1/p_t, & p_t>0,\[4pt] \infty, & p_t\le 0. \end{cases}$3

the indirect effect is

$\mathrm{NNT} \equiv g(p_t) = \begin{cases} 1/p_t, & p_t>0,\[4pt] \infty, & p_t\le 0. \end{cases}$4

the total effect is

$\mathrm{NNT} \equiv g(p_t) = \begin{cases} 1/p_t, & p_t>0,\[4pt] \infty, & p_t\le 0. \end{cases}$5

and the overall effect is

$\mathrm{NNT} \equiv g(p_t) = \begin{cases} 1/p_t, & p_t>0,\[4pt] \infty, & p_t\le 0. \end{cases}$6

For binary outcomes, each supports a corresponding NNT by reciprocal transformation of the relevant risk difference (Aronow et al., 2020).

Within-group randomized experiments can also be parameterized through an exposure mapping $\mathrm{NNT} \equiv g(p_t) = \begin{cases} 1/p_t, & p_t>0,\[4pt] \infty, & p_t\le 0. \end{cases}$7, often exchangeability, under which potential outcomes are indexed by own treatment $\mathrm{NNT} \equiv g(p_t) = \begin{cases} 1/p_t, & p_t>0,\[4pt] \infty, & p_t\le 0. \end{cases}$8 and number of treated peers $\mathrm{NNT} \equiv g(p_t) = \begin{cases} 1/p_t, & p_t>0,\[4pt] \infty, & p_t\le 0. \end{cases}$9. Average direct effects are

$i=1,\ldots,N$0

and spillover effects are

$i=1,\ldots,N$1

This yields direct-effect NNTs at fixed peer exposure and spillover NNTs for changes in peer treatment while holding own treatment fixed (Vazquez-Bare, 2017).

3. Direct and indirect NNT via nested potential outcomes

A distinct formulation treats “indirect” not as between-person interference but as an effect transmitted through a mediator. With $i=1,\ldots,N$2 the exposure, $i=1,\ldots,N$3 a mediator, and $i=1,\ldots,N$4 the outcome, the relevant counterfactuals are $i=1,\ldots,N$5, $i=1,\ldots,N$6, and $i=1,\ldots,N$7. The total effect is

$i=1,\ldots,N$8

the natural direct effect is

$i=1,\ldots,N$9

and the natural indirect effect is

$z=(z_1,\ldots,z_N)'$0

Their marginal versions are $z=(z_1,\ldots,z_N)'$1, $z=(z_1,\ldots,z_N)'$2, and $z=(z_1,\ldots,z_N)'$3, and on the risk-difference scale they satisfy

$z=(z_1,\ldots,z_N)'$4

Applying the transformation $z=(z_1,\ldots,z_N)'$5 produces the total NNT, direct NNT, and indirect NNT, namely NNT, DNNT, and INNT (Vancak et al., 23 Apr 2025).

The corresponding indices are

$z=(z_1,\ldots,z_N)'$6

with group-specific analogues DNNE, DEIN, INNE, IEIN, NNE, and EIN. This framework formalizes an “NNT profile” in which overall benefit per treated is decomposed into non-mediated and mediated components. The data explicitly note that the paper does not study spillover between different individuals, but spillover from treatment to outcome through an intermediate variable; methodologically, this is the standard mediation setting (Vancak et al., 23 Apr 2025).

Identification requires sequential ignorability,

$z=(z_1,\ldots,z_N)'$7

together with consistency and positivity. Under effect homogeneity, the paper derives observable formulas such as

$z=(z_1,\ldots,z_N)'$8

and then estimates the resulting risk differences through mediator and outcome models $z=(z_1,\ldots,z_N)'$9 and $y_i(z)=y_i(z_i;z_{-i})$0, followed by plug-in or M-estimation and a sandwich covariance matrix (Vancak et al., 23 Apr 2025).

This mediated formulation is narrower than general network interference. A plausible implication is that it is best treated as one member of the broader spillover-adjusted NNT family, useful when the indirect pathway is represented through a mediator rather than peer treatment.

4. Dynamic and cross-group spillover-adjusted NNT

A substantially different notion appears in deterministic epidemic modeling. In a three-group HIV model with MSM, heterosexual females, and heterosexual males, PrEP uptake in group $y_i(z)=y_i(z_i;z_{-i})$1 changes infections in group $y_i(z)=y_i(z_i;z_{-i})$2 through nonlinear transmission dynamics. The paper defines incidence as

$y_i(z)=y_i(z_i;z_{-i})$3

coverage among susceptibles as

$y_i(z)=y_i(z_i;z_{-i})$4

and sensitivity of the infected population $y_i(z)=y_i(z_i;z_{-i})$5 to coverage in group $y_i(z)=y_i(z_i;z_{-i})$6 as $y_i(z)=y_i(z_i;z_{-i})$7. Using the chain rule,

$y_i(z)=y_i(z_i;z_{-i})$8

so $y_i(z)=y_i(z_i;z_{-i})$9 has units “change in $f:\Omega\times\Theta\to\Delta$0 per additional person on PrEP in group $f:\Omega\times\Theta\to\Delta$1” (Piazzola et al., 11 Aug 2025).

Starting from

$f:\Omega\times\Theta\to\Delta$2

the paper considers a small increment $f:\Omega\times\Theta\to\Delta$3 in PrEP users over $f:\Omega\times\Theta\to\Delta$4: $f:\Omega\times\Theta\to\Delta$5 A first-order Taylor expansion and an approximation that neglects the $f:\Omega\times\Theta\to\Delta$6 term yield the main expression

$f:\Omega\times\Theta\to\Delta$7

This is spillover-adjusted in two senses stated explicitly in the paper: first, $f:\Omega\times\Theta\to\Delta$8 need not equal $f:\Omega\times\Theta\to\Delta$9, so one can compute the NNT for preventing infections in another group; second, $\tau(d_k,d_l)=\mu(d_k)-\mu(d_l),$0 comes from the full nonlinear, interacting ODE system, so it reflects both direct and indirect epidemic effects over time (Piazzola et al., 11 Aug 2025).

The same formula covers direct and spillover effects simply by choice of $\tau(d_k,d_l)=\mu(d_k)-\mu(d_l),$1. For $\tau(d_k,d_l)=\mu(d_k)-\mu(d_l),$2 it is direct, as in $\tau(d_k,d_l)=\mu(d_k)-\mu(d_l),$3. For $\tau(d_k,d_l)=\mu(d_k)-\mu(d_l),$4 it is spillover, as in $\tau(d_k,d_l)=\mu(d_k)-\mu(d_l),$5. The paper reports that, in the Georgia-calibrated model, the cumulative infections averted in HETF per additional MSM on PrEP by 10 years is more than five times the infections averted in HETF per additional HETF on PrEP, implying a much smaller spillover-adjusted NNT for HETF infections from MSM PrEP than from direct HETF PrEP (Piazzola et al., 11 Aug 2025).

The paper also gives concrete magnitudes. At 10 years, MSM PrEP has NNT $\tau(d_k,d_l)=\mu(d_k)-\mu(d_l),$6 to avert one MSM infection, $\tau(d_k,d_l)=\mu(d_k)-\mu(d_l),$7 to avert one HETF infection via spillover, and $\tau(d_k,d_l)=\mu(d_k)-\mu(d_l),$8 to avert one HETM infection via spillover. HETF-directed PrEP has NNT $\tau(d_k,d_l)=\mu(d_k)-\mu(d_l),$9 for HETF infections, while HETM-directed PrEP has NNT $\mu(d_k)=N^{-1}\sum_i y_i(d_k)$0 for HETM infections and $\mu(d_k)=N^{-1}\sum_i y_i(d_k)$1 for HETF through spillover. In the risk-structured extension, high-risk HETF targeting substantially lowers NNTs for both HETF and HETM relative to low-risk HETF or HETM strategies (Piazzola et al., 11 Aug 2025).

This dynamic formulation differs sharply from static reciprocal-of-risk-difference definitions. It is local, first-order, and time-dependent. The paper states explicitly that the spillover equations and NNT formula are local approximations around baseline coverage levels and that if PrEP coverage becomes large, the linear approximation may degrade (Piazzola et al., 11 Aug 2025).

5. Lower-bound and prevalence-based spillover-adjusted NNT

Another strand does not estimate infections prevented at all. Instead, it measures how many units were causally affected by others’ treatment. In a finite population of $\mu(d_k)=N^{-1}\sum_i y_i(d_k)$2 units with arbitrary interference,

$\mu(d_k)=N^{-1}\sum_i y_i(d_k)$3

the key counterfactual keeps unit $\mu(d_k)=N^{-1}\sum_i y_i(d_k)$4’s own treatment status as observed while setting all others to control: $\mu(d_k)=N^{-1}\sum_i y_i(d_k)$5 The prevalence of indirect effects is then

$\mu(d_k)=N^{-1}\sum_i y_i(d_k)$6

the number of units whose outcomes differ from what they would have been under isolated treatment. The prevalence proportion is $\mu(d_k)=N^{-1}\sum_i y_i(d_k)$7 (Choi, 2023).

The paper emphasizes that $\mu(d_k)=N^{-1}\sum_i y_i(d_k)$8 does not quantify how big the indirect effect is in outcome units; it only records whether the effect is zero versus nonzero for each unit. It is therefore a count of how many units are touched at all by others’ treatment. Under randomization alone, the paper develops conservative point estimation and one-sided interval estimation for $\mu(d_k)=N^{-1}\sum_i y_i(d_k)$9, using a statistic

$Y(z,g)$0

with

$Y(z,g)$1

a variance surrogate

$Y(z,g)$2

and a one-sided lower confidence bound

$Y(z,g)$3

A backup optimization based on the deterministic constraint $Y(z,g)$4 is also introduced to guarantee coverage when the CLT variance condition fails (Choi, 2023).

This setup does not identify a classical “events prevented” NNT. The data explicitly state that under these assumptions the only rigorous NNT-style object is “NNT per affected unit,” not an “NNT per prevented event.” If $Y(z,g)$5 units are treated, a spillover-only analogue is

$Y(z,g)$6

and because only a lower bound is known,

$Y(z,g)$7

Its interpretation is that, in the realized randomized allocation, treating at most $Y(z,g)$8 units sufficed to change the outcome, in some direction, of at least one other unit, with confidence level $Y(z,g)$9 (Choi, 2023).

The Kenya deworming illustration makes this concrete. With $p_t = \mathbb E[I_1-I_0]$00 students, $p_t = \mathbb E[I_1-I_0]$01 when at least half of nearby schools within 6 km were treated, and $p_t = \mathbb E[I_1-I_0]$02 for any parasitic infection in 1999, the conservative point estimate is $p_t = \mathbb E[I_1-I_0]$03, implying about $p_t = \mathbb E[I_1-I_0]$04 of students had outcomes depending on whether other schools were treated. The one-sided lower bounds on $p_t = \mathbb E[I_1-I_0]$05 are $p_t = \mathbb E[I_1-I_0]$06 at 90% and $p_t = \mathbb E[I_1-I_0]$07 at 95%. The notes then give an illustrative, not exact, translation to an NNT-like metric: if $p_t = \mathbb E[I_1-I_0]$08, the point estimate yields $p_t = \mathbb E[I_1-I_0]$09, while the conservative 95% bound gives $p_t = \mathbb E[I_1-I_0]$10 (Choi, 2023).

6. Identification, estimation, and major limitations

Across formulations, spillover-adjusted NNT inherits all identification conditions of the underlying causal effect. In exposure-mapping and network designs, identification depends on the exposure mapping fully characterizing interference and on non-zero exposure probabilities $p_t = \mathbb E[I_1-I_0]$11. Horvitz–Thompson and Hájek estimators then identify exposure-specific mean potential outcomes, with the Hájek estimator having substantially lower variance than Horvitz–Thompson with negligible bias increase in the simulations summarized in the chapter. Because NNT inverts a risk difference, variance inflation near zero is especially consequential (Aronow et al., 2020).

In nonparametric estimation under partial interference, standard regressions can be misleading. Difference in means between treated and controls identifies a combination of direct and spillover effects entering with different signs, while the reduced-form linear-in-means coefficient identifies a weighted sum of spillover effects with some negative weights. The paper proposes nonparametric estimators for average direct and spillover effects and shows they are consistent and asymptotically normal under restrictions tying the number of parameters, total sample size, and assignment probabilities (Vazquez-Bare, 2017).

When cluster randomization is combined with noncompliance, the conventional instrumental-variables ratio does not identify the usual complier average causal effect in the presence of spillovers. Under one-sided noncompliance and stratified interference, the Wald estimand is a mixture of a total effect among compliers and a spillover effect among never-takers. The paper further states that no analysis of CRT data can unbiasedly estimate local network causal effects, and develops bounds under a non-harm assumption. For binary outcomes, those bounds yield interval-identified NNTs rather than point estimates for type-specific direct or spillover components (Kang et al., 2018).

Several limitations recur across the literature.

Limitation	Consequence for NNT
Exposure mapping misspecification	The risk difference being inverted may be biased (Aronow et al., 2020)
Sparse exposure cells or rare assignments	NNT becomes unstable because it is the reciprocal of a noisy risk difference (Vazquez-Bare, 2017)
Strong mediation assumptions	DNNT and INNT require sequential ignorability, consistency, positivity, and effect homogeneity (Vancak et al., 23 Apr 2025)

A further limitation is conceptual. The term “spillover-adjusted NNT” can refer to at least three distinct objects already present in the literature: an inverse risk difference under interference, a dynamic person-years-per-infection-averted quantity from a transmission model, or an upper bound on treated units per person causally affected by spillovers. These are not interchangeable. This suggests that any use of the term should state explicitly the estimand, exposure contrast, target population, time horizon, and whether the denominator counts treated persons, treated peers, person-years on treatment, or treated units in a policy perturbation.

7. Interpretation and applications

The principal application domains in the cited work are infectious disease transmission, network-based public health interventions, and mediated causal processes. In HIV prevention, the dynamic NNT framework is used to compare PrEP allocation strategies across MSM, HETF, HETM, and risk-stratified HETF, showing that the efficient group to treat may differ from the group in which infections are measured because $p_t = \mathbb E[I_1-I_0]$12 (Piazzola et al., 11 Aug 2025). In egocentric network trials, the overall effect

$p_t = \mathbb E[I_1-I_0]$13

provides a direct basis for a spillover-adjusted NNT per treated ego when one treated seed affects both self and alters (Fang et al., 2023). In finite-population randomization inference, prevalence-of-indirect-effects analysis offers an assumption-lean way to quantify whether spillovers are widespread even when no outcome-scale NNT is identified (Choi, 2023).

A common misconception is that spillover-adjusted NNT is simply the classical NNT computed after adding a peer-treatment covariate. The interference literature directly rejects that simplification: reduced-form linear-in-means coefficients generally do not isolate spillover effects, and naive treated-versus-control contrasts mix direct and spillover components with different weights and signs (Vazquez-Bare, 2017). Another misconception is that any IV-based NNT in a cluster trial with noncompliance remains a complier-only measure; with spillovers, the identified estimand is a mixture unless stronger assumptions or different designs are imposed (Kang et al., 2018).

Taken together, the literature presents spillover-adjusted NNT as a rigorously defined but design-dependent extension of classical NNT. Its essential feature is not a new reciprocal transformation, but the replacement of the no-interference risk difference by an interference-aware causal contrast: $p_t = \mathbb E[I_1-I_0]$14, $p_t = \mathbb E[I_1-I_0]$15, $p_t = \mathbb E[I_1-I_0]$16, $p_t = \mathbb E[I_1-I_0]$17, $p_t = \mathbb E[I_1-I_0]$18, $p_t = \mathbb E[I_1-I_0]$19, or $p_t = \mathbb E[I_1-I_0]$20, depending on the scientific problem (Aronow et al., 2020).