Causal Directed Acyclic Graphs Research Articles

Effective Causal Discovery under Identifiable Heteroscedastic Noise Model

Capturing the underlying structural causal relations represented by Directed Acyclic Graphs (DAGs) has been a fundamental task in various AI disciplines. Causal DAG learning via the continuous optimization framework has recently achieved promising performance in terms of accuracy and efficiency. However, most methods make strong assumptions of homoscedastic noise, i.e., exogenous noises have equal variances across variables, observations, or even both. The noises in real data usually violate both assumptions due to the biases introduced by different data collection processes. To address the heteroscedastic noise issue, we introduce relaxed implementable sufficient conditions and prove the identifiability of a general class of SEM subject to those conditions. Based on the identifiable general SEM, we propose a novel formulation for DAG learning which accounts for the noise variance variation across variables and observations. We then propose an effective two-phase iterative DAG learning algorithm to address the increasing optimization difficulties and learn a causal DAG from data with heteroscedastic variables noise under varying variance. We show significant empirical gains of the proposed approaches over state-of-the-art methods on both synthetic data and real data.

Identifiability of causal effects in test-negative design studies.

Causal directed acyclic graphs (DAGs) are often used to select variables in a regression model to identify causal effects. Outcome-based sampling studies, such as the 'test-negative design' used to assess vaccine effectiveness, present unique challenges that are not addressed by the common back-door criterion. Here we discuss intuitive, graphical approaches to explain why the common back-door criterion cannot be used for identification of population average causal effects with outcome-based sampling studies. We also describe graphical rules that can be used instead in outcome-based sampling studies when the objective is limited to determining if the causal odds ratio is identifiable, and illustrate recent changes to the free online software Dagitty which incorporate these principles.

A Primer on Structural Equation Model Diagrams and Directed Acyclic Graphs: When and How to Use Each in Psychological and Epidemiological Research

Many psychological researchers use some form of a visual diagram in their research processes. Model diagrams used with structural equation models (SEMs) and causal directed acyclic graphs (DAGs) can guide causal-inference research. SEM diagrams and DAGs share visual similarities, often leading researchers familiar with one to wonder how the other differs. This article is intended to serve as a guide for researchers in the psychological sciences and psychiatric epidemiology on the distinctions between these methods. We offer high-level overviews of SEMs and causal DAGs using a guiding example. We then compare and contrast the two methodologies and describe when each would be used. In brief, SEM diagrams are both a conceptual and statistical tool in which a model is drawn and then tested, whereas causal DAGs are exclusively conceptual tools used to help guide researchers in developing an analytic strategy and interpreting results. Causal DAGs are explicitly tools for causal inference, whereas the results of a SEM are only sometimes interpreted causally. A DAG may be thought of as a “qualitative schematic” for some SEMs, whereas SEMs may be thought of as an “algebraic system” for a causal DAG. As psychology begins to adopt more causal-modeling concepts and psychiatric epidemiology begins to adopt more latent-variable concepts, the ability of researchers to understand and possibly combine both of these tools is valuable. Using an applied example, we provide sample analyses, code, and write-ups for both SEM and causal DAG approaches.

Modeling COVID-19 disease processes by remote elicitation of causal Bayesian networks from medical experts

BackgroundCOVID-19 is a new multi-organ disease causing considerable worldwide morbidity and mortality. While many recognized pathophysiological mechanisms are involved, their exact causal relationships remain opaque. Better understanding is needed for predicting their progression, targeting therapeutic approaches, and improving patient outcomes. While many mathematical causal models describe COVID-19 epidemiology, none have described its pathophysiology.MethodsIn early 2020, we began developing such causal models. The SARS-CoV-2 virus’s rapid and extensive spread made this particularly difficult: no large patient datasets were publicly available; the medical literature was flooded with sometimes conflicting pre-review reports; and clinicians in many countries had little time for academic consultations. We used Bayesian network (BN) models, which provide powerful calculation tools and directed acyclic graphs (DAGs) as comprehensible causal maps. Hence, they can incorporate both expert opinion and numerical data, and produce explainable, updatable results. To obtain the DAGs, we used extensive expert elicitation (exploiting Australia’s exceptionally low COVID-19 burden) in structured online sessions. Groups of clinical and other specialists were enlisted to filter, interpret and discuss the literature and develop a current consensus. We encouraged inclusion of theoretically salient latent (unobservable) variables, likely mechanisms by extrapolation from other diseases, and documented supporting literature while noting controversies. Our method was iterative and incremental: systematically refining and validating the group output using one-on-one follow-up meetings with original and new experts. 35 experts contributed 126 hours face-to-face, and could review our products.ResultsWe present two key models, for the initial infection of the respiratory tract and the possible progression to complications, as causal DAGs and BNs with corresponding verbal descriptions, dictionaries and sources. These are the first published causal models of COVID-19 pathophysiology.ConclusionsOur method demonstrates an improved procedure for developing BNs via expert elicitation, which other teams can implement to model emergent complex phenomena. Our results have three anticipated applications: (i) freely disseminating updatable expert knowledge; (ii) guiding design and analysis of observational and clinical studies; (iii) developing and validating automated tools for causal reasoning and decision support. We are developing such tools for the initial diagnosis, resource management, and prognosis of COVID-19, parameterized using the ISARIC and LEOSS databases.

Nested Markov properties for acyclic directed mixed graphs

Conditional independence models associated with directed acyclic graphs (DAGs) may be characterized in at least three different ways: via a factorization, the global Markov property (given by the d-separation criterion), and the local Markov property. Marginals of DAG models also imply equality constraints that are not conditional independences; the well-known “Verma constraint” is an example. Constraints of this type are used for testing edges, and in a computationally efficient marginalization scheme via variable elimination. We show that equality constraints like the “Verma constraint” can be viewed as conditional independences in kernel objects obtained from joint distributions via a fixing operation that generalizes conditioning and marginalization. We use these constraints to define, via ordered local and global Markov properties, and a factorization, a graphical model associated with acyclic directed mixed graphs (ADMGs). We prove that marginal distributions of DAG models lie in this model, and that a set of these constraints given by Tian provides an alternative definition of the model. Finally, we show that the fixing operation used to define the model leads to a particularly simple characterization of identifiable causal effects in hidden variable causal DAG models.

Deep neural networks with knockoff features identify nonlinear causal relations and estimate effect sizes in complex biological systems.

Learning the causal structure helps identify risk factors, disease mechanisms, and candidate therapeutics for complex diseases. However, although complex biological systems are characterized by nonlinear associations, existing bioinformatic methods of causal inference cannot identify the nonlinear relationships and estimate their effect size. To overcome these limitations, we developed the first computational method that explicitly learns nonlinear causal relations and estimates the effect size using a deep neural network approach coupled with the knockoff framework, named causal directed acyclic graphs using deep learning variable selection (DAG-deepVASE). Using simulation data of diverse scenarios and identifying known and novel causal relations in molecular and clinical data of various diseases, we demonstrated that DAG-deepVASE consistently outperforms existing methods in identifying true and known causal relations. In the analyses, we also illustrate how identifying nonlinear causal relations and estimating their effect size help understand the complex disease pathobiology, which is not possible using other methods. With these advantages, the application of DAG-deepVASE can help identify driver genes and therapeutic agents in biomedical studies and clinical trials.

A local method for identifying causal relations under Markov equivalence

Causality is important for designing interpretable and robust methods in artificial intelligence research. We propose a local approach to identify whether a variable is a cause of a given target under the framework of causal graphical models of directed acyclic graphs (DAGs). In general, the causal relation between two variables may not be identifiable from observational data as many causal DAGs encoding different causal relations are Markov equivalent. In this paper, we first introduce a sufficient and necessary graphical condition to check the existence of a causal path from a variable to a target in every Markov equivalent DAG. Next, we provide local criteria for identifying whether a variable is a cause/non-cause of a target based only on the local structure instead of the entire graph. Finally, we propose a local learning algorithm for this causal query via learning the local structure of the variable and some additional statistical independence tests related to the target. Simulation studies show that our local algorithm is efficient and effective, compared with other state-of-art methods.

Structural learning and estimation of joint causal effects among network-dependent variables

Bayesian networks in the form of Directed Acyclic Graphs (DAGs) represent an effective tool for modeling and inferring dependence relations among variables, a process known as structural learning. In addition, when equipped with the notion of intervention, a causal DAG model can be adopted to quantify the causal effect on a response due to a hypothetical intervention on some variable. Observational data cannot distinguish between DAGs encoding the same set of conditional independencies (Markov equivalent DAGs), which however can be different from a causal perspective. In addition, because causal effects depend on the underlying network structure, uncertainty around the DAG generating model crucially affects the causal estimation results. We propose a Bayesian methodology which combines structural learning of Gaussian DAG models and inference of causal effects as arising from simultaneous interventions on any given set of variables in the system. Our approach fully accounts for the uncertainty around both the network structure and causal relationships through a joint posterior distribution over DAGs, DAG parameters and then causal effects.

Causal Diagrams: Pitfalls and Tips.

Graphical models are useful tools in causal inference, and causal directed acyclic graphs (DAGs) are used extensively to determine the variables for which it is sufficient to control for confounding to estimate causal effects. We discuss the following ten pitfalls and tips that are easily overlooked when using DAGs: 1) Each node on DAGs corresponds to a random variable and not its realized values; 2) The presence or absence of arrows in DAGs corresponds to the presence or absence of individual causal effect in the population; 3) “Non-manipulable” variables and their arrows should be drawn with care; 4) It is preferable to draw DAGs for the total population, rather than for the exposed or unexposed groups; 5) DAGs are primarily useful to examine the presence of confounding in distribution in the notion of confounding in expectation; 6) Although DAGs provide qualitative differences of causal structures, they cannot describe details of how to adjust for confounding; 7) DAGs can be used to illustrate the consequences of matching and the appropriate handling of matched variables in cohort and case-control studies; 8) When explicitly accounting for temporal order in DAGs, it is necessary to use separate nodes for each timing; 9) In certain cases, DAGs with signed edges can be used in drawing conclusions about the direction of bias; and 10) DAGs can be (and should be) used to describe not only confounding bias but also other forms of bias. We also discuss recent developments of graphical models and their future directions.

Discovering Causal Structures in Bayesian Gaussian Directed Acyclic Graph Models

SummaryCausal directed acyclic graphs (DAGs) are naturally tailored to represent biological signalling pathways. However, a causal DAG is only identifiable up to Markov equivalence if only observational data are available. Interventional data, based on exogenous perturbations of the system, can greatly improve identifiability. Since the gain of an intervention crucially depends on the intervened variables, a natural issue is devising efficient strategies for optimal causal discovery. We present a Bayesian active learning procedure for Gaussian DAGs which requires no subjective specification on the side of the user, explicitly takes into account the uncertainty on the space of equivalence classes (through the posterior distribution) and sequentially proposes the choice of the optimal intervention variable. In simulation experiments our method, besides surpassing designs based on a random choice of intervention nodes, shows decisive improvements over currently available algorithms and is competitive with the best alternative benchmarks. An important reason behind this strong performance is that, unlike non-Bayesian algorithms, our utility function naturally incorporates graph estimation uncertainty through the posterior edge inclusion probability. We also reanalyse the Sachs data on protein signalling pathways from an active learning perspective and show that DAG identification can be achieved by using only a subset of the available intervention samples.

Counting and Sampling Markov Equivalent Directed Acyclic Graphs

Exploring directed acyclic graphs (DAGs) in a Markov equivalence class is pivotal to infer causal effects or to discover the causal DAG via appropriate interventional data. We consider counting and uniform sampling of DAGs that are Markov equivalent to a given DAG. These problems efficiently reduce to counting the moral acyclic orientations of a given undirected connected chordal graph on n vertices, for which we give two algorithms. Our first algorithm requires O(2nn4) arithmetic operations, improving a previous superexponential upper bound. The second requires O(k!2kk2n) operations, where k is the size of the largest clique in the graph; for bounded-degree graphs this bound is linear in n. After a single run, both algorithms enable uniform sampling from the equivalence class at a computational cost linear in the graph size. Empirical results indicate that our algorithms are superior to previously presented algorithms over a range of inputs; graphs with hundreds of vertices and thousands of edges are processed in a second on a desktop computer.

The Obesity Paradox in Cardiovascular Diseases

The Obesity Paradox in Cardiovascular Diseases

Conditioning on Parity in Studies of Perfluoroalkyl Acids and Time to Pregnancy: An Example from the Danish National Birth Cohort.

Background:Previous studies have investigated the associations between perfluoroalkyl acids (PFAAs) in women and time to pregnancy (TTP). Inconsistent results may be explained by differences in conditioning on parity.Objectives:We used causal directed acyclic graphs to illustrate potential confounding related to previous pregnancies and exposure measurement error due to differences in the interpregnancy interval in pregnancy-based studies that include parous women. We exemplified the potential importance of these issues using data from the Danish National Birth Cohort.Methods:We used discrete time survival models to estimate associations between maternal plasma PFAAs in early pregnancy and TTP in 638 nulliparous and 613 parous women.Results:PFAA quartiles were not associated with the TTP in nulliparous women. In parous women, higher PFAA quartiles were associated with longer TTP. The strongest associations were estimated for perfluorohexane sulfonate and perfluorooctane sulfonate. PFAA concentrations were higher in women with longer interpregnancy intervals. Accounting for the interpregnancy interval attenuated the estimated associations.Conclusions:Associations between PFAAs and TTP in parous women may be biased by confounders related to previous pregnancies and exposure measurement error. To avoid these biases, studies that include parous women may need to condition on a) common causes of PFAAs and the TTP in the index pregnancy, b) previous births (a descendant of a collider), c) PFAA levels or common causes of PFAA levels and the TTP in the previous pregnancy (to alleviate collider stratification bias caused by conditioning on previous births), and d) the interpregnancy interval (in pregnancy-based studies). Alternatives would be to restrict studies to nulliparous women or to use toxicokinetic modeling to correct exposure estimates in parous women. These recommendations may be extended to studies of other chemicals with similar toxicokinetic properties. https://doi.org/10.1289/EHP1493

Directed acyclic graphs in statistical modelling of epidemiological studies

Confusion in observational epidemiological studies distorts the relationship between exposure and event. "Step by step" regression models, diverts the decision to a statistical algorithm with little causal basis. Directed Acyclic Graphs (DAGs), qualitatively and visually assess the confusion. They can complement the decision on confounder control during statistical modeling. To evaluate the minimum set of confounders to be controlled in a cause-effect relationship with the use of "step-by-step regression" and DAGs, in a study of arsenic exposure. We worked with data from Cáceres et al., 2010 in 66 individuals from northern Chile. The interindividual variability in the urinary excretion of dimethyl arsenic acid attributable to the GSTT1 polymorphism was estimated. A causal DAG was constructed using DAGitty v2.3 with the list of variables. A multiple linear regression model with the step-by-step backwards methodology was carried out. The causal diagram included 12 non-causal open pathways. The minimum adjustment set corresponded to the variables "sex", "body mass index" and "fish and seafood ingest". Confusion retention of the multivariate model included normal and overweight status, gender and the interaction between "water intake" and GSTT1. The use of DAG prior to the modeling would allow a more comprehensive, coherent and biologically plausible analysis of causal relationships in public health.

Data-based Structure Learning to Identify Confounders – Air Pollution and Progression of Subclinical Atherosclerosis

Background/Aim: Identifying confounders based on causal directed acyclic graphs (DAGs) is usually based solely on expert knowledge, although epidemiological studies provide observational data. We applied data-based structure learning algorithms for the specific purpose of identifying confounders, investigating the research question whether particulate matter with an aerodynamic diameter ≤2.5 µg/m3 (PM2.5) affects progression of atherosclerosis, measured as yearly change in carotid intima media thickness (ΔcIMT). Methods: We used 5-year follow-up data of the population-based Heinz Nixdorf Recall Study. Due to computational limitations, potentially relevant variables were restricted to continuously measured data and hence included age, BMI, smoking, blood pressure, HbA1c, LDL/HDL, education, income, neighborhood unemployment (nSES) rate, physical activity and alcohol consumption. Prior knowledge was based on causal chain assumptions regarding major causes of ischaemic heart disease1, and translated to prohibited arcs, i.e. given conditions (e.g. age) are unaffected by lifestyle variables (e.g. smoking) and lifestyle variables are unaffected by subclinical medical determinants (e.g. laboratory measures). We applied a revised structure learning algorithm to learn a causal DAG, including the score-based hill-climbing algorithm, a bootstrap-check to validate the conditional independence structure between variables, and a clustering of minimal sufficient adjustment set (MSAS)-equivalent subsets within the Markov equivalence-class. Results: Based on 2279 participants, the revised structure learning resulted in one MSAS-equivalent cluster, including one DAG of 59 unambiguous arcs. The only biasing path from PM2.5 to ΔcIMT emanated from nSES. Suggested adjustment consisted of four confounder-equivalent MSAS ({nSES}; {age, physical activity, blood pressure}; {age, alcohol consumption, BMI, physical activity, smoking}; {age, BMI, education, income, physical activity, smoking}), yielding an identical exposure effect estimate of 0.002 mm (95%-CI: -0.002, 0.007) increase in cIMT per year per 5 µg/m3 increase in PM2.5. Conclusions: Data-based structure learning returned one unambiguous DAG widely matching general believes. Identical exposure effect estimates of confounder-equivalent MSAS underline a stable (causal) structure.

Causality in objective world: Directed Acyclic Graphs-based structural parsing

The overall details of causality frames in the objective world remain obscure, which poses difficulty for causality research. Based on the temporality of cause and effect, the objective world is divided into three time zones and two time points, in which the causal relationships of the variables are parsed by using Directed Acyclic Graphs (DAGs). Causal DAGs of the world (or causal web) is composed of two parts. One is basic or core to the whole DAGs, formed by the combination of any one variable originating from each time unit mentioned above. Cause effect is affected by the confounding only. The other is an internal DAGs within each time unit representing a parent-child or ancestor-descendant relationship, which exhibits a structure similar to the confounding. This paper summarizes the construction of causality frames for objective world research (causal DAGs), and clarify a structural basis for the control of the confounding in effect estimate.

Compartmental Model Diagrams as Causal Representations in Relation to DAGs.

Compartmental model diagrams have been used for nearly a century to depict causal relationships in infectious disease epidemiology. Causal directed acyclic graphs (DAGs) have been used more broadly in epidemiology since the 1990s to guide analyses of a variety of public health problems. Using an example from chronic disease epidemiology, the effect of type 2 diabetes on dementia incidence, we illustrate how compartmental model diagrams can represent the same concepts as causal DAGs, including causation, mediation, confounding, and collider bias. We show how to use compartmental model diagrams to explicitly depict interaction and feedback cycles. While DAGs imply a set of conditional independencies, they do not define conditional distributions parametrically. Compartmental model diagrams parametrically (or semiparametrically) describe state changes based on known biological processes or mechanisms. Compartmental model diagrams are part of a long-term tradition of causal thinking in epidemiology and can parametrically express the same concepts as DAGs, as well as explicitly depict feedback cycles and interactions. As causal inference efforts in epidemiology increasingly draw on simulations and quantitative sensitivity analyses, compartmental model diagrams may be of use to a wider audience. Recognizing simple links between these two common approaches to representing causal processes may facilitate communication between researchers from different traditions.

Can we believe the DAGs? A comment on the relationship between causal DAGs and mechanisms

Directed acyclic graphs (DAGs) play a large role in the modern approach to causal inference. DAGs describe the relationship between measurements taken at various discrete times including the effect of interventions. The causal mechanisms, on the other hand, would naturally be assumed to be a continuous process operating over time in a cause–effect fashion. How does such immediate causation, that is causation occurring over very short time intervals, relate to DAGs constructed from discrete observations? We introduce a time-continuous model and simulate discrete observations in order to judge the relationship between the DAG and the immediate causal model. We find that there is no clear relationship; indeed the Bayesian network described by the DAG may not relate to the causal model. Typically, discrete observations of a process will obscure the conditional dependencies that are represented in the underlying mechanistic model of the process. It is therefore doubtful whether DAGs are always suited to describe causal relationships unless time is explicitly considered in the model. We relate the issues to mechanistic modeling by using the concept of local (in)dependence. An example using data from the Swiss HIV Cohort Study is presented.

Invited commentary: every good randomization deserves observation.

Preexposure prophylaxis (PrEP) is a promising approach to prevention of human immunodeficiency virus (HIV) transmission, in which an HIV-negative individual takes a single daily dose of an antiretroviral drug so that, if exposed to HIV, an active antiretroviral drug is already present in his or her system. In this issue of the Journal, Murnane et al. (Am J Epidemiol. 2015;182(10):848-856) use data from the Partners PrEP Study (Kenya and Uganda, 2008-2011) to estimate the efficacy of PrEP under perfect adherence. We discuss Murnane et al.'s work and then examine the larger issues of generalizability or transportability of findings from a randomized trial to a new setting when adherence to an intervention determines its effectiveness. We discuss sufficient conditions for generalizability and use causal directed acyclic graphs to show how these assumptions might play out when trials are used to make inferences about the effect of PrEP in current and future real-world target populations.

Do sugar-sweetened beverages cause adverse health outcomes in adults? A systematic review protocol

BackgroundChronic diseases, such as cardiovascular disease and type 2 diabetes, impose significant burden to public health. Most chronic diseases are associated with underlying preventable risk factors, such as elevated blood pressure, blood glucose, and lipids, physical inactivity, excessive sedentary behaviours, overweight and obesity, and tobacco usage. Sugar-sweetened beverages are known to be significant sources of additional caloric intake, and given recent attention to their contribution in the development of chronic diseases, a systematic review is warranted. We will assess whether the consumption of sugar-sweetened beverages in adults is associated with adverse health outcomes and what the potential moderating factors are.Methods/DesignOf interest are studies addressing sugar-sweetened beverage consumption, taking a broad perspective. Both direct consumption studies as well as those evaluating interventions that influence consumption (e.g. school policy, educational) will be relevant. Non-specific or multi-faceted behavioural, educational, or policy interventions may also be included subject to the level of evidence that exists for the other interventions/exposures. Comparisons of interest and endpoints of interest are pre-specified. We will include randomized controlled trials, controlled clinical trials, interrupted time series studies, controlled before-after studies, prospective and retrospective comparative cohort studies, case-control studies, and nested case-control designs. The MEDLINE®, Embase, The Cochrane Library, CINAHL, ERIC, and PsycINFO® databases and grey literature sources will be searched. The processes for selecting studies, abstracting data, and resolving conflicts are described. We will assess risk of bias using design-specific tools. To determine sets of confounding variables that should be adjusted for, we have developed causal directed acyclic graphs and will use those to inform our risk of bias assessments. Meta-analysis will be conducted where appropriate; parameters for exploring statistical heterogeneity and effect modifiers are pre-specified. The Grading of Recommendations Assessment, Development and Evaluation (GRADE) approach will be used for determining the quality of evidence for outcomes.Systematic review registrationPROSPERO CRD42014009638