Bernal Arzola, Victor

Bischoff, Rainer

Grzegorczyk, Marco

Guryev, Victor

Horvatovich, Peter

English

Gaussian Graphical Models (GGMs) are important probabilistic graphical models in Statistics. Inferring a GGM’s structure from data implies computing the inverse of the covariance matrix (i.e. the precision matrix). When the number of variables p is larger than the sample size n, the (sample) covariance estimator is not invertible and therefore another estimator is required. Covariance estimators based on shrinkage are more stable (and invertible), however, classical hypothesis testing for the ”shrunk” coefficients is an open challenge. In this paper, we present an exact null-density that naturally includes the shrinkage, and allows an accurate parametric significance test that is accurate and computationally efficient.<br/

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   University of GroningenSignificance Tests for Gaussian Graphical Models Based on Shrunken DensitiesBernal Arzola, Victor; Guryev, Victor; Bischoff, Rainer; Horvatovich, Peter; Grzegorczyk,MarcoPublished in: proceedings  of  the  33rd  Inter- national Workshop on Statistical Modelling (IWSM), University of Bristol, UK,16-20 July 2018IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionFinal author's version (accepted by publisher, after peer review)Publication date:2018Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Bernal Arzola, V., Guryev, V., Bischoff, R., Horvatovich, P., & Grzegorczyk, M. (2018). Significance Testsfor Gaussian Graphical Models Based on Shrunken Densities. In proceedings of the 33rd Inter- nationalWorkshop on Statistical Modelling (IWSM), University of Bristol, UK, 16-20 July 2018 (Vol. 2, pp. 27).University of Bristol.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 28-12-2022Significance Tests for Gaussian GraphicalModels Based on Shrunken DensitiesVictor Bernal 12, Victor Guryev3, Rainer Bischoff2, PeterHorvatovich2, Marco Grzegorczyk 11 Johann Bernoulli Institute, University of Groningen, Groningen, NL.2 Department of Pharmacy, Analytical Biochemistry, University of Groningen,Groningen, NL.3 Universitair Medisch Centrum Groningen (UMCG), ERIBA, University of Gronin-gen, Groningen, NL.E-mail for correspondence: v.a.bernal.arzola@rug.nlAbstract: Gaussian Graphical Models (GGMs) are important probabilistic graph-ical models in Statistics. Inferring a GGM’s structure from data implies comput-ing the inverse of the covariance matrix (i.e. the precision matrix). When thenumber of variables p is larger than the sample size n, the (sample) covarianceestimator is not invertible and therefore another estimator is required. Covari-ance estimators based on shrinkage are more stable (and invertible), however,classical hypothesis testing for the ”shrunk” coefficients is an open challenge. Inthis paper we present an exact null-density that naturally includes the shrinkage,and allows an accurate parametric significance test that is accurate and compu-tationally efficient.Keywords: Gaussian Graphical Models; Shrinkage; Genetic Networks, ”small n,large ” problem.1 IntroductionGaussian Graphical Models (GGMs) are important network models inStatistics. A GGM is represented as a network where each variable is anode, and an edge is present between a pair of nodes if their respectivepartial correlation is (statistically) significant. Partial correlations measurelinear dependences between a pair of variables adjusted for all other nodes.Inferring the matrix of pair-wise partial correlations (i.e. the GGM’s struc-ture) demands the estimation of the covariance matrix Ĉ and its inverse,This paper was published as a part of the proceedings of the 33rd Inter-national Workshop on Statistical Modelling (IWSM), University of Bristol, UK,16-20 July 2018. The copyright remains with the author(s). Permission to repro-duce or extract any parts of this abstract should be requested from the author(s).2 Significance Tests for Gaussian Graphical Modelstherefore the importance that the covariance estimator is invertible, andwell-conditioned (i.e. numerical errors are not magnified). The sample co-variance estimator Ĉsm with p variables and n samples is not invertibleif n p, thus another estimator must be employed. This is a commonscenario in systems biology (e.g large number of genes with few measure-ments), and is usually refered to as the ”small n, large p” problem, sym-bolically n p.Covariance estimators based on shrinkage produce a more stable estimatorby using a (convex) linear combination of Ĉsm with a target estimator T(e.g. a diagonal matrix). The result is a well-conditioned estimator, and itsinverse can be used to compute the ”shrunk” partial correlations. A sig-nificance test have been developed by Schäfer, J. and Strimmer, K. (2005)but it does not take the shinkage intensity into account. This is an openand important challenge as the reconstruction is a multiple testing prob-lem (testing p(p−1)2 edges), thus an slight bias would translate into an errorrepeated systematically during the inference. In this work we aim to obtainan exact density that includes the shrinkage effects. Our empirical resultsin Section 3 demonstrate that this leads to a substantial improvement overthe earlier approach.2 Shrinkage based Gaussian Graphical ModelsPartial correlations are a measure of linear dependence between two vari-ables adjusting the effects coming from all other variables. GGMs are undi-rected graphical models represented by a matrix of partial correlations. Thematrix entry ρij in a GGM represents the partial correlation between thevariables i and j and can be computed from the inverse C−1 of the covari-ance matrix C,ρij = −C−1ij2√C−1ii2√C−1jj(1)where C needs to be estimated from the data. However, when n ≤ p thesample covariance estimator Ĉsm is ill-conditioned and cannot be used.Instead, the shrinkage based estimator Ĉ[λ] is a linear combination ofĈsm with a target matrix T in the form Ĉ[λ] = λT + (1 − λ)Ĉsm, whereλ ∈ [0, 1]. The resulting estimator is well-conditioned, and is implementedin the widely used R package GeneNet (see. Schäfer, J. and Strimmer,K. (2005)) where λ is chosen following an optimization criteria. Moreover,significance is tested with the density of the standard partial correlationf(ρ, k).In the same way the correlation matrix R (i.e. the standarized Ĉsm) canbe combined with (or shrunk towards) the identity matrix I. In this caseBernal et al. 3the diagonal elements of R (i.e. the variances) remains equal to 1, andthe off-diagonal rij (i.e. the pair-wise correlations coefficients) are scaledby a factor of (1− λ). Therefore, the probability density function (pdf) ofthe ”shrunk” correlation r[λ] can be found via the transformation r[λ] =(1− λ)r,f(r[λ], k) =[(1− λ)2 − (r[λ])2]( k−32 )Beta( 12 ,k−12 )(1− λ)(1− λ)k−32(2)where k denotes the degrees of freedom. We now use a classical result fromFisher (1924) to obtain the probability density of the ”shrunk” partialcorrelation f(r[λ], k). Here it was proved that ρ and r have the same densitydiffering only in the value of k. The main idea is to study the problem insubject space where each random variable is represented with a vector, andprobabilistic relationships can be interpreted geometrically (see Wickens,T. D. (2014)). For the purpose of illustration, lets consider three randomvariables X, Y , and Z with expectation equal to zero (i.e. E[X] = E[Y ] =E[Z] = 0). Given n data points for each variable, the corresponding randomvectors ~x, ~y, and ~z are in an space of dimension n. The correlation betweenX and Y can be writen asr =∑ni=1 xiyi√∑ni=1 x2i√∑nj=1 y2j= cos(∠~x, ~y) (3)where the last equality comes from the product ~x · ~y = ||~x||||~y||cos(∠~x, ~y)under the Euclidean norm. The rationale behind the proof is that r is re-lated to the angle between the vectors (see Eq 3), and that this angle isinvariant under rotations of the coordinate axes. Therefore, the rotationcan be peformed in such a way that one of the axis coincides with ~z, andconditioning on the random variable Z is equivalent to decreasing k byone. This procedure can be continued by rotating again, and conditioningover a new variable. The same idea can be used for r[λ] (as it is a scaledcorrelation), andf(ρ[λ], k) is the probability density of ρ[λ].To test the null hypothesis H0 : (the ”shrunk” partial correlation is zero)with f(ρ[λ], k) we propose the following approach: Suppose the data Dconsists of p variables and sample size n.1. For D find the optimal shrinkage λopt, and estimate ρ[λopt]ij (Schäfer,J. and Strimmer, K. (2005)).2. Estimate k under H0:(a) Simulate data of length n from H0 (i.e. the precision matrixis the p x p identity), and using λopt (from step 1) infer thenull-hypothetic coefficients ρ[λopt]0ij.(b) Find k̂ by maximizing the likelihood of the ρ[λopt]0ijwith Eq 2.4 Significance Tests for Gaussian Graphical Models3. Test the coefficients ρ[λopt]ij from step 1 with f(ρ[λopt]ij , k̂).We will refer to this approach as ”Shrunk MLE” in the following section.3 ResultsIn this section we provide empirical evidence that the proposed ”ShrunkMLE” approach is significantly superior to GeneNet 1.2.13. We cross-compare the methods on synthetic, and real gene expression data by testingthe null hypothesis H0 : (the ”shrunk” partial correlation is zero). The Pos-itive Predictive Values (PPVs) are compared on (i) syntethic data were thetrue structure is known, and (ii) on real data were we use MC (a computa-tionally expensive approach) to generate a reliable goldstandard network.To simulate GGMs with a fixed percentage of edges δ we used GeneNet (forthe algorithm see Schäfer, J. and Strimmer, K. (2005)). Figure 1 shows thePPV = TP(TP+FP ) for different samples sizes n.0.000.250.500.751.0050 100 150 200nPPV0.000.250.500.751.0050 100 150 200nGeneNetMethodMCShrunk𝛼 = 0.05 (BH) 𝛼 = 0.05FIGURE 1. Positive predictive value. On the left : GGM simulation forp = 100, and n ranging from 10 to 200 in steps of size 10. The simulated GGMstructure has 148 correlations (i.e. δ = 0.03). The Positive predictive values(PPV) are computed using p-values at α = 0.05. Dots (and bars) represent theaverage PPV (± 2 standard errors) over 25 repeated simulations, and MC wasperformed 15 times. Three methods are displayed:GeneNet (in dashed red), MC(green), and Shrunk MLE (thick blue). Note that the green and blue curvesare superposed.On the left : The PPV are computed using Benjamini Hochbergadjusted p-values at α = 0.05.The results show a close agreement between the PPV obtained by MC,and with ”Shrunk MLE”. On the other hand, GeneNet has a lower PPVsuggesting that it learns too many False Positives (FPs) Figure 2. We alsoBernal et al. 5analyzed Escherichia Coli microarray data from Schmidt-Heck, W. et al.(2004), consisting of stress temporal response of 102 genes in 9 time pointsafter IPTG (induction of the recombinant protein SOD). Figure 2 showsa Venn diagram for the edges found by each method, here we can observethat ”Shrunk MLE” learns nearly the same edges as MC.10020030050 100 150 200nGeneNetMethodFPGeneNet Shrunk MLEMC 4673000220200238MCShrunk𝛼 = 0.05FIGURE 2. False positives and Empirical results. On the left : GGM simu-lation for p = 100, and n ranging from 10 to 200 in steps of size 10. The simulatedGGM structure has 148 correlations (i.e. δ = 0.03). The False Positives (FPs) arecomputed using p-values at α = 0.05. Dots (and bars) represent the average FPs(± 2 standard errors) over 25 repeated simulations, and MC was performed 15times. Three methods are displayed: GeneNet (in dashed red), MC (green), andShrunk MLE (thick blue). Note that the green and blue curves are superposed.On the right : Venn diagram for the inferred edges in E. coli. Taking MC as agold standard GeneNet ’s sensitivity is 258/258=1, with a low PPV of 258/478≈ 0.54. Shrunk MLE has a slightly decreased sensitivity of 238/258 ≈ 0.92, butyields a perfect PPV of 1.A GO enrichment analysis (http://geneontology.org/) with False Discov-ery Rate (FDR< 0.05) shows that the connected genes belong signifi-cantly to stress response (FDR= 2.02E−02), in contrast with GeneNet(FDR=7.73E−02). This suggests a dillution of the GO’s significance duehigher rate of FPs. The strongest connections were lacA–lacZ, lacY– lacZ,and lacA–lacY related to the lac operon (known to be triggered by IPTG).4 ConclusionsGaussian Graphical Models (GGMs) are an important tool for networklearning. Reconstructing the network demands the estimation of the co-variance matrix, which is ill-conditioned if the sample size is smaller thanthe number of variables. Covariance estimators based in shrinkage make the6 Significance Tests for Gaussian Graphical Modelscovariance matrix invertible, however, for an accurate (parametric) signifi-canc tests the shrinkage value needs to be included, otherwise the inferencewill have a systematic error (e.g. biased p-values). In this paper a newshrunk density was introduced, and to our knowledge is the only test thatincludes the regularization effects. In the ”small n, large p” scenario thenew density allows an accurate inference for any shrinkage value.ReferencesFisher, R. (1924). The Distribution of the Partial Correlation Coefficient.Metron, 3, 329 – 332.Schäfer, J. and Strimmer, K. (2005). A Shrinkage Approach to Large-ScaleCovariance Matrix Estimation and Implications for Funtional Ge-nomics. Statistical Applications in Genetics and Molecular Biology,4, 1 – 30.Schmidt-Heck, W. et al. (2004). Reverse Engineering of the Stress Responseduring Expression of a Recombinant Protein. Proceedings of the EU-NITE symposium, Aachen, 10 – 12.Wickens, T. D. (2014). The Geometry of Multivariate Statistics. Psychol-ogy Press.

Significance Tests for Gaussian Graphical Models Based on Shrunken Densities

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Significance Tests for Gaussian Graphical Models Based on Shrunken Densities

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