A complete graphical criterion for the adjustment formula in mediation analysis

Various assumptions have been used in the literature to identify natural direct and indirect effects in mediation analysis. These effects are of interest because they allow for effect decomposition of a total effect into a direct and indirect effect even in the presence of interactions or non-linear models. In this paper, we consider the relation and interpretation of various identification assumptions in terms of causal diagrams interpreted as a set of non-parametric structural equations. We show that for such causal diagrams, two sets of assumptions for identification that have been described in the literature are in fact equivalent in the sense that if either set of assumptions holds for all models inducing a particular causal diagram, then the other set of assumptions will also hold for all models inducing that diagram. We moreover build on prior work concerning a complete graphical identification criterion for covariate adjustment for total effects to provide a complete graphical criterion for using covariate adjustment to identify natural direct and indirect effects. Finally, we show that this criterion is equivalent to the two sets of independence assumptions used previously for mediation analysis.     

PDZBase: a protein-protein interaction database for PDZ-domains

SUMMARY: PDZBase is a database that aims to contain all known PDZ-domain-mediated protein-protein interactions. Currently, PDZBase contains approximately 300 such interactions, which have been manually extracted from > 200 articles. The database can be queried through both sequence motif and keyword-based searches, and the sequences of interacting proteins can be visually inspected through alignments (for the comparison of several interactions), or as residue-based diagrams including schematic secondary structure information (for individual complexes).     

Towards mapping the late Quaternary vegetation change of Europe

The number of well-dated pollen diagrams in Europe has increased considerably over the last 30 years and many of them have been submitted to the European Pollen Database (EPD). This allows for the construction of increasingly precise maps of Holocene vegetation change across the continent. Chronological information in the EPD has been expressed in uncalibrated radiocarbon years, and most chronologies to date are based on this time scale. Here we present new chronologies for most of the datasets stored in the EPD based on calibrated radiocarbon years. Age information associated with pollen diagrams is often derived from the pollen stratigraphy itself or from other sedimentological information. We reviewed these chronological tie points and assigned uncertainties to them. The steps taken to generate the new chronologies are described and the rationale for a new classification system for age uncertainties is introduced. The resulting chronologies are fit for most continental-scale questions. They may not provide the best age model for particular sites, but may be viewed as general purpose chronologies. Taxonomic particularities of the data stored in the EPD are explained. An example is given of how the database can be queried to select samples with appropriate age control as well as the suitable taxonomic level to answer a specific research question.     

A new criterion for confounder selection

Summary We propose a new criterion for confounder selection when the underlying causal structure is unknown and only limited knowledge is available. We assume all covariates being considered are pretreatment variables and that for each covariate it is known (i) whether the covariate is a cause of treatment, and (ii) whether the covariate is a cause of the outcome. The causal relationships the covariates have with one another is assumed unknown. We propose that control be made for any covariate that is either a cause of treatment or of the outcome or both. We show that irrespective of the actual underlying causal structure, if any subset of the observed covariates suffices to control for confounding then the set of covariates chosen by our criterion will also suffice. We show that other, commonly used, criteria for confounding control do not have this property. We use formal theory concerning causal diagrams to prove our result but the application of the result does not rely on familiarity with causal diagrams. An investigator simply need ask, “Is the covariate a cause of the treatment?” and “Is the covariate a cause of the outcome?” If the answer to either question is “yes” then the covariate is included for confounder control. We discuss some additional covariate selection results that preserve unconfoundedness and that may be of interest when used with our criterion.     

When two become one: the limits of causality analysis of brain dynamics

Biological systems often consist of multiple interacting subsystems, the brain being a prominent example. To understand the functions of such systems it is important to analyze if and how the subsystems interact and to describe the effect of these interactions. In this work we investigate the extent to which the cause-and-effect framework is applicable to such interacting subsystems. We base our work on a standard notion of causal effects and define a new concept called natural causal effect. This new concept takes into account that when studying interactions in biological systems, one is often not interested in the effect of perturbations that alter the dynamics. The interest is instead in how the causal connections participate in the generation of the observed natural dynamics. We identify the constraints on the structure of the causal connections that determine the existence of natural causal effects. In particular, we show that the influence of the causal connections on the natural dynamics of the system often cannot be analyzed in terms of the causal effect of one subsystem on another. Only when the causing subsystem is autonomous with respect to the rest can this interpretation be made. We note that subsystems in the brain are often bidirectionally connected, which means that interactions rarely should be quantified in terms of cause-and-effect. We furthermore introduce a framework for how natural causal effects can be characterized when they exist. Our work also has important consequences for the interpretation of other approaches commonly applied to study causality in the brain. Specifically, we discuss how the notion of natural causal effects can be combined with Granger causality and Dynamic Causal Modeling (DCM). Our results are generic and the concept of natural causal effects is relevant in all areas where the effects of interactions between subsystems are of interest.     

Covariate selection strategies for causal inference: Classification and comparison

When causal effects are to be estimated from observational data, we have to adjust for confounding. A central aim of covariate selection for causal inference is therefore to determine a set that is sufficient for confounding adjustment, but other aims such as efficiency or robustness can be important as well. In this paper, we review six general approaches to covariate selection that differ in the targeted type of adjustment set. We discuss and illustrate their advantages and disadvantages using causal diagrams. Moreover, the approaches and different ways of implementing them are compared empirically in an extensive simulation study. We conclude that there are considerable differences between the approaches but none of them is uniformly best, with performance depending on the chosen adjustment method as well as the true confounding structure. Any prior structural knowledge on the causal relations is helpful to choose the most appropriate method.     

Bayesian inference of causal effects from observational data in Gaussian graphical models

We assume that multivariate observational data are generated from a distribution whose conditional independencies are encoded in a Directed Acyclic Graph (DAG). For any given DAG, the causal effect of a variable onto another one can be evaluated through intervention calculus. A DAG is typically not identifiable from observational data alone. However, its Markov equivalence class (a collection of DAGs) can be estimated from the data. As a consequence, for the same intervention a set of causal effects, one for each DAG in the equivalence class, can be evaluated. In this paper, we propose a fully Bayesian methodology to make inference on the causal effects of any intervention in the system. Main features of our method are: (a) both uncertainty on the equivalence class and the causal effects are jointly modeled; (b) priors on the parameters of the modified Cholesky decomposition of the precision matrices across all DAG models are constructively assigned starting from a unique prior on the complete (unrestricted) DAG; (c) an efficient algorithm to sample from the posterior distribution on graph space is adopted; (d) an objective Bayes approach, requiring virtually no user specification, is used throughout. We demonstrate the merits of our methodology in simulation studies, wherein comparisons with current state‐of‐the‐art procedures turn out to be highly satisfactory. Finally we examine a real data set of gene expressions for Arabidopsis thaliana.     

Fluorescence studies of chlorophyll a incorporated into lipid mixtures, and the interpretation of "phase" diagrams.

The fluorescence of chlorophyll a incorporated into liposomes of mixtures of phosphatidylcholines and phosphatidylethanolamines is reported. Plots of fluorescence intensities against temperature show breaks at characteristic temperatures which can be attributed to the onset and completion of solid phase lipid formation. These temperatures can be plotted to give diagrams analogous to the phase diagrams obtained for macroscopic systems. Complications due to "small-system effects" are discussed, and the experimental diagrams are compared with theoretical phase diagrams calculated for ideal mixing. Introduction of cholesterol leads to a reduction in fluorescence intensity, most readily explained by a 1:1 lipid:cholesterol interaction with exclusion of monomeric, fluorescent, chlorophyll a. Interaction of divalent ions with mixtures of dipalmitoyl phosphatidylcholine and dipalmitoyl phosphatidylserine leads to exclusion of chlorophyll a from the phosphatidylserine.     

Generalised Chromaticity Diagrams for Animals with <Emphasis Type="Italic">n</Emphasis>-chromatic Colour Vision

Chromaticity diagrams for tri- and tetrachromatic animals (with three and four cone classes in their retina, respectively, contributing to colour perception) are widely used in studies of animal colour vision. These diagrams not only allow the graphical representation of perceived colours, but the coordinates of colours plotted within these diagrams can be used to extract colour metrics, such as hue and chroma, or can be used directly in statistical analyses, and therefore aid our understanding of vision-mediated behaviour. However, many invertebrate species have more than four cone classes in their retina, and may therefore have pentachromatic or hexachromatic (or greater) vision. This paper describes an extension to the triangular and tetrahedral chromaticity diagrams commonly used for tri- and tetrachromats, respectively, that allows colour coordinates (and hence colour metrics) to be calculated for animals with more than four cone classes. Because the resulting chromaticity diagrams have more than three dimensions, meaningful ways to visualise the spatial position of plotted colours are discussed.     

Polydesigns and causal inference

In an increasingly common class of studies, the goal is to evaluate causal effects of treatments that are only partially controlled by the investigator. In such studies there are two conflicting features: (1) a model on the full cohort design and data can identify the causal effects of interest, but can be sensitive to extreme regions of that design's data, where model specification can have more impact; and (2) models on a reduced design (i.e., a subset of the full data), for example, conditional likelihood on matched subsets of data, can avoid such sensitivity, but do not generally identify the causal effects. We propose a framework to assess how inference is sensitive to designs by exploring combinations of both the full and reduced designs. We show that using such a "polydesign" framework generates a rich class of methods that can identify causal effects and that can also be more robust to model specification than methods using only the full design. We discuss implementation of polydesign methods, and provide an illustration in the evaluation of a needle exchange program.     

Principal stratification in causal inference

Many scientific problems require that treatment comparisons be adjusted for posttreatment variables, but the estimands underlying standard methods are not causal effects. To address this deficiency, we propose a general framework for comparing treatments adjusting for posttreatment variables that yields principal effects based on principal stratification. Principal stratification with respect to a posttreatment variable is a cross-classification of subjects defined by the joint potential values of that posttreatment variable tinder each of the treatments being compared. Principal effects are causal effects within a principal stratum. The key property of principal strata is that they are not affected by treatment assignment and therefore can be used just as any pretreatment covariate. such as age category. As a result, the central property of our principal effects is that they are always causal effects and do not suffer from the complications of standard posttreatment-adjusted estimands. We discuss briefly that such principal causal effects are the link between three recent applications with adjustment for posttreatment variables: (i) treatment noncompliance, (ii) missing outcomes (dropout) following treatment noncompliance. and (iii) censoring by death. We then attack the problem of surrogate or biomarker endpoints, where we show, using principal causal effects, that all current definitions of surrogacy, even when perfectly true, do not generally have the desired interpretation as causal effects of treatment on outcome. We go on to forrmulate estimands based on principal stratification and principal causal effects and show their superiority.     

The Sign of the Unmeasured Confounding Bias under Various Standard Populations

Unmeasured confounders are a common problem in drawing causal inferences in observational studies. VanderWeele (Biometrics 2008, 64, 702–706) presented a theorem that allows researchers to determine the sign of the unmeasured confounding bias when monotonic relationships hold between the unmeasured confounder and the treatment, and between the unmeasured confounder and the outcome. He showed that his theorem can be applied to causal effects with the total group as the standard population, but he did not mention the causal effects with treated and untreated groups as the standard population. Here, we extend his results to these causal effects, and apply our theorems to an observational study. When researchers have a sense of what the unmeasured confounder may be, conclusions can be drawn about the sign of the bias.     

Minimizing Cognitive Errors in Site-Specific Causal Assessments

Interest in causal investigations in aquatic systems has been a natural outgrowth of the increased use of biological monitoring to characterize the condition of resources. Although biological monitoring approaches are critical tools for detecting whether effects are occurring, they do not identify the cause of the observed effects. Formal approaches to causal evaluation can provide a mechanism to build on expert knowledge, increasing the likelihood that remedial efforts will achieve the desired environmental improvement. This paper examines how formal approaches to causal investigations minimize common errors. We reviewed common cognitive errors reported in the literature, and compared them with considerations suggested for strength-of-evidence approaches. Many of the causal considerations are directed toward distinguishing spurious correlations from true causal relationships. However, this is only one type of error; others include hypothesis dependence, confirmation bias, hypothesis tenacity and anchoring. We suggest three general principles for minimizing error in site-specific investigations: (1) Conduct the causal evaluation as a fair, transparent comparison among alternatives; (2) Carefully describe and quantify the conjunction of cause and effect; and (3) Consider that conjunction between cause and effect is spurious, or that a real conjunction was masked.     

Fluoroscence studies of chloropyll α incorporated into lipid mixtures,and the interpretation of “phase” diagrams

The fluorescence of chloropyll α incorporated into liposomes of mixtures of phosphatidylcholines and phosphatidylethanolamines is reprted. Plots of fluorescence intensities against temperature show breaks at characteristic temperatures which can be attributed to the onset and completion of solid phase lipid formation. These temperatures can be plotted to give diagrams analogous to the phase diagrams obtained for macroscopic systems. Complications due to “small-system effects” are discussed, and the experimental diagrams are compared with theoretical phase digrams calculated for ideal mixing. Introduction of cholesterol leads to a reduction in fluorescence intensity, most readily explained by a1 : 1 lipid :cholesterol interaction with exclution of monomeric, fluorescent, chlorophyll a. Interaction of divalent ions with mixtures of dipalmitoyl phosphatidylcholine and dipalmitoyl phosphatidylserine leads to exclution oc chlorophyll a from the phosphatidylserine.     

From Tree to Map: Using Cognitive Learning Theory to Suggest Alternative Ways to Visualize Macroevolution

Diagrams can be important tools for communicating about evolution. One of the most common visual metaphors that unites a variety of diagrams that describe macroevolution is a tree. Tree-based diagrams are designed to provide a phylogenetic framework for thinking about evolutionary pattern. As is the case with any other metaphor, however, misunderstandings about evolution may either arise from or be perpetuated by how we depict the tree of life. Researchers have tried various approaches to create tree-based diagrams that communicate evolution more accurately. This paper addresses the conceptual limitations of the tree as a visual metaphor for evolution and explores the ways we can use digital tools to extend our visual metaphors for evolution communication. The theory of distributed cognition provides a framework to aid in the analysis of the conceptual affordances and constraints of tree-based diagrams, and develop new ways to visualize evolution. By combining a new map-based visual metaphor for macroevolution with the interactive properties of digital technology, a new method of visualizing evolution called the dynamic evolutionary map is proposed. This paper concludes by comparing the metaphoric affordances and constraints of tree diagrams and the dynamic evolutionary map, and discussing the potential applications of the latter as an educational tool.     

The Causal Meaning of Genomic Predictors and How It Affects Construction and Comparison of Genome-Enabled Selection Models

The term “effect” in additive genetic effect suggests a causal meaning. However, inferences of such quantities for selection purposes are typically viewed and conducted as a prediction task. Predictive ability as tested by cross-validation is currently the most acceptable criterion for comparing models and evaluating new methodologies. Nevertheless, it does not directly indicate if predictors reflect causal effects. Such evaluations would require causal inference methods that are not typical in genomic prediction for selection. This suggests that the usual approach to infer genetic effects contradicts the label of the quantity inferred. Here we investigate if genomic predictors for selection should be treated as standard predictors or if they must reflect a causal effect to be useful, requiring causal inference methods. Conducting the analysis as a prediction or as a causal inference task affects, for example, how covariates of the regression model are chosen, which may heavily affect the magnitude of genomic predictors and therefore selection decisions. We demonstrate that selection requires learning causal genetic effects. However, genomic predictors from some models might capture noncausal signal, providing good predictive ability but poorly representing true genetic effects. Simulated examples are used to show that aiming for predictive ability may lead to poor modeling decisions, while causal inference approaches may guide the construction of regression models that better infer the target genetic effect even when they underperform in cross-validation tests. In conclusion, genomic selection models should be constructed to aim primarily for identifiability of causal genetic effects, not for predictive ability.     

Quantifying the causal pathways contributing to natural selection

The consequences of natural selection can be understood from a purely statistical perspective. In contrast, an explicitly causal approach is required to understand why trait values covary with fitness. In particular, key evolutionary constructs, such as sexual selection, fecundity selection, and so on, are best understood as selection via particular fitness components. To formalize and operationalize these concepts, we must disentangle the various causal pathways contributing to selection. Such decompositions are currently only known for linear models, where they are sometimes referred to as “Wright's rules.” Here, we provide a general framework, based on path analysis, for partitioning selection among its contributing causal pathways. We show how the extended selection gradient—which represents selection arising from a trait's causal effects on fitness—can be decomposed into path-specific selection gradients, which correspond to distinct causal mechanisms of selection. This framework allows for nonlinear effects and nonadditive interactions among variables, which may be estimated using standard statistical methods (e.g., generalized linear [mixed] models or generalized additive models). We thus provide a generalization of Wright's path rules that accommodates the nonlinear and nonadditive mechanisms by which natural selection commonly arises.     

Top-down Causation Without Top-down Causes

We argue that intelligible appeals to interlevel causes (top-down and bottom-up) can be understood, without remainder, as appeals to mechanistically mediated effects. Mechanistically mediated effects are hybrids of causal and constitutive relations, where the causal relations are exclusively intralevel. The idea of causation would have to stretch to the breaking point to accommodate interlevel causes. The notion of a mechanistically mediated effect is preferable because it can do all of the required work without appealing to mysterious interlevel causes. When interlevel causes can be translated into mechanistically mediated effects, the posited relationship is intelligible and should raise no special philosophical objections. When they cannot, they are suspect.