ANCOM-BC2 Tutorial

Huang Lin\(^1\)

\(^1\)NICHD, 6710B Rockledge Dr, Bethesda, MD 20892

October 18, 2022

1. Introduction

Analysis of Compositions of Microbiomes with Bias Correction 2 (ANCOM-BC2) is a methodology for performing differential abundance (DA) analysis of microbiome count data. This version extends and refines the previously published Analysis of Compositions of Microbiomes with Bias Correction (ANCOM-BC) methodology (Lin and Peddada 2020) in several ways as follows:

  1. Bias correction: ANCOM-BC2 estimates and corrects both the sample-specific (sampling fraction) as well as taxon-specific (sequencing efficiency) biases.

  2. Regularization of variance: Inspired by Significance Analysis of Microarrays (SAM) (Tusher, Tibshirani, and Chu 2001) methodology, a small positive constant is added to the denominator of ANCOM-BC2 test statistic corresponding to each taxon to avoid the significance due to extremely small standard errors, especially for rare taxa. By default, we used the 5-th percentile of the distribution of standard errors for each fixed effect as the regularization factor.

  3. Sensitivity analysis for the pseudo-count addition: Like other differential abundance analysis methods, ANCOM-BC2 log transforms the observed counts. However, to deal with zero counts, a pseudo-count is added before the log transformation. Several studies have shown that differential abundance results could be sensitive to the choice of pseudo-count (Costea et al. 2014; Paulson, Bravo, and Pop 2014), resulting in an inflated false positive rate. To avoid such false positives, we conduct a sensitivity analysis and provide a sensitivity score for each taxon to determine if a particular taxon is sensitive to the choice of pseudo-count. The larger the score, the more likely the significant result is a false positive.

  4. Multi-group comparisons and repeated measurements: The ANCOM-BC2 methodology extends ANCOM-BC for multiple groups and repeated measurements as follows:

    • Multiple pairwise directional test: When performning pairwise directional test, the mixed directional false discover rate (mdFDR) should be taken into account. The mdFDR is the combination of false discovery rate due to multiple testing, multiple pairwise comparisons, and directional tests within each pairwise comparison. For example, suppose we have five taxa and three experimental groups: g1, g2, and g3. Thus, we are performing five tests corresponding to five taxa. For each taxon, we are also conducting three pairwise comparisons (g1 vs. g2, g2 vs. g3, and g1 vs. g3). Within each pairwise comparison, we wish to determine if the abundance has increased or decreased or did not change (direction of the effect size). Errors could occur in each step. The overall false discovery rate is controlled by the mdFDR methodology we adopted from (Guo, Sarkar, and Peddada 2010; Grandhi, Guo, and Peddada 2016).

    • Multiple pairwise directional test against a pre-specified group (e.g., Dunnett’s type of test): We use the same set-up as in the multiple pairwise directional test but use the Dunnett-type modification described in (Grandhi, Guo, and Peddada 2016) to control the mdFDR which is more powerful.

    • Trend test for ordered groups: In some instances, researchers are interested in discovering abundance patterns of each taxon over the ordered groups, for example, groups based on the health condition of subjects (e.g., lean, overweight, obese). In such cases, in addition to pairwise comparison, one may be interested in identifying taxa whose abundances are increasing or decreasing or have other patterns over the groups. We adopted methodologies from (Jelsema and Peddada 2016) to perform trend test under the ANCOM-BC2 framework.

A clarification regarding Structural zeros: A taxon is considered to have structural zeros in some (>=1) groups if it is completely (or nearly completely) missing in these groups. For instance, suppose there are three groups: g1, g2, and g3. If the counts of taxon A in g1 are 0, but they are nonzero in g2 and g3, then taxon A will be considered to contain structural zeros in g1. In this example, taxon A is declared to be differentially abundant between g1 and g2, g1 and g3, and consequently, it is globally differentially abundant with respect to this group variable. Such taxa are not further analyzed using ANCOM-BC2, but the results are summarized in the overall summary.

2. Installation

Download the package.

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

Load the package.

library(ANCOMBC)

3. Run ANCOM-BC2 on a simulated dataset

3.1 Generate simulated data

  1. The simulated data contain: one continuous covariate: cont_cov, and one categorical covariate: cat_cov. The categorical covariate has three levels: “1”, “2”, and “3”. Let’s focus on the discussions on the group variable: cat_cov.

  2. The true abundances are generated using the Poisson lognormal (PLN) model based on the mechanism described in the LDM paper (Hu and Satten 2020). The PLN model relates the abundance vector \(Y_i\) with a Gaussian latent vector \(Z_i\) for taxon \(i\) as follows: \[ \text{latent layer: } Z_i \sim N(\mu_i, \Sigma_i) \\ \text{observation layer: } Y_i|Z_i \sim POI(N \exp(Z_i)) \] where \(N\) is the scaling factor. Because of the presence of a latent layer, the PLN model displays a larger variance than the Poisson model (over-dispersion). Also, the covariance (correlation) between abundances has the same sign as the covariance (correlation) between the corresponding latent variables. This property gives enormous flexibility in modeling the variance-covariance structure of microbial abundances since it is easy to specify different variance-covariance matrices in the multivariate Gaussian distribution.

  3. Instead of specifying the variance-covariance matrix, we choose to estimate the variance-covariance matrix from a real dataset: the Quantitative Microbiome Project (QMP) data (Vandeputte et al. 2017). This dataset contains quantitative microbiome count data of 106 samples and 91 OTUs.

data(QMP)
set.seed(12345)
n = 150
d = ncol(QMP)
diff_prop = 0.1
lfc_cont = -1
lfc_cat2_vs_1 = -2
lfc_cat3_vs_1 = 1

# Generate the true abundances
abn_data = sim_plnm(abn_table = QMP, taxa_are_rows = FALSE, prv_cut = 0.05, 
                    n = n, scale_mean = 1e8, scale_sd = 1e8/3)
log_abn_data = log(abn_data + 1e-5)
rownames(log_abn_data) = paste0("T", seq_len(d))
colnames(log_abn_data) = paste0("S", seq_len(n))

# Generate the sample and feature meta data
# Sampling fractions are set to differ by batches
smd = data.frame(samp_frac = log(c(runif(n/3, min = 1e-4, max = 1e-3),
                                   runif(n/3, min = 1e-3, max = 1e-2),
                                   runif(n/3, min = 1e-2, max = 1e-1))),
                 cont_cov = rnorm(n),
                 cat_cov = as.factor(rep(seq_len(3), each = n/3)))
rownames(smd) = paste0("S", seq_len(n))
                      
fmd = data.frame(seq_eff = log(runif(d, min = 0.1, max = 1)),
                 lfc_cont = sample(c(0, lfc_cont), 
                                   size = d,
                                   replace = TRUE,
                                   prob = c(1 - diff_prop, diff_prop)),
                 lfc_cat2_vs_1 = sample(c(0, lfc_cat2_vs_1), 
                                        size = d,
                                        replace = TRUE,
                                        prob = c(1 - diff_prop, diff_prop)),
                 lfc_cat3_vs_1 = sample(c(0, lfc_cat3_vs_1), 
                                        size = d,
                                        replace = TRUE,
                                        prob = c(1 - diff_prop, diff_prop))) %>%
    mutate(lfc_cat3_vs_2 = lfc_cat3_vs_1 - lfc_cat2_vs_1)
rownames(fmd) = paste0("T", seq_len(d))

# Add effect sizes of covariates to the true abundances
dmy = caret::dummyVars(" ~ cat_cov", data = smd)
smd_dmy = data.frame(predict(dmy, newdata = smd))

log_abn_data = log_abn_data + outer(fmd$lfc_cont, smd$cont_cov)
log_abn_data = log_abn_data + outer(fmd$lfc_cat2_vs_1, smd_dmy$cat_cov.2)
log_abn_data = log_abn_data + outer(fmd$lfc_cat3_vs_1, smd_dmy$cat_cov.3)

# Add sample- and taxon-specific biases
log_otu_data = t(t(log_abn_data) + smd$samp_frac)
log_otu_data = log_otu_data + fmd$seq_eff
otu_data = round(exp(log_otu_data))

# Create the tse object
assays = SimpleList(counts = otu_data)
smd = DataFrame(smd)
tse = TreeSummarizedExperiment(assays = assays, colData = smd)

3.2 Run ancombc2 function

3.3 Sensitivity scores for “cat_cov”

  1. Extreme values (>5) of the sensitivity score are observed.

  2. Significant taxa with extreme sensitivity scores are likely false positives.

  3. We empirically set the threshold for the sensitivity score as 5. Taxa with sensitivity scores larger than the specified threshold will be considered nonsignificant.

3.4 Power and FDR

# Not considering the effect of pseudo-count addition
lfc = res_prim %>%
    dplyr::select(starts_with("lfc"))
diff = res_prim %>%
    dplyr::select(starts_with("diff"))
res_merge = (lfc * diff) %>%
    rownames_to_column("tax_id") %>%
    left_join(fmd %>%
                  rownames_to_column("tax_id"), 
              by = "tax_id") %>%
    transmute(
        tax_id = tax_id,
        est_cov2 = case_when(
            lfc_cat_cov2 > 0 ~ 1,
            lfc_cat_cov2 < 0 ~ -1,
            TRUE ~ 0),
        est_cov3 = case_when(
            lfc_cat_cov3 > 0 ~ 1,
            lfc_cat_cov3 < 0 ~ -1,
            TRUE ~ 0),
        true_cat2 = case_when(
            lfc_cat2_vs_1 > 0 ~ 1,
            lfc_cat2_vs_1 < 0 ~ -1,
            TRUE ~ 0),
        true_cat3 = case_when(
            lfc_cat3_vs_1 > 0 ~ 1,
            lfc_cat3_vs_1 < 0 ~ -1,
            TRUE ~ 0)
        )

# Cat 2 vs Cat 1
lfc_est = res_merge$est_cov2
lfc_true = res_merge$true_cat2
tp = sum(lfc_true != 0 & lfc_est != 0)
fp = sum(lfc_true == 0 & lfc_est != 0)
fn = sum(lfc_true != 0 & lfc_est == 0)
power1_nosens = tp/(tp + fn)
fdr1_nosens = fp/(tp + fp)

# Cat 3 vs Cat 1
lfc_est = res_merge$est_cov3
lfc_true = res_merge$true_cat3
tp = sum(lfc_true != 0 & lfc_est != 0)
fp = sum(lfc_true == 0 & lfc_est != 0)
fn = sum(lfc_true != 0 & lfc_est == 0)
power2_nosens = tp/(tp + fn)
fdr2_nosens = fp/(tp + fp)

# Considering the effect of pseudo-count addition
res_merge2 = res_merge %>%
    left_join(tab_sens %>%
                  rownames_to_column("tax_id"),
              by = "tax_id")

# Cat 2 vs Cat 1
lfc_est = res_merge2$est_cov2 * (res_merge2$cat_cov2 < sens_cut1)
lfc_true = res_merge2$true_cat2
tp = sum(lfc_true != 0 & lfc_est != 0)
fp = sum(lfc_true == 0 & lfc_est != 0)
fn = sum(lfc_true != 0 & lfc_est == 0)
power1_sens = tp/(tp + fn)
fdr1_sens = fp/(tp + fp)

# Cat 3 vs Cat 1
lfc_est = res_merge2$est_cov3 * (res_merge2$cat_cov3 < sens_cut2)
lfc_true = res_merge2$true_cat3
tp = sum(lfc_true != 0 & lfc_est != 0)
fp = sum(lfc_true == 0 & lfc_est != 0)
fn = sum(lfc_true != 0 & lfc_est == 0)
power2_sens = tp/(tp + fn)
fdr2_sens = fp/(tp + fp)

tab_summ1 = data.frame(Comparison = c("Not considering the effect of pseudo-count addition", 
                                      "Considering the effect of pseudo-count addition"),
                       Power = round(c(power1_nosens, power1_sens), 2),
                       FDR = round(c(fdr1_nosens, fdr1_sens), 2))

tab_summ2 = data.frame(Comparison = c("Not considering the effect of pseudo-count addition", 
                                      "Considering the effect of pseudo-count addition"),
                       Power = round(c(power2_nosens, power2_sens), 2),
                       FDR = round(c(fdr2_nosens, fdr2_sens), 2))
tab_summ1 %>%
    datatable(caption = "Cat 2 vs Cat 1")

4. Run ANCOM-BC2 on a real cross-sectional dataset

4.1 Import example data

The HITChip Atlas dataset contains genus-level microbiota profiling with HITChip for 1006 western adults with no reported health complications, reported in (Lahti et al. 2014). The dataset is also available via the microbiome R package (Lahti et al. 2017) in phyloseq (McMurdie and Holmes 2013) format. In this tutorial, we consider the following covariates:

class: TreeSummarizedExperiment 
dim: 130 873 
metadata(0):
assays(1): counts
rownames(130): Actinomycetaceae Aerococcus ... Xanthomonadaceae
  Yersinia et rel.
rowData names(3): Phylum Family Genus
colnames(873): Sample-1 Sample-2 ... Sample-1005 Sample-1006
colData names(12): age sex ... bmi region
reducedDimNames(0):
mainExpName: NULL
altExpNames(0):
rowLinks: NULL
rowTree: NULL
colLinks: NULL
colTree: NULL

4.2 Run ancombc2 function

4.3 Structural zeros

4.4 Sensitivity scores

4.5 ANCOM-BC2 primary analysis

Result from the ANCOM-BC2 methodology to determine taxa that are differentially abundant according to the covariate of interest. Results contain: 1) log fold changes, 2) standard errors, 3) test statistics, 4) p-values, 5) adjusted p-values, 6) indicators of whether the taxon is differentially abundant (TRUE) or not (FALSE).

Results for age

Results for bmi

4.6 ANCOM-BC2 global test

ANCOM-BC2 global test aims to determine taxa that are differentially abundant between at least two groups across three or more experimental groups.

In this example, we want to identify taxa that are differentially abundant between at least two groups across “lean”, “overweight”, and “obese”. The result contains: 1) test statistics, 2) p-values, 3) adjusted p-values, 4) indicators of whether the taxon is differentially abundant (TRUE) or not (FALSE).

4.7 ANCOM-BC2 pairwise directional test

ANCOM-BC2 pairwise directional test aims to determine taxa that are differentially abundant between any pair of two groups across three or more experimental groups, while controlling the mdFDR.

In this example, we want to identify taxa that are differentially abundant between any pair of two groups across “lean”, “overweight”, and “obese”. The result contains: 1) log fold changes, 2) standard errors, 3) test statistics, 4) p-values, 5) adjusted p-values, 6) indicators of whether the taxon is differentially abundant (TRUE) or not (FALSE).

4.8 ANCOM-BC2 Dunnett’s type of test

The Dunnett’s test (Dunnett 1955; Dunnett and Tamhane 1991, 1992) is designed for making comparisons of several experimental groups with the control or the reference group. ANCOM-BC2 Dunnett’s type of test adopts the framework of Dunnett’s test while controlling the mdFDR. Of note is that the ANCOM-BC2 primary results do not control the mdFDR for the comparison of multiple groups.

In this example, we want to identify taxa that are differentially abundant between “lean”, “overweight”, and the reference group “obese”. The result contains: 1) log fold changes, 2) standard errors, 3) test statistics, 4) p-values, 5) adjusted p-values, 6) indicators of whether the taxon is differentially abundant (TRUE) or not (FALSE).

4.9 ANCOM-BC2 trend test

Sometimes the experimental groups are intrinsically ordered (e.g., in a dose-response study), and we would like to see if the microbial abundances show some trends accordingly. Examples of trends include: monotonically increasing, monotonically decreasing, and umbrella shape.

ANCOM-BC2 is able to identify potential trends by testing the contrast: \[Ax \ge 0\] where \(A\) is the contrast matrix and \(x\) is the vector of parameters.

For instance, in this example, we want to identify taxa that are monotonically increasing across “obese”, “overweight”, and “lean”. Note that “obese” is the reference group, and the parameters we can estimate are the differences as compared to the reference group, i.e., \[x = (\text{overweight - obese}, \text{lean - obese})^T\] To test the monotonically increasing trend: \[H_0: \text{obese} = \text{overweight} = \text{lean} \\ H_1: \text{obese} \le \text{overweight} \le \text{lean} \quad \text{with at least one strict inequality}\] We shall specify the contrast matrix \(A\) as \[A = \begin{bmatrix} 1 & 0 \\ -1 & 1 \end{bmatrix}\] Similarly, to test for the monotonically decreasing trend: \[H_0: \text{obese} = \text{overweight} = \text{lean} \\ H_1: \text{obese} \ge \text{overweight} \ge \text{lean} \quad \text{with at least one strict inequality}\] We shall specify the contrast matrix \(A\) as \[A = \begin{bmatrix} -1 & 0 \\ 1 & -1 \end{bmatrix}\] To test for monotonic trend (increasing or decreasing), one should specify the node parameter in trend_control as the last position of x. In this example, the vector of parameters \(x\) is of length 2, thus, the last position is 2. For testing umbrella shape, for instance, in this example: \[H_0: \text{obese} = \text{overweight} = \text{lean} \\ H_1: \text{obese} \le \text{overweight} \ge \text{lean} \quad \text{with at least one strict inequality}\] one should set node as the position of the turning point of x. In this example, the turning position is overweight, thus, node = 1.

We will test both the monotonically increasing and decreasing trends in this example. The result contains: 1) log fold changes, 2) standard errors, 3) test statistics, 4) p-values, 5) adjusted p-values, 6) indicators of whether the taxon is differentially abundant (TRUE) or not (FALSE).

5. Run ANCOM-BC2 on a real longitudinal dataset

5. 1 Import example data

A two-week diet swap study between western (USA) and traditional (rural Africa) diets (Lahti et al. 2014). The dataset is also available via the microbiome R package (Lahti et al. 2017) in phyloseq (McMurdie and Holmes 2013) format.

class: TreeSummarizedExperiment 
dim: 130 222 
metadata(0):
assays(1): counts
rownames(130): Actinomycetaceae Aerococcus ... Xanthomonadaceae
  Yersinia et rel.
rowData names(3): Phylum Family Genus
colnames(222): Sample-1 Sample-2 ... Sample-221 Sample-222
colData names(8): subject sex ... timepoint.within.group bmi_group
reducedDimNames(0):
mainExpName: NULL
altExpNames(0):
rowLinks: NULL
rowTree: NULL
colLinks: NULL
colTree: NULL

5.2 Run ancombc2 function

In this tutorial, we consider the following fixed effects:

and the following random effects:

Procedures of ANCOM-BC2 global test, pairwise directional test, Dunnett’s type of test, and trend test are the same as those for the cross-sectional data shown above.

Session information

sessionInfo()
R version 4.2.1 (2022-06-23)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: Ubuntu 20.04.5 LTS

Matrix products: default
BLAS:   /home/biocbuild/bbs-3.15-bioc/R/lib/libRblas.so
LAPACK: /home/biocbuild/bbs-3.15-bioc/R/lib/libRlapack.so

locale:
 [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C              
 [3] LC_TIME=en_GB              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       

attached base packages:
[1] stats4    stats     graphics  grDevices utils     datasets  methods  
[8] base     

other attached packages:
 [1] caret_6.0-93                   lattice_0.20-45               
 [3] DT_0.25                        mia_1.4.0                     
 [5] MultiAssayExperiment_1.22.0    TreeSummarizedExperiment_2.4.0
 [7] Biostrings_2.64.1              XVector_0.36.0                
 [9] SingleCellExperiment_1.18.1    SummarizedExperiment_1.26.1   
[11] Biobase_2.56.0                 GenomicRanges_1.48.0          
[13] GenomeInfoDb_1.32.4            IRanges_2.30.1                
[15] S4Vectors_0.34.0               BiocGenerics_0.42.0           
[17] MatrixGenerics_1.8.1           matrixStats_0.62.0            
[19] forcats_0.5.2                  stringr_1.4.1                 
[21] dplyr_1.0.10                   purrr_0.3.5                   
[23] readr_2.1.3                    tidyr_1.2.1                   
[25] tibble_3.1.8                   ggplot2_3.3.6                 
[27] tidyverse_1.3.2                ANCOMBC_1.6.4                 

loaded via a namespace (and not attached):
  [1] estimability_1.4.1          ModelMetrics_1.2.2.2       
  [3] coda_0.19-4                 bit64_4.0.5                
  [5] knitr_1.40                  irlba_2.3.5.1              
  [7] multcomp_1.4-20             DelayedArray_0.22.0        
  [9] data.table_1.14.4           rpart_4.1.16               
 [11] hardhat_1.2.0               RCurl_1.98-1.9             
 [13] doParallel_1.0.17           generics_0.1.3             
 [15] ScaledMatrix_1.4.1          TH.data_1.1-1              
 [17] RSQLite_2.2.18              proxy_0.4-27               
 [19] future_1.28.0               bit_4.0.4                  
 [21] tzdb_0.3.0                  xml2_1.3.3                 
 [23] lubridate_1.8.0             assertthat_0.2.1           
 [25] DirichletMultinomial_1.38.0 viridis_0.6.2              
 [27] gargle_1.2.1                gower_1.0.0                
 [29] xfun_0.33                   hms_1.1.2                  
 [31] jquerylib_0.1.4             evaluate_0.17              
 [33] fansi_1.0.3                 dbplyr_2.2.1               
 [35] readxl_1.4.1                DBI_1.1.3                  
 [37] htmlwidgets_1.5.4           googledrive_2.0.0          
 [39] Rmpfr_0.8-9                 CVXR_1.0-10                
 [41] ellipsis_0.3.2              crosstalk_1.2.0            
 [43] backports_1.4.1             energy_1.7-10              
 [45] permute_0.9-7               deldir_1.0-6               
 [47] sparseMatrixStats_1.8.0     vctrs_0.4.2                
 [49] cachem_1.0.6                withr_2.5.0                
 [51] checkmate_2.1.0             emmeans_1.8.1-1            
 [53] vegan_2.6-4                 treeio_1.20.2              
 [55] cluster_2.1.4               gsl_2.1-7.1                
 [57] ape_5.6-2                   lazyeval_0.2.2             
 [59] crayon_1.5.2                recipes_1.0.2              
 [61] pkgconfig_2.0.3             labeling_0.4.2             
 [63] nlme_3.1-160                vipor_0.4.5                
 [65] nnet_7.3-18                 rlang_1.0.6                
 [67] globals_0.16.1              lifecycle_1.0.3            
 [69] sandwich_3.0-2              modelr_0.1.9               
 [71] rsvd_1.0.5                  cellranger_1.1.0           
 [73] rngtools_1.5.2              Matrix_1.5-1               
 [75] boot_1.3-28                 zoo_1.8-11                 
 [77] reprex_2.0.2                base64enc_0.1-3            
 [79] beeswarm_0.4.0              googlesheets4_1.0.1        
 [81] png_0.1-7                   viridisLite_0.4.1          
 [83] rootSolve_1.8.2.3           bitops_1.0-7               
 [85] pROC_1.18.0                 blob_1.2.3                 
 [87] DelayedMatrixStats_1.18.2   doRNG_1.8.2                
 [89] decontam_1.16.0             parallelly_1.32.1          
 [91] jpeg_0.1-9                  DECIPHER_2.24.0            
 [93] beachmat_2.12.0             scales_1.2.1               
 [95] memoise_2.0.1               magrittr_2.0.3             
 [97] plyr_1.8.7                  zlibbioc_1.42.0            
 [99] compiler_4.2.1              RColorBrewer_1.1-3         
[101] lme4_1.1-30                 cli_3.4.1                  
[103] lmerTest_3.1-3              listenv_0.8.0              
[105] htmlTable_2.4.1             Formula_1.2-4              
[107] MASS_7.3-58.1               mgcv_1.8-40                
[109] tidyselect_1.2.0            stringi_1.7.8              
[111] highr_0.9                   yaml_2.3.6                 
[113] BiocSingular_1.12.0         latticeExtra_0.6-30        
[115] ggrepel_0.9.1               grid_4.2.1                 
[117] sass_0.4.2                  tools_4.2.1                
[119] lmom_2.9                    future.apply_1.9.1         
[121] parallel_4.2.1              rstudioapi_0.14            
[123] foreach_1.5.2               foreign_0.8-83             
[125] gridExtra_2.3               gld_2.6.5                  
[127] prodlim_2019.11.13          farver_2.1.1               
[129] digest_0.6.30               lava_1.6.10                
[131] Rcpp_1.0.9                  broom_1.0.1                
[133] scuttle_1.6.3               httr_1.4.4                 
[135] Rdpack_2.4                  colorspace_2.0-3           
[137] rvest_1.0.3                 fs_1.5.2                   
[139] splines_4.2.1               yulab.utils_0.0.5          
[141] tidytree_0.4.1              expm_0.999-6               
[143] scater_1.24.0               Exact_3.2                  
[145] xtable_1.8-4                gmp_0.6-6                  
[147] jsonlite_1.8.2              nloptr_2.0.3               
[149] timeDate_4021.106           ipred_0.9-13               
[151] R6_2.5.1                    Hmisc_4.7-1                
[153] pillar_1.8.1                htmltools_0.5.3            
[155] glue_1.6.2                  fastmap_1.1.0              
[157] minqa_1.2.4                 BiocParallel_1.30.4        
[159] BiocNeighbors_1.14.0        class_7.3-20               
[161] codetools_0.2-18            mvtnorm_1.1-3              
[163] utf8_1.2.2                  bslib_0.4.0                
[165] numDeriv_2016.8-1.1         ggbeeswarm_0.6.0           
[167] DescTools_0.99.46           interp_1.1-3               
[169] survival_3.4-0              rmarkdown_2.17             
[171] munsell_0.5.0               e1071_1.7-11               
[173] GenomeInfoDbData_1.2.8      iterators_1.0.14           
[175] haven_2.5.1                 reshape2_1.4.4             
[177] gtable_0.3.1                rbibutils_2.2.9

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