Partial correlation matrix estimation using ridge penalty followed by thresholding and re‐estimation

Ha, Min Jin; Sun, Wei

doi:10.1111/biom.12186

Cited by 19 publications

(31 citation statements)

References 27 publications

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“…In these situations, the covariance matrix cannot be inverted due to singularity (Hartlap, Simon, & Schneider, ), which is overcome by the glasso method. Accordingly, most of the simulation work has focused on high‐dimensional settings ( n < p ), where model selection consistency is not typically evaluated in more common asymptotic settings ( n → ∞; Ha & Sun, ; Heinävaara, Leppä‐aho, Corander, & Honkela, ; Peng, Wang, Zhou, & Zhu, ). Further, in behavioural science applications, the majority of network models are fitted in low‐dimensional settings ( p ≪ n ; Costantini et al ., ; Rhemtulla et al ., ).…”

Section: Introductionmentioning

confidence: 99%

Back to the basics: Rethinking partial correlation network methodology

Williams

Rast

2019

Brit J Math & Statis

132

106

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The Gaussian graphical model (GGM) is an increasingly popular technique used in psychology to characterize relationships among observed variables. These relationships are represented as elements in the precision matrix. Standardizing the precision matrix and reversing the sign yields corresponding partial correlations that imply pairwise dependencies in which the effects of all other variables have been controlled for. The graphical lasso (glasso) has emerged as the default estimation method, which uses ' 1based regularization. The glasso was developed and optimized for high-dimensional settings where the number of variables (p) exceeds the number of observations (n), which is uncommon in psychological applications. Here we propose to go 'back to the basics', wherein the precision matrix is first estimated with non-regularized maximum likelihood and then Fisher Z transformed confidence intervals are used to determine non-zero relationships. We first show the exact correspondence between the confidence level and specificity, which is due to 1 minus specificity denoting the false positive rate (i.e., a). With simulations in low-dimensional settings (p ( n), we then demonstrate superior performance compared to the glasso for detecting the non-zero effects. Further, our results indicate that the glasso is inconsistent for the purpose of model selection and does not control the false discovery rate, whereas the proposed method converges on the true model and directly controls error rates. We end by discussing implications for estimating GGMs in psychology.

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Section: Introductionmentioning

confidence: 99%

Back to the basics: Rethinking partial correlation network methodology

Williams

Rast

2019

Brit J Math & Statis

132

106

View full text Add to dashboard Cite

show abstract

“…In these situations, the covariance matrix cannot be inverted due to singularity (Hartlap, Simon, & Schneider, 2007), which is overcome by the glasso method. Accordingly, most of the simulation work has focused on high-dimensional settings (n < p), where model selection consistency is not typically evaluated in more common asymptotic settings (n → ∞) (Ha & Sun, 2014;Heinävaara, Leppä-aho, Corander, & Honkela, 2016;Peng, Wang, Zhou, & Zhu, 2009). Further, in behavioral science applications, the majority of network models are t in low-dimensional settings (p n) (Costantini et al, 2015;Rhemtulla et al, 2016).…”

Section: Introductionmentioning

confidence: 99%

Back to the Basics: Rethinking Partial Correlation Network Methodology

Williams¹,

Rast²

2018

Preprint

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Gaussian graphical models are an increasingly popular technique in psychology to characterize relationships among observed variables. These relationships are represented as covariances in the precision matrix. Standardizing this covariance matrix and reversing the sign yields corresponding partial correlations that imply pairwise dependencies in which the e ects of all other variables have been controlled for. In order to estimate the precision matrix, the graphical lasso (glasso) has emerged as the default estimation method, which uses 1 -based regularization. Glasso was developed and optimized for high dimensional settings where the number of variables (p) exceeds the number of observations (n) which are uncommon in psychological applications. Here we propose to go "back to the basics", wherein the precision matrix is rst estimated with non-regularized maximum likelihood and then Fisher Z-transformed con dence intervals are used to determine non-zero relationships. We rst show the exact correspondence between the con dence level and speci city, which is due to 1 -speci city denoting the false positive rate (i.e., α). With simulations in low-dimensional settings (p n), we then demonstrate superior performance compared to glasso for determining conditional relationships, in addition to frequentist risk measured with various loss functions. Further, our results indicate that glasso is inconsistent for the purpose of model selection, whereas the proposed method converged on the true model with a probability that approached 100%. We end by discussing implications for estimating Gaussian graphical models in psychology.

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“…There are alternative forms of regularization that could improve predictive performance (e.g., non-convex penalties, Kim, Lee, & Kwon, 2018). For large p/n ratios (with small n) it could be advantageous to employ 2regularization (Ha & Sun, 2014;Kuismin, Kemppainen, & Sillanpää, 2017;Van Wieringen & Peeters, 2016). On the one hand, the ridge penalty does not push values all the way to zero and thus overcomes the issue of Glasso EBIC selecting empty networks (see Experiment 1).…”

Section: Limitationsmentioning

confidence: 99%

Why Overfitting is Not (Usually) a Problem in Partial Correlation Networks

Dr¹,

Rodriguez²

2020

Preprint

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Network psychometrics is undergoing a time of methodological reflection. In part, this was spurred by the revelation that l1-regularization does not reduce spurious associations in partial correlation networks. In this work, we address another motivation for the widespread use of regularized estimation: the thought that it is needed to mitigate overfitting. We first clarify important aspects of overfitting and the bias-variance tradeoff that are especially relevant for the network literature, where the number of nodes or items in a psychometric scale are not largecompared to the number of observations (i.e., a low p/n ratio). This revealed that bias and especially variance are most problematic in p=n ratios rarely encountered. We then introduce a nonregularized method, based on classical hypothesis testing, that fulfills two desiderata: (1) reducing or controlling the false positives rate and (2) quelling concerns of overfitting by providing accurate predictions. These were the primary motivations for initially adopting the graphical lasso (glasso). In several simulation studies, our nonregularized method provided more than competitive predictive performance, and, in many cases, outperformed glasso. Itappears to be nonregularized, as opposed to regularized estimation, that best satisfies these desiderata. We then provide insights into using our methodology. Here we discuss the multiple comparisons problem in relation to prediction: stringent alpha levels, resulting in a sparse network, can deteriorate predictive accuracy. We end by emphasizing key advantages of our approach that make it ideal for both inference and prediction in network analysis.

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Partial correlation matrix estimation using ridge penalty followed by thresholding and re‐estimation

Cited by 19 publications

References 27 publications

Back to the basics: Rethinking partial correlation network methodology

Back to the basics: Rethinking partial correlation network methodology

Back to the Basics: Rethinking Partial Correlation Network Methodology

Why Overfitting is Not (Usually) a Problem in Partial Correlation Networks

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