On Nonregularized Estimation of Psychological Networks

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 ($p\ll n$). 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.

Keywords:

• Network models
• partial correlation
• nonregularized
• multiple regression
• model selection
• ${\ell }_1$-regularization

Article Information

Conflict of interest disclosures: Each author signed a form for disclosure of potential conflicts of interest. No authors reported any financial or other conflicts of interest in relation to the work described.

Ethical principles: The authors affirm having followed professional ethical guidelines in preparing this work. These guidelines include obtaining informed consent from human participants, maintaining ethical treatment and respect for the rights of human or animal participants, and ensuring the privacy of participants and their data, such as ensuring that individual participants cannot be identified in reported results or from publicly available original or archival data.

Funding: This work was supported by Grant R01AG050720 from the National Institute on Aging of the National Institutes of Health (to P.R.), The National Academies of Sciences, Engineering, and Medicine FORD foundation pre-doctoral fellowship (to D.R.W.P.), and The National Science Foundation Graduate Research Fellowship (to D.R.W.).

Role of the funders/sponsors: None of the funders or sponsors of this research had any role in the design and conduct of the study; collection, management, analysis, and interpretation of data; preparation, review, or approval of the manuscript; or decision to submit the manuscript for publication.

Acknowledgments: The authors thank the three reviewers and the editor for their comments on prior versions of this manuscript. The ideas and opinions expressed herein are those of the authors alone, and endorsement by the authors’ institution, the National Institute on Aging or the National Institutes of Health is not intended and should not be inferred.

Notes

Notes

1 F – 1 score = $\frac{2TP}{2TP+\mathit{FP}+\mathit{FN}}.$ TP and FP denote the number of true and false positives, whereas FN is the number of false negatives.

2 The same procedure applies to BIC.

3 The FDR is defined as $\frac{\text{FP}}{\text{TP}+\text{FP}}.$ This was computed by approximating the number of false and true positives, which was done in the preceding two paragraphs, then solving the FDR equation.

4 We have implemented both best subset selection and backwards elimination in the accompanying R package.

5 To understand the inferential challenges for Lasso estimates, and recently proposed methods for inference, we recommend chapter six (Statistical Inference) of Hastie et al. (2015).