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Network Trees: A Method for Recursively Partitioning Covariance Structures

Published online by Cambridge University Press:  01 January 2025

Payton J. Jones Open the ORCID record for Payton J. Jones [Opens in a new window] ,
Patrick Mair Open the ORCID record for Patrick Mair [Opens in a new window] ,
Thorsten Simon Open the ORCID record for Thorsten Simon [Opens in a new window]  and
Achim Zeileis Open the ORCID record for Achim Zeileis [Opens in a new window]
Payton J. Jones*
Affiliation:
Harvard University
Patrick Mair
Affiliation:
Harvard University
Thorsten Simon
Affiliation:
Universität Innsbruck
Achim Zeileis
Affiliation:
Universität Innsbruck
*
Correspondence should be made to Payton J. Jones, Harvard University, Cambridge, MA, USA. Email: paytonjjones@gmail.com, payton_jones@g.harvard.edu; URL: https://scholar.harvard.edu/paytonjones/home
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Abstract

In many areas of psychology, correlation-based network approaches (i.e., psychometric networks) have become a popular tool. In this paper, we propose an approach that recursively splits the sample based on covariates in order to detect significant differences in the structure of the covariance or correlation matrix. Psychometric networks or other correlation-based models (e.g., factor models) can be subsequently estimated from the resultant splits. We adapt model-based recursive partitioning and conditional inference tree approaches for finding covariate splits in a recursive manner. The empirical power of these approaches is studied in several simulation conditions. Examples are given using real-life data from personality and clinical research.

Keywords

network analysiscorrelation networksrecursive partitioningdecision treesconditional inference
Type
Theory and Methods
Information
Psychometrika , Volume 85 , Issue 4 , December 2020 , pp. 926 - 945
Copyright
Copyright © 2020 The Psychometric Society

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References

American Psychiatric Association.(2013). Diagnostic and statistical manual of mental disorders (DSM-5 ® \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textregistered $$\end{document} ). Washington: American Psychiatric Publishing.Google Scholar
Andrews, D. W. K(1993). Tests for parameter instability and structural change with unknown change point.Econometrica, 61,821–856.CrossRefGoogle Scholar
Boker, S. M., & Martin, M.(2018). A conversation between theory, methods, and data.Multivariate Behavioral Research, 53 (6), 806–819.CrossRefGoogle ScholarPubMed
Borsboom, D.(2017). A network theory of mental disorders.World Psychiatry, 16,5–13.CrossRefGoogle ScholarPubMed
Brandmaier, A. M., von Oertzen, T., McArdle, J. J., & Lindenberger, U.(2013). Structural equation model trees.Psychological Methods, 18(1), 71.CrossRefGoogle ScholarPubMed
Breiman, L., Friedman, J. H., Olshen, R. A., & Stone, C. J.(1984). Classification and regression trees,New York:Chapman & Hall/CRC.Google Scholar
Brown, T. A., Chorpita, B. F., Korotitsch, W., & Barlow, D. H.(1997). Psychometric properties of the depression anxiety stress scales (DASS) in clinical samples.Behaviour Research and Therapy, 35,79–89.CrossRefGoogle ScholarPubMed
Cabrieto, J., Tuerlinckx, F., Kuppens, P., Wilhelm, F. H., Liedlgruber, M., & Ceulemans, E.(2018). Capturing correlation changes by applying kernel change point detection on the running correlations.Information Sciences, 447,117–139.CrossRefGoogle Scholar
Costantini, G., Richetin, J., Preti, E., Casini, E., Epskamp, S., & Perugini, M.(2019). Stability and variability of personality networks. A tutorial on recent developments in network psychometrics.Personality and Individual Differences, 136,68–78.CrossRefGoogle Scholar
Dalege, J., Borsboom, D., van Harreveld, F., van den Berg, H., Conner, M., & van der Maas, H. L.(2016). Toward a formalized account of attitudes: The causal attitude network (CAN) model. Psychological Review, 123,2–22.CrossRefGoogle Scholar
Drasgow, F.(1986). Polychoric and polyserial correlations. Kotz, S., & Johnson, N. L. Encyclopedia of statistical sciences,New York:Wiley 68–74.Google Scholar
Epskamp, S., Borsboom, D., & Fried, E. I.(2018). Estimating psychological networks and their accuracy: A tutorial paper.Behavior Research Methods, 50,195–212.CrossRefGoogle ScholarPubMed
Epskamp, S., Cramer, A. O. J, Waldorp, L. J., Schmittmann, V. D., & Borsboom, D.(2012). qgraph: Network visualizations of relationships in psychometric data.Journal of Statistical Software, 48(4), 1–18.CrossRefGoogle Scholar
Epskamp, S., Rhemtulla, M., & Borsboom, D.(2017). Generalized network psychometrics: Combining network and latent variable models.Psychometrika, 82,904–927.CrossRefGoogle ScholarPubMed
Fokkema, M., Smits, N., Zeileis, A., Hothorn, T., & Kelderman, H.(2018). Detecting treatment-subgroup interactions in clustered data with generalized linear mixed-effects model trees.Behavior Research Methods, 50,2016–2034.CrossRefGoogle ScholarPubMed
Fried, E. I., & Nesse, R. M.(2015). Depression is not a consistent syndrome: An investigation of unique symptom patterns in the STAR*D study.Journal of Affective Disorders, 172,96–102.CrossRefGoogle ScholarPubMed
Friedman, J., Hastie, T., & Tibshirani, R.(2008). Sparse inverse covariance estimation with the graphical lasso.Biostatistics, 9,432–441.CrossRefGoogle ScholarPubMed
Fritz, J., Fried, E. I., Goodyer, I. M., Wilkinson, P. O., & van Harmelen, A.-L.(2018). A network model of resilience factors for adolescents with and without exposure to childhood adversity.Scientific Reports, 8,15774.CrossRefGoogle ScholarPubMed
Gosling, S. D., Rentfrow, P. J., & Swann, W. B. Jr(2003). A very brief measure of the Big-Five personality domains.Journal of Research in Personality, 37,504–528.CrossRefGoogle Scholar
Hanley, G. P., Iwata, B. A., & McCord, B. E.(2003). Functional analysis of problem behavior: A review.Journal of Applied Behavior Analysis, 36,147–185.CrossRefGoogle ScholarPubMed
Hansen, B. E.Approximate asymptotic p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p$$\end{document} values for structural-change tests. (1997). Journal of Business & Economic Statistics, 15,60–67.Google Scholar
Haslbeck, J., & Fried, E. I.(2017). How predictable are symptoms in psychopathological networks? A reanalysis of 18 published datasets.Psychological Medicine, 47,2767–2776.CrossRefGoogle ScholarPubMed
Haslbeck, J. M. B., Borsboom, D., & Waldorp, L. J. (2019). Moderated network models. Multivariate Behavioral Research. https://doi.org/10.1080/00273171.2019.1677207.CrossRefGoogle Scholar
Hjort, N. L., & Koning, A.(2002). Tests for constancy of model parameters over time.Nonparametric Statistics, 14,113–132.CrossRefGoogle Scholar
Hothorn, T., Hornik, K., van de Wiel, M. A., & Zeileis, A.(2006). A lego system for conditional inference.The American Statistician, 60,257–263.CrossRefGoogle Scholar
Hothorn, T., Hornik, K., & Zeileis, A.(2006). Unbiased recursive partitioning: A conditional inference framework.Journal of Computational and Graphical Statistics, 15,651–674.CrossRefGoogle Scholar
Hothorn, T., & Zeileis, A.(2015). partykit: A modular toolkit for recursive partytioning in R.Journal of Machine Learning Research, 16,3905–3909.Google Scholar
Jones, P., Simon, T., & Zeileis, A. (2018). networktree: Recursive Partitioning of Network Models. R package version 1.0.0.Google Scholar
Jones, P. J., Heeren, A., & McNally, R. J.(2017). Commentary: A network theory of mental disorders.Frontiers in Psychology, 8,1305.CrossRefGoogle ScholarPubMed
Komboz, B., Strobl, C., & Zeileis, A.(2018). Tree-based global model tests for polytomous Rasch models.Educational and Psychological Measurement, 78,128–166.CrossRefGoogle ScholarPubMed
Mair, P., & De Leeuw, J.(2010). A general framework for multivariate analysis with optimal scaling: The R package aspect.Journal of Statistical Software, 32,1–23.CrossRefGoogle Scholar
Marsman, M., Borsboom, D., Kruis, J., Epskamp, S., van Bork, R., Waldorp, L., van der Maas, H., & Maris, G.(2018). An introduction to network psychometrics: Relating ising network models to item response theory models.Multivariate Behavioral Research, 53,15–35.CrossRefGoogle ScholarPubMed
McNally, R. J.(2019). The network takeover reaches psychopathology.Behavioral and Brain Sciences, 42,e15.CrossRefGoogle Scholar
Merkle, E. C., Fan, J., & Zeileis, A.(2014). Testing for measurement invariance with respect to an ordinal variable.Psychometrika, 79,569–584.CrossRefGoogle Scholar
Merkle, E. C., & Shaffer, V. A.(2011). Binary recursive partitioning methods with application to psychology.British Journal of Mathematical and Statistical Psychology, 64,161–181.CrossRefGoogle ScholarPubMed
Merkle, E. C., & Zeileis, A.(2013). Tests of measurement invariance without subgroups: A generalization of classical methods.Psychometrika, 78,59–82.CrossRefGoogle ScholarPubMed
Molenaar, P. C.(2004). A manifesto on psychology as idiographic science: Bringing the person back into scientific psychology, this time forever.Measurement, 2,201–218.Google Scholar
Park, J. H., & Sohn, Y.(2019). Detecting structural changes in longitudinal network data.Bayesian Analysis, 15,133–157.Google Scholar
R Development Core Team. (2020). R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing. ISBN 3-900051-07-0.Google Scholar
Salters-Pedneault, K., Tull, M. T., & Roemer, L.(2004). The role of avoidance of emotional material in the anxiety disorders.Applied and Preventive Psychology, 11,95–114.CrossRefGoogle Scholar
Schaefer, J., Opgen-Rhein, R., & Strimmer, K.(2015). GeneNet: Modeling and inferring gene networks.R package version, 1(2), 13.Google Scholar
Schaefer, J., & Strimmer, K.(2004). An empirical bayes approach to inferring large-scale gene association networks.Bioinformatics, 21,754–764.CrossRefGoogle Scholar
Schlosser, L., Hothorn, T., and Zeileis, A. (2019). A unifying view of CTree, MOB, and GUIDE. arXiv:1906.10179, E-Print Archive.Google Scholar
Seibold, H., Zeileis, A., & Hothorn, T.(2016). Model-based recursive partitioning for subgroup analyses.The International Journal of Biostatistics, 12,45–63.CrossRefGoogle ScholarPubMed
Strasser, H., & Weber, C.(1999). On the asymptotic theory of permutation tests.Mathematical Methods of Statistics, 8,220–250.Google Scholar
Strobl, C., Kopf, J., & Zeileis, A.(2015). Rasch trees: A new method for detecting differential item functioning in the Rasch model.Psychometrika, 80,289–316.CrossRefGoogle Scholar
Strobl, C., Malley, J., & Tutz, G.(2009). An introduction to recursive partitioning: Rationale, application, and characteristics of classification and regression trees, bagging, and random forests.Psychological Methods, 14,323–348.CrossRefGoogle ScholarPubMed
Strobl, C., Wickelmaier, F., & Zeileis, A.(2011). Accounting for individual differences in Bradley–Terry models by means of recursive partitioning.Journal of Educational and Behavioral Statistics, 36,135–153.CrossRefGoogle Scholar
van Borkulo, C. D., Boschloo, L., Kossakowski, J. J., Tio, P., Schoevers, R. A., Borsboom, D., & Waldorp, L. J. (2017). Comparing network structures on three aspects: A permutation test. Journal of Statistical Software. Forthcoming.Google Scholar
Wang, T., Merkle, E. C., & Zeileis, A.(2014). Score-based tests of measurement invariance: Use in practice.Frontiers in Psychology, 5,1–11.CrossRefGoogle ScholarPubMed
Wickelmaier, F., & Zeileis, A.(2018). Using recursive partitioning to account for parameter heterogeneity in multinomial processing tree models.Behavior Research Methods, 50,1217–1233.CrossRefGoogle ScholarPubMed
Williams, D. R., Rast, P., Pericchi, L. R., & Mulder, J.(2020). Comparing Gaussian graphical models with the posterior predictive distribution and Bayesian model selection.Psychological Methods, 25,653–672.CrossRefGoogle ScholarPubMed
Zeileis, A.(2006). Implementing a class of structural change tests: An econometric computing approach.Computational Statistics & Data Analysis, 50,2987–3008.CrossRefGoogle Scholar
Zeileis, A., Hornik, K.Generalized M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M$$\end{document}-fluctuation tests for parameter instability. (2007). Statistica Neerlandica, 61,488–508.CrossRefGoogle Scholar
Zeileis, A., Hothorn, T., & Hornik, K.(2008). Model-based recursive partitioning.Journal of Computational and Graphical Statistics, 17,492–514.CrossRefGoogle Scholar