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Most work in causal inference considers deterministic interventions that set each unit's treatment to some fixed value. However, under positivity violations these interventions can lead to nonidentification, inefficiency, and effects with little practical relevance. Further, corresponding effects in longitudinal studies are highly sensitive to the curse of dimensionality, resulting in widespread use of unrealistic parametric models. We propose a novel solution to these problems: incremental interventions that shift propensity score values rather than set treatments to fixed values. Incremental interventions have several crucial advantages. First, they avoid positivity assumptions entirely. Second, they require no parametric assumptions and yet still admit a simple characterization of longitudinal effects, independent of the number of timepoints. For example, they allow longitudinal effects to be visualized with a single curve instead of lists of coefficients. After characterizing incremental interventions and giving identifying conditions for corresponding effects, we also develop general efficiency theory, propose efficient nonparametric estimators that can attain fast convergence rates even when incorporating flexible machine learning, and propose a bootstrap-based confidence band and simultaneous test of no treatment effect. Finally, we explore finite-sample performance via simulation, and apply the methods to study time-varying sociological effects of incarceration on entry into marriage. Supplementary materials for this article are available online.

Mediation analysis in causal inference has traditionally focused on binary exposures and deterministic interventions, and a decomposition of the average treatment effect in terms of direct and indirect effects.We present an analogous decomposition of the population intervention effect, defined through stochastic interventions on the exposure. Population intervention effects provide a generalized framework in which a variety of interesting causal contrasts can be defined, including effects for continuous and categorical exposures. We show that identification of direct and indirect effects for the population intervention effect requires weaker assumptions than its average treatment effect counterpart, under the assumption of no mediator–outcome confounders affected by exposure. In particular, identification of direct effects is guaranteed in experiments that randomize the exposure and the mediator.We propose various estimators of the direct and indirect effects, including substitution, reweighted and efficient estimators based on flexible regression techniques, allowing for multivariate mediators. Our efficient estimator is asymptotically linear under a condition requiring n 1/4-consistency of certain regression functions. We perform a simulation study in which we assess the finite sample properties of our proposed estimators.We present the results of an illustrative study where we assess the effect of participation in a sports team on the body mass index among children, using mediators such as exercise habits, daily consumption of snacks and overweight status.

Estimating the causal effect of an intervention on a population typically involves defining parameters in a nonparametric structural equation model (Pearl, 2000, Causality: Models, Reasoning, and Inference) in which the treatment or exposure is deterministically assigned in a static or dynamic way. We define a new causal parameter that takes into account the fact that intervention policies can result in stochastically assigned exposures. The statistical parameter that identifies the causal parameter of interest is established. Inverse probability of treatment weighting (IPTW), augmented IPTW (A‐IPTW), and targeted maximum likelihood estimators (TMLE) are developed. A simulation study is performed to demonstrate the properties of these estimators, which include the double robustness of the A‐IPTW and the TMLE. An application example using physical activity data is presented.

The COVE trial randomized participants to receive two doses of mRNA-1273 vaccine or placebo on Days 1 and 29 (D1, D29). Anti-SARS-CoV-2 Spike IgG binding antibodies (bAbs), anti-receptor binding domain IgG bAbs, 50% inhibitory dilution neutralizing antibody (nAb) titers, and 80% inhibitory dilution nAb titers were measured at D29 and D57. We assessed these markers as correlates of protection (CoPs) against COVID-19 using stochastic interventional vaccine efficacy (SVE) analysis and principal surrogate (PS) analysis, frameworks not used in our previous COVE immune correlates analyses. By SVE analysis, hypothetical shifts of the D57 Spike IgG distribution from a geometric mean concentration (GMC) of 2737 binding antibody units (BAU)/mL (estimated vaccine efficacy (VE): 92.9% (95% CI: 91.7%, 93.9%)) to 274 BAU/mL or to 27,368 BAU/mL resulted in an overall estimated VE of 84.2% (79.0%, 88.1%) and 97.6% (97.4%, 97.7%), respectively. By binary marker PS analysis of Low and High subgroups (cut-point: 2094 BAU/mL), the ignorance interval (IGI) and estimated uncertainty interval (EUI) for VE were [85%, 90%] and (78%, 93%) for Low compared to [95%, 96%] and (92%, 97%) for High. By continuous marker PS analysis, the IGI and 95% EUI for VE at the 2.5th percentile (519.4 BAU/mL) vs. at the 97.5th percentile (9262.9 BAU/mL) of D57 Spike IgG concentration were [92.6%, 93.4%] and (89.2%, 95.7%) vs. [94.3%, 94.6%] and (89.7%, 97.0%). Results were similar for other D29 and D57 markers. Thus, the SVE and PS analyses additionally support all four markers at both time points as CoPs.

BACKGROUND:Evidence suggests that aspects of the neighborhood environment may influence risk of problematic drug use among adolescents. Our objective was to examine mediating roles of aspects of the school and peer environments on the effect of receiving a Section 8 housing voucher and using it to move out of public housing on adolescent substance use outcomes. METHODS:We used data from the Moving to Opportunity (MTO) experiment that randomized receipt of a Section 8 housing voucher. Hypothesized mediators included school climate, safety, peer drug use, and participation in an after-school sport or club. We applied a doubly robust, semiparametric estimator to longitudinal MTO data to estimate stochastic direct and indirect effects of randomization on cigarette use, marijuana use, and problematic drug use. Stochastic direct and indirect effects differ from natural direct and indirect effects in that they do not require assuming no posttreatment confounder of the mediator–outcome relationship. Such an assumption would be at odds with any causal model that reflects an intervention affecting a mediator and outcome through adherence to treatment assignment. RESULTS:Having friends who use drugs and involvement in after-school sports or clubs partially mediated the effect of housing voucher receipt on adolescent substance use (e.g., stochastic indirect effect 0.45% [95% confidence interval0.12%, 0.79%] for having friends who use drugs and 0.04% [95% confidence interval−0.02%, 0.10%] for involvement in after-school sports or clubs mediating the relationship between housing voucher receipt and marijuana use among boys). However, these mediating effects were small, contributing only fractions of a percent to the effect of voucher receipt on probability of substance use. No school environment variables were mediators. CONCLUSIONS:Measured school- and peer-environment variables played little role in mediating the effect of housing voucher receipt on subsequent adolescent substance use.

Assessing the causal effect of an exposure often involves the definition of counterfactual outcomes in a hypothetical world in which the stochastic nature of the exposure is modified. Although stochastic interventions are a powerful tool to measure the causal effect of a realistic intervention that intends to alter the population distribution of an exposure, their importance to answer questions about plausible policy interventions has been obscured by the generalized use of deterministic interventions. In this article, we follow the approach described in Díaz and van der Laan (2012) to define and estimate the effect of an intervention that is expected to cause a truncation in the population distribution of the exposure. The observed data parameter that identifies the causal parameter of interest is established, as well as its efficient influence function under the non-parametric model. Inverse probability of treatment weighted (IPTW), augmented IPTW and targeted minimum loss-based estimators (TMLE) are proposed, their consistency and efficiency properties are determined. An extension to longitudinal data structures is presented and its use is demonstrated with a real data example.

Young, Hernán, and Robins consider the mean outcome under a dynamic intervention that may rely on the natural value of treatment. They first identify this value with a statistical target parameter, and then show that this statistical target parameter can also be identified with a causal parameter which gives the mean outcome under a stochastic intervention. The authors then describe estimation strategies for these quantities. Here we augment the authors’ insightful discussion by sharing our experiences in situations where two causal questions lead to the same statistical estimand, or the newer problem that arises in the study of data adaptive parameters, where two statistical estimands can lead to the same estimation problem. Given a statistical estimation problem, we encourage others to always use a robust estimation framework where the data generating distribution truly belongs to the statistical model. We close with a discussion of a framework which has these properties.

Background In the early stages of a novel infectious disease outbreak, when vaccines, treatments, and herd immunity are lacking, non-pharmaceutical interventions—particularly self-reporting and contact tracing—play a critical role in suppressing transmission. During the first wave of COVID-19 in Korea, a large outbreak centered around a religious community in Daegu rapidly escalated, thus highlighting the transmission risks associated with a closed and low-reporting high-risk group. This study aimed to quantitatively assess the effectiveness of self-reporting and contact tracing strategies across heterogeneous risk groups. Methods The population of Daegu was stratified into two groups: a high-risk group characterized by high transmissibility and low reporting rate, and a low-risk group with lower transmissibility and higher reporting compliance. We developed a stochastic model and applied a modified Gillespie algorithm incorporating both Markovian and non-Markovian processes. Scenario-based simulations were conducted to evaluate the impact of changes in self-reporting rates and delays in contact tracing. We simulated each scenario 10,000 times to estimate the mean and 95% credible intervals for the number of infections. Results When the self-reporting rate in the high-risk group was lowered to 0.1, the total infections increased by approximately 22%, while unreported infections rose by 164% compared to the baseline. Conversely, increasing the self-reporting rate in the high-risk group to 0.8 reduced the total cases by approximately 21% and unreported infections by 86%. Notably, unreported infections in the low-risk group increased by approximately 416% when their reporting rate declined to 0.4, although this group had a lower transmission potential. Even a modest contact tracing delay of 4–7 d resulted in an 85% increase in unreported cases, with diminishing returns for longer delays, highlighting the critical importance of timely tracing in outbreak control. Conclusions In situations with heterogeneous risk groups, improving the self-reporting behavior of high-risk populations and maintaining high compliance in the low-risk group are essential for effective outbreak control. Contact tracing should be completed within 1–4 d to prevent further spread. Our study which accounts for behavioral heterogeneity, provides a scientific foundation for designing group-specific intervention strategies in future outbreaks of emerging infectious diseases.

Objectives Criminal networks tend to recover after a disruption, and this recovery may trigger negative unintended consequences by strengthening network cohesion. This study uses a real-world street gang network as a basis for simulating the effect of disruption and subsequent recovery on network structure. Methods This study utilises cohesion and centrality measures to describe the network and to simulate nine network disruptions. Stationary stochastic actor-oriented models are used to identify relational mechanisms in this network and subsequently to simulate network recovery in five scenarios. Results Removing the most central and the highest-ranking actors have the largest immediate impact on the network. In the long-term recovery simulation, networks become more compact (substantially so when increasing triadic closure), while the structure disintegrates when preferential attachment decreases. Conclusion These results indicate that the mechanisms driving network recovery are more important than the immediate impact of disruption due to network recovery.

Background Semi-field experiments with human landing catch (HLC) measure as the outcome are an important step in the development of novel vector control interventions against outdoor transmission of malaria since they provide good estimates of personal protection. However, it is often infeasible to determine whether the reduction in HLC counts is due to mosquito mortality or repellency, especially considering that spatial repellents based on volatile pyrethroids might induce both. Due to the vastly different impact of repellency and mortality on transmission, the community-level impact of spatial repellents can not be estimated from such semi-field experiments. Methods We present a new stochastic model that is able to estimate for any product inhibiting outdoor biting, its repelling effect versus its killing and disarming (preventing host-seeking until the next night) effects, based only on time-stratified HLC data from controlled semi-field experiments. For parameter inference, a Bayesian hierarchical model is used to account for nightly variation of semi-field experimental conditions. We estimate the impact of the products on the vectorial capacity of the given Anopheles species using an existing mathematical model. With this methodology, we analysed data from recent semi-field studies in Kenya and Tanzania on the impact of transfluthrin-treated eave ribbons, the odour-baited Suna trap and their combination (push–pull system) on HLC of Anopheles arabiensis in the peridomestic area. Results Complementing previous analyses of personal protection, we found that the transfluthrin-treated eave ribbons act mainly by killing or disarming mosquitoes. Depending on the actual ratio of disarming versus killing, the vectorial capacity of An. arabiensis is reduced by 41 to 96% at 70% coverage with the transfluthrin-treated eave ribbons and by 38 to 82% at the same coverage with the push–pull system, under the assumption of a similar impact on biting indoors compared to outdoors. Conclusions The results of this analysis of semi-field data suggest that transfluthrin-treated eave ribbons are a promising tool against malaria transmission by An. arabiensis in the peridomestic area, since they provide both personal and community protection. Our modelling framework can estimate the community-level impact of any tool intervening during the mosquito host-seeking state using data from only semi-field experiments with time-stratified HLC. Graphical Abstract