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The causal inference literature has provided a clear formal definition of confounding expressed in terms of counterfactual independence. The literature has not, however, come to any consensus on a formal definition of a confounder, as it has given priority to the concept of confounding over that of a confounder. We consider a number of candidate definitions arising from various more informal statements made in the literature. We consider the properties satisfied by each candidate definition, principally focusing on (i) whether under the candidate definition control for all \"confounders\" suffices to control for \"confounding\" and (ii) whether each confounder in some context helps eliminate or reduce confounding bias. Several of the candidate definitions do not have these two properties. Only one candidate definition of those considered satisfies both properties. We propose that a \"confounder\" be defined as a pre-exposure covariate C for which there exists a set of other covariates X such that effect of the exposure on the outcome is unconfounded conditional on (X, C) but such that for no proper subset of (X, C) is the effect of the exposure on the outcome unconfounded given the subset. We also provide a conditional analogue of the above definition; and we propose a variable that helps reduce bias but not eliminate bias be referred to as a \"surrogate confounder.\" These definitions are closely related to those given by Robins and Morgenstern [Comput. Math. Appl. 14 (1987) 869-916]. The implications that hold among the various candidate definitions are discussed.

Recent discussions of model selection and multimodel inference highlight a general challenge for researchers: how to convey the explanatory content of a hypothesized model or set of competing models clearly. The advice from statisticians for scientists employing multimodel inference is to develop a well-thought-out set of candidate models for comparison, though precise instructions for how to do that are typically not given. A coherent body of knowledge, which falls under the general term causal analysis, now exists for examining the explanatory scientific content of candidate models. Much of the literature on causal analysis has been recently developed, and we suspect may not be familiar to many ecologists. This body of knowledge comprises a set of graphical tools and axiomatic principles to support scientists in their endeavors to create “well-formed hypotheses,” as statisticians are asking them to do. Causal analysis is complementary to methods such as structural equation modeling, which provides the means for evaluation of proposed hypotheses against data. In this paper, we summarize and illustrate a set of principles that can guide scientists in their quest to develop explanatory hypotheses for evaluation. The principles presented in this paper have the capacity to close the communication gap between statisticians, who urge scientists to develop well-thought-out coherent models, and scientists, who would like some practical advice for exactly how to do that.

Reintroducing locally extinct/extirpated species has been considered as an approach for restoring ecosystems. Although such projects share the same goals of rebuilding previously affected ecosystems, the overall impacts that such reintroductions generate on both ecosystems and human society, i.e., on the social-ecological system, are difficult to measure. We propose a system dynamics approach, a platform on which both natural and social scientists could collaborate to identify the social-ecological impacts of species reintroduction as well as factors that affect such decision making. We use cases in Japan to demonstrate the potential applicability of system dynamics in terms of (1) understanding the impacts of a previously reintroduced species, the Oriental Stork (Ciconia boyciana), and (2) predicting the impacts of reintroduction of wolves (Canis lupus). We present a causal loop diagram of the social and ecological effects of Oriental Stork reintroduction, and we discuss how the relationships between factors could be articulated based on empirical data and ongoing projects in Japan. The model demonstrates how local residents began to appreciate the rich biodiversity, including the Oriental Stork, following its reintroduction, and how public support toward such reintroduction enhanced further projects to reintroduce these species in different parts of Japan. A similar diagram, created to illustrate the social and ecological effects of the potential reintroduction of wolves to Japan, demonstrates how social factors such as environmental education and public attitudes could affect decision making as well as ecological factors such as predator-prey dynamics and overall biodiversity. Further, human-wolf conflicts could negatively affect the overall loop. Creating causal loop diagrams can help managers and stakeholders understand that species reintroduction projects need to be considered via an interdisciplinary approach. The models illustrate that these problems are dynamic and that the factors affecting or affected by such projects change over time, implying the importance of both the spatial and temporal scales in managing reintroduction projects.

Covariate adjustment is integral to the validity of observational studies assessing causal effects. It is common practice to adjust for as many variables as possible in observational studies in the hopes of reducing confounding by other variables. However, indiscriminate adjustment for variables using standard regression models may actually lead to biased estimates. In this paper, we differentiate between confounders, mediators, colliders, and effect modifiers. We will discuss that while confounders should be adjusted for in the analysis, one should be wary of adjusting for colliders. Mediators should not be adjusted for when examining the total effect of an exposure on an outcome. Automated statistical programs should not be used to decide which variables to include in causal models. Using a case scenario in cardiology, we will demonstrate how to identify confounders, colliders, mediators and effect modifiers and the implications of adjustment or non-adjustment for each of them.

The term \"interference\" has been used to describe any setting in which one subject's exposure may affect another subject's outcome. We use causal diagrams to distinguish among three causal mechanisms that give rise to interference. The first causal mechanism by which interference can operate is a direct causal effect of one individual's treatment on another individual's outcome; we call this direct interference. Interference by contagion is present when one individual's outcome may affect the outcomes of other individuals with whom he comes into contact. Then giving treatment to the first individual could have an indirect effect on others through the treated individual's outcome. The third pathway by which interference may operate is allocational interference. Treatment in this case allocates individuals to groups; through interactions within a group, individuals may affect one another's outcomes in any number of ways. In many settings, more than one type of interference will be present simultaneously. The causal effects of interest differ according to which types of interference are present, as do the conditions under which causal effects are identifiable. Using causal diagrams for interference, we describe these differences, give criteria for the identification of important causal effects, and discuss applications to infectious diseases.

Effect modification and interaction are key concepts within epidemiologic research. They refer to situations where the magnitude and/or direction of the causal effect of some exposure variable on an outcome depends on the level of a second variable (effect modification) or on the effect of a second variable (interaction). Interest in case of effect modification is primarily in one exposure variable, with its effects varying across subgroups, whereas primary interest in case of interaction is in the interplay of effects of two exposure variables. Distinctions between the concepts of effect modification and interaction are subtle. The goal of this article is to clarify these distinctions by using the concept of mediation, which focuses on elucidating mechanisms of causal effects.

Structural equation modeling (SEM) is increasingly being chosen by researchers as a framework for gaining scientific insights from the quantitative analyses of data. New ideas and methods emerging from the study of causality, influences from the field of graphical modeling, and advances in statistics are expanding the rigor, capability, and even purpose of SEM. Guidelines for implementing the expanded capabilities of SEM are currently lacking. In this paper we describe new developments in SEM that we believe constitute a third-generation of the methodology. Most characteristic of this new approach is the generalization of the structural equation model as a causal graph. In this generalization, analyses are based on graph theoretic principles rather than analyses of matrices. Also, new devices such as metamodels and causal diagrams, as well as an increased emphasis on queries and probabilistic reasoning, are now included. Estimation under a graph theory framework permits the use of Bayesian or likelihood methods. The guidelines presented start from a declaration of the goals of the analysis. We then discuss how theory frames the modeling process, requirements for causal interpretation, model specification choices, selection of estimation method, model evaluation options, and use of queries, both to summarize retrospective results and for prospective analyses. The illustrative example presented involves monitoring data from wetlands on Mount Desert Island, home of Acadia National Park. Our presentation walks through the decision process involved in developing and evaluating models, as well as drawing inferences from the resulting prediction equations. In addition to evaluating hypotheses about the connections between human activities and biotic responses, we illustrate how the structural equation (SE) model can be queried to understand how interventions might take advantage of an environmental threshold to limit Typha invasions. The guidelines presented provide for an updated definition of the SEM process that subsumes the historical matrix approach under a graph-theory implementation. The implementation is also designed to permit complex specifications and to be compatible with various estimation methods. Finally, they are meant to foster the use of probabilistic reasoning in both retrospective and prospective considerations of the quantitative implications of the results.

The divergence between agricultural water use and the annual supply of water resources (water gap) has been increasing for decades. The forecast is that this water gap will continue to widen, compromising the water security of a large share of the global population. On the one hand, the increase in demand is attributed to an ever-growing population that, in addition, is adopting a high-water consumption per capita lifestyle (e.g., meat-rich diet, increased use of biofuels and of irrigated agriculture). On the other hand, climate change is increasing aridification and the spatio-temporal heterogeneity of precipitation worldwide. The water gap is particularly acute in drylands, where development and food security has been based on the massive exploitation of water resources, particularly groundwater. Here we analyze the mechanisms underlying this water gap, which is mainly driven by water use in agriculture, and suggest suitable solutions that can help to close it. Using causal diagrams, we show how population generates different demands that create a water gap that prevailing supply-side solutions cannot close. Indeed, it has been widening over the years because water consumption has grown exponentially. This behaviour is explained by a series of mechanisms that it is necessary to understand to realize the complexity of water scarcity problems. For solving the water gap, we propose and exemplify eight lines of action that can be combined and tailored to each territory. Our analyses corroborate the urgent need to plan an integral management of water resources to avoid widespread scenarios of water scarcity under future climatic conditions.

Recent developments in computer science have substantially advanced the use of observational causal inference under Pearl's structural causal model (SCM) framework. A key tool in the application of SCM is the use of casual diagrams, used to visualize the causal structure of a system or process under study. Here, we show how causal diagrams can be extended to ensure proper study design under quasi‐experimental settings, including propensity score analysis, before‐after‐control‐impact studies, regression discontinuity design, and instrumental variables. Causal diagrams represent a unified approach to variable selection across methodologies and should be routinely applied in ecology research with causal implications.