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BMDA

Bayesian Modeling of Metagenomic Sequencing Data for Discovering Microbial Biomarkers in Colorectal Cancer Detection

Introduction

The following script is a tutorial for performing differential abundance analysis for microbiome count data using the proposed Zero-inflated Negative Binomial model with the Dirichlet Process Prior (ZINB-DPP) for data normalization in the manuscript Q.Li et al. 2019 https://arxiv.org/abs/1902.08741.

Contents

Data Generation

To mimic the real microbiome data that contains taxa from different taxonomic levels, we simulate the data following the tree structure of the real data analyzed in the paper. The tree contains 492 taxa from species to kingdom levels.

Taxonomic tree structure

We first load the real taxonomic tree information for data simulation. Suppose we have p_0 = 20 discriminating species as highlighted by the colored dots in the cladogram shown below.

source("user_functions.R")
load("data/simulation_realtree.Rdata")

# print the names of the discriminating species 
print(da_taxa_mild)
##  [1] "s__Bacteroides_dorei"            "s__Bacteroides_cellulosilyticus"
##  [3] "s__Bacteroides_plebeius"         "s__Bacteroides_vulgatus"        
##  [5] "s__Bacteroides_sp_2_1_22"        "s__Bacteroides_clarus"          
##  [7] "s__Bacteroides_ovatus"           "s__Bacteroides_fragilis"        
##  [9] "s__Bacteroides_faecis"           "s__Bacteroides_eggerthii"       
## [11] "s__Enterorhabdus_caecimuris"     "s__Methanosphaera_stadtmanae"   
## [13] "s__Porcine_type_C_oncovirus"     "s__Coprobacillus_sp_29_1"       
## [15] "s__Ruminococcus_obeum"           "s__Butyricicoccus_pullicaecorum"
## [17] "s__Clostridium_bartlettii"       "s__Coprococcus_comes"           
## [19] "s__Adlercreutzia_equolifaciens"  "s__Alloscardovia_omnicolens"

# randomly set the direction of enrichment (direction_t) of these discriminating species 
set.seed(20201201)
direction_t = rep(1, 20)
neg_idx = sample(x = seq(1, 20), size = round(0.5*20) )
direction_t[neg_idx] = -1

# check the cladogram 
tree.check(controln = da_taxa_mild[direction_t == -1],
           casen = da_taxa_mild[direction_t == 1],
           hl_size = 2)

Here, the two different colors used for the discriminating species indicate the enrichment direction. The pink dots are species that are more abundant in group 1, while the blue dots are species enriched in group 2.

Count simulation

Based on the tree structure above, we generate the count table by assuming there are n=24 subjects with equal group sizes. The function gen_zinb_raw and gen_zinb_info generate the data from a ZINB distribution

# set group size 
n_group = 20
# set the phenotype information
group_index = rep(c(0,1), each = n_group)
# set the number of discriminating and nondiscriminating species 
p_diff = 20; p_nondiff = length(taxa_name) - 20
temp_file = gen_zinb_raw(zt = group_index, p_diff = 20, 
                       p_remain = p_nondiff, 
                       direction_t = direction_t,
                       seed = 1234, 
                       low = 1, high = 2)
temp_dat = gen_zinb_info(da_taxa_mild, taxa_name, info_list = temp_file)

Next, we get the adjacent matrix based on the taxonomic tree structure

G_mat = S2adj(Y = temp_dat$Y, S = S_mat)

Model Fitting

Next, we fit the ZINB-DPP model based on the default setting:

library(Rcpp)
sourceCpp("core_zinb_x5.cpp")

# model input & initial settings
Ymat = temp_dat$Y
n_sample = n_group * 2
s_ini = rep(1, n_sample)
Niter = 10000

# MCMC 
model_fit = zinb_model_estimator(Y = Ymat, z = group_index, s = s_ini, 
                                 iter = Niter, 
                                 DPP = TRUE, 
                                 S = S_mat, 
                                 aggregate = TRUE, 
                                 store = TRUE,
                                 MRF = TRUE, 
                                 G = G_mat)
## 0% has been done
## 10% has been done
## 20% has been done
## 30% has been done
## 40% has been done
## 50% has been done
## 60% has been done
## 70% has been done
## 80% has been done
## 90% has been done

Evaluation

Now, we evaluate our model by visualizing the result of the discriminating species detection.

# get the species level result
p_bottom = ncol(Ymat)
specis_idx = model_fit$flag[1:p_bottom]
gamma_bottom = model_fit[["gamma_ppi"]][1:p_bottom]
gamma_bottom = gamma_bottom[specis_idx == 0]

# truly discrimintating species
gamma_true = temp_dat$gamma_true
gamma_true = gamma_true[specis_idx == 0]
DA_true = which(gamma_true == 1)
DA_gamma = gamma_bottom[DA_true]

# detect discrimintating species by controlling the Bayesian false discovery rate 
cut_off = BayFDR(model_fit[["gamma_ppi"]][model_fit$flag == 0], 0.05)
DA_idx = which(gamma_bottom > cut_off)
plot(gamma_bottom, type = 'h', 
     ylim = c(0, 1), ylab = "Posterior probability of inclusion (PPI)", xlab = "Taxon index")
abline(h = cut_off, col = 'red',lty = 2)
points(x = DA_true , y = DA_gamma, pch = 16, col = 'red')

Each vertial line is the posterior proability of inclusion (PPI) of a species. The dashed line represents the threshold that controls the Bayesian false discovery rate to be less than 0.05. The species whose PPIs exceed the dashed line would be selected to be differentially abundant.

Notice that the red dots are true discriminating taxa. According to the model fitting, we have

  • Number of true positive = 18

  • Number of false positive = 0

  • Number of true negative = 251

  • Number of false negative = 2

Citation

Qiwei Li, Shuang Jiang, Guanghua Xiao, Andrew Y. Koh, Xiaowei Zhan (2019), Bayesian Modeling of Microbiome Data for Differential Abundance Analysis, arXiv:1902.08741.

Contact

Qiwei Li, Department of Mathematical Sciences, The University of Texas at Dallas, Richardson, Texas liqiwei2000@gmail.com