Bayesian hypothesis testing for Gaussian graphical models: Conditional independence and order constraints
TL;DR: This work introduces exploratory and confirmatory Bayesian tests for partial correlations in Gaussian graphical models and describes the novel matrix- F prior distribution that provides increased flexibility in specification compared to the Wishart prior.
About: This article is published in Journal of Mathematical Psychology.The article was published on 2020-12-01 and is currently open access. It has received 27 citations till now. The article focuses on the topics: Conditional independence & Graphical model.
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19 Aug 2021
TL;DR: This Primer provides an anatomy of network analysis techniques, describes the current state of the art and discusses open problems, as well as assessment techniques to evaluate network robustness and replicability.
Abstract: In recent years, network analysis has been applied to identify and analyse patterns of statistical association in multivariate psychological data. In these approaches, network nodes represent variables in a data set, and edges represent pairwise conditional associations between variables in the data, while conditioning on the remaining variables. This Primer provides an anatomy of these techniques, describes the current state of the art and discusses open problems. We identify relevant data structures in which network analysis may be applied: cross-sectional data, repeated measures and intensive longitudinal data. We then discuss the estimation of network structures in each of these cases, as well as assessment techniques to evaluate network robustness and replicability. Successful applications of the technique in different research areas are highlighted. Finally, we discuss limitations and challenges for future research. Network analysis allows the investigation of complex patterns and relationships by examining nodes and the edges connecting them. Borsboom et al. discuss the adoption of network analysis in psychological research.
206 citations
TL;DR: This article describes the glasso method in the context of the fields where it was developed, and demonstrates that the advantages of regularization diminish in settings where psychological networks are often fitted, and introduces nonregularized methods based on multiple regression and a nonparametric bootstrap strategy.
Abstract: An important goal for psychological science is developing methods to characterize relationships between variables. Customary approaches use structural equation models to connect latent factors to a number of observed measurements, or test causal hypotheses between observed variables. More recently, regularized partial correlation networks have been proposed as an alternative approach for characterizing relationships among variables through off-diagonal elements in the precision matrix. While the graphical Lasso (glasso) has emerged as the default network estimation method, it was optimized in fields outside of psychology with very different needs, such as high dimensional data where the number of variables (p) exceeds the number of observations (n). In this article, we describe the glasso method in the context of the fields where it was developed, and then we demonstrate that the advantages of regularization diminish in settings where psychological networks are often fitted ( p≪n ). We first show that improved properties of the precision matrix, such as eigenvalue estimation, and predictive accuracy with cross-validation are not always appreciable. We then introduce nonregularized methods based on multiple regression and a nonparametric bootstrap strategy, after which we characterize performance with extensive simulations. Our results demonstrate that the nonregularized methods can be used to reduce the false-positive rate, compared to glasso, and they appear to provide consistent performance across sparsity levels, sample composition (p/n), and partial correlation size. We end by reviewing recent findings in the statistics literature that suggest alternative methods often have superior performance than glasso, as well as suggesting areas for future research in psychology. The nonregularized methods have been implemented in the R package GGMnonreg.
74 citations
Cites methods from "Bayesian hypothesis testing for Gau..."
...These reviews do not include Bayesian methods, but these can be found in Mohammadi and Wit (2015a) and Williams and Mulder (2019)....
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TL;DR: A formal statistical analysis of three popular claims in the metascientific literature is presented, showing how the use and benefits of such formalism can inform and shape debates about such methodological claims.
Abstract: Current attempts at methodological reform in sciences come in response to an overall lack of rigor in methodological and scientific practices in experimental sciences. However, most methodological reform attempts suffer from similar mistakes and over-generalizations to the ones they aim to address. We argue that this can be attributed in part to lack of formalism and first principles. Considering the costs of allowing false claims to become canonized, we argue for formal statistical rigor and scientific nuance in methodological reform. To attain this rigor and nuance, we propose a five-step formal approach for solving methodological problems. To illustrate the use and benefits of such formalism, we present a formal statistical analysis of three popular claims in the metascientific literature: (a) that reproducibility is the cornerstone of science; (b) that data must not be used twice in any analysis; and (c) that exploratory projects imply poor statistical practice. We show how our formal approach can inform and shape debates about such methodological claims.
51 citations
Cites methods from "Bayesian hypothesis testing for Gau..."
...…flexibility in revealing patterns, such as graphical evaluation of data (Behrens, 1997; Tukey, 1980), exploratory factor analysis (Behrens, 1997; Haig, 2005), principal components regression (Massy, 1965), and Bayesian methods to generate EDA graphs (Gelman, 2003, 2004; Williams and Mulder, 2020)....
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TL;DR: The present study compared the existing suite of methods for maximizing and quantifying the stability and consistency of PMRF networks with a set of metrics for directly comparing the detailed network characteristics interpreted in the literature and concluded that the limited reliability of the detailed characteristics of networks observed here is likely to be common in practice, but overlooked by current methods.
Abstract: Pairwise Markov random field networks—including Gaussian graphical models (GGMs) and Ising models—have become the “state-of-the-art” method for psychopathology network analyses. Recent research has...
48 citations
Cites methods from "Bayesian hypothesis testing for Gau..."
...(p. 10) aBased on jointly estimated Gaussian graphical models (GGMs) derived from polychoric correlations using fused graphical lasso selecting tuning parameters using k-fold cross-validation. bBased on individually estimated GGMs derived from polychoric correlations. cBased on individually estimated GGMs derived from Pearson correlations, excluding the 0.3–3.8% of cases with missing data. dBased on jointly estimated Gaussian graphical models (GGMs) derived from polychoric correlations using fused graphical lasso selecting tuning parameters using information criteria....
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...The depression and anxiety symptom networks in the primary analyses were estimated as Gaussian graphical models (GGMs; i.e., PMRFs for ordinal or continuous data) separately at each wave using graphical LASSO regularization with EBIC, as described above....
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...For internal consistency in the results as well as continuity with the methods in the present study and methods currently applied in the literature, we report Fried et al.’s (2018) original results below, but also reestimated coefficients of similarity, centrality estimates, and calculated all direct metrics of consistency based on the individually estimated GGMs using graphical LASSO regularization with EBIC (i.e., in line with the bootnet and NCT results)....
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...In GGMs based on ordinal data, the edges connecting symptoms represent regularized and fully partialled polychoric correlations....
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...Another promising direction for comparing estimated network structures is in the emergence of methods for Bayesian hypothesis testing in GGMs (Williams & Mulder, 2019)....
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TL;DR: A Bayesian re-analysis that quantifies uncertainty and compares relative evidence for replication and nonreplication versus nonequivalence for each network edge provides a principled roadmap for future assessments of network replicability.
Abstract: Forbes, Wright, Markon, and Krueger claim that psychopathology network characteristics have “limited” or “poor” replicability, supporting their argument primarily with data from two waves of an obs...
31 citations
Cites background or methods from "Bayesian hypothesis testing for Gau..."
...Using the BGGM R package (Williams & Mulder, 2019a), we computed Bayes Factors (H1 ¼ equivalence, H2 ¼ nonequivalence) for each pairwise partial correlation in the depression and anxiety samples furnished by Forbes et al.2 These methods are introduced 1As a side note, the expected sampling…...
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...2We focused on the depression and anxiety sample data because it was indeed sampled from the same population, albeit at different time points. in greater detail in Williams and Mulder (2019b) and Williams et al. (2020)....
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References
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TL;DR: A convenient, although not comprehensive, presentation of required sample sizes is providedHere the sample sizes necessary for .80 power to detect effects at these levels are tabled for eight standard statistical tests.
Abstract: One possible reason for the continued neglect of statistical power analysis in research in the behavioral sciences is the inaccessibility of or difficulty with the standard material. A convenient, although not comprehensive, presentation of required sample sizes is provided here. Effect-size indexes and conventional values for these are given for operationally defined small, medium, and large effects. The sample sizes necessary for .80 power to detect effects at these levels are tabled for eight standard statistical tests: (a) the difference between independent means, (b) the significance of a product-moment correlation, (c) the difference between independent rs, (d) the sign test, (e) the difference between independent proportions, (f) chi-square tests for goodness of fit and contingency tables, (g) one-way analysis of variance, and (h) the significance of a multiple or multiple partial correlation.
38,291 citations
Book•
01 Jan 1995TL;DR: Detailed notes on Bayesian Computation Basics of Markov Chain Simulation, Regression Models, and Asymptotic Theorems are provided.
Abstract: FUNDAMENTALS OF BAYESIAN INFERENCE Probability and Inference Single-Parameter Models Introduction to Multiparameter Models Asymptotics and Connections to Non-Bayesian Approaches Hierarchical Models FUNDAMENTALS OF BAYESIAN DATA ANALYSIS Model Checking Evaluating, Comparing, and Expanding Models Modeling Accounting for Data Collection Decision Analysis ADVANCED COMPUTATION Introduction to Bayesian Computation Basics of Markov Chain Simulation Computationally Efficient Markov Chain Simulation Modal and Distributional Approximations REGRESSION MODELS Introduction to Regression Models Hierarchical Linear Models Generalized Linear Models Models for Robust Inference Models for Missing Data NONLINEAR AND NONPARAMETRIC MODELS Parametric Nonlinear Models Basic Function Models Gaussian Process Models Finite Mixture Models Dirichlet Process Models APPENDICES A: Standard Probability Distributions B: Outline of Proofs of Asymptotic Theorems C: Computation in R and Stan Bibliographic Notes and Exercises appear at the end of each chapter.
16,079 citations
TL;DR: This article reviewed the literature on such tests, pointed out some statistics that should be avoided, and presented a variety of techniques that can be used safely with medium to large samples, and several illustrative numerical examples are provided.
Abstract: In a variety of situations in psychological research, it is desirable to be able to make statistical comparisons between correlation coefficients measured on the same individuals. For example, an experimenter may wish to assess whether two predictors correlate equally with a criterion variable. In another situation, the experimenter may wish to test the hypothesis that an entire matrix of correlations has remained stable over time. The present article reviews the literature on such tests, points out some statistics that should be avoided, and presents a variety of techniques that can be used safely with medium to large samples. Several illustrative numerical examples are provided.
4,245 citations
TL;DR: It is shown that neighborhood selection with the Lasso is a computationally attractive alternative to standard covariance selection for sparse high-dimensional graphs and is hence equivalent to variable selection for Gaussian linear models.
Abstract: The pattern of zero entries in the inverse covariance matrix of a multivariate normal distribution corresponds to conditional independence restrictions between variables. Covariance selection aims at estimating those structural zeros from data. We show that neighborhood selection with the Lasso is a computationally attractive alternative to standard covariance selection for sparse high-dimensional graphs. Neighborhood selection estimates the conditional independence restrictions separately for each node in the graph and is hence equivalent to variable selection for Gaussian linear models. We show that the proposed neighborhood selection scheme is consistent for sparse high-dimensional graphs. Consistency hinges on the choice of the penalty parameter. The oracle value for optimal prediction does not lead to a consistent neighborhood estimate. Controlling instead the probability of falsely joining some distinct connectivity components of the graph, consistent estimation for sparse graphs is achieved (with exponential rates), even when the number of variables grows as the number of observations raised to an arbitrary power.
3,793 citations
TL;DR: An examines methodologies suited to identify such symptom networks and discusses network analysis techniques that may be used to extract clinically and scientifically useful information from such networks (e.g., which symptom is most central in a person's network).
Abstract: In network approaches to psychopathology, disorders result from the causal interplay between symptoms (e.g., worry → insomnia → fatigue), possibly involving feedback loops (e.g., a person may engage in substance abuse to forget the problems that arose due to substance abuse). The present review examines methodologies suited to identify such symptom networks and discusses network analysis techniques that may be used to extract clinically and scientifically useful information from such networks (e.g., which symptom is most central in a person's network). The authors also show how network analysis techniques may be used to construct simulation models that mimic symptom dynamics. Network approaches naturally explain the limited success of traditional research strategies, which are typically based on the idea that symptoms are manifestations of some common underlying factor, while offering promising methodological alternatives. In addition, these techniques may offer possibilities to guide and evaluate therape...
1,824 citations