Secondary Total Treatment Effect (STTE)

• STTE is a family of causal estimands that quantify indirect or spillover treatment effects among secondary or non-primary units in structured experiments.
• It is estimated using tailored methods such as IPW, TMLE, and ensemble machine learning to address challenges in mediation, interference, and sequential designs.
• STTE provides actionable insights for evaluating spillover effects and informing policy decisions in fields like public health, adaptive trials, and network interventions.

The Secondary Total Treatment Effect (STTE) is a family of causal estimands designed to quantify the indirect or subgroup-specific total effects of an intervention, often in the presence of mediation, interference, bipartite structure, partial eligibility, or principal strata. STTE formalizes effects among non-primary, secondary units—whether defined by post-treatment outcomes (e.g., survivors, compliers), network position (e.g., ineligible nodes in a bipartite graph), or response at earlier stages in adaptive interventions. The construct generalizes existing frameworks such as the Survivor Average Causal Effect (SACE), provides unconfounded estimands in structured experiments, and plays a central role in modern designs with spillovers, sequenced treatments, and post-treatment stratification.

1. Formal Definitions and Contexts

Multiple instantiations of the STTE exist, tailored to distinct experimental designs and scientific questions:

Principal Stratification Context:

In settings with post-treatment variables $M$ (e.g., compliance, survival, implementation), the STTE is often defined as the average causal effect among units that would satisfy $M(1)=M(0)=1$ (always-takers). Formally,

$\mathrm{STTE} = E[Y(1) - Y(0) \mid M(1)=M(0)=1]$

This corresponds to the SACE or the survivor effect in classical post-treatment subgroups (Hazlett et al., 10 May 2025).

Network and Bipartite Designs:

In bipartite experiments with partial eligibility—where only a subset of units can be treated but all interact—the STTE quantifies the total effect of treating all eligible units on outcomes attributed to ineligible (secondary) units (Tan et al., 14 Nov 2025). Notationally, with $\mathcal{T}_{\rm sec}$ denoting ineligible nodes and $Y_{j,\rm sec}(\mathbf{Z})$ their outcome: $$\mathrm{STTE}_{\rm treatment} = \frac{1}{|\mathcal{T}_{\rm sec}|} \sum_{j \in \mathcal{T}_{\rm sec}} \{ E[Y_{j,\rm sec}(\mathbf{Z}^{(1)})] - E[Y_{j,\rm sec}(\mathbf{Z}^{(0)})] \}$$

Sequential Multiple Assignment Randomized Trials (SMARTs):

For sequential treatments, the STTE assesses whether the effect of a later intervention depends on response to earlier interventions. It is often operationalized as the interaction term in a marginal structural model relating the effect of the second-stage treatment to the first-stage conditional average treatment effect (CATE) (Montoya et al., 26 Aug 2024): $\Psi^f(P_{U,X}) \equiv \beta_3$ where $\beta_3$ is the interaction coefficient in

$$\mathrm{logit}\, m_\beta(a(2),B_n(H(r))) = \beta_0 + \beta_1 a(2) + \beta_2 B_n(H(r)) + \beta_3 a(2) B_n(H(r))$$

Household Transmission Studies:

In infectious disease, the STTE can denote the effect of prior intervention (e.g., vaccination) in a household on the secondary attack rate (SAR) among non-index individuals, comparing different vaccination timing regimes (House et al., 2021).

2. Identification Assumptions

Identification of the STTE is highly context-dependent:

• Principal Stratification: Requires monotonicity (no defiers, e.g., all $M(1) \ge M(0)$), no unmeasured confounding between $M$ and $Y$, and explainable nonrandom differences between strata. For the TRACE estimand (a related concept), Hazlett et al. relax these, partially identifying effects through bounding (Hazlett et al., 10 May 2025).
• Network/Bipartite Experiments: Requires exogenous (non-endogenous) network structure, randomization within the eligible set, weak unconfoundedness (node-level exposure independent of outcome given covariates), overlap (positivity), and linear additive outcome decomposability (Tan et al., 14 Nov 2025).
• SMARTs: Key identification follows from sequential randomization at both treatment stages, ensuring ignorability of stage assignments given timely covariate history, and positivity at each decision point. Data-adaptive parameters require machine learning–based estimation of first-stage blip functions (Montoya et al., 26 Aug 2024).
• Household Analysis: Assumes partial interference (within but not between households), full outcome ascertainment via regular testing, and (crucially) exchangeability of strata by treatment timing (House et al., 2021). No adjustment is made for possible confounders outside stratification.

3. Statistical Estimands and Estimation Techniques

The following table summarizes principal instantiations of the STTE across select designs:

Study Context	Formal STTE Estimand	Estimation Approach
Principal strat.	$E[Y(1)-Y(0) \mid M(1)=M(0)=1]$	Plug-in/IPW, bounding, monotonicity
Bipartite exp.	$\sum_{j\in \mathcal{T}_{\rm sec}}$ diff.	ML (XGB, KRR), exposure mapping, proj.
SMART	$\beta_3$ in MSM linking 2nd stage effect	TMLE, SuperLearner for CATE
Household SARS	$SAR(1)-SAR(0)$ on secondary non-index	Bootstrap, stratified SAR computation

Estimation Strategies:

• Plug-in and IPW methods use randomization and observed means/propensities for principal stratification (Hazlett et al., 10 May 2025).
• Ensemble machine learning (random forests, boosting, kernel ridge regression) is employed in high-dimensional bipartite settings to flexibly map exposures to outcomes, augmented with counterfactual imputation and projection operators under additivity (Tan et al., 14 Nov 2025).
• Targeted Maximum Likelihood Estimation (TMLE) efficiently targets the STTE as a data-adaptive parameter in SMARTs, incorporating machine-learned blip estimates and efficient influence curve–based inference (Montoya et al., 26 Aug 2024).
• Bootstrap and Jeffreys-interval resampling quantify uncertainty in SAR-based household transmission analyses (House et al., 2021).

4. Applications and Empirical Findings

• In the ONS COVID-19 Infection Survey, the STTE operationalized as the difference in SARs demonstrated that prior household vaccination approximately halved the SAR, from 23.5% to 12.5% (STTE_RD = -11.0 percentage points, relative reduction ≈47%), under strong ascertainment and stratification assumptions (House et al., 2021).
• In bipartite rider–driver simulations, ensemble estimators for STTE precisely recovered ground truth secondary effects, and projection-based computation achieved substantial speedups. Ignoring spillovers resulted in substantial bias and incorrect inferences regarding secondary outcomes (Tan et al., 14 Nov 2025).
• Sequential intervention studies (e.g., HIV retention under conditional cash transfers in Kenya) found that discontinuing treatment was most harmful for participants with the largest estimated first-stage benefits, as shown by a significant positive STTE (β₃ > 0 in marginal structural models), indicating heterogeneity in secondary effects (Montoya et al., 26 Aug 2024).
• Comparative analyses of TRACE, STTE, and SACE illuminate distinctions between estimands: TRACE includes "compliers" and "always-takers," STTE is restricted to "always-takers," and each requires careful attention to identification under potential unmeasured confounding (Hazlett et al., 10 May 2025).

5. Connections to Related Estimands and Methodological Issues

• TRACE vs. STTE vs. SACE: TRACE targets $E[Y(1)-Y(0)|M(1)=1]$, averaging over both always-takers and compliers. The STTE (and synonymously SACE in biomedical applications) restricts to always-takers ($M(1)=M(0)=1$), often yielding numerically distinct conclusions unless monotonicity and exclusion restrictions hold (Hazlett et al., 10 May 2025).
• Partial identification and bounding: When effects in the non-reactive subgroup cannot be separately identified, bounding techniques (by varying plausible values of the effect in that group) are used to generate sharp confidence intervals for the STTE (Hazlett et al., 10 May 2025).
• Interference and network effects: In complex interaction networks, STTE quantifies indirect causal effects on units not directly eligible for or assigned treatment, requiring explicit handling of secondary exposure and spillover channels (Tan et al., 14 Nov 2025).
• Programmatic versus direct/indirect effects: In practice, STTE is frequently interpreted as the total (not direct or mediator-specific) effect of program deployment (e.g., vaccination, cash transfers) with no attempt to partition mediation or pathway-specific effects (House et al., 2021, Montoya et al., 26 Aug 2024).

6. Impact, Simulation Results, and Limitations

• Simulation studies confirm that model-based and projection estimators recover STTE with low bias and variance when design assumptions (randomization, additivity, network exogeneity) hold, while unmodeled interference can cause sign flips in naïve estimators (Tan et al., 14 Nov 2025).
• Empirical applications show that STTE-aware metrics can overturn business decisions based solely on primary effects, underscoring the operational significance of secondary treatment inference in A/B testing and field trials (Tan et al., 14 Nov 2025).
• Limitations include reliance on unverifiable assumptions in principal stratification (monotonicity, no unobserved post-treatment confounding), and potential bias if network structure or exposure mapping is misspecified. In SMARTs and adaptive trials, STTE estimation rests on the fidelity of machine-learned blip estimates and correct specification of marginal structural working models (Montoya et al., 26 Aug 2024).
• Interpretation requires care: the STTE is a contextually-defined estimand, and its programmatic or policy relevance depends on whether secondary units represent a scientifically meaningful or decision-relevant group.

7. Comparative Table of STTE Definitions

Setting	STTE Conditioning Set	Target Units	Identification Path
Principal stratification	$M(1)=M(0)=1$	Always-takers	Monotonicity, no conf.
Bipartite experiment	Ineligible units, full eligible assignment	Secondary units	Additivity, randomiz.
Sequential randomized trial	Blip/CATE at earlier stage modifies later effect	Blip-modified population	Sequential randomiz.
Household transmission	Household stratum by vaccine timing	Non-index household cases	Stratification, ascert.

The STTE provides a unified conceptual tool for evaluating causal effects in subpopulations, secondary groups, or spillover contexts, bridging principal stratification, interference-aware experimental analysis, and adaptive trial methodology. Its estimation must be aligned with the identification structure imposed by randomization, network, or sequential design, and its policy relevance is dictated by the scientific or operational importance of secondary and spillover impacts.

Key references: (House et al., 2021, Montoya et al., 26 Aug 2024, Hazlett et al., 10 May 2025, Tan et al., 14 Nov 2025).