Processing math: 100%
  • Empirical Power and False Discovery Rate for different target thresholds

Last updated: 2024-06-18

Checks: 6 1

Knit directory: zinck-website/

This reproducible R Markdown analysis was created with workflowr (version 1.7.1). The Checks tab describes the reproducibility checks that were applied when the results were created. The Past versions tab lists the development history.


The R Markdown is untracked by Git. To know which version of the R Markdown file created these results, you’ll want to first commit it to the Git repo. If you’re still working on the analysis, you can ignore this warning. When you’re finished, you can run wflow_publish to commit the R Markdown file and build the HTML.

Great job! The global environment was empty. Objects defined in the global environment can affect the analysis in your R Markdown file in unknown ways. For reproduciblity it’s best to always run the code in an empty environment.

The command set.seed(20240617) was run prior to running the code in the R Markdown file. Setting a seed ensures that any results that rely on randomness, e.g. subsampling or permutations, are reproducible.

Great job! Recording the operating system, R version, and package versions is critical for reproducibility.

Nice! There were no cached chunks for this analysis, so you can be confident that you successfully produced the results during this run.

Great job! Using relative paths to the files within your workflowr project makes it easier to run your code on other machines.

Great! You are using Git for version control. Tracking code development and connecting the code version to the results is critical for reproducibility.

The results in this page were generated with repository version 6e66a4e. See the Past versions tab to see a history of the changes made to the R Markdown and HTML files.

Note that you need to be careful to ensure that all relevant files for the analysis have been committed to Git prior to generating the results (you can use wflow_publish or wflow_git_commit). workflowr only checks the R Markdown file, but you know if there are other scripts or data files that it depends on. Below is the status of the Git repository when the results were generated:


Untracked files:
    Untracked:  analysis/CRC.Rmd
    Untracked:  analysis/Heatmaps.Rmd
    Untracked:  analysis/IBD.Rmd
    Untracked:  analysis/simulation.Rmd

Unstaged changes:
    Modified:   analysis/index.Rmd

Note that any generated files, e.g. HTML, png, CSS, etc., are not included in this status report because it is ok for generated content to have uncommitted changes.


There are no past versions. Publish this analysis with wflow_publish() to start tracking its development.


Empirical Power and False Discovery Rate for different target thresholds

We take the colon cancer (CRC) dataset which encompasses five distinct studies each originating from different countries. We randomly select roughly half of the samples from each metagenomic study, which are pooled together for further analysis. For the sake of this example, we retain the top 100 most abundant species according to the average proportions. Among these species, we randomly selected 30 biomarkers and simulated both continuous and binary outcomes using the symmetric reformulation of the log-contrast model:

g(E(yi))=pj=1log(ˉXij)βjsubject topj=1βj=0 where g(.) is the link function and ˉXij is the observed proportions based on the count matrix with zero replaced by 0.5 to avoid taking log on zero values. For simulating the continuous responses, the link function g(.) is the identity function, and for simulating the binary responses, the link function g(.) is the logit link function. We adopt the signal strengths in such a way that the zero-sum constraint is satisfied. We replicate a single iteration of this setting for both continuous and binary outcomes.

load("/Users/Patron/Documents/zinLDA research/count.Rdata")
load("/Users/Patron/Documents/zinLDA research/meta.RData")
library(rstan)
Loading required package: StanHeaders
Loading required package: ggplot2
rstan (Version 2.21.8, GitRev: 2e1f913d3ca3)
For execution on a local, multicore CPU with excess RAM we recommend calling
options(mc.cores = parallel::detectCores()).
To avoid recompilation of unchanged Stan programs, we recommend calling
rstan_options(auto_write = TRUE)
library(glmnet)
Loading required package: Matrix
Loaded glmnet 4.1-7
library(knockoff)
library(zinck)
library(kosel)
library(dplyr)

Attaching package: 'dplyr'
The following objects are masked from 'package:stats':

    filter, lag
The following objects are masked from 'package:base':

    intersect, setdiff, setequal, union
library(reshape2)

generate_data <- function(p,seed){
  dcount <- count[,order(decreasing=T,colSums(count,na.rm=T),
                         apply(count,2L,paste,collapse=''))] ## ordering the columns w/ decreasing abundance
  ####### Randomly sampling patients from 574 observations #######
  set.seed(seed)
  sel_index <- rbinom(nrow(meta),size=1,prob=0.5)
  selected_samples <- which(sel_index==1)
  meta_selected <- meta[selected_samples,]
  X <- dcount[selected_samples,]
  if(p == 100){
    X1 <- X[,1:p]
    n = nrow(X1)
    
    Five1 = c(-3,3,2.5,-1, -1.5)
    Five2 = c(3,3,-2,-2,-2)
    Five3 = c(1,-1,3,-2,-1)
    Five4 = c(-1,1,2,-1,-1)
    Five5 = c(3,3,-3,-2,-1)
    Five6 = c(-1,1,-2,1,1)
    Five_all <- c(Five1,Five2,Five3,Five4,Five5,Five6)  ## Signals satisfy sum-to-zero constraint ##
    randBeta <- rep(0,p)
    set.seed(1)
    rand_indices <- sample(1:p,size=30,replace=FALSE)  ## Randomly injecting 30 signals out of p=100 ##
    set.seed(1)
    randBeta[rand_indices] <- sample(Five_all,size=30,replace=FALSE)
    
    W1 <- log_normalize(X1) ## log-normalizing counts to make it compositional
    
    ##### Generating continuous responses ######
    set.seed(1)
    eps=rnorm(n,mean = 0, sd=1)
    Y1 <- W1 %*% randBeta + eps
    
    ##### Generating binary responses #####
    set.seed(1)
    pr = 1/(1+exp(-W1 %*% randBeta))
    Y1_bin = rbinom(n,1,pr)
  } else if (p %in% c(200,300,400)) {
    X1 <- X[,1:p]
    n = nrow(X1)
    
    Five1 = c(-3,3,2.5,-1, -1.5)
    Five2 = c(3,3,-2,-2,-2)
    Five3 = c(1,-1,3,-2,-1)
    Five4 = c(-1,1,2,-1,-1)
    Five5 = c(3,3,-3,-2,-1)
    Five6 = c(-1,1,-2,1,1)
    Five_all <- c(Five1,Five2,Five3,Five4,Five5,Five6)  ## Signals satisfy sum-to-zero constraint ##
    randBeta <- rep(0,p)
    set.seed(1)
    rand_indices <- sample(1:200,size=30,replace=FALSE)  ## Randomly injecting 30 signals out of p=200 ##
    set.seed(1)
    randBeta[rand_indices] <- sample(Five_all,size=30,replace=FALSE)
    
    W1 <- log_normalize(X1) ## log-normalizing counts to make it compositional
    
    ##### Generating continuous responses ######
    set.seed(1)
    eps=rnorm(n,mean = 0, sd=1)
    Y1 <- W1 %*% randBeta + eps
    
    ##### Generating binary responses #####
    set.seed(1)
    pr = 1/(1+exp(-W1 %*% randBeta))
    Y1_bin = rbinom(n,1,pr)
  }

  return(list(Y1 = Y1, X1 = X1,  W1 = W1, Y1_bin = Y1_bin, index = rand_indices))
}

Now, let’s look at the case when p=100. On extracting the sample taxa count matrix X1, we train the zinck model using ADVI. Plugging in the learnt parameters into the generative framework of zinck, we generate the knockoff matrix ˜X1.

X1 <- generate_data(p=100, seed=2)$X1
fit1 <- zinck::fit.zinck(X1,num_clusters = 6, method="ADVI", seed = 1)
Trying to compile a simple C file
Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
clang -mmacosx-version-min=10.13 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppEigen/include/unsupported"  -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/StanHeaders/include/src/"  -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppParallel/include/"  -I"/Library/Frameworks/R.framework/Versions/4.1/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -DBOOST_DISABLE_ASSERTS  -DBOOST_PENDING_INTEGER_LOG2_HPP  -DSTAN_THREADS  -DBOOST_NO_AUTO_PTR  -include '/Library/Frameworks/R.framework/Versions/4.1/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp'  -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1   -I/usr/local/include   -fPIC  -Wall -g -O2  -c foo.c -o foo.o
In file included from <built-in>:1:
In file included from /Library/Frameworks/R.framework/Versions/4.1/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
In file included from /Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppEigen/include/Eigen/Dense:1:
In file included from /Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppEigen/include/Eigen/Core:88:
/Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name 'namespace'
namespace Eigen {
^
/Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:16: error: expected ';' after top level declarator
namespace Eigen {
               ^
               ;
In file included from <built-in>:1:
In file included from /Library/Frameworks/R.framework/Versions/4.1/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
In file included from /Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppEigen/include/Eigen/Dense:1:
/Library/Frameworks/R.framework/Versions/4.1/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
#include <complex>
         ^~~~~~~~~
3 errors generated.
make: *** [foo.o] Error 1
Chain 1: ------------------------------------------------------------
Chain 1: EXPERIMENTAL ALGORITHM:
Chain 1:   This procedure has not been thoroughly tested and may be unstable
Chain 1:   or buggy. The interface is subject to change.
Chain 1: ------------------------------------------------------------
Chain 1: 
Chain 1: 
Chain 1: 
Chain 1: Gradient evaluation took 0.013091 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 130.91 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1: 
Chain 1: 
Chain 1: Begin eta adaptation.
Chain 1: Iteration:   1 / 250 [  0%]  (Adaptation)
Chain 1: Iteration:  50 / 250 [ 20%]  (Adaptation)
Chain 1: Iteration: 100 / 250 [ 40%]  (Adaptation)
Chain 1: Iteration: 150 / 250 [ 60%]  (Adaptation)
Chain 1: Iteration: 200 / 250 [ 80%]  (Adaptation)
Chain 1: Iteration: 250 / 250 [100%]  (Adaptation)
Chain 1: Success! Found best value [eta = 0.1].
Chain 1: 
Chain 1: Begin stochastic gradient ascent.
Chain 1:   iter             ELBO   delta_ELBO_mean   delta_ELBO_med   notes 
Chain 1:    100    -15973492.527             1.000            1.000
Chain 1:    200     -8690390.660             0.919            1.000
Chain 1:    300     -5775030.814             0.781            0.838
Chain 1:    400     -4505025.584             0.656            0.838
Chain 1:    500     -3902911.200             0.556            0.505
Chain 1:    600     -3632699.792             0.476            0.505
Chain 1:    700     -3505002.928             0.413            0.282
Chain 1:    800     -3433489.203             0.364            0.282
Chain 1:    900     -3389828.201             0.325            0.154
Chain 1:   1000     -3356212.590             0.293            0.154
Chain 1:   1100     -3327328.773             0.194            0.074
Chain 1:   1200     -3302681.095             0.111            0.036
Chain 1:   1300     -3281220.678             0.061            0.021
Chain 1:   1400     -3260946.832             0.034            0.013
Chain 1:   1500     -3237468.494             0.019            0.010
Chain 1:   1600     -3216932.441             0.012            0.009   MEDIAN ELBO CONVERGED
Chain 1: 
Chain 1: Drawing a sample of size 1000 from the approximate posterior... 
Chain 1: COMPLETED.
theta <- fit1$theta
beta <- fit1$beta
X1_tilde <- generateKnockoff(X1,theta,beta,seed=1)

Now that we have generated the knockoff copy, we will fit a model associating the response Y1 (both continuous and binary) with the augmented set of covariates [X1,˜X1]D×2p. We fit a glmnet model separately for each outcome type and compute the importance statistics for each feature. Then, we detect the non-zero features for four different FDR thresholds 0.05, 0.1, 0.15, and 0.2.

W1 <- generate_data(p=100, seed=2)$W1 ## Log-Normalized Sample Taxa matrix 
W_tilde1 <- log_normalize(X1_tilde) ## Log-Normalized Knockoff Matrix
Y1 <- generate_data(p=100, seed=2)$Y1 ## Continuous Responses
Y1_bin <- generate_data(p = 100, seed = 2)$Y1_bin ## Binary Responses
index <- generate_data(p = 100, seed = 2)$index ## True signals

################### Continuous Outcomes #########################
############## FDR = 0.05 ###############
index_est <- zinck.filter(W1,W_tilde1,Y1,model="glmnet",fdr=0.05)

Selected variables:
 [1]  1  7 14 20 21 28 33 34 35 37 38 39 42 43 44 51 54 59 68 70 73 74 79 82 83
[26] 84 85 87 96 97
# ### Evaluation metrics ###
FN <- sum(index %in% index_est == FALSE) ## False Negatives
FP <- sum(index_est %in% index == FALSE) ## False Positives
TP <- sum(index_est %in% index == TRUE) ## True Positives

estimated_FDR <- FP / (FP + TP) # Evaluating the empirical False Discovery Rate
estimated_power <- TP / (TP + FN) # Evaluating the empirical Power or TPR

print(paste("Estimated FDR for Target 0.05 (Continuous case):", estimated_FDR))
[1] "Estimated FDR for Target 0.05 (Continuous case): 0"
print(paste("Estimated Power for Target 0.05 (Continuous case):", estimated_power))
[1] "Estimated Power for Target 0.05 (Continuous case): 1"
############## FDR = 0.1 ###############
index_est <- zinck.filter(W1,W_tilde1,Y1,model="glmnet",fdr=0.1)

Selected variables:
 [1]  1  7 14 20 21 28 33 34 35 37 38 39 42 43 44 51 54 59 68 70 73 74 79 82 83
[26] 84 85 87 96 97
# ### Evaluation metrics ###
FN <- sum(index %in% index_est == FALSE) ## False Negatives
FP <- sum(index_est %in% index == FALSE) ## False Positives
TP <- sum(index_est %in% index == TRUE) ## True Positives

estimated_FDR <- FP / (FP + TP) # Evaluating the empirical False Discovery Rate
estimated_power <- TP / (TP + FN) # Evaluating the empirical Power or TPR

print(paste("Estimated FDR for Target 0.1 (Continuous case):", estimated_FDR))
[1] "Estimated FDR for Target 0.1 (Continuous case): 0"
print(paste("Estimated Power for Target 0.1 (Continuous case):", estimated_power))
[1] "Estimated Power for Target 0.1 (Continuous case): 1"
############## FDR = 0.15 ###############
index_est <- zinck.filter(W1,W_tilde1,Y1,model="glmnet",fdr=0.15)

Selected variables:
 [1]  1  7 14 20 21 28 33 34 35 37 38 39 42 43 44 51 54 59 68 70 73 74 79 82 83
[26] 84 85 87 96 97
# ### Evaluation metrics ###
FN <- sum(index %in% index_est == FALSE) ## False Negatives
FP <- sum(index_est %in% index == FALSE) ## False Positives
TP <- sum(index_est %in% index == TRUE) ## True Positives

estimated_FDR <- FP / (FP + TP) # Evaluating the empirical False Discovery Rate
estimated_power <- TP / (TP + FN) # Evaluating the empirical Power or TPR

print(paste("Estimated FDR for Target 0.15 (Continuous case):", estimated_FDR))
[1] "Estimated FDR for Target 0.15 (Continuous case): 0"
print(paste("Estimated Power for Target 0.15 (Continuous case):", estimated_power))
[1] "Estimated Power for Target 0.15 (Continuous case): 1"
############## FDR = 0.2 ###############
index_est <- zinck.filter(W1,W_tilde1,Y1,model="glmnet",fdr=0.2)

Selected variables:
 [1]  1  7 14 20 21 28 33 34 35 37 38 39 42 43 44 51 54 59 68 70 73 74 79 82 83
[26] 84 85 87 96 97
# ### Evaluation metrics ###
FN <- sum(index %in% index_est == FALSE) ## False Negatives
FP <- sum(index_est %in% index == FALSE) ## False Positives
TP <- sum(index_est %in% index == TRUE) ## True Positives

estimated_FDR <- FP / (FP + TP) # Evaluating the empirical False Discovery Rate
estimated_power <- TP / (TP + FN) # Evaluating the empirical Power or TPR

print(paste("Estimated FDR for Target 0.2 (Continuous case):", estimated_FDR))
[1] "Estimated FDR for Target 0.2 (Continuous case): 0"
print(paste("Estimated Power for Target 0.2 (Continuous case):", estimated_power))
[1] "Estimated Power for Target 0.2 (Continuous case): 1"
################### Binary Outcomes #########################
############## FDR = 0.05 ###############
index_est <- zinck.filter(W1,W_tilde1,Y1_bin,model="glmnet",fdr=0.05)

Selected variables:
 [1]  7 20 28 33 34 35 37 38 43 44 54 59 68 70 73 74 82 87 97
# ### Evaluation metrics ###
FN <- sum(index %in% index_est == FALSE) ## False Negatives
FP <- sum(index_est %in% index == FALSE) ## False Positives
TP <- sum(index_est %in% index == TRUE) ## True Positives

estimated_FDR <- FP / (FP + TP) # Evaluating the empirical False Discovery Rate
estimated_power <- TP / (TP + FN) # Evaluating the empirical Power or TPR

print(paste("Estimated FDR for Target 0.05 (Binary case):", estimated_FDR))
[1] "Estimated FDR for Target 0.05 (Binary case): 0"
print(paste("Estimated Power for Target 0.05 (Binary case):", estimated_power))
[1] "Estimated Power for Target 0.05 (Binary case): 0.633333333333333"
############## FDR = 0.1 ###############
index_est <- zinck.filter(W1,W_tilde1,Y1_bin,model="glmnet",fdr=0.1)

Selected variables:
 [1]  1  7 20 28 33 34 35 37 38 43 44 54 59 68 70 73 74 82 87 97
# ### Evaluation metrics ###
FN <- sum(index %in% index_est == FALSE) ## False Negatives
FP <- sum(index_est %in% index == FALSE) ## False Positives
TP <- sum(index_est %in% index == TRUE) ## True Positives

estimated_FDR <- FP / (FP + TP) # Evaluating the empirical False Discovery Rate
estimated_power <- TP / (TP + FN) # Evaluating the empirical Power or TPR

print(paste("Estimated FDR for Target 0.1 (Binary case):", estimated_FDR))
[1] "Estimated FDR for Target 0.1 (Binary case): 0"
print(paste("Estimated Power for Target 0.1 (Binary case):", estimated_power))
[1] "Estimated Power for Target 0.1 (Binary case): 0.666666666666667"
############## FDR = 0.15 ###############
index_est <- zinck.filter(W1,W_tilde1,Y1_bin,model="glmnet",fdr=0.15)

Selected variables:
 [1]  1  7 20 21 28 33 34 35 37 38 43 44 54 59 63 68 70 73 74 82 84 87 97
# ### Evaluation metrics ###
FN <- sum(index %in% index_est == FALSE) ## False Negatives
FP <- sum(index_est %in% index == FALSE) ## False Positives
TP <- sum(index_est %in% index == TRUE) ## True Positives

estimated_FDR <- FP / (FP + TP) # Evaluating the empirical False Discovery Rate
estimated_power <- TP / (TP + FN) # Evaluating the empirical Power or TPR

print(paste("Estimated FDR for Target 0.15 (Binary case):", estimated_FDR))
[1] "Estimated FDR for Target 0.15 (Binary case): 0.0434782608695652"
print(paste("Estimated Power for Target 0.15 (Binary case):", estimated_power))
[1] "Estimated Power for Target 0.15 (Binary case): 0.733333333333333"
############## FDR = 0.2 ###############
index_est <- zinck.filter(W1,W_tilde1,Y1_bin,model="glmnet",fdr=0.2)

Selected variables:
 [1]  1  7 20 21 28 33 34 35 37 38 43 44 51 54 59 63 68 70 73 74 82 84 87 97
# ### Evaluation metrics ###
FN <- sum(index %in% index_est == FALSE) ## False Negatives
FP <- sum(index_est %in% index == FALSE) ## False Positives
TP <- sum(index_est %in% index == TRUE) ## True Positives

estimated_FDR <- FP / (FP + TP) # Evaluating the empirical False Discovery Rate
estimated_power <- TP / (TP + FN) # Evaluating the empirical Power or TPR

print(paste("Estimated FDR for Target 0.2 (Binary case):", estimated_FDR))
[1] "Estimated FDR for Target 0.2 (Binary case): 0.0416666666666667"
print(paste("Estimated Power for Target 0.2 (Binary case):", estimated_power))
[1] "Estimated Power for Target 0.2 (Binary case): 0.766666666666667"

The users are encouraged to play with other values of p, such as p=200,300,400.


sessionInfo()
R version 4.1.3 (2022-03-10)
Platform: x86_64-apple-darwin17.0 (64-bit)
Running under: macOS Big Sur/Monterey 10.16

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/4.1/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.1/Resources/lib/libRlapack.dylib

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
 [1] reshape2_1.4.4       dplyr_1.1.2          kosel_0.0.1         
 [4] zinck_0.0.0.9000     knockoff_0.3.6       glmnet_4.1-7        
 [7] Matrix_1.5-1         rstan_2.21.8         ggplot2_3.4.2       
[10] StanHeaders_2.21.0-7 workflowr_1.7.1     

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.10          lattice_0.21-8       prettyunits_1.1.1   
 [4] getPass_0.2-2        ps_1.7.5             rprojroot_2.0.3     
 [7] digest_0.6.31        foreach_1.5.2        utf8_1.2.3          
[10] plyr_1.8.8           R6_2.5.1             stats4_4.1.3        
[13] evaluate_0.21        httr_1.4.6           pillar_1.9.0        
[16] rlang_1.1.1          rstudioapi_0.14      whisker_0.4.1       
[19] callr_3.7.3          jquerylib_0.1.4      ordinalNet_2.12     
[22] rmarkdown_2.22       splines_4.1.3        stringr_1.5.0       
[25] loo_2.6.0            munsell_0.5.0        compiler_4.1.3      
[28] httpuv_1.6.11        xfun_0.39            pkgconfig_2.0.3     
[31] pkgbuild_1.4.2       shape_1.4.6          htmltools_0.5.5     
[34] tidyselect_1.2.0     tibble_3.2.1         gridExtra_2.3       
[37] codetools_0.2-19     matrixStats_0.63.0   randomForest_4.7-1.1
[40] fansi_1.0.4          crayon_1.5.2         withr_2.5.0         
[43] later_1.3.1          grid_4.1.3           jsonlite_1.8.5      
[46] gtable_0.3.3         lifecycle_1.0.3      DBI_1.1.3           
[49] git2r_0.32.0         magrittr_2.0.3       scales_1.2.1        
[52] RcppParallel_5.1.7   cli_3.6.1            stringi_1.7.12      
[55] cachem_1.0.8         fs_1.6.2             promises_1.2.0.1    
[58] doParallel_1.0.17    bslib_0.5.0          generics_0.1.3      
[61] vctrs_0.6.5          iterators_1.0.14     tools_4.1.3         
[64] glue_1.6.2           processx_3.8.1       parallel_4.1.3      
[67] fastmap_1.1.1        survival_3.5-5       yaml_2.3.7          
[70] inline_0.3.19        colorspace_2.1-0     knitr_1.43          
[73] sass_0.4.6