IFAA offers a robust approach to make inference on the association of covariates with the absolute abundance (AA) of microbiome in an ecosystem. It can be also directly applied to relative abundance (RA) data to make inference on AA because the ratio of two RA is equal ratio of their AA. This algorithm can estimate and test the associations of interest while adjusting for potential confounders. High-dimensional covariates are handled with regularization. The estimates of this method have easy interpretation like a typical regression analysis. High-dimensional covariates are handled with regularization and it is implemented by parallel computing. False discovery rate is automatically controlled by this approach. Zeros do not need to be imputed by a positive value for the analysis. The IFAA package also offers the ‘MZILN’ function for estimating and testing associations of abundance ratios with covariates.
To model the association, the following equation is used: $$ \log(\mathcal{Y}_i^k)|\mathcal{Y}_i^k>0=\beta^{0k}+X_i^T\beta^k+W_i^T\gamma^k+Z_i^Tb_i+\epsilon_i^k,\hspace{0.2cm}k=1,...,K+1, $$ where
𝒴ik is the AA of taxa k in subject i in the entire ecosystem.
Xi is the covariate matrix.
Wi is the confounder matrix.
Zi is the design matrix for random effects.
βk
is the regression coefficients that will be estimated and tested with
the IFAA() function.
The challenge in microbiome analysis is that we can not oberve 𝒴ik. What is observed is its small proportion: Yik = Ci𝒴ik where Ci is an unknown number between 0 and 1 that denote the observed proportion. The IFAA method successfuly addressed this challenge.
To install, type the following command in R console:
The package could be also installed from GitHub using the following code:
Most of the time, users just need to feed these three inputs to the
function: experiment_dat, testCov and
ctrlCov. All other inputs can just take their default
values. Below are all the inputs of the functions
experiment_dat: A SummarizedExperiment object
containing microbiome data and covariates (see example on how to create
a SummarizedExperiment object). The microbiome data can be absolute
abundance or relative abundance with each column per sample and each row
per taxon/OTU/ASV (or any other unit). No imputation is needed for
zero-valued data points. The covariates data contains covariates and
confounders with each row per sample and each column per variable. The
covariates data has to be numeric or binary. Categorical variables
should be converted into binary dummy variables.
microbVar: This takes a single or vector of
microbiome variable names (e.g., taxa, OTU and ASV names) of interest.
Default is “all” meaning all microbiome variables will be analyzed. If a
subset of microbiome variables is specified, the output will only
contain the specified variables, and p-value adjustment for multiple
testing will only be applied to the subset.
testCov: Covariates that are of primary interest for
testing and estimating the associations. It corresponds to Xi in the
equation. Default is NULL which means all covariates are
testCov.
ctrlCov: Potential confounders that will be adjusted
in the model. It corresponds to Wi in the
equation. Default is NULL which means all covariates except
those in testCov are adjusted as confounders.
sampleIDname: Name of the sample ID variable in the
data. In the case that the data does not have an ID variable, this can
be ignored. Default is NULL.
testMany: This takes logical value TRUE
or FALSE. If TRUE, the testCov
will contain all the variables in CovData provided
testCov is set to be NULL. The default value
is TRUE which does not do anything if testCov
is not NULL.
ctrlMany: This takes logical value TRUE
or FALSE. If TRUE, all variables except
testCov are considered as control covariates provided
ctrlCov is set to be NULL. The default value
is FALSE.
nRef: The number of randomly picked reference taxa
used in phase 1. Default number is 40.
nRefMaxForEsti: The maximum number of final
reference taxa used in phase 2. The default is 2.
refTaxa: A vector of taxa names. These are reference
taxa specified by the user to be used in phase 1 if the user believe
these taxa are indepenent of the covariates. If the number of reference
taxa is less than ‘nRef’, the algorithm will randomly pick extra
reference taxa to make up ‘nRef’. The default is NULL since
the algorithm will pick reference taxa randomly.
adjust_method: The adjusting method used for p value
adjustment. Default is “BY” for dependent FDR adjustment. It can take
any adjustment method for p.adjust function in R.
fdrRate: The false discovery rate for identifying
taxa/OTU/ASV associated with testCov. Default is
0.05.
paraJobs: If sequentialRun is
FALSE, this specifies the number of parallel jobs that will
be registered to run the algorithm. If specified as NULL,
it will automatically detect the cores to decide the number of parallel
jobs. Default is NULL.
bootB: Number of bootstrap samples for obtaining
confidence interval of estimates in phase 2 for the high dimensional
regression. The default is 500.
standardize: This takes a logical value
TRUE or FALSE. If TRUE, the
design matrix for X will be standardized in the analyses and the
results. Default is FALSE.
sequentialRun: This takes a logical value
TRUE or FALSE. Default is FALSE.
This argument could be useful for debug.
refReadsThresh: The threshold of proportion of
non-zero sequencing reads for choosing the reference taxon in phase 2.
The default is 0.2 which means at least 20% non-zero
sequencing reads.
taxDropThresh: The threshold of number of non-zero
sequencing reads for each taxon to be dropped from the analysis. The
default is 0 which means taxon without any sequencing reads
will be dropped from the analysis.
SDThresh: The threshold of standard deviations of
sequencing reads for been chosen as the reference taxon in phase 2. The
default is 0.05 which means the standard deviation of
sequencing reads should be at least 0.05 in order to be
chosen as reference taxon.
SDquantilThresh: The threshold of the quantile of
standard deviation of sequencing reads, above which could be selected as
reference taxon. The default is 0.
balanceCut: The threshold of the proportion of
non-zero sequencing reads in each group of a binary variable for
choosing the final reference taxa in phase 2. The default number is
0.2 which means at least 20% non-zero sequencing reads in
each group are needed to be eligible for being chosen as a final
reference taxon.
A list containing 2 elements
full_results: The main results for IFAA containing
the estimation and testing results for all associations between all taxa
and all test covariates in testCov. It is a dataframe with
each row representing an association, and eight columns named as
“taxon”, “cov”, “estimate”, “SE.est”, “CI.low”, “CI.up”, “adj.p.value”,
and “sig_ind”. The columns correspond to taxon name, covariate name,
association estimates, standard error estimates, lower bound and upper
bound of the 95% confidence interval, adjusted p value, and the
indicator showing whether the association is significant after multiple
testing adjustment.
metadata: The metadata is a list containing the
following items: covariatesData: A dataset containing
covariates and confounders used in the analyses.
final_ref_taxon: The final 2 reference taxa used for
analysis. ref_taxon_count: The counts of selection for the
associations of all taxa with test covariates in Phase 1.
ref_taxon_est: The average magnitude estimates for the
associations of all taxa with test covariates in Phase 1.
totalTimeMins: Total time used for the entire analysis.
fdrRate: FDR rate used for the analysis.
adjust_method: Multiple testing adjust method used for the
analysis.
The example generates an example data from scratch, with 10 taxon, 40 subjects, and 3 covariates.
library(IFAA)
set.seed(1)
## create an ID variable for the example data
ID <- seq_len(20)
## generate three covariates x1, x2, and x3, with x2 binary
x1 <- rnorm(20)
x2 <- rbinom(20, 1, 0.5)
x3 <- rnorm(20)
dataC <- data.frame(cbind(ID, x1, x2, x3))
## Coefficients for x1, x2, and x3 among 10 taxa.
beta_1 <- c(0.1, rep(0, 9))
beta_2 <- c(0, 0.2, rep(0, 8))
beta_3 <- rnorm(10)
beta_mat <- cbind(beta_1, beta_2, beta_3)
## Generate absolute abundance for 10 taxa in ecosystem.
dataM_eco <- floor(exp(10 + as.matrix(dataC[, -1]) %*% t(beta_mat) + rnorm(200, sd = 0.05)))
## Generate sequence depth and generate observed abundance
Ci <- runif(20, 0.01, 0.05)
dataM <- floor(apply(dataM_eco, 2, function(x) x * Ci))
colnames(dataM) <- paste0("rawCount", 1:10)
## Randomly introduce 0 to make 25% sparsity level.
dataM[sample(seq_len(length(dataM)), length(dataM) / 4)] <- 0
dataM <- data.frame(cbind(ID, dataM))
## The following steps are to create a SummarizedExperiment object.
## If you already have a SummarizedExperiment format data, you can
## ignore the following steps and directly feed it to the IFAA function.
## Merge two dataset by ID variable
data_merged <- merge(dataM, dataC, by = "ID", all = FALSE)
## Seperate microbiome data and covariate data, drop ID variable from microbiome data
dataM_sub <- data_merged[, colnames(dataM)[!colnames(dataM) %in% c("ID")]]
dataC_sub <- data_merged[, colnames(dataC)]
## Create SummarizedExperiment object
test_dat <- SummarizedExperiment::SummarizedExperiment(
assays = list(MicrobData = t(dataM_sub)), colData = dataC_sub
)If you already have a SummarizedExperiment format data, you can
ignore the above steps for creating a SummarizedExperiment object. In
the generated data, rawCount1 is associated with x1, rawCount2 is
associated with x2, and the sparsity level is 25%. Next we analyze the
data to test the association between microbiome and the variable
"x1", "x2" while adjusting for the variable (potential
confounder) "x3".
set.seed(123) # For full reproducibility
results <- IFAA(
experiment_dat = test_dat,
testCov = c("x1", "x2"),
ctrlCov = c("x3"),
sampleIDname = "ID",
fdrRate = 0.05,
nRef = 2,
paraJobs = 2
)
#> Data dimensions (after removing missing data if any):
#> 20 samples
#> 10 taxa/OTU/ASV
#> 2 testCov variables in the analysis
#> These are the testCov variables:
#> x1, x2
#> 1 ctrlCov variables in the analysis
#> These are the ctrlCov variables:
#> x3
#> 1 binary covariates in the analysis
#> These are the binary covariates:
#> x2
#> 25 percent of microbiome sequencing reads are zero
#> Start Phase 1 analysis
#> 2 parallel jobs are registered for the analysis.
#> 33 percent of phase 1 analysis has been done and it took 0.16 minutes
#> 2 parallel jobs are registered for the analysis.
#> 100 percent of phase 1 analysis has been done and it took 0.32 minutes
#> Start Phase 2 parameter estimation
#> 2 parallel jobs are registered for analyzing reference taxa in Phase 2
#> 50 percent of Phase 2 is done and it took 0.119 minutes
#> 2 parallel jobs are registered for analyzing reference taxa in Phase 2
#> Entire Phase 2 parameter estimation done and took 0.238 minutes.
#> The entire analysis took 0.56 minutesNotice that this example used “nRef = 2” and “paraJobs = 2” for the sake of speed in this example. Typically one should use the default values for those two arguments.
In this example, we are only interested in testing the associations
with "x1" and "x2" which is why
testCov=c("x1","x2). The variables "x3" are
adjusted as potential confounders in the analyses. The final analysis
results are saved in the list full_result and the
significant results can be extracted as follows:
summary_res <- results$full_results
sig_results <- subset(summary_res, sig_ind == TRUE)
sig_results
#> DataFrame with 2 rows and 9 columns
#> taxon cov estimate SE.est CI.low CI.up unadj.p.value
#> <character> <character> <numeric> <numeric> <numeric> <numeric> <numeric>
#> 1 rawCount1 x1 0.0778196 0.0204586 0.0377207 0.117918 1.42517e-04
#> 2 rawCount2 x2 0.1842519 0.0387891 0.1082253 0.260278 2.03319e-06
#> adj.p.value sig_ind
#> <numeric> <logical>
#> 1 3.09872e-03 TRUE
#> 2 4.42074e-05 TRUEAfter adjusting for multiple testing, the results found
"rawCount1" associated with "x1", and
"rawCount2" associated with "x2", while
adjusting for "x3". The regression coefficients and their
95% confidence intervals are provided. These coefficients correspond to
βk in the
model equation.
The interpretation is:
"x1" is associated with
approximately 7.8% increase in the absolute abundance of
"rawCount1" in the entire ecosystem of interest (while
controlling for "x2" and "x3"); Every unit
increase in "x2" is associated with approximately 18.4%
increase in the absolute abundance of "rawCount2" in the
entire ecosystem of interest (while controlling for "x1"
and "x3"), .If only interested in certain taxa, say
"rawCount1", "rawCount2", and "rawCount3", one can do:
set.seed(123) # For full reproducibility
results <- IFAA(
experiment_dat = test_dat,
microbVar = c("rawCount1", "rawCount2", "rawCount3"),
testCov = c("x1", "x2"),
ctrlCov = c("x3"),
sampleIDname = "ID",
fdrRate = 0.05,
nRef = 2,
paraJobs = 2
)Again, notice that this example used “nRef = 2” and “paraJobs = 2” for the sake of speed in this example. Typically one should use the default values for those two arguments.
Li et al.(2021) IFAA: Robust association identification and Inference For Absolute Abundance in microbiome analyses. Journal of the American Statistical Association. 116(536):1595-1608
The IFAA package can also implement the Multivariate Zero-Inflated
Logistic Normal (MZILN) regression model for estimating and testing the
association of abundance ratios with covariates. The
MZILN() function estimates and tests the associations of
user-specified abundance ratios with covariates. When the denominator
taxon of the ratio is independent of the covariates, ‘MZILN()’ should
generate similar results as ‘IFAA()’. The regression model of ‘MZILN()’
can be expressed as follows: $$
\log\bigg(\frac{\mathcal{Y}_i^k}{\mathcal{Y}_i^{K+1}}\bigg)|\mathcal{Y}_i^k>0,\mathcal{Y}_i^{K+1}>0=\alpha^{0k}+\mathcal{X}_i^T\alpha^k+\epsilon_i^k,\hspace{0.2cm}k=1,...,K,
$$ where
𝒴ik is the AA of taxa k in subject i in the entire ecosystem.
𝒴iK + 1 is the reference taxon (specified by user).
𝒳i is the covariate matrix for all covariates including confounders.
αk is the regression coefficients that will be estimated and tested.
Most of the time, users just feed the first four inputs to the
function: experiment_dat, microbVar,
refTaxa and allCov. All other inputs can just
take their default values. All the inputs for ‘MZILN()’ are:
experiment_dat: A SummarizedExperiment object
containing microbiome data and covariates (see example on how to create
a SummarizedExperiment object). The microbiome data can be absolute
abundance or relative abundance with each column per sample and each row
per taxon/OTU/ASV (or any other unit). No imputation is needed for
zero-valued data points. The covariates data contains covariates and
confounders with each row per sample and each column per variable. The
covariates data has to be numeric or binary. Categorical variables
should be converted into binary dummy variables.
microbVar: This takes a single or vector of
microbiome variable names (e.g., taxa, OTU and ASV names) of interest
for the numerator of the ratios. Default is “all” meaning all microbiome
variables (except denominator) will be analyzed as numerators. If a
subset of microbiome variables is specified, the output will only
contain the specified ratios, and p-value adjustment for multiple
testing will only be applied to the subset.
refTaxa: Denominator taxa names specified by the
user for the targeted ratios. This could be a vector of names.
allCov: All covariates of interest (including
confounders) for estimating and testing their associations with the
targeted ratios. Default is ‘NULL’ meaning that all covariates in
covData are of interest.
sampleIDname: Name of the sample ID variable in the
data. In the case that the data does not have an ID variable, this can
be ignored. Default is NULL.
adjust_method: The adjusting method for p value
adjustment. Default is “BY” for dependent FDR adjustment. It can take
any adjustment method for p.adjust function in R.
fdrRate The false discovery rate for identifying
ratios associated with allCov. Default is
0.05.
paraJobs: If sequentialRun is
FALSE, this specifies the number of parallel jobs that will
be registered to run the algorithm. If specified as NULL,
it will automatically detect the cores to decide the number of parallel
jobs. Default is NULL.
bootB: Number of bootstrap samples for obtaining
confidence interval of estimates for the high dimensional regression.
The default is 500.
taxDropThresh: The threshold of number of non-zero
sequencing reads for each taxon to be dropped from the analysis. The
default is 0 which means taxon without any sequencing reads
will be dropped from the analysis.
standardize: This takes a logical value
TRUE or FALSE. If TRUE, the
design matrix for X will be standardized in the analyses and the
results. Default is FALSE.
sequentialRun: This takes a logical value
TRUE or FALSE. Default is TRUE.
It can be set to be “FALSE” to increase speed if there are multiple taxa
in the argument ‘refTaxa’.
A list with two elements:
full_results: The main results for MZILN containing
the estimation and testing results for all associations between all taxa
ratios with refTaxan being the denominator and all covariates in
allCov. It is a dataframe with each row representing an
association, and ten columns named as “ref_tax”, “taxon”, “cov”,
“estimate”, “SE.est”, “CI.low”, “CI.up”, “adj.p.value”, “unadj.p.value”
and “sig_ind”. The columns correspond to the denominator taxon,
numerator taxon, covariate name, association estimates, standard error
estimates, lower bound and upper bound of the 95% confidence interval,
adjusted p value, and the indicator showing whether the association is
significant after multiple testing adjustment.
metadata: The metadata is a list containing a
dataset of covariates and confounders used in the analyses, total time
used in minutes, FDR rate, and multiple testing adjustment method
used.
The example used data generated from IFAA example, with 10 taxon, 40 subjects, and 3 covariates.
Next we analyze the data to test the associations between the ratios
“rawCount5/rawCount10” and “rawCount8/rawCount10” and all the three
variables "x1", "x2" and "x3" in
a multivariate model where all "x1", "x2" and
"x3" are independent variables simultaneously.
set.seed(123) # For full reproducibility
resultsRatio <- MZILN(
experiment_dat = test_dat,
microbVar = c("rawCount5","rawCount8"),
refTaxa = c("rawCount10"),
allCov = c("x1", "x2", "x3"),
sampleIDname = "ID",
fdrRate = 0.05,
paraJobs = 2
)
#> Data dimensions (after removing missing data if any):
#> 20 samples
#> 10 taxa/OTU/ASV
#> 3 testCov variables in the analysis
#> These are the testCov variables:
#> x1, x2, x3
#> 0 ctrlCov variables in the analysis
#> 1 binary covariates in the analysis
#> These are the binary covariates:
#> x2
#> 25 percent of microbiome sequencing reads are zero
#> 2 parallel jobs are registered for analyzing reference taxa in Phase 2
#> Estimation done for the 1th denominator taxon: rawCount10 and it took 0.12 minutes
#> The entire analysis took 0.12 minutesAgain, notice that this example used “nRef = 2” and “paraJobs = 2” for the sake of speed in this example. Typically one should use the default values for those two arguments.
The final analysis results can be extracted as follows:
summary_res_ratio <- resultsRatio$full_results
summary_res_ratio
#> DataFrame with 6 rows and 10 columns
#> ref_tax taxon cov estimate SE.est CI.low
#> <character> <character> <character> <numeric> <numeric> <numeric>
#> 1 rawCount10 rawCount5 x1 0.00997480 0.0228440 -0.0347994
#> 2 rawCount10 rawCount5 x2 -0.09323061 0.0386687 -0.1690213
#> 3 rawCount10 rawCount5 x3 1.58194615 0.0276923 1.5276693
#> 4 rawCount10 rawCount8 x1 -0.00681531 0.0211971 -0.0483617
#> 5 rawCount10 rawCount8 x2 -0.00445645 0.0394673 -0.0818123
#> 6 rawCount10 rawCount8 x3 -0.91801508 0.0297290 -0.9762839
#> CI.up unadj.p.value adj.p.value sig_ind
#> <numeric> <numeric> <numeric> <logical>
#> 1 0.0547490 0.6623657 1.0000000 FALSE
#> 2 -0.0174399 0.0159084 0.0477253 TRUE
#> 3 1.6362230 0.0000000 0.0000000 TRUE
#> 4 0.0347310 0.7478158 1.0000000 FALSE
#> 5 0.0728994 0.9100979 1.0000000 FALSE
#> 6 -0.8597462 0.0000000 0.0000000 TRUEThe regression coefficients and their 95% confidence intervals are provided. These coefficients correspond to αk in the model equation.
The results show that the ratio “rawCount5/rawCount10” is
significantly associated with "x2" and "x3"
and the ratio “rawCount8/rawCount10” is significantly associated with
"x3" after multiple testing adjustment. The interpretation
is:
"x2" is associated with
approximately 9.3% reduction in the abundance ratio
"rawCount5/rawCount10" (while controlling for
"x1" and "x3"); Every unit increase in
"x3" is associated with approximately 386.4%
(=exp(1.582)-1) increase in the abundance ratio
"rawCount5/rawCount10" (while controlling for
"x1" and "x2"). Every unit increase in
"x3" is associated with approximately 60.1%
(=1-exp(-0.918)) decrease in the abundance ratio
"rawCount8/rawCount10" (while controlling for
"x1" and "x2").If the purpose is to explore all taxa ratios with “rawCount10” being the denominator, one can do:
set.seed(123) # For full reproducibility
resultsAllRatio <- MZILN(
experiment_dat = test_dat,
refTaxa = c("rawCount10"),
allCov = c("x1", "x2", "x3"),
sampleIDname = "ID",
fdrRate = 0.05,
paraJobs = 2
)
#> Data dimensions (after removing missing data if any):
#> 20 samples
#> 10 taxa/OTU/ASV
#> 3 testCov variables in the analysis
#> These are the testCov variables:
#> x1, x2, x3
#> 0 ctrlCov variables in the analysis
#> 1 binary covariates in the analysis
#> These are the binary covariates:
#> x2
#> 25 percent of microbiome sequencing reads are zero
#> 2 parallel jobs are registered for analyzing reference taxa in Phase 2
#> Estimation done for the 1th denominator taxon: rawCount10 and it took 0.12 minutes
#> The entire analysis took 0.12 minutesAgain, notice that this example used “nRef = 2” and “paraJobs = 2” for the sake of speed in this example. Typically one should use the default values for those two arguments.
The results for the ratio “rawCount5/rawCount10” can extracted as follows:
summary_res_ratio <- resultsAllRatio$full_results
summary_res_ratio[summary_res_ratio$taxon == "rawCount5", , drop = FALSE]
#> DataFrame with 3 rows and 10 columns
#> ref_tax taxon cov estimate SE.est CI.low
#> <character> <character> <character> <numeric> <numeric> <numeric>
#> 1 rawCount10 rawCount5 x1 0.0099748 0.0228440 -0.0347994
#> 2 rawCount10 rawCount5 x2 -0.0932306 0.0386687 -0.1690213
#> 3 rawCount10 rawCount5 x3 1.5819462 0.0276923 1.5276693
#> CI.up unadj.p.value adj.p.value sig_ind
#> <numeric> <numeric> <numeric> <logical>
#> 1 0.0547490 0.6623657 1.00000 FALSE
#> 2 -0.0174399 0.0159084 0.20252 FALSE
#> 3 1.6362230 0.0000000 0.00000 TRUEThe interpretation of the association estimates remain the same.
Notice that the significance dropped for the association between the
ratio “rawCount5/rawCount10” and "x2" due to the correction
for multiple testing (i.e., 0.2025>0.05). It is because this
exploratory analysis included all microbiome variables and thus the
p-value adjustment is more stringent for multiple testing which led to
the larger adjusted p-value.
The results for the ratio “rawCount8/rawCount10” can be extracted similarly.
To extract all significant ratios with "rawCount10"
being the denominator,
one can do: subset(summary_res_ratio,sig_ind==TRUE).
Li et al.(2018) Conditional Regression Based on a Multivariate Zero-Inflated Logistic-Normal Model for Microbiome Relative Abundance Data. Statistics in Biosciences 10(3): 587-608
Session Info
sessionInfo()
#> R version 4.4.1 (2024-06-14)
#> Platform: x86_64-pc-linux-gnu
#> Running under: Ubuntu 24.04 LTS
#>
#> Matrix products: default
#> BLAS: /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3
#> LAPACK: /usr/lib/x86_64-linux-gnu/openblas-pthread/libopenblasp-r0.3.26.so; LAPACK version 3.12.0
#>
#> locale:
#> [1] LC_CTYPE=en_US.UTF-8 LC_NUMERIC=C
#> [3] LC_TIME=en_US.UTF-8 LC_COLLATE=C
#> [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: Etc/UTC
#> tzcode source: system (glibc)
#>
#> attached base packages:
#> [1] stats graphics grDevices utils datasets methods base
#>
#> other attached packages:
#> [1] IFAA_1.7.0
#>
#> loaded via a namespace (and not attached):
#> [1] SummarizedExperiment_1.35.1 gld_2.6.6
#> [3] shape_1.4.6.1 xfun_0.46
#> [5] bslib_0.8.0 Biobase_2.65.0
#> [7] lattice_0.22-6 mathjaxr_1.6-0
#> [9] tools_4.4.1 stats4_4.4.1
#> [11] parallel_4.4.1 proxy_0.4-27
#> [13] Matrix_1.7-0 data.table_1.15.4
#> [15] rngtools_1.5.2 S4Vectors_0.43.2
#> [17] readxl_1.4.3 lifecycle_1.0.4
#> [19] GenomeInfoDbData_1.2.12 rootSolve_1.8.2.4
#> [21] stringr_1.5.1 compiler_4.4.1
#> [23] Exact_3.3 RhpcBLASctl_0.23-42
#> [25] codetools_0.2-20 GenomeInfoDb_1.41.1
#> [27] htmltools_0.5.8.1 sys_3.4.2
#> [29] buildtools_1.0.0 DescTools_0.99.55
#> [31] class_7.3-22 sass_0.4.9
#> [33] yaml_2.3.10 glmnet_4.1-8
#> [35] crayon_1.5.3 jquerylib_0.1.4
#> [37] MASS_7.3-61 DelayedArray_0.31.11
#> [39] cachem_1.1.0 doRNG_1.8.6
#> [41] iterators_1.0.14 abind_1.4-5
#> [43] boot_1.3-30 foreach_1.5.2
#> [45] parallelly_1.38.0 digest_0.6.36
#> [47] stringi_1.8.4 HDCI_1.0-2
#> [49] mvtnorm_1.2-5 slam_0.1-52
#> [51] maketools_1.3.0 splines_4.4.1
#> [53] fastmap_1.2.0 grid_4.4.1
#> [55] SparseArray_1.5.31 lmom_3.0
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