HDTD: Analyzing High-Dimensional Transposable Data

Installation

You can install the release version of HDTD:

if (!requireNamespace("BiocManager", quietly = TRUE)) install.packages("BiocManager")
BiocManager::install("HDTD")

The source code for the release version of HDTD is available on Bioconductor at:

• http://bioconductor.org/packages/release/bioc/html/HDTD.html

Or you can install the development version of HDTD:

if (!requireNamespace("devtools", quietly = TRUE)) install.packages("devtools")
devtools::install_github("AnestisTouloumis/HDTD")

To use HDTD, you should load the package as follows:

library("HDTD")

Usage

This package offers functions to estimate and test the matrix parameters of transposable data in high-dimensional settings. The term transposable data refers to datasets that are structured in a matrix form such that both the rows and columns correspond to variables of interest and dependencies are expected to occur among rows, among columns and between rows and columns. For example, consider microarray studies in genetics where multiple RNA samples across different tissues are available per subject. In this case, a data matrix can be created with row variables the genes, column variables the tissues and measurements the corresponding expression levels. We expect dependencies to occur among genes, among tissues and between genes and tissues. For more examples of transposable data see references in Touloumis, Marioni and Tavaré (2021), Touloumis, Tavaré and Marioni (2015) and Touloumis, Marioni and Tavaré (2016).

There are four core functions:

• meanmat.hat to estimate the mean matrix of the transposable data,
• meanmat.ts to test the overall mean of the row (column) variables across groups of column (row) variables,
• covmat.hat to estimate the row and column covariance matrix,
• covmat.ts to test the sphericity, identity and diagonality hypothesis test for the row/column covariance matrix.

There are also three utility functions:

• transposedata for interchanging the role of rows and columns,
• centerdata for centering the transposable data around their mean matrix,
• orderdata for rearranging the order of the row and/or column variables.

Example

We replicate the analysis that can be found in the vignette based on the mouse dataset

data(VEGFmouse)

This dataset contains expression levels for 40 mice. For each mouse, the expression levels of 46 genes (rows) that belong to the vascular endothelial growth factor signalling pathway were measured across 9 tissues (adrenal gland, cerebrum, hippocampus, kidney, lung, muscle, spinal cord, spleen and thymus) that are displayed in the columns.

One can estimate the mean relationship of the gene expression levels across the 9 tissues

sample_mean <- meanmat.hat(datamat = VEGFmouse, N = 40)
sample_mean
#> ESTIMATION OF THE MEAN MATRIX 
#> Sample size      = 40 
#> Row variables    = 46 
#> Column variables = 9 
#> 
#> Estimated mean matrix [1:5, 1:5] =
#>       adrenal.1 cerebrum.1 hippocampus.1 kidney.1  lung.1
#> Akt1     0.8399     1.2157        1.0597   1.1469  1.2673
#> Akt2    -0.2333    -0.6201       -0.3881  -0.5524 -0.5359
#> Akt3    -1.0856    -0.4351       -0.5490  -0.2534 -0.6091
#> Arnt     0.1089     0.1898        0.0968   0.2551 -0.1171
#> Casp9    0.0877     0.2600        0.4812   0.3203  0.7416

and test whether the overall gene expression is constant across the 9 tissues:

tissue_mean_test <- meanmat.ts(datamat = VEGFmouse, N = 40, group.sizes = 9)
tissue_mean_test
#> MEAN MATRIX TEST 
#> Sample size      = 40 
#> Row variables    = 46 
#> Column variables = 9 
#> 
#> H_0: a constant mean vector across columns 
#> H_1: not H_0 
#> 
#> Test statistic = 373.5277, p-value < 0.0001

In this case, the overall gene expression is not conserved.

To analyze the gene-wise and tissue-wise dependence structure, one needs to estimate the two covariance matrices:

est_cov_mat <- covmat.hat(datamat = VEGFmouse, N = 40)
est_cov_mat
#> ESTIMATION OF THE ROW AND/OR COLUMN COVARIANCE MATRIX 
#> Sample size      = 40 
#> Row variables    = 46 
#> Column variables = 9 
#> Shrinking        = Both sets of variables 
#> Centered data    = FALSE 
#> 
#> ROW VARIABLES
#> Estimated optimal intensity = 0.0115 
#> Estimated covariance matrix [1:5, 1:5] =
#>          Akt1    Akt2    Akt3    Arnt   Casp9
#> Akt1   0.4139 -0.0248  0.0420 -0.0010  0.1084
#> Akt2  -0.0248  0.3341 -0.0240 -0.0029 -0.0151
#> Akt3   0.0420 -0.0240  0.6954  0.1733 -0.0168
#> Arnt  -0.0010 -0.0029  0.1733  0.4746  0.0850
#> Casp9  0.1084 -0.0151 -0.0168  0.0850  0.5337
#> 
#> COLUMN VARIABLES
#> Estimated optimal intensity = 0.3341 
#> Estimated covariance matrix [1:5, 1:5] =
#>               adrenal.1 cerebrum.1 hippocampus.1 kidney.1  lung.1
#> adrenal.1        0.0368    -0.0006        0.0001  -0.0006  0.0010
#> cerebrum.1      -0.0006     0.0432       -0.0002   0.0000 -0.0034
#> hippocampus.1    0.0001    -0.0002        0.0266   0.0019  0.0000
#> kidney.1        -0.0006     0.0000        0.0019   0.0317  0.0012
#> lung.1           0.0010    -0.0034        0.0000   0.0012  0.0809

Finally, the package allows users to perform hypothesis tests for the covariance matrix of the genes

genes_cov_test <- covmat.ts(VEGFmouse, N = 40)
genes_cov_test
#> HYPOTHESES TESTS FOR THE ROW COVARIANCE MATRIX 
#> Sample size      = 40 
#> Row variables    = 46 
#> Column variables = 9 
#> Centered data    = FALSE 
#> 
#> Diagonality hypothesis test:
#> Test Statistic = 8.6324, p-value < 0.0001
#> 
#> Sphericity hypothesis test:
#> Test Statistic = 132.8086, p-value < 0.0001
#> 
#> Identity hypothesis test:
#> Test Statistic = 30.3864, p-value < 0.0001

and of the tissues:

tissues_cov_test <- covmat.ts(VEGFmouse, N = 40, voi = "columns")
tissues_cov_test
#> HYPOTHESES TESTS FOR THE COLUMN COVARIANCE MATRIX 
#> Sample size      = 40 
#> Row variables    = 46 
#> Column variables = 9 
#> Centered data    = FALSE 
#> 
#> Diagonality hypothesis test:
#> Test Statistic = 1.4866, p-value = 0.0686
#> 
#> Sphericity hypothesis test:
#> Test Statistic = 10.2122, p-value < 0.0001
#> 
#> Identity hypothesis test:
#> Test Statistic = 38.0811, p-value < 0.0001

At a 5% significance level, it appears that the genes are correlated but we do not have enough evidence to reject the hypothesis that the tissues are uncorrelated.

Getting help

The statistical methods implemented in HDTD are described in Touloumis, Marioni and Tavaré (2021), Touloumis, Tavaré and Marioni (2015) and Touloumis, Marioni and Tavaré (2016). Detailed examples of HDTD can be found in Touloumis, Marioni and Tavaré (2016) or in the vignette:

browseVignettes("HDTD")

How to cite

Please use the following guidelines for citing `HDTD' in publication:

To cite the mean matrix hypothesis testing methodology, please use

  Touloumis, A., Tavar\'{e}, S. and Marioni, J.C. (2015). Testing the
  Mean Matrix in High-Dimensional Transposable Data. Biometrics 71 (1),
  157-166

A BibTeX entry for LaTeX users is

  @Article{,
    title = {Testing the Mean Matrix in High-Dimensional Transposable Data},
    author = {Anestis Touloumis and Simon Tavar\'{e} and John C. Marioni},
    journal = {Biometrics},
    year = {2015},
    volume = {71},
    issue = {1},
    pages = {157--166},
    url = {http://onlinelibrary.wiley.com/doi/10.1111/biom.12257/full},
  }

To cite the covariance matrix hypothesis testing methodology, please
use

  Touloumis, A., Marioni, J.C. and Tavar\'{e}, S. (2019+). Hypothesis
  Testing for the Covariance Matrix in High-Dimensional Transposable
  Data with Kronecker Product Dependence Structure. Statistica Sinica

A BibTeX entry for LaTeX users is

  @Article{,
    title = {Hypothesis Testing for the Covariance Matrix in 
         High-Dimensional Transposable Data with Kronecker Product Dependence Structure},
    author = {Anestis Touloumis and John C. Marioni and Simon Tavar\'{e}},
    journal = {Statistica Sinica},
    year = {2019+},
    url = {http://www3.stat.sinica.edu.tw/ss_newpaper/SS-2018-0268_na.pdf},
  }

To cite HDTD or the estimation method for the covariance matrices,
please use

  Touloumis, A., Marioni, J.C. and Tavar\'{e}, S. (2016). HDTD:
  Analyzing multi-tissue gene expression data. Bioinformatics 32 (14),
  2193-2195

A BibTeX entry for LaTeX users is

  @Article{,
    title = {HDTD: Analyzing multi-tissue gene expression data},
    author = {Anestis Touloumis and John C. Marioni and Simon Tavar\'{e}},
    journal = {Bioinformatics},
    year = {2016},
    volume = {32},
    issue = {14},
    pages = {2193--2195},
    url = {https://doi.org/10.1093/bioinformatics/btw224},
  }

References

Touloumis, A., Marioni, J.C. and Tavaré, S. (2016) HDTD: Analyzing Multi-tissue Gene Expression Data. Bioinformatics, 32, 2193–2195.

Touloumis, A., Marioni, J.C. and Tavaré, S. (2021) Hypothesis Testing for the Covariance Matrix in High-Dimensional Transposable Data with Kronecker Product Dependence Structure. Statistica Sinica, 31, 1309–1329.

Touloumis, A., Tavaré, S. and Marioni, J.C. (2015) Testing the Mean Matrix in High-Dimensional Transposable Data. Biometrics, 71, 157–166.