A key challenge in neuroscience and, in particular, neuroimaging, is to move beyond identification of regional activations toward the characterization of functional circuits underpinning perception, ...cognition, behavior, and consciousness. Granger causality (G-causality) analysis provides a powerful method for achieving this, by identifying directed functional ("causal") interactions from time-series data. G-causality implements a statistical, predictive notion of causality whereby causes precede, and help predict, their effects. It is defined in both the time and frequency domains, and it allows for the conditioning out of common causal influences. In this paper we explain the theoretical basis and computational implementation of G-causality analysis in neuroimaging and, more broadly, in neurophysiology, noting both its exciting potential and the assumptions that govern its application and interpretation.
Correlation in the broadest sense is a measure of an association between variables. In correlated data, the change in the magnitude of 1 variable is associated with a change in the magnitude of ...another variable, either in the same (positive correlation) or in the opposite (negative correlation) direction. Most often, the term correlation is used in the context of a linear relationship between 2 continuous variables and expressed as Pearson product-moment correlation. The Pearson correlation coefficient is typically used for jointly normally distributed data (data that follow a bivariate normal distribution). For nonnormally distributed continuous data, for ordinal data, or for data with relevant outliers, a Spearman rank correlation can be used as a measure of a monotonic association. Both correlation coefficients are scaled such that they range from –1 to +1, where 0 indicates that there is no linear or monotonic association, and the relationship gets stronger and ultimately approaches a straight line (Pearson correlation) or a constantly increasing or decreasing curve (Spearman correlation) as the coefficient approaches an absolute value of 1. Hypothesis tests and confidence intervals can be used to address the statistical significance of the results and to estimate the strength of the relationship in the population from which the data were sampled. The aim of this tutorial is to guide researchers and clinicians in the appropriate use and interpretation of correlation coefficients.This is an open-access article distributed under the terms of the Creative Commons Attribution-Non Commercial-No Derivatives License 4.0 (CCBY-NC-ND), where it is permissible to download and share the work provided it is properly cited. The work cannot be changed in any way or used commercially without permission from the journal.
Directed acyclic graphs (DAGs), which offer systematic representations of causal relationships, have become an established framework for the analysis of causal inference in epidemiology, often being ...used to determine covariate adjustment sets for minimizing confounding bias. DAGitty is a popular web application for drawing and analysing DAGs. Here we introduce the R package 'dagitty', which provides access to all of the capabilities of the DAGitty web application within the R platform for statistical computing, and also offers several new functions. We describe how the R package 'dagitty' can be used to: evaluate whether a DAG is consistent with the dataset it is intended to represent; enumerate 'statistically equivalent' but causally different DAGs; and identify exposure-outcome adjustment sets that are valid for causally different but statistically equivalent DAGs. This functionality enables epidemiologists to detect causal misspecifications in DAGs and make robust inferences that remain valid for a range of different DAGs. The R package 'dagitty' is available through the comprehensive R archive network (CRAN) at https://cran.r-project.org/web/packages/dagitty/. The source code is available on github at https://github.com/jtextor/dagitty. The web application 'DAGitty' is free software, licensed under the GNU general public licence (GPL) version 2 and is available at http://dagitty.net/.
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