Effect partitioning under interference in two-stage randomized vaccine trials

G-formula for observational studies under stratified interference, with application to bed net use on malaria

Assessing population-level effects of vaccines and other infectious disease prevention measures is important to the field of public health. In infectious disease studies, one person's treatment may affect another individual's outcome, that is, there may be interference between units. For example, the use of bed nets to prevent malaria by one individual may have an indirect effect on other individuals living in close proximity. In some settings, individuals may form groups or clusters where interference only occurs within groups, that is, there is partial interference. Inverse probability weighted estimators have previously been developed for observational studies with partial interference. Unfortunately, these estimators are not well suited for studies with large clusters. Therefore, in this paper, the parametric g-formula is extended to allow for partial interference. G-formula estimators are proposed for overall effects, effects when treated, and effects when untreated. The proposed estimators can accommodate large clusters and do not suffer from the g-null paradox that may occur in the absence of interference. The large sample properties of the proposed estimators are derived assuming no unmeasured confounders and that the partial interference takes a particular form (referred to as 'weak stratified interference'). Simulation studies are presented demonstrating the finite-sample performance of the proposed estimators. The Demographic and Health Survey from the Democratic Republic of the Congo is then analyzed using the proposed g-formula estimators to assess the effects of bed net use on malaria.

Propensity Score in the Face of Interference: Discussion of

Rosenbaum and Rubin's (1983) propensity score revolutionized the field of causal inference and has emerged as a standard tool when researchers reason about cause-and-effect relationship across many disciplines. This discussion centers around the key "no interference" assumption in Rosenbaum and Rubin's original development of the propensity score and reviews some recent advances in extending the propensity score to studies involving dependent happenings.

Randomization for the susceptibility effect of an infectious disease intervention

Randomized trials of infectious disease interventions, such as vaccines, often focus on groups of connected or potentially interacting individuals. When the pathogen of interest is transmissible between study subjects, interference may occur: individual infection outcomes may depend on treatments received by others. Epidemiologists have defined the primary parameter of interest-called the "susceptibility effect"-as a contrast in infection risk under treatment versus no treatment, while holding exposure to infectiousness constant. A related quantity-the "direct effect"-is defined as an unconditional contrast between the infection risk under treatment versus no treatment. The purpose of this paper is to show that under a widely recommended randomization design, the direct effect may fail to recover the sign of the true susceptibility effect of the intervention in a randomized trial when outcomes are contagious. The analytical approach uses structural features of infectious disease transmission to define the susceptibility effect. A new probabilistic coupling argument reveals stochastic dominance relations between potential infection outcomes under different treatment allocations. The results suggest that estimating the direct effect under randomization may provide misleading conclusions about the effect of an intervention-such as a vaccine-when outcomes are contagious. Investigators who estimate the direct effect may wrongly conclude an intervention that protects treated individuals from infection is harmful, or that a harmful treatment is beneficial.

Estimating sibling spillover effects with unobserved confounding using gain-scores

PURPOSE

A growing area of research in epidemiology is the identification of health-related sibling spillover effects, or the effect of one individual's exposure on their sibling's outcome. The health within families may be confounded by unobserved factors, rendering identification of sibling spillovers challenging.

METHODS

We demonstrate a gain-score (fixed effects) regression method for identifying exposure-to-outcome spillover effects within sibling pairs in linear models. The method identifies the exposure-to-outcome spillover effect if only one sibling's exposure affects the other's outcome, and it identifies the difference between the spillover effects if both siblings' exposures affect the others' outcomes. The method fails with outcome-to-exposure spillover or with outcome-to-outcome spillover. Analytic results, Monte Carlo simulations, and a brief application demonstrate the method and its limitations.

RESULTS

We estimate the spillover effect of a child's preterm birth on an older sibling's literacy skills, measured by the Phonological Awareness Literacy Screening-Kindergarten test. We analyze 20,010 sibling pairs from a population-wide, Wisconsin-based (United States) birth cohort. Without covariate adjustment, we estimate that preterm birth modestly decreases an older sibling's test score.

CONCLUSIONS

Gain-scores are a promising strategy for identifying exposure-to-outcome spillover effects in sibling pairs while controlling for sibling-invariant unobserved confounding.

From Model Organisms to Humans, the Opportunity for More Rigor in Methodologic and Statistical Analysis, Design, and Interpretation of Aging and Senescence Research

This review identifies frequent design and analysis errors in aging and senescence research and discusses best practices in study design, statistical methods, analyses, and interpretation. Recommendations are offered for how to avoid these problems. The following issues are addressed: (a) errors in randomization, (b) errors related to testing within-group instead of between-group differences, (c) failing to account for clustering, (d) failing to consider interference effects, (e) standardizing metrics of effect size, (f) maximum life-span testing, (g) testing for effects beyond the mean, (h) tests for power and sample size, (i) compression of morbidity versus survival curve squaring, and (j) other hot topics, including modeling high-dimensional data and complex relationships and assessing model assumptions and biases. We hope that bringing increased awareness of these topics to the scientific community will emphasize the importance of employing sound statistical practices in all aspects of aging and senescence research.

AVERAGE TREATMENT EFFECTS IN THE PRESENCE OF UNKNOWN INTERFERENCE

We investigate large-sample properties of treatment effect estimators under unknown interference in randomized experiments. The inferential target is a generalization of the average treatment effect estimand that marginalizes over potential spillover effects. We show that estimators commonly used to estimate treatment effects under no interference are consistent for the generalized estimand for several common experimental designs under limited but otherwise arbitrary and unknown interference. The rates of convergence depend on the rate at which the amount of interference grows and the degree to which it aligns with dependencies in treatment assignment. Importantly for practitioners, the results imply that if one erroneously assumes that units do not interfere in a setting with limited, or even moderate, interference, standard estimators are nevertheless likely to be close to an average treatment effect if the sample is sufficiently large. Conventional confidence statements may, however, not be accurate.

Estimands and inference in cluster-randomized vaccine trials

Cluster-randomized trials are often conducted to assess vaccine effects. Defining estimands of interest before conducting a trial is integral to the alignment between a study's objectives and the data to be collected and analyzed. This paper considers estimands and estimators for overall, indirect, and total vaccine effects in trials, where clusters of individuals are randomized to vaccine or control. The scenario is considered where individuals self-select whether to participate in the trial, and the outcome of interest is measured on all individuals in each cluster. Unlike the overall, indirect, and total effects, the direct effect of vaccination is shown in general not to be estimable without further assumptions, such as no unmeasured confounding. An illustrative example motivated by a cluster-randomized typhoid vaccine trial is provided.

Interpretation of the Individual Effect Under Treatment Spillover

Some interventions are intended to benefit both individuals and the groups to which they belong. When a treatment given to one person exerts a causal effect on others, the treatment is said to exhibit spillover, dissemination, or interference. However, defining meaningful causal effects under spillover can be challenging. In this commentary, we discuss the meaning of the "individual effect," a quantity proposed to summarize the effect of treatment on the person who receives it, when spillover may be present.

Spillover effects in epidemiology: parameters, study designs and methodological considerations

Many public health interventions provide benefits that extend beyond their direct recipients and impact people in close physical or social proximity who did not directly receive the intervention themselves. A classic example of this phenomenon is the herd protection provided by many vaccines. If these 'spillover effects' (i.e. 'herd effects') are present in the same direction as the effects on the intended recipients, studies that only estimate direct effects on recipients will likely underestimate the full public health benefits of the intervention. Causal inference assumptions for spillover parameters have been articulated in the vaccine literature, but many studies measuring spillovers of other types of public health interventions have not drawn upon that literature. In conjunction with a systematic review we conducted of spillovers of public health interventions delivered in low- and middle-income countries, we classified the most widely used spillover parameters reported in the empirical literature into a standard notation. General classes of spillover parameters include: cluster-level spillovers; spillovers conditional on treatment or outcome density, distance or the number of treated social network links; and vaccine efficacy parameters related to spillovers. We draw on high quality empirical examples to illustrate each of these parameters. We describe study designs to estimate spillovers and assumptions required to make causal inferences about spillovers. We aim to advance and encourage methods for spillover estimation and reporting by standardizing spillover parameter nomenclature and articulating the causal inference assumptions required to estimate spillovers.

Optimal treatment allocations in space and time for on-line control of an emerging infectious disease

A key component in controlling the spread of an epidemic is deciding where, when and to whom to apply an intervention. We develop a framework for using data to inform these decisions in realtime. We formalize a treatment allocation strategy as a sequence of functions, one per treatment period, that map up-to-date information on the spread of an infectious disease to a subset of locations where treatment should be allocated. An optimal allocation strategy optimizes some cumulative outcome, e.g. the number of uninfected locations, the geographic footprint of the disease or the cost of the epidemic. Estimation of an optimal allocation strategy for an emerging infectious disease is challenging because spatial proximity induces interference between locations, the number of possible allocations is exponential in the number of locations, and because disease dynamics and intervention effectiveness are unknown at out-break. We derive a Bayesian on-line estimator of the optimal allocation strategy that combines simulation-optimization with Thompson sampling. The estimator proposed performs favourably in simulation experiments. This work is motivated by and illustrated using data on the spread of white nose syndrome, which is a highly fatal infectious disease devastating bat populations in North America.

Spillover effects on health outcomes in low- and middle-income countries: a systematic review

Background

Many interventions delivered to improve health may benefit not only direct recipients but also people in close physical or social proximity. Our objective was to review all published literature about the spillover effects of interventions on health outcomes in low-middle income countries and to identify methods used in estimating these effects.

Methods

We searched 19 electronic databases for articles published before 2014 and hand-searched titles from 2010 to 2013 in five relevant journals. We adapted the Cochrane Collaboration’s quality grading tool for spillover estimation and rated the quality of evidence.

Results

A total of 54 studies met inclusion criteria. We found a wide range of terminology used to describe spillovers, a lack of standardization among spillover methods and poor reporting of spillovers in many studies. We identified three primary mechanisms of spillovers: reduced disease transmission, social proximity and substitution of resources within households. We found the strongest evidence for spillovers through reduced disease transmission, particularly vaccines and mass drug administration. In general, the proportion of a population receiving an intervention was associated with improved health. Most studies were of moderate or low quality. We found evidence of publication bias for certain spillover estimates but not for total or direct effects. To facilitate improved reporting and standardization in future studies, we developed a reporting checklist adapted from the CONSORT framework specific to reporting spillover effects.

Conclusions

We found the strongest evidence for spillovers from vaccines and mass drug administration to control infectious disease. There was little high quality evidence of spillovers for other interventions.

On inverse probability-weighted estimators in the presence of interference

We consider inference about the causal effect of a treatment or exposure in the presence of interference, i.e., when one individual’s treatment affects the outcome of another individual. In the observational setting where the treatment assignment mechanism is not known, inverse probability-weighted estimators have been proposed when individuals can be partitioned into groups such that there is no interference between individuals in different groups. Unfortunately this assumption, which is sometimes referred to as partial interference, may not hold, and moreover existing weighted estimators may have large variances. In this paper we consider weighted estimators that could be employed when interference is present. We first propose a generalized inverse probability-weighted estimator and two Hájek-type stabilized weighted estimators that allow any form of interference. We derive their asymptotic distributions and propose consistent variance estimators assuming partial interference. Empirical results show that one of the Hájek estimators can have substantially smaller finite-sample variance than the other estimators. The different estimators are illustrated using data on the effects of rotavirus vaccination in Nicaragua.

Dependent Happenings: A Recent Methodological Review

One hundred years ago Sir Ronald Ross published his treatise on a general Theory of Happenings. Dependent happenings are those in which the frequency depends on the number already affected. When there is dependency of events, interventions can have different types of effects. Interventions such as vaccination can have direct protective effects for the person receiving the treatment, as well as indirect/spillover effects for others in the population. Causal inference is a framework for carefully defining the causal effect of a treatment, exposure, or policy, and then determining conditions under which such effects can be estimated from the observed data. We consider here scenarios in which the potential outcomes of an individual can depend on the treatment of other individuals in the population, known as causal inference with interference. Much of the research so far has assumed the population is divided into groups or clusters, and individuals can interfere with others within their clusters but not across clusters. Recent developments have assumed more general forms of interference. We review some of the different types of effects that have been defined for dependent happenings, particularly using the methods of causal inference with interference. Many of the methods are applicable across disciplines, such as infectious diseases, social sciences, and economics.

Statistical Reasoning and Methods in Epidemiology to Promote Individualized Health: In Celebration of the 100th Anniversary of the Johns Hopkins Bloomberg School of Public Health

Epidemiology is concerned with determining the distribution and causes of disease. Throughout its history, epidemiology has drawn upon statistical ideas and methods to achieve its aims. Because of the exponential growth in our capacity to measure and analyze data on the underlying processes that define each person's state of health, there is an emerging opportunity for population-based epidemiologic studies to influence health decisions made by individuals in ways that take into account the individuals' characteristics, circumstances, and preferences. We refer to this endeavor as "individualized health." The present article comprises 2 sections. In the first, we describe how graphical, longitudinal, and hierarchical models can inform the project of individualized health. We propose a simple graphical model for informing individual health decisions using population-based data. In the second, we review selected topics in causal inference that we believe to be particularly useful for individualized health. Epidemiology and biostatistics were 2 of the 4 founding departments in the world's first graduate school of public health at Johns Hopkins University, the centennial of which we honor. This survey of a small part of the literature is intended to demonstrate that the 2 fields remain just as inextricably linked today as they were 100 years ago.

Interference and Sensitivity Analysis

Causal inference with interference is a rapidly growing area. The literature has begun to relax the "no-interference" assumption that the treatment received by one individual does not affect the outcomes of other individuals. In this paper we briefly review the literature on causal inference in the presence of interference when treatments have been randomized. We then consider settings in which causal effects in the presence of interference are not identified, either because randomization alone does not suffice for identification, or because treatment is not randomized and there may be unmeasured confounders of the treatment-outcome relationship. We develop sensitivity analysis techniques for these settings. We describe several sensitivity analysis techniques for the infectiousness effect which, in a vaccine trial, captures the effect of the vaccine of one person on protecting a second person from infection even if the first is infected. We also develop two sensitivity analysis techniques for causal effects in the presence of unmeasured confounding which generalize analogous techniques when interference is absent. These two techniques for unmeasured confounding are compared and contrasted.

Large sample randomization inference of causal effects in the presence of interference

Recently, increasing attention has focused on making causal inference when interference is possible. In the presence of interference, treatment may have several types of effects. In this paper, we consider inference about such effects when the population consists of groups of individuals where interference is possible within groups but not between groups. A two stage randomization design is assumed where in the first stage groups are randomized to different treatment allocation strategies and in the second stage individuals are randomized to treatment or control conditional on the strategy assigned to their group in the first stage. For this design, the asymptotic distributions of estimators of the causal effects are derived when either the number of individuals per group or the number of groups grows large. Under certain homogeneity assumptions, the asymptotic distributions provide justification for Wald-type confidence intervals (CIs) and tests. Empirical results demonstrate the Wald CIs have good coverage in finite samples and are narrower than CIs based on either the Chebyshev or Hoeffding inequalities provided the number of groups is not too small. The methods are illustrated by two examples which consider the effects of cholera vaccination and an intervention to encourage voting.

Causal Inference Under Multiple Versions of Treatment

Abstract: In this article, we discuss causal inference when there are multiple versions of treatment. The potential outcomes framework, as articulated by Rubin, makes an assumption of no multiple versions of treatment, and here we discuss an extension of this potential outcomes framework to accommodate causal inference under violations of this assumption. A variety of examples are discussed in which the assumption may be violated. Identification results are provided for the overall treatment effect and the effect of treatment on the treated when multiple versions of treatment are present and also for the causal effect comparing a version of one treatment to some other version of the same or a different treatment. Further identification and interpretative results are given for cases in which the version precedes the treatment as when an underlying treatment variable is coarsened or dichotomized to create a new treatment variable for which there are effectively “multiple versions”. Results are also given for effects defined by setting the version of treatment to a prespecified distribution. Some of the identification results bear resemblance to identification results in the literature on direct and indirect effects. We describe some settings in which ignoring multiple versions of treatment, even when present, will not lead to incorrect inferences.

Estimating the Effect of a Community-Based Intervention with Two Communities

Due to the need to evaluate the effectiveness of community-based programs in practice, there is substantial interest in methods to estimate the causal effects of community-level treatments or exposures on individual level outcomes. The challenge one is confronted with is that different communities have different environmental factors affecting the individual outcomes, and all individuals in a community share the same environment and intervention. In practice, data are often available from only a small number of communities, making it difficult if not impossible to adjust for these environmental confounders. In this paper we consider an extreme version of this dilemma, in which two communities each receives a different level of the intervention, and covariates and outcomes are measured on a random sample of independent individuals from each of the two populations; the results presented can be straightforwardly generalized to settings in which more than two communities are sampled. We address the question of what conditions are needed to estimate the causal effect of the intervention, defined in terms of an ideal experiment in which the exposed level of the intervention is assigned to both communities and individual outcomes are measured in the combined population, and then the clock is turned back and a control level of the intervention is assigned to both communities and individual outcomes are measured in the combined population. We refer to the difference in the expectation of these outcomes as the marginal (overall) treatment effect. We also discuss conditions needed for estimation of the treatment effect on the treated community. We apply a nonparametric structural equation model to define these causal effects and to establish conditions under which they are identified. These identifiability conditions provide guidance for the design of studies to investigate community level causal effects and for assessing the validity of causal interpretations when data are only available from a few communities. When the identifiability conditions fail to hold, the proposed statistical parameters still provide nonparametric treatment effect measures (albeit non-causal) whose statistical interpretations do not depend on model specifications. In addition, we study the use of a matched cohort sampling design in which the units of different communities are matched on individual factors. Finally, we provide semiparametric efficient and doubly robust targeted MLE estimators of the community level causal effect based on i.i.d. sampling and matched cohort sampling.

Components of the indirect effect in vaccine trials: identification of contagion and infectiousness effects

Vaccination of one person may prevent the infection of another either because the vaccine prevents the first from being infected and from infecting the second, or because, even if the first person is infected, the vaccine may render the infection less infectious. We might refer to the first of these mechanisms as a contagion effect and the second as an infectiousness effect. In the simple setting of a randomized vaccine trial with households of size two, we use counterfactual theory under interference to provide formal definitions of a contagion effect and an unconditional infectiousness effect. Using ideas analogous to mediation analysis, we show that the indirect effect (the effect of one person's vaccine on another's outcome) can be decomposed into a contagion effect and an unconditional infectiousness effect on the risk difference, risk ratio, odds ratio, and vaccine efficacy scales. We provide identification assumptions for such contagion and unconditional infectiousness effects and describe a simple statistical technique to estimate these effects when they are identified. We also give a sensitivity analysis technique to assess how inferences would change under violations of the identification assumptions. The concepts and results of this paper are illustrated with hypothetical vaccine trial data.

The Minicommunity Design to Assess Indirect Effects of Vaccination

We propose the minicommunity design to estimate indirect effects of vaccination. Establishing indirect effects of vaccination in unvaccinated subpopulations could have important implications for global vaccine policies. In the minicommunity design, the household or other small transmission unit serves as the cluster in which to estimate indirect effects of vaccination, similar to studies in larger communities to estimate indirect, total, and overall effects. Examples from the literature include studies in small transmission units to estimate indirect effects of pertussis, pneumococcal, influenza, and cholera vaccines. We characterize the minicommunity design by several methodologic considerations, including the assignment mechanism, ascertainment, the role of transmission outside the transmission unit, and the relation of the size of the transmission unit to number of people vaccinated. The minicommunity study for indirect effects is contrasted with studies to estimate vaccine effects on infectiousness and protective effects under conditions of household exposure within small transmission units. The minicommunity design can be easily implemented in individually randomized studies by enrolling and following-up members of households of the randomized individuals. The methodology for the minicommunity design for estimating indirect effects of vaccination deserves much future research.

Mediation analysis with multiple versions of the mediator

The causal inference literature has provided definitions of direct and indirect effects based on counterfactuals that generalize the approach found in the social science literature. However, these definitions presuppose well-defined hypothetical interventions on the mediator. In many settings, there may be multiple ways to fix the mediator to a particular value, and these various hypothetical interventions may have very different implications for the outcome of interest. In this paper, we consider mediation analysis when multiple versions of the mediator are present. Specifically, we consider the problem of attempting to decompose a total effect of an exposure on an outcome into the portion through the intermediate and the portion through other pathways. We consider the setting in which there are multiple versions of the mediator but the investigator has access only to data on the particular measurement, not information on which version of the mediator may have brought that value about. We show that the quantity that is estimated as a natural indirect effect using only the available data does indeed have an interpretation as a particular type of mediated effect; however, the quantity estimated as a natural direct effect, in fact, captures both a true direct effect and an effect of the exposure on the outcome mediated through the effect of the version of the mediator that is not captured by the mediator measurement. The results are illustrated using 2 examples from the literature, one in which the versions of the mediator are unknown and another in which the mediator itself has been dichotomized.

A mapping between interactions and interference: implications for vaccine trials

In this paper, we discuss relationships between causal interactions within the counterfactual framework and interference in which the exposure of one person may affect the outcomes of another. We show that the empirical tests for causal interactions can, in fact, all be adapted to empirical tests for particular forms of interference. In the context of interference, by recoding the response as some function of the outcomes of the various persons within a cluster, a wide range of different forms of interference can potentially be detected. The correspondence between causal interactions and forms of interference extends to encompass n-way causal interactions, interference between n persons within a cluster, and multivalued exposures. The theory for causal interactions provides a complete conceptual apparatus for assessing interference as well. The results are illustrated using data from a hypothetical vaccine trial to reason about specific forms of interference and spillover effects that may be present in this vaccine setting. We discuss the implications of this correspondence for our conceptualizations of interaction and for application to vaccine trials and many other settings in which spillover effects may be present.