discordant 1.28.0
Discordant is an R package that identifies pairs of features that correlate differently between phenotypic groups, with application to -omics data sets. Discordant uses a mixture model that “bins” molecular feature pairs based on their type of coexpression or coabbundance. More information on the algorithm can be found in (Siska, Bowler, and Kechris (2015), Siska. and Kechris (2016)). The final output are posterior probabilities of differential correlation. This package can be used to determine differential correlation within one –omics data set or between two –omics data sets (provided that both –omics data sets were taken from the same samples). Also, the type of data can be any type of –omics with normal or non-normal distributions. Some examples are metabolomics, transcriptomic, proteomics, etc.
The functions in the Discordant package provide a simple pipeline for intermediate R users to determine differentially correlated pairs. The final output is a table of molecular feature pairs and their respective posterior probabilities. Functions have been written to allow flexibility for users in how they interpret results, which will be discussed further. Currently, the package only supports the comparison between two phenotypic groups (e.g., disease versus control, mutant versus wildtype).
Discordant is originally derived from the Concordant algorithm written by (Lai et al. (2007), Lai et al. (2014)). It was used to determine concordance between microarrays. We have applied it to determine differential correlation of features between groups (Siska, Bowler, and Kechris (2015), Siska. and Kechris (2016)).
Using a three component mixture model and the Expectation Maximization (EM) algorithm, the model predicts if the correlation coefficients in phenotypic groups 1 and 2 for a molecular feature pair are dissimilar (Siska, Bowler, and Kechris (2015)). The correlation coefficients are generated for all possible molecular feature pairs witin an -omics dataset or between two -omics data sets. The correlation coefficients are transformed into z scores using Fisher’s transformation. The three components are -, + and 0 which correspond respectively to a negative, positive or no correlation. Molecular features that have correlation coefficients in different components are considered differentially correlated, as opposed to when correlation coefficients are in the same component then they are equivalently correlated.
\[ \begin{array}{c|c c c} \text{} & \text{0} & \text{-} & \text{+}\\ \hline 0 & 1 & 2 & 3 \\ - & 4 & 5 & 6 \\ + & 7 & 8 & 9 \\ \end{array} \]
The class matrix (above) contains the classes that represent all possible paired-correlation scenarios. These scenarios are based off the components in the mixture models. Molecular features that have correlation coefficients in different components are considered differentially correlated, as opposed to when correlation coefficients are in the same component they are equivalently correlated. This can be visualized in the class matrix, where the rows represent the components for group 1 and the columns represent the components for group 2. The classes on the diagonal represent equivalent correlation (1, 5 and 9), and classes in the off-diagonal represent differential correlation (2, 3, 4, 6, 8).
After running the EM algorithm, we have 9 posterior probabilities for each molecular feature pair that correspond to the 9 classes in the class matrix. Since we want to summarize the probability that the molecular feature pair is differentially correlated, we sum the posterior probabilities representing the off-diagonal classes in the class matrix.
The following data sets are provided by discordant and will be used in the examples which follow. All data sets are originally from the Cancer Genome Atlas (TCGA) and can be found at http://cancergenome.nih.gov/.
The data sets are provided as described above with no other modifications.
Throughout this vignette we will use the
data sets TCGA_GBM_miRNA_microarray and TCGA_GBM_transcript_microarray to
demonstrate discordant’s essential functionality. They are
loaded as follows:
“Within” –omics refers to when Discordant analysis is performed within one –omics dataset where all molecular features within a -omics dataset are paired to each other (e.g. transcript-transcript pairs in a transcriptomics experiment).
“Between” -omics refers to analysis of two -omics data sets. Molecular feature pairs analyzed are between the two -omics, (e.g. transcript-protein, protein- metabolite) are paired.
discordant provides tools for both within and between -omics analysis that will be described in the sections that follow.
In our work, we found that features with outliers would skew correlation and
cause false positives. Our approach was to filter out features that had large
outliers. With normal data, such as in gene expression data from microarrays,
Grubbs’ test can be used.
The null hypothesis is that there are no outliers in the data, and so features
with p-value \(\ge\) 0.05 are kept. A simple R function is found in the
outliers R package as grubbs.test().
Determining outliers in non-normal data is more complicated. We used the median absolute deviation (MAD). Normally, features are filtered if they are outside 2 or 3 MADs from the median (Leys et al. (2013)). This is not completely applicable to sequencing data, because sequencing data has large variance and a non- symmetrical distribution. Therefore we used the ‘split MAD’ approach (Magwene et al. (2011)). A left MAD is determined based on data left to the median and a right MAD is determined based on data to the right of the median. If there are any feature outside a factor of the left or right MAD from the median, they are filtered out.
discordant provides splitMADOutlier() to identify features
with outliers using MAD. The
number of MAD outside of the median can be changed with option
threshold. Another option is filter0 which if TRUE
will filter out any feature with at least one 0. Arguments returned are
mat.filtered, which is the filtered matrix and index which
is the index of features that are retained in mat.filtered.
To run the Discordant algorithm, correlation vectors respective to each group
are necessary for input, which are easy to create using the function
createVectors(). Each correlation coefficient represents the correlation
between two molecular features. The type of molecular feature pairs depends if
a within -omics or between -omics analysis is performed. Correlation between
molecular features in the same -omics dataset is within -omics, and correlation
between molecular features in two different -omics datasets is between -omics.
Whether or not within -omics or between -omics analysis is performed depends on
whether one or two matrices are parameters for this function. The arguments for
createVectors() are:
createVectors() is then run as follows:
groups <- c(rep(1,10), rep(2,20))
# Within -omics
wthn_vectors <- createVectors(x = TCGA_GBM_transcript_microarray,
groups = groups)
# Between -omics
btwn_vectors <- createVectors(x = TCGA_GBM_miRNA_microarray,
y = TCGA_GBM_transcript_microarray,
groups = groups)createVectors() returns a list with two elements, v1 and v2, which are the
correlation vectors of molecular feature pairs corresponding to
samples labeled 1 and 2 using the groups argument, respectively. Each
vector is a numeric named vector with names indicating each feature in the pair
separated by an underscore. Below are the first few correlations for each group,
first from the within -omics analysis and second from the between -omics
analysis.
## A_23_P138644_A_23_P24296 A_23_P138644_A_24_P345312 A_23_P138644_A_24_P571870
## 0.36969697 -0.26060606 -0.13939394
## A_23_P138644_A_32_P71885 A_23_P138644_A_32_P82889 A_23_P138644_A_23_P105264
## -0.09090909 -0.04863244 -0.69696970## A_23_P138644_A_23_P24296 A_23_P138644_A_24_P345312 A_23_P138644_A_24_P571870
## 0.216541353 -0.433082707 -0.254135338
## A_23_P138644_A_32_P71885 A_23_P138644_A_32_P82889 A_23_P138644_A_23_P105264
## -0.285714286 -0.009022556 -0.154887218## hsa-miR-19b-5p_A_23_P138644 hsa-miR-19b-5p_A_23_P24296
## -0.4303030 -0.6000000
## hsa-miR-19b-5p_A_24_P345312 hsa-miR-19b-5p_A_24_P571870
## 0.1878788 -0.3939394
## hsa-miR-19b-5p_A_32_P71885 hsa-miR-19b-5p_A_32_P82889
## 0.3818182 -0.4060606## hsa-miR-19b-5p_A_23_P138644 hsa-miR-19b-5p_A_23_P24296
## 0.44060150 -0.02857143
## hsa-miR-19b-5p_A_24_P345312 hsa-miR-19b-5p_A_24_P571870
## -0.03308271 0.26015038
## hsa-miR-19b-5p_A_32_P71885 hsa-miR-19b-5p_A_32_P82889
## -0.08571429 -0.37744361The function createVectors() provides several options for correlation metrics
using the argument cor.method. The methods provided include "spearman" (the
default metric), "pearson", "bwmc", and "sparcc". For information and
comparison of Spearman, Pearson and biweight midcorrelation (bwmc) see
Song et al
(Song, Langfelder, and Horvath (2012)). We have also investigated correlation metrics in Discordant in relation
to sequencing data and found Spearman’s correlation had the best performance
(Siska. and Kechris (2016)).
The algorithm for SparCC was introduced by Friedman et al. (Friedman and Alm (2012)), and we use code provided by Huaying Fang (Fang et al. (2015)).
The Discordant Algorithm is implemented in the the function discordantRun()
which requires two correlation vectors and the original data. If the user wishes
to generate their own correlation vector before inputting the data set,
they can do so. However, the function will return an error message if the
dimensions of the data sets inserted do not match the correlation vector.
discordantRun() is called as follows:
# Within -omics
wthn_result <- discordantRun(v1 = wthn_vectors$v1,
v2 = wthn_vectors$v2,
x = TCGA_GBM_transcript_microarray)
# Between -omics
btwn_result <- discordantRun(v1 = btwn_vectors$v1,
v2 = btwn_vectors$v2,
x = TCGA_GBM_miRNA_microarray,
y = TCGA_GBM_transcript_microarray)The posterior probability output of the Discordant algorithm are the
differential correlation posterior probabilities (the sum of the off-diagonal of
the class matrix described above). If the user wishes to observe more
detailed information, alternative outputs are available. discordantRun() has
six outputs:
discordPPMatrix
Matrix of differential correlation posterior probabilities where rows and columns reflect features. If only x was inputted, then the number of rows and columns are number of features in x. The rows and column names are the feature names, and the upper diagonal of the matrix are NAs to avoid repeating results. If x and y are inputted, the number of rows is the feature size of x and the number of columns the feature size of y. The row names are features from x and the column names are features from y.
## A_23_P138644 A_23_P24296 A_24_P345312 A_24_P571870
## A_23_P138644 NA NA NA NA
## A_23_P24296 0.2878537 NA NA NA
## A_24_P345312 0.8912767 0.60018437 NA NA
## A_24_P571870 0.5241415 0.14509976 0.223684 NA
## A_32_P71885 0.6669908 0.09268763 0.808103 0.2945575## A_23_P138644 A_23_P24296 A_24_P345312 A_24_P571870
## hsa-miR-19b-5p 0.97651996 0.38597448 0.54913355 0.30547498
## hsa-miR-206-5p 0.86697876 0.03064989 0.11030081 0.21411118
## hsa-miR-369-5p 0.04519821 0.08113338 0.90726304 0.04050697
## hsa-miR-374-5p 0.70594948 0.03692205 0.07830365 0.58323497
## hsa-miR-376a-5p 0.21071194 0.04581947 0.90287297 0.05971681discordPPVector
Vector of differential correlation posterior probabilities. The length is the number of feature pairs. The names of the vector are the feature pairs.
## A_23_P138644_A_23_P24296 A_23_P138644_A_24_P345312 A_23_P138644_A_24_P571870
## 0.28785373 0.89127669 0.52414148
## A_23_P138644_A_32_P71885 A_23_P138644_A_32_P82889 A_23_P138644_A_23_P105264
## 0.66699081 0.03938042 0.89322774## hsa-miR-19b-5p_A_23_P138644 hsa-miR-19b-5p_A_23_P24296
## 0.97651996 0.86697876
## hsa-miR-19b-5p_A_24_P345312 hsa-miR-19b-5p_A_24_P571870
## 0.04519821 0.70594948
## hsa-miR-19b-5p_A_32_P71885 hsa-miR-19b-5p_A_32_P82889
## 0.21071194 0.47601903classMatrix
Matrix of classes with the highest posterior probability for each pair.
Row and column names are the same as in discordPPMatrix and determined by
whether only x is inputted or both x and y.
## A_23_P138644 A_23_P24296 A_24_P345312 A_24_P571870
## A_23_P138644 NA NA NA NA
## A_23_P24296 1 NA NA NA
## A_24_P345312 4 3 NA NA
## A_24_P571870 4 1 1 NA
## A_32_P71885 4 1 3 1## A_23_P138644 A_23_P24296 A_24_P345312 A_24_P571870
## hsa-miR-19b-5p 7 1 2 1
## hsa-miR-206-5p 2 1 9 1
## hsa-miR-369-5p 1 1 2 1
## hsa-miR-374-5p 7 1 1 2
## hsa-miR-376a-5p 1 1 2 1classVector
Vector of class with the highest posterior probability for each pair. The
length is the number of feature pairs. Names of vector correspond to the
feature pairs, similar to discordPPVector.
## A_23_P138644_A_23_P24296 A_23_P138644_A_24_P345312 A_23_P138644_A_24_P571870
## 1 4 4
## A_23_P138644_A_32_P71885 A_23_P138644_A_32_P82889 A_23_P138644_A_23_P105264
## 4 1 2## hsa-miR-19b-5p_A_23_P138644 hsa-miR-19b-5p_A_23_P24296
## 7 2
## hsa-miR-19b-5p_A_24_P345312 hsa-miR-19b-5p_A_24_P571870
## 1 7
## hsa-miR-19b-5p_A_32_P71885 hsa-miR-19b-5p_A_32_P82889
## 1 4probMatrix
Matrix of all posterior probabilities, where the number of rows is
the number of feature pairs and the columns represent the class within the
class matrix. The number of columns can be 9 or 25, depending on how many
mixture components are chosen (discussed later). The values across each row
add up to 1. Posterior probabilities in discordPPMatrix and
discordPPVector are the summation of columns that correspond to
differential correlation classes (described above). Each column corresponds
to the respectively numbered element from the class matrix above for three
components or the class matrix described below for five components.
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
## A_23_P138644_A_23_P24296 0.44 0.00 0.24 0.00 0.00 0 0.04 0.00 0.27
## A_23_P138644_A_24_P345312 0.10 0.00 0.00 0.89 0.01 0 0.00 0.00 0.00
## A_23_P138644_A_24_P571870 0.48 0.00 0.01 0.51 0.00 0 0.00 0.00 0.00
## A_23_P138644_A_32_P71885 0.33 0.00 0.01 0.66 0.00 0 0.00 0.00 0.00
## A_23_P138644_A_32_P82889 0.96 0.00 0.03 0.00 0.00 0 0.01 0.00 0.00
## A_23_P138644_A_23_P105264 0.05 0.88 0.00 0.00 0.05 0 0.00 0.01 0.00## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
## hsa-miR-19b-5p_A_23_P138644 0.02 0.02 0.00 0.00 0.00 0 0.73 0.22 0.00
## hsa-miR-19b-5p_A_23_P24296 0.13 0.86 0.00 0.00 0.00 0 0.00 0.01 0.00
## hsa-miR-19b-5p_A_24_P345312 0.95 0.00 0.02 0.00 0.00 0 0.02 0.00 0.00
## hsa-miR-19b-5p_A_24_P571870 0.29 0.16 0.00 0.00 0.00 0 0.45 0.09 0.00
## hsa-miR-19b-5p_A_32_P71885 0.78 0.00 0.19 0.01 0.00 0 0.01 0.00 0.01
## hsa-miR-19b-5p_A_32_P82889 0.17 0.11 0.00 0.37 0.35 0 0.00 0.00 0.00loglik
The log likelihood from the model fit.
## [1] 1199.92## [1] 1266.161Subsampling is an option to run the EM algorithm with a random sample of independent feature pairs. This is repeated for a number of samplings, and then the average of these parameters are used to maximize posterior probabilities for all feature pairs. This option was introduced to speed up Discordant method and to also address the independence assumption.
The argument subsampling must be set to TRUE for subsampling to be used. By
default, the number of independent feature pairs to be subsampled is half the
total number of features divided by two for within -omics analysis and the
number of features in the data set with fewer features for between -omics
analysis. This number may be altered by users using the subSize argument, but
the value set by users cannot exceed the default value, as the default value
is the maximum number of independent sample pairs possible for a given analysis.
The number of random samplings to be run is set by the argument
iter which has a default value of 100.
As discussed in the next section, the discordant method requires a sufficient
number of features to estimate components, and using subsampling reduces the
quantity of features used for analysis, so subsampling should be reserved for
larger data sets. For some data sets, certain random samples will be sufficient,
while others may not be. For those data sets, the subsampling algorithm will
allow up to 10% of iterations to fail and be repeated. If more than 10% of
iterations fail, discordantRun() will throw an error with potential solutions,
as shown below.
# Between -omics
btwn_result <- discordantRun(v1 = btwn_vectors$v1,
v2 = btwn_vectors$v2,
x = TCGA_GBM_miRNA_microarray,
y = TCGA_GBM_transcript_microarray,
components = 3,
subsampling = TRUE)## Warning in total_mu + pd$mu_sigma[1, ]: longer object length is not a multiple
## of shorter object length## Warning in total_sigma + pd$mu_sigma[2, ]: longer object length is not a
## multiple of shorter object length## Warning in total_nu + pd$nu_tau[1, ]: longer object length is not a multiple of
## shorter object length## Warning in total_tau + pd$nu_tau[2, ]: longer object length is not a multiple
## of shorter object length## Warning in total_pi + pd$pi: longer object length is not a multiple of shorter
## object length## Error in discordantRun(v1 = btwn_vectors$v1, v2 = btwn_vectors$v2, x = TCGA_GBM_miRNA_microarray, :
## Insufficient data for subsampling. Increase number of
## features, reduce numberof components used, or increase
## subSize if not at default value. Alternatively, set
## subsampling=FALSE.Given the limited number of features in the TCGA_GBM data sets, they are not
a suitable candidate for subsampling, so we will instead use the
TCGA_Breast_miRNASeq and TCGA_Breast_RNASeq data sets to demonstrate
a multi -omics analysis with subsampling. Note that a seed is set prior to
calling discordantRun() with subsampling = TRUE, as there is a randomness
involved in drawing subsamples, and results may differ using different seeds.
# Load Data
data(TCGA_Breast_miRNASeq_voom)
data(TCGA_Breast_RNASeq_voom)
# Prepare groups
groups <- c(rep(1, 15), rep(2, 42))
# Create correlation vectors
sub_vectors <- createVectors(x = TCGA_Breast_miRNASeq_voom,
y = TCGA_Breast_RNASeq_voom,
groups = groups)
# Run analysis with subsampling
set.seed(126)
sub_result <- discordantRun(sub_vectors$v1, sub_vectors$v2,
x = TCGA_Breast_miRNASeq_voom,
y = TCGA_Breast_RNASeq_voom,
components = 3, subsampling = TRUE)
# Results
round(head(sub_result$probMatrix), 2)## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
## hsa-mir-1247_MAPK8IP1|9479 0.12 0.00 0.02 0.00 0.00 0 0.53 0 0.34
## hsa-mir-1247_PPWD1|23398 0.98 0.01 0.00 0.00 0.00 0 0.00 0 0.00
## hsa-mir-1247_CEACAM22P|388550 0.94 0.03 0.00 0.00 0.02 0 0.00 0 0.00
## hsa-mir-1247_HTR2A|3356 0.94 0.00 0.01 0.00 0.00 0 0.02 0 0.03
## hsa-mir-1247_SAMD12|401474 0.00 0.00 0.00 0.53 0.47 0 0.00 0 0.00
## hsa-mir-1247_INTS7|25896 0.35 0.00 0.00 0.58 0.07 0 0.00 0 0.00We also provide the option to increase component size from three to five in the mixture model. The number of classes in the class matrix increases, as seen in the table below. Incorporating the extra components means that it is possible to identify elevated differential correlation, which is when there are associations in both groups in the same direction but one is more extreme. Using this option introduces more parameters, which does have an effect on run-time. We also found that using the five mixture component mixture model reduces performance compared to the three component mixture model(Siska. and Kechris (2016)). However, the option is available if users wish to explore more types of differential correlation.
\[ \begin{array}{c|c c c c c} \text{} & \text{0} & \text{-} & \text{--} & \text{+} & \text{++} \\ \hline 0 & 1 & 2 & 3 & 4 & 5 \\ - & 6 & 7 & 8 & 9 & 10 \\ -- & 11 & 12 & 13 & 14 & 15 \\ + & 16 & 17 & 18 & 19 & 20 \\ ++ & 21 & 22 & 23 & 24 & 25 \end{array} \]
By default, discordantRun() uses a three component mixture model, but this may
be changed to a five component mixture model by setting the argument
components = 5. A greater amount of data (specifically a greater number of
features) is necessary to accurately estimate 5 components compared to 3.
If an insufficient amount of data is used, discordantRun() will throw an error
suggesting the user increase the number of features or reduce the chosen
number of components. The data used for the within -omics analysis above does
not have enough features to estimate 5 components, so an error is thrown below.
# Within -omics
wthn_result <- discordantRun(v1 = wthn_vectors$v1,
v2 = wthn_vectors$v2,
x = TCGA_GBM_transcript_microarray,
components = 5)## Error in discordantRun(v1 = wthn_vectors$v1, v2 = wthn_vectors$v2, x = TCGA_GBM_transcript_microarray, :
## Insufficient data for component estimation. Increase
## number of features or reduce number of components used.However, the between -omics analysis above is a suitable candidate for further analysis using five components, so an example of such an analysis is provided below.
# Between -omics
btwn_result <- discordantRun(v1 = btwn_vectors$v1,
v2 = btwn_vectors$v2,
x = TCGA_GBM_miRNA_microarray,
y = TCGA_GBM_transcript_microarray,
components = 5)
# Between -omics
round(head(btwn_result$probMatrix), 2)## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
## hsa-miR-19b-5p_A_23_P138644 0.02 0.01 0.00 0.00 0 0.0 0.00 0 0 0
## hsa-miR-19b-5p_A_23_P24296 0.14 0.86 0.00 0.01 0 0.0 0.00 0 0 0
## hsa-miR-19b-5p_A_24_P345312 0.98 0.00 0.01 0.00 0 0.0 0.00 0 0 0
## hsa-miR-19b-5p_A_24_P571870 0.27 0.10 0.00 0.00 0 0.0 0.00 0 0 0
## hsa-miR-19b-5p_A_32_P71885 0.79 0.00 0.21 0.00 0 0.0 0.00 0 0 0
## hsa-miR-19b-5p_A_32_P82889 0.18 0.08 0.00 0.00 0 0.3 0.44 0 0 0
## [,11] [,12] [,13] [,14] [,15] [,16] [,17] [,18]
## hsa-miR-19b-5p_A_23_P138644 0.48 0.18 0 0 0 0 0 0
## hsa-miR-19b-5p_A_23_P24296 0.00 0.00 0 0 0 0 0 0
## hsa-miR-19b-5p_A_24_P345312 0.00 0.00 0 0 0 0 0 0
## hsa-miR-19b-5p_A_24_P571870 0.50 0.11 0 0 0 0 0 0
## hsa-miR-19b-5p_A_32_P71885 0.00 0.00 0 0 0 0 0 0
## hsa-miR-19b-5p_A_32_P82889 0.00 0.00 0 0 0 0 0 0
## [,19] [,20] [,21] [,22] [,23] [,24] [,25]
## hsa-miR-19b-5p_A_23_P138644 0 0 0.24 0.06 0 0 0
## hsa-miR-19b-5p_A_23_P24296 0 0 0.00 0.00 0 0 0
## hsa-miR-19b-5p_A_24_P345312 0 0 0.00 0.00 0 0 0
## hsa-miR-19b-5p_A_24_P571870 0 0 0.01 0.00 0 0 0
## hsa-miR-19b-5p_A_32_P71885 0 0 0.00 0.00 0 0 0
## hsa-miR-19b-5p_A_32_P82889 0 0 0.00 0.00 0 0 0## R version 4.4.0 beta (2024-04-15 r86425)
## Platform: x86_64-pc-linux-gnu
## Running under: Ubuntu 22.04.4 LTS
##
## Matrix products: default
## BLAS: /home/biocbuild/bbs-3.19-bioc/R/lib/libRblas.so
## LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.10.0
##
## locale:
## [1] LC_CTYPE=en_US.UTF-8 LC_NUMERIC=C
## [3] LC_TIME=en_US.UTF-8 LC_COLLATE=en_US.UTF-8
## [5] LC_MONETARY=en_US.UTF-8 LC_MESSAGES=en_US.UTF-8
## [7] LC_PAPER=en_US.UTF-8 LC_NAME=C
## [9] LC_ADDRESS=C LC_TELEPHONE=C
## [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C
##
## time zone: America/New_York
## tzcode source: system (glibc)
##
## attached base packages:
## [1] stats graphics grDevices utils datasets methods base
##
## other attached packages:
## [1] discordant_1.28.0 BiocStyle_2.32.0
##
## loaded via a namespace (and not attached):
## [1] vctrs_0.6.5 cli_3.6.2 knitr_1.46
## [4] rlang_1.1.3 xfun_0.43 DEoptimR_1.1-3
## [7] generics_0.1.3 biwt_1.0.1 jsonlite_1.8.8
## [10] gtools_3.9.5 glue_1.7.0 htmltools_0.5.8.1
## [13] sass_0.4.9 fansi_1.0.6 rmarkdown_2.26
## [16] Biobase_2.64.0 tibble_3.2.1 evaluate_0.23
## [19] jquerylib_0.1.4 MASS_7.3-60.2 fastmap_1.1.1
## [22] yaml_2.3.8 lifecycle_1.0.4 bookdown_0.39
## [25] BiocManager_1.30.22 compiler_4.4.0 dplyr_1.1.4
## [28] robustbase_0.99-2 pkgconfig_2.0.3 Rcpp_1.0.12
## [31] digest_0.6.35 R6_2.5.1 tidyselect_1.2.1
## [34] utf8_1.2.4 pillar_1.9.0 magrittr_2.0.3
## [37] bslib_0.7.0 tools_4.4.0 BiocGenerics_0.50.0
## [40] cachem_1.0.8Fang, Huaying, Chengcheng Huang, Hongyu Zhao, and Minghua Deng. 2015. “CCLasso: Correlation Inference for Compositional Data Through Lasso.” Bioinformatics 31 (19): 3172–80.
Friedman, Jonathan, and Eric J. Alm. 2012. “Inferring Correlation Networks from Genomic Survey Data.” PLoS Computational Biology.
Lai, Yinglei, Bao-ling Adam, Robert Podolsky, and Jin-Xiong She. 2007. “A Mixture Model Approach to the Tests of Concordance and Discordance Between Two Large-Scale Experiments with Two-Sample Groups.” Bioinformatics 23 (10): 1243–50.
Lai, Yinglei, Fanni Zhang, Tapan K. Naya, Reza Modarres, Norman H. Lee, and Timonthy A. McCaffrey. 2014. “Concordant Integrative Gene Set Enrichment Analysis of Multiple Large-Scale Two-Sample Expression Data Sets.” BMC Genomics 15.
Leys, Christophe, Olivier Klein, Philippe Bernard, and Laurent Licata. 2013. “Detecting Outliers: Do Not Use Standard Deviation Around the Mean, Use Absolute Deviation Around the Median.” Journal of Experimental Social Psychology 49 (4).
Magwene, Paul M., John H. Willis, John K. Kelley, and Adam Siepel. 2011. “The Statistics of Bulk Segregant Analysis Using Next Generation Sequencing.” PLoS Computational Biology 7 (11).
Siska, Charlotte, Russ Bowler, and Katerina Kechris. 2015. “The Discordant Method: A Novel Approach for Differential Correlation.” Bioinformatics 32 (5): 690–96.
Siska., Charlotte, and Katerina Kechris. 2016. “Differential Correlation for Sequencing Data.”
Song, Lin, Peter Langfelder, and Steve Horvath. 2012. “Comparison of Co-Expression Measures: Mutual Information, Correlation, and Model Based Indices.” BMC Bioinformatics 13 (328).