Fast Bayesian Inference in Large Gaussian Graphical Models

Description

The package enables inference of marginal and conditional dependencies from high-dimensional using the method of Leday and Richardson (2019). Inference is carried out by multiple testing of hypotheses about pairwise (marginal or conditional) independence using closed-form Bayes factors. Exact tail probabilities are obtained from the null distributions of the Bayes factors to help address the multiplicity problem and control desired error rates for incorrect edge inclusion. The method is computationally very efficient and allows to address problems with hundreds or thousands of variables.

Details

1. The main function of the package is beam which carries out shrinkage estimation of the (inverse) covariance and compute the (scaled) Bayes factors as well as the tail probabilities (p-values). The function returns an (S4) object of class beam-class that is associated with the following methods:

- summary,beam-method:
provides a summary of inferred (marginal and/or conditional) associations.

- marg,beam-method:
returns a data.frame with marginal correlations, Bayes factors and/or tail probabilities.

- cond,beam-method:
returns a data.frame with partial correlations, Bayes factors and/or tail probabilities.

- mcor,beam-method:
return marginal correlation matrix (scaled posterior expectation of the covariance matrix).

- pcor,beam-method:
return partial correlation matrix (scaled posterior expectation of the inverse covariance matrix).

- plotML,beam-method:
plot log-marginal likelihood of the Gaussian conjugate model as a function of shrinkage parameter.

- plotCor,beam-method:
plot heatmap of marginal (upper triangle) and/or partial (lower triangle) correlation estimates.

2. The function beam.select takes as input an object of class beam-class and carries out edge selection by multiple testing of hypotheses about pairwise (marginal or conditional) independence. The function helps address the multiplicity problem and control different types of error rates (e.g. false discovery rate, family-wise error rate, ...). beam.select returns an (S4) object of class beam.select-class that is associated with the following methods:

- summary,beam.select-method:
provides a summary of inferred (marginal or conditional) associations.

- marg,beam.select-method:
returns a data.frame with marginal correlations, Bayes factors and/or tail probabilities for selected edges.

- cond,beam.select-method:
returns a data.frame with partial correlations, Bayes factors and/or tail probabilities for selected edges.

- bgraph,beam.select-method:
return an igraph object containing the marginal (in)dependence graph.

- ugraph,beam.select-method:
return an igraph object containing the conditional (in)dependence graph.

Author(s)

Authors: Gwenael G.R. Leday and Ilaria Speranza

Maintainer: Gwenael G.R. Leday <gwenael.leday@proton.me>

References

Leday, G.G.R. and Richardson, S. (2019). Fast Bayesian inference in large Gaussian graphical models. Biometrics. 75(4), 1288–1298.

Protein expression data.

Description

Level 3 normalized proteomic data (v3.0) from The Cancer Proteome Alas (http://tcpaportal.org/tcpa). The data comprise the measurements of 189 antibodies obtained from 164 tumor tissue samples (prostate adenocarcinoma) using reverse phase protein arrays (RPPA).

Usage

TCPAprad

Format

A 164 by 189 matrix

Source

The Cancer Proteome Alas (http://tcpaportal.org/tcpa)

References

Li J, Lu Y, Akbani R, Ju Z, Roebuck PL, Liu W, Yang J-Y, Broom BM, Verhaak RGW, Kane DW, Wakefield C, Weinstein JN, Mills GB, Liang H. (2013). TCPA: A Resource for Cancer Functional Proteomics Data. Nature Methods 10(11), 1046-1047.

Examples

data(TCPAprad)
dim(TCPAprad)
TCPAprad[1:5, 1:5]

Bayesian inference in large Gaussian graphical models

Description

This function carries out covariance and inverse covariance estimation within the Gaussian conjugate model. The scale matrix parameter of the inverse-Wishart is set to the identity (default), whereas the degree of freedom parameter is estimated by maximization of the marginal likelihood. The function also computes the Bayes factor and tail probability (p-values) to test the marginal or conditional independence between all pairs of variables.

Usage

beam(X, type = "conditional", return.only = c("cor", "BF", "prob"),
verbose=TRUE, D=NULL)

Arguments

Details

The arguments type and return.only have essentially been introduced for computational and memory savings. Using argument type the user may indicate whether the marginal dependencies ("marginal"), the conditional dependencies ("conditional") or both ("both") are to be inferred. On the other hand, the argument return.only is used to indicate whether the correlations ("cor"), Bayes factors ("BF") or tail probabilities ("prob") should be returned. Default is to return all three quantities for conditional dependencies.

Value

An object of class beam-class

Author(s)

Gwenael G.R. Leday and Ilaria Speranza

References

Leday, G.G.R. and Richardson, S. (2019). Fast Bayesian inference in large Gaussian graphical models. Biometrics.

Examples

# Load data
data(TCPAprad)

# beam
fit <- beam(X = TCPAprad, type="both") 

# Print summary
summary(fit)

# Extract matrix of marginal correlations
mcor(fit)[1:5, 1:5]

# Extract matrix of partial correlations
pcor(fit)[1:5, 1:5]

# Plot log-marginal likelihood of the Gaussian conjugate model
plotML(fit)

# Plot heatmap of marginal (upper triangle) and/or
# partial (lower triangle) correlation estimates
plotCor(fit)

Class beam

Description

An S4 class representing the output of the beam function.

Usage

  ## S4 method for signature 'beam'
print(x, ...)

  ## S4 method for signature 'beam'
show(object)

  ## S4 method for signature 'beam'
summary(object, ...)

  ## S4 method for signature 'beam'
marg(object)

  ## S4 method for signature 'beam'
cond(object)

  ## S4 method for signature 'beam'
mcor(object)

  ## S4 method for signature 'beam'
pcor(object)
  
  ## S4 method for signature 'beam'
postExpSigma(object, vars.method="eb")
  
  ## S4 method for signature 'beam'
postExpOmega(object, vars.method="eb")

  ## S4 method for signature 'beam'
plotML(object, ...)

  ## S4 method for signature 'beam'
plotCor(object, type = object@type, order = 'original', by = "marginal")

  ## S4 method for signature 'beam'
bgraph(object)

  ## S4 method for signature 'beam'
ugraph(object)

Arguments

Slots

table

dat.frame. A data.frame containing marginal and/or partial correlation estimates, Bayes factors and tail probabilities for each edge.

deltaOpt

numeric. Empirical Bayes estimate of hyperparameter delta.

alphaOpt

numeric. Empirical Bayes estimate of hyperparameter alpha.

dimX

numeric. Dimension of the input data matrix X.

type

character. Input argument.)

varlabs

character. Column labels of X.

gridAlpha

matrix. A matrix containing the log-marginal likelihood of the Gaussian conjugate model as a function of a grid of values of alpha and delta.

valOpt

numeric. Maximum value of the log-marginal likelihood of the Gaussian conjugate model.

return.only

character. Input argument.

time

numeric. Running time (in seconds).

TinvStdev

numeric. Square root of partial variances.

s

numeric. Sample variances.

rzij

numeric. Statistics.

Author(s)

Gwenael G.R. Leday and Ilaria Speranza

Edge selection with multiple testing and error control

Description

Infer graphical structures by multiple testing

Usage

beam.select(object, thres = 0.1, method = "BH",
return.only = c(object@return.only, "adj"))

Arguments

Details

The argument method allows to adjust the tail probabilities obtained from the null distributions of the Bayes factors for multiple comparisons. Possible choices are: "holm", "bonferroni", "BH", "BY" and "HC". Apart from "HC", these are passed onto the R function p.adjust from package stats and we refer the user to its documentation for details. The method "HC" provides an optimal decision threshold based on the Higher Criticism score which is computed using the R function hc.thresh from package fdrtool. Again, we refer to the associated documentation for details.

The argument return.only allows to decide which quantities have to be in the output: it could be any subvector of c('cor', 'BF', 'prob', 'adj') (provided that the requested quantities have been computed in the beam object, except for adjusted probabilities). It can also be set to NULL: in this case, only the selected edges will be returned without any additional information. The default value for this argument are the columns present in the beam object plus the adjusted probabilities.

Value

An object of class beam.select-class

Author(s)

Gwenael G.R. Leday and Ilaria Speranza

References

Drton, M., & Perlman, M. D. (2007). Multiple testing and error control in Gaussian graphical model selection. Statistical Science, 430-449.
Goeman, J. J., & Solari, A. (2014). Multiple hypothesis testing in genomics. Statistics in medicine, 33(11), 1946-1978.
Donoho, D., & Jin, J. (2015). Higher criticism for large-scale inference, especially for rare and weak effects. Statistical Science, 30(1), 1-25.
Klaus, B., & Strimmer, K. (2012). Signal identification for rare and weak features: higher criticism or false discovery rates?. Biostatistics, 14(1), 129-143.

Class beam.select

Description

An S4 class representing the output of the beam.select function.

Usage

  ## S4 method for signature 'beam.select'
print(x, ...)

  ## S4 method for signature 'beam.select'
show(object)

  ## S4 method for signature 'beam.select'
summary(object, ...)
  
  ## S4 method for signature 'beam.select'
marg(object)

  ## S4 method for signature 'beam.select'
cond(object)

  ## S4 method for signature 'beam.select'
mcor(object)
  
  ## S4 method for signature 'beam.select'
pcor(object)

  ## S4 method for signature 'beam.select'
plotML(object, ...)

  ## S4 method for signature 'beam.select'
plotAdj(object, type=object@type, order = "original")

  ## S4 method for signature 'beam.select'
bgraph(object)

  ## S4 method for signature 'beam.select'
ugraph(object)

Arguments

Slots

marginal

data.frame. A data.frame containing the marginal correlation estimates, Bayes factors and tail probabilities for the selected edges only.

conditional

data.frame. A data.frame containing the partial correlation estimates, Bayes factors and tail probabilities for the selected edges only.

dimX

numeric. Dimension of the input data matrix X.

type

character. Input type (marginal, conditional or both)

varlabs

character. Column labels of X.

alphaOpt

numeric. Empirical Bayes estimates of hyperparameter alpha.

gridAlpha

matrix. A matrix containing the log-marginal likelihood of the Gaussian conjugate model as a function of a grid of values of alpha and delta.

valOpt

numeric. Maximum value of the log-marginal likelihood of the Gaussian conjugate model

method

character. Input method.

thres

numeric. Input threshold

Author(s)

Gwenael G.R. Leday and Ilaria Speranza

Fast inference of a conditional independence graph

Description

Fast and memory efficient reconstruction of large conditional independence networks.

Usage

lightbeam(X, thres = 0.1, method = "holm", verbose=TRUE)

Arguments

Details

The function returns a sparse upper triangular matrix containing the (adjusted) tails probabilities that are below the given threshold (i.e. significant).

Value

An object of class dgCMatrix-class

Author(s)

Gwenael G.R. Leday

References

Leday, G.G.R. and Richardson, S. (2019). Fast Bayesian inference in large Gaussian graphical models. Biometrics.

Examples

# Load data
data(TCPAprad)

# beam
res <- lightbeam(X = TCPAprad, thres=0.1)