---
title: "ANCOM-BC2 Tutorial"
author: 
  - Huang Lin$^1$
  - $^1$University of Maryland, College Park, MD 20742
date: '`r format(Sys.Date(), "%B %d, %Y")`'
output: rmarkdown::html_vignette
bibliography: bibliography.bib
vignette: >
  %\VignetteIndexEntry{ANCOM-BC2 Tutorial}
  %\VignetteEngine{knitr::rmarkdown}
  \usepackage[utf8]{inputenc}
---

```{r setup, message = FALSE, warning = FALSE, comment = NA}
knitr::opts_chunk$set(message = FALSE, warning = FALSE, comment = NA, 
                      fig.width = 6.25, fig.height = 5)
library(ANCOMBC)
library(tidyverse)
library(DT)
options(DT.options = list(
  initComplete = JS("function(settings, json) {",
  "$(this.api().table().header()).css({'background-color': 
  '#000', 'color': '#fff'});","}")))

# It appears to be a package compatibility issue between the release version of 
# phyloseq and lme4, a fresh installation of phyloseq might be needed
# See this post: https://github.com/lme4/lme4/issues/743
# remotes::install_github("joey711/phyloseq", force = TRUE)
```

# 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 [@lin2020analysis] 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) [@tusher2001significance] 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 applies a log transformation 
to the observed counts. However, the presence of zero counts poses a challenge, 
and researchers often consider adding a pseudo-count before the log 
transformation. However, it has been shown that the choice of pseudo-count can 
impact the results and lead to an inflated false positive 
rate [@costea2014fair; @paulson2014reply]. To address this issue, 
we conduct a sensitivity analysis to assess the impact of different 
pseudo-counts on zero counts for each taxon. This analysis involves adding a 
series of pseudo-counts (ranging from 0.01 to 0.5 in increments of 0.01) to 
the zero counts of each taxon. Linear regression models are then performed on 
the bias-corrected log abundance table using the different pseudo-counts. The 
sensitivity score for each taxon is calculated as the proportion of times 
that the p-value exceeds the specified significance level (alpha). If all 
p-values consistently show significance or nonsignificance across different 
pseudo-counts and are consistent with the results obtained without adding 
pseudo-counts to zero counts (using the complete data), then the taxon is 
considered not sensitive to the pseudo-count addition. 

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

    + Multiple pairwise comparisons: When performning multiple pairwise 
    comparisons, 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 [@guo2010controlling; @grandhi2016multiple].
    
    + Multiple pairwise comparisons against a pre-specified group 
    (e.g., Dunnett's type of test): We use the same set-up as in the multiple 
    pairwise comparisons but use the Dunnett-type modification described 
    in [@grandhi2016multiple] to control the mdFDR which is more powerful.
    
    + Pattern analysis 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 [@jelsema2016clme] to 
    perform pattern analysis 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 a separate table. 

# 2. Installation

Download the package. 

```{r getPackage, eval=FALSE}
if (!requireNamespace("BiocManager", quietly = TRUE))
    install.packages("BiocManager")
BiocManager::install("ANCOMBC")
```

Load the package. 

```{r load, eval=FALSE}
library(ANCOMBC)
```

# 3. Run ANCOM-BC2 on a simulated dataset {.tabset}

## 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 [@hu2020testing]. 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 [@vandeputte2017quantitative]. This dataset contains 
quantitative microbiome count data of 106 samples and 91 OTUs.

```{r}
data(QMP, package = "ANCOMBC")
set.seed(123)
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, lib_mean = 1e8, disp = 0.5)
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(taxon = paste0("T", seq_len(d)),
                 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)

# Add effect sizes of covariates to the true abundances
smd_dmy = model.matrix(~ 0 + cont_cov + cat_cov, data = smd)
log_abn_data = log_abn_data + outer(fmd$lfc_cont, smd_dmy[, "cont_cov"] )
log_abn_data = log_abn_data + outer(fmd$lfc_cat2_vs_1, smd_dmy[, "cat_cov2"])
log_abn_data = log_abn_data + outer(fmd$lfc_cat3_vs_1, smd_dmy[, "cat_cov3"])

# 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))
```

## 3.2 Run ancombc2 function

```{r}
set.seed(123)
output = ancombc2(data = otu_data, meta_data = smd, 
                  fix_formula = "cont_cov + cat_cov", rand_formula = NULL,
                  p_adj_method = "holm", pseudo_sens = TRUE,
                  prv_cut = 0.10, lib_cut = 1000, s0_perc = 0.05,
                  group = "cat_cov", struc_zero = FALSE, neg_lb = FALSE,
                  alpha = 0.05, n_cl = 2, verbose = TRUE,
                  global = FALSE, pairwise = TRUE, 
                  dunnet = FALSE, trend = FALSE,
                  iter_control = list(tol = 1e-5, max_iter = 20, 
                                      verbose = FALSE),
                  em_control = list(tol = 1e-5, max_iter = 100),
                  lme_control = NULL, 
                  mdfdr_control = list(fwer_ctrl_method = "holm", B = 100), 
                  trend_control = NULL)

res_prim = output$res
res_pair = output$res_pair
```

## 3.3 Power and FDR

```{r}
res_merge1 = res_pair %>%
  dplyr::transmute(taxon, 
                   lfc_est1 = lfc_cat_cov2 * diff_cat_cov2,
                   lfc_est2 = lfc_cat_cov3 * diff_cat_cov3,
                   lfc_est3 = lfc_cat_cov3_cat_cov2 * diff_cat_cov3_cat_cov2) %>%
  dplyr::left_join(fmd %>%
                     dplyr::transmute(taxon, 
                                      lfc_true1 = lfc_cat2_vs_1,
                                      lfc_true2 = lfc_cat3_vs_1,
                                      lfc_true3 = lfc_cat3_vs_2),
                   by = "taxon") %>%
  dplyr::transmute(taxon, 
                   lfc_est1 = case_when(lfc_est1 > 0 ~ 1,
                                        lfc_est1 < 0 ~ -1,
                                        TRUE ~ 0),
                   lfc_est2 = case_when(lfc_est2 > 0 ~ 1,
                                        lfc_est2 < 0 ~ -1,
                                        TRUE ~ 0),
                   lfc_est3 = case_when(lfc_est3 > 0 ~ 1,
                                        lfc_est3 < 0 ~ -1,
                                        TRUE ~ 0),
                   lfc_true1 = case_when(lfc_true1 > 0 ~ 1,
                                         lfc_true1 < 0 ~ -1,
                                         TRUE ~ 0),
                   lfc_true2 = case_when(lfc_true2 > 0 ~ 1,
                                         lfc_true2 < 0 ~ -1,
                                         TRUE ~ 0),
                   lfc_true3 = case_when(lfc_true3 > 0 ~ 1,
                                         lfc_true3 < 0 ~ -1,
                                         TRUE ~ 0))
lfc_est1 = res_merge1$lfc_est1
lfc_true1 = res_merge1$lfc_true1
lfc_est2 = res_merge1$lfc_est2
lfc_true2 = res_merge1$lfc_true2
lfc_est3 = res_merge1$lfc_est3
lfc_true3 = res_merge1$lfc_true3

tp1 = sum(lfc_true1 == 1 & lfc_est1 == 1) +
  sum(lfc_true1 == -1 & lfc_est1 == -1)
fp1 = sum(lfc_true1 == 0 & lfc_est1 != 0) +
  sum(lfc_true1 == 1 & lfc_est1 == -1) +
  sum(lfc_true1 == -1 & lfc_est1 == 1)
fn1 = sum(lfc_true1 != 0 & lfc_est1 == 0)

tp2 = sum(lfc_true2 == 1 & lfc_est2 == 1) +
  sum(lfc_true2 == -1 & lfc_est2 == -1)
fp2 = sum(lfc_true2 == 0 & lfc_est2 != 0) +
  sum(lfc_true2 == 1 & lfc_est2 == -1) +
  sum(lfc_true2 == -1 & lfc_est2 == 1)
fn2 = sum(lfc_true2 != 0 & lfc_est2 == 0)

tp3 = sum(lfc_true3 == 1 & lfc_est3 == 1) +
  sum(lfc_true3 == -1 & lfc_est3 == -1)
fp3 = sum(lfc_true3 == 0 & lfc_est3 != 0) +
  sum(lfc_true3 == 1 & lfc_est3 == -1) +
  sum(lfc_true3 == -1 & lfc_est3 == 1)
fn3 = sum(lfc_true3 != 0 & lfc_est3 == 0)

tp = tp1 + tp2 + tp3
fp = fp1 + fp2 + fp3
fn = fn1 + fn2 + fn3

power1 = tp/(tp + fn)
fdr1 = fp/(tp + fp)

res_merge2 = res_pair %>%
  dplyr::transmute(taxon, 
                   lfc_est1 = lfc_cat_cov2 * diff_cat_cov2 * passed_ss_cat_cov2,
                   lfc_est2 = lfc_cat_cov3 * diff_cat_cov3 * passed_ss_cat_cov3,
                   lfc_est3 = lfc_cat_cov3_cat_cov2 * diff_cat_cov3_cat_cov2* passed_ss_cat_cov3_cat_cov2) %>%
  dplyr::left_join(fmd %>%
                     dplyr::transmute(taxon, 
                                      lfc_true1 = lfc_cat2_vs_1,
                                      lfc_true2 = lfc_cat3_vs_1,
                                      lfc_true3 = lfc_cat3_vs_2),
                   by = "taxon") %>%
  dplyr::transmute(taxon, 
                   lfc_est1 = case_when(lfc_est1 > 0 ~ 1,
                                        lfc_est1 < 0 ~ -1,
                                        TRUE ~ 0),
                   lfc_est2 = case_when(lfc_est2 > 0 ~ 1,
                                        lfc_est2 < 0 ~ -1,
                                        TRUE ~ 0),
                   lfc_est3 = case_when(lfc_est3 > 0 ~ 1,
                                        lfc_est3 < 0 ~ -1,
                                        TRUE ~ 0),
                   lfc_true1 = case_when(lfc_true1 > 0 ~ 1,
                                         lfc_true1 < 0 ~ -1,
                                         TRUE ~ 0),
                   lfc_true2 = case_when(lfc_true2 > 0 ~ 1,
                                         lfc_true2 < 0 ~ -1,
                                         TRUE ~ 0),
                   lfc_true3 = case_when(lfc_true3 > 0 ~ 1,
                                         lfc_true3 < 0 ~ -1,
                                         TRUE ~ 0))
lfc_est1 = res_merge2$lfc_est1
lfc_true1 = res_merge2$lfc_true1
lfc_est2 = res_merge2$lfc_est2
lfc_true2 = res_merge2$lfc_true2
lfc_est3 = res_merge2$lfc_est3
lfc_true3 = res_merge2$lfc_true3

tp1 = sum(lfc_true1 == 1 & lfc_est1 == 1) +
  sum(lfc_true1 == -1 & lfc_est1 == -1)
fp1 = sum(lfc_true1 == 0 & lfc_est1 != 0) +
  sum(lfc_true1 == 1 & lfc_est1 == -1) +
  sum(lfc_true1 == -1 & lfc_est1 == 1)
fn1 = sum(lfc_true1 != 0 & lfc_est1 == 0)

tp2 = sum(lfc_true2 == 1 & lfc_est2 == 1) +
  sum(lfc_true2 == -1 & lfc_est2 == -1)
fp2 = sum(lfc_true2 == 0 & lfc_est2 != 0) +
  sum(lfc_true2 == 1 & lfc_est2 == -1) +
  sum(lfc_true2 == -1 & lfc_est2 == 1)
fn2 = sum(lfc_true2 != 0 & lfc_est2 == 0)

tp3 = sum(lfc_true3 == 1 & lfc_est3 == 1) +
  sum(lfc_true3 == -1 & lfc_est3 == -1)
fp3 = sum(lfc_true3 == 0 & lfc_est3 != 0) +
  sum(lfc_true3 == 1 & lfc_est3 == -1) +
  sum(lfc_true3 == -1 & lfc_est3 == 1)
fn3 = sum(lfc_true3 != 0 & lfc_est3 == 0)

tp = tp1 + tp2 + tp3
fp = fp1 + fp2 + fp3
fn = fn1 + fn2 + fn3

power2 = tp/(tp + fn)
fdr2 = fp/(tp + fp)

tab_summ = data.frame(Comparison = c("Without sensitivity score filter", 
                                     "With sensitivity score filter"),
                      Power = round(c(power1, power2), 2),
                      FDR = round(c(fdr1, fdr2), 2))
tab_summ %>%
    datatable(caption = "Power/FDR Comparison")
```

# 4. Benchmark the performance of ANCOM-BC2 on a null 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 [@lahti2014tipping]. The dataset is available via the 
`microbiome` R package [@lahti2017tools] in `phyloseq` [@mcmurdie2013phyloseq] 
format. In this tutorial, we consider the following covariates:

* Continuous covariates: "age"

* Categorical covariates: "region", "bmi"

* The group variable of interest: "bmi"

    + Three groups: "lean", "overweight", "obese"
    
    + The reference group: "obese"

```{r}
data(atlas1006, package = "microbiome")

# Subset to baseline
pseq = phyloseq::subset_samples(atlas1006, time == 0)

# Re-code the bmi group
meta_data = microbiome::meta(pseq)
meta_data$bmi = recode(meta_data$bmi_group,
                       obese = "obese",
                       severeobese = "obese",
                       morbidobese = "obese")

# Note that by default, levels of a categorical variable in R are sorted 
# alphabetically. In this case, the reference level for `bmi` will be 
# `lean`. To manually change the reference level, for instance, setting `obese`
# as the reference level, use:
meta_data$bmi = factor(meta_data$bmi, levels = c("obese", "overweight", "lean"))
# You can verify the change by checking:
# levels(meta_data$bmi)

# Create the region variable
meta_data$region = recode(as.character(meta_data$nationality),
                          Scandinavia = "NE", UKIE = "NE", SouthEurope = "SE", 
                          CentralEurope = "CE", EasternEurope = "EE",
                          .missing = "unknown")

phyloseq::sample_data(pseq) = meta_data

# Subset to lean, overweight, and obese subjects
pseq = phyloseq::subset_samples(pseq, bmi %in% c("lean", "overweight", "obese"))
# Discard "EE" as it contains only 1 subject
# Discard subjects with missing values of region
pseq = phyloseq::subset_samples(pseq, ! region %in% c("EE", "unknown"))

print(pseq)
```

## 4.2 Permute the `bmi` label

After permutation, the data can be considered null for `bmi`, meaning that a DA method that effectively controls false positives should produce p-values that are roughly uniformly distributed.

```{r}
set.seed(123)

pseq_perm = pseq
meta_data_perm = microbiome::meta(pseq_perm)
meta_data_perm$bmi = sample(meta_data_perm$bmi)

phyloseq::sample_data(pseq_perm) = meta_data_perm
```

## 4.3 Run `ancombc2` function 

```{r}
set.seed(123)
output = ancombc2(data = pseq_perm, tax_level = "Genus",
                  fix_formula = "bmi", rand_formula = NULL,
                  p_adj_method = "holm", pseudo_sens = TRUE,
                  prv_cut = 0, lib_cut = 1000, s0_perc = 0.05,
                  group = "bmi", struc_zero = TRUE, neg_lb = TRUE)
```

## 4.4 Visualization

The p-values from ANCOM-BC2, both with and without sensitivity analysis, exhibit a nearly uniform distribution. 

Notably, enabling sensitivity analysis significantly reduces the number of significant p-values, thereby decreasing the risk of false positives.

We **strongly recommend** that users activate the sensitivity analysis and incorporate it into their final assessment of taxon significance.

For researchers aiming to benchmark ANCOM-BC2 against new methods in terms of false positive control, it is **essential** to account for the sensitivity analysis feature to ensure a fair comparison.

```{r}
res_perm = output$res %>%
    dplyr::transmute(taxon = taxon,
                     p = p_bmioverweight,
                     ss_pass = passed_ss_bmioverweight) %>%
  rowwise() %>%
  mutate(
      p_update = case_when(
          # For taxa with significant p-values that failed the sensitivity analysis,
          # we assign a random value uniformly drawn from the range [0.05, 1].
          p < 0.05 & ss_pass == FALSE ~ runif(1, min = 0.05, max = 1),
    TRUE ~ p 
  )) %>%
  ungroup()

df_p = res_perm %>%
  dplyr::select(p, p_update) %>%
  pivot_longer(cols = p:p_update, names_to = "type", values_to = "value") 
df_p$type = recode(df_p$type, 
                   `p` = "ANCOM-BC2 (Without Sensitivity Analysis)", 
                   `p_update` = "ANCOM-BC2 (With Sensitivity Analysis)")

num_bins = 30
fig_p = df_p %>%
  ggplot(aes(x = value, fill = type)) +
  geom_histogram(position = "identity", alpha = 0.5, bins = num_bins) +
  geom_hline(yintercept = phyloseq::ntaxa(pseq_perm)/num_bins, linetype = "dashed") +
  theme_minimal() +
  labs(title = "Distribution of P-values",
       x = "P-value",
       y = "Frequency",
       fill = NULL,
       color = NULL) +
    theme_bw() + 
    theme(plot.title = element_text(hjust = 0.5),
          panel.grid.minor.y = element_blank(),
          legend.position = "bottom")
fig_p
```

# 5. Run ANCOM-BC2 on a real cross-sectional dataset

## 5.1 Import example data

We continue using the HITChip Atlas dataset as shown above.
    
## 5.2 Run `ancombc2` function using the `phyloseq` object

To control the FDR arising from multiple testing, we opt for the Holm-Bonferroni 
method over the Benjamini-Hochberg (BH) procedure, especially when dealing with 
large sample sizes where statistical power isn't the primary concern. The 
Holm-Bonferroni method, accommodating any dependence structure among p-values, 
is known to be robust against inaccuracies in p-values, an issue often seen in 
DA analysis. Figures below display only results significant after the 
Holm-Bonferroni adjustment.

```{r}
set.seed(123)
# It should be noted that we have set the number of bootstrap samples (B) equal 
# to 10 in the 'trend_control' function for computational expediency. 
# However, it is recommended that users utilize the default value of B, 
# which is 100, or larger values for optimal performance.
output = ancombc2(data = pseq, tax_level = "Family",
                  fix_formula = "age + region + bmi", rand_formula = NULL,
                  p_adj_method = "holm", pseudo_sens = TRUE,
                  prv_cut = 0.10, lib_cut = 1000, s0_perc = 0.05,
                  group = "bmi", struc_zero = TRUE, neg_lb = TRUE,
                  alpha = 0.05, n_cl = 2, verbose = TRUE,
                  global = TRUE, pairwise = TRUE, dunnet = TRUE, trend = TRUE,
                  iter_control = list(tol = 1e-2, max_iter = 20, 
                                      verbose = TRUE),
                  em_control = list(tol = 1e-5, max_iter = 100),
                  lme_control = lme4::lmerControl(),
                  mdfdr_control = list(fwer_ctrl_method = "holm", B = 100),
                  trend_control = list(contrast = list(matrix(c(1, 0, -1, 1),
                                                              nrow = 2, 
                                                              byrow = TRUE),
                                                       matrix(c(-1, 0, 1, -1),
                                                              nrow = 2, 
                                                              byrow = TRUE),
                                                       matrix(c(1, 0, 1, -1),
                                                              nrow = 2, 
                                                              byrow = TRUE)),
                                       node = list(2, 2, 1),
                                       solver = "ECOS",
                                       B = 10))
```

Tests for interactions are supported by specifying the interactions in the `fix_formula`. 
Please ensure that interactions between variables are included using the `*` operator, 
as the `:` operator is not recognized by the `ancombc2` function and will result in an error.

Additionally, when the `group` variable contains interaction terms, only the main effect 
will be considered in multi-group comparisons.

```{r}
set.seed(123)
output_interact = ancombc2(data = pseq, tax_level = "Family",
                          fix_formula = "age * bmi + region", rand_formula = NULL,
                          p_adj_method = "holm", pseudo_sens = TRUE,
                          prv_cut = 0.10, lib_cut = 1000, s0_perc = 0.05,
                          group = "bmi", struc_zero = TRUE, neg_lb = TRUE,
                          alpha = 0.05, n_cl = 2, verbose = TRUE)

head(output_interact$res) %>%
    dplyr::select(starts_with("lfc_age")) %>%
    mutate_if(is.numeric, function(x) round(x, 2)) %>%
    datatable(caption = "Example of including interaction terms")
```

## 5.3 Structural zeros (taxon presence/absence)

```{r}
tab_zero = output$zero_ind
tab_zero %>%
    datatable(caption = "The detection of structural zeros")
```

## 5.4 ANCOM-BC2 primary analysis {.tabset}

The primary output of the ANCOM-BC2 methodology identifies taxa with 
differential abundance based on the chosen covariate. The results include: 
1) log fold changes, 2) standard errors, 3) test statistics, 4) p-values, 
5) adjusted p-values, 6) indicators denoting whether the taxon is differentially 
abundant (TRUE) or not (FALSE), and 7) indicators denoting whether the taxon 
passed the sensitivity analysis (TRUE) or not (FALSE).

```{r}
res_prim = output$res
```

### Results for age 

In the subsequent waterfall plot, each bar represents a log fold-change 
(in natural log) value. Any taxon highlighted in green indicates its successful 
passage through the sensitivity analysis for pseudo-count addition.

```{r}
df_age = res_prim %>%
    dplyr::select(taxon, ends_with("age")) 
df_fig_age = df_age %>%
    dplyr::filter(diff_age == 1) %>% 
    dplyr::arrange(desc(lfc_age)) %>%
    dplyr::mutate(direct = ifelse(lfc_age > 0, "Positive LFC", "Negative LFC"),
                  color = ifelse(passed_ss_age == 1, "aquamarine3", "black"))
df_fig_age$taxon = factor(df_fig_age$taxon, levels = df_fig_age$taxon)
df_fig_age$direct = factor(df_fig_age$direct, 
                           levels = c("Positive LFC", "Negative LFC"))
  
fig_age = df_fig_age %>%
    ggplot(aes(x = taxon, y = lfc_age, fill = direct)) + 
    geom_bar(stat = "identity", width = 0.7, color = "black", 
             position = position_dodge(width = 0.4)) +
    geom_errorbar(aes(ymin = lfc_age - se_age, ymax = lfc_age + se_age), 
                  width = 0.2, position = position_dodge(0.05), color = "black") + 
    labs(x = NULL, y = "Log fold change", 
         title = "Log fold changes as one unit increase of age") + 
    scale_fill_discrete(name = NULL) +
    scale_color_discrete(name = NULL) +
    theme_bw() + 
    theme(plot.title = element_text(hjust = 0.5),
          panel.grid.minor.y = element_blank(),
          axis.text.x = element_text(angle = 60, hjust = 1,
                                     color = df_fig_age$color))
fig_age
```

### Results for bmi

In the subsequent heatmap, each cell represents a log fold-change 
(in natural log) value. Entries highlighted in green have successfully passed 
the sensitivity analysis for pseudo-count addition.

```{r}
df_bmi = res_prim %>%
    dplyr::select(taxon, contains("bmi")) 
df_fig_bmi1 = df_bmi %>%
    dplyr::filter(diff_bmilean == 1 | 
                    diff_bmioverweight == 1) %>%
    dplyr::mutate(lfc1 = ifelse(diff_bmioverweight == 1, 
                                round(lfc_bmioverweight, 2), 0),
                  lfc2 = ifelse(diff_bmilean == 1, 
                                round(lfc_bmilean, 2), 0)) %>%
    tidyr::pivot_longer(cols = lfc1:lfc2, 
                        names_to = "group", values_to = "value") %>%
    dplyr::arrange(taxon)

df_fig_bmi2 = df_bmi %>%
    dplyr::filter(diff_bmilean == 1 | 
                    diff_bmioverweight == 1) %>%
    dplyr::mutate(lfc1 = ifelse(passed_ss_bmioverweight == 1 & diff_bmioverweight == 1, 
                                "aquamarine3", "black"),
                  lfc2 = ifelse(passed_ss_bmilean == 1 & diff_bmilean == 1, 
                                "aquamarine3", "black")) %>%
    tidyr::pivot_longer(cols = lfc1:lfc2, 
                        names_to = "group", values_to = "color") %>%
    dplyr::arrange(taxon)

df_fig_bmi = df_fig_bmi1 %>%
    dplyr::left_join(df_fig_bmi2, by = c("taxon", "group"))

df_fig_bmi$group = recode(df_fig_bmi$group, 
                          `lfc1` = "Overweight - Obese",
                          `lfc2` = "Lean - Obese")
df_fig_bmi$group = factor(df_fig_bmi$group, 
                          levels = c("Overweight - Obese",
                                     "Lean - Obese"))
  
lo = floor(min(df_fig_bmi$value))
up = ceiling(max(df_fig_bmi$value))
mid = (lo + up)/2
fig_bmi = df_fig_bmi %>%
  ggplot(aes(x = group, y = taxon, fill = value)) + 
  geom_tile(color = "black") +
  scale_fill_gradient2(low = "blue", high = "red", mid = "white", 
                       na.value = "white", midpoint = mid, limit = c(lo, up),
                       name = NULL) +
  geom_text(aes(group, taxon, label = value, color = color), size = 4) +
  scale_color_identity(guide = "none") +
  labs(x = NULL, y = NULL, title = "Log fold changes as compared to obese subjects") +
  theme_minimal() +
  theme(plot.title = element_text(hjust = 0.5))
fig_bmi
```

## 5.5 ANCOM-BC2 global test

The primary goal of the ANCOM-BC2 global test is to discern taxa that 
demonstrate differential abundance between a minimum of two groups when 
analyzing three or more experimental groups.

To illustrate, in our current example, the objective is to pinpoint taxa with 
differential abundance across the "lean", "overweight", and "obese" categories. 
The results encompass: 1) test statistics, 2) p-values, 3) adjusted p-values, 
4) indicators denoting whether the taxon is differentially abundant (TRUE) or 
not (FALSE), and 5) indicators denoting whether the taxon passed the sensitivity 
analysis (TRUE) or not (FALSE).

In the subsequent heatmap, each cell represents a log fold-change (in natural 
log) value. Taxa marked in green have successfully passed the sensitivity 
analysis for pseudo-count addition.

```{r}
res_global = output$res_global
df_bmi = res_prim %>%
    dplyr::select(taxon, contains("bmi")) 
df_fig_global = df_bmi %>%
    dplyr::left_join(res_global %>%
                       dplyr::transmute(taxon, 
                                        diff_bmi = diff_abn, 
                                        passed_ss = passed_ss)) %>%
    dplyr::filter(diff_bmi == 1) %>%
    dplyr::mutate(lfc_overweight = lfc_bmioverweight,
                  lfc_lean = lfc_bmilean,
                  color = ifelse(passed_ss == 1, "aquamarine3", "black")) %>%
    dplyr::transmute(taxon,
                     `Overweight - Obese` = round(lfc_overweight, 2),
                     `Lean - Obese` = round(lfc_lean, 2), 
                     color = color) %>%
    tidyr::pivot_longer(cols = `Overweight - Obese`:`Lean - Obese`, 
                        names_to = "group", values_to = "value") %>%
    dplyr::arrange(taxon)

df_fig_global$group = factor(df_fig_global$group, 
                             levels = c("Overweight - Obese",
                                        "Lean - Obese"))
  
lo = floor(min(df_fig_global$value))
up = ceiling(max(df_fig_global$value))
mid = (lo + up)/2
fig_global = df_fig_global %>%
  ggplot(aes(x = group, y = taxon, fill = value)) + 
  geom_tile(color = "black") +
  scale_fill_gradient2(low = "blue", high = "red", mid = "white", 
                       na.value = "white", midpoint = mid, limit = c(lo, up),
                       name = NULL) +
  geom_text(aes(group, taxon, label = value), color = "black", size = 4) +
  labs(x = NULL, y = NULL, title = "Log fold changes for globally significant taxa") +
  theme_minimal() +
  theme(plot.title = element_text(hjust = 0.5),
        axis.text.y = element_text(color = df_fig_global %>%
                                       dplyr::distinct(taxon, color) %>%
                                       .$color))
fig_global
```

## 5.6 ANCOM-BC2 multiple pairwise comparisons

The ANCOM-BC2 methodology for multiple pairwise comparisons is designed to 
identify taxa that exhibit differential abundance between any two groups within 
a set of three or more experimental groups, all while maintaining control over 
the mdFDR.

For instance, in our analysis focusing on the categories "lean", "overweight", 
and "obese", the output provides: 1) log fold changes, 2) standard errors, 
3) test statistics, 4) p-values, 5) adjusted p-values, 6) indicators denoting 
whether the taxon is differentially abundant (TRUE) or not (FALSE), and 
7) indicators denoting whether the taxon passed the sensitivity analysis (TRUE) 
or not (FALSE).

In the subsequent heatmap, each cell represents a log fold-change 
(in natural log) value. Entries highlighted in green have successfully passed 
the sensitivity analysis for pseudo-count addition.

```{r}
res_pair = output$res_pair

df_fig_pair1 = res_pair %>%
    dplyr::filter(diff_bmioverweight == 1 |
                      diff_bmilean == 1 | 
                      diff_bmilean_bmioverweight == 1) %>%
    dplyr::mutate(lfc1 = ifelse(diff_bmioverweight == 1, 
                                round(lfc_bmioverweight, 2), 0),
                  lfc2 = ifelse(diff_bmilean == 1, 
                                round(lfc_bmilean, 2), 0),
                  lfc3 = ifelse(diff_bmilean_bmioverweight == 1, 
                                round(lfc_bmilean_bmioverweight, 2), 0)) %>%
    tidyr::pivot_longer(cols = lfc1:lfc3, 
                        names_to = "group", values_to = "value") %>%
    dplyr::arrange(taxon)

df_fig_pair2 = res_pair %>%
    dplyr::filter(diff_bmioverweight == 1 |
                      diff_bmilean == 1 | 
                      diff_bmilean_bmioverweight == 1) %>%
    dplyr::mutate(lfc1 = ifelse(passed_ss_bmioverweight == 1 & diff_bmioverweight == 1, 
                                "aquamarine3", "black"),
                  lfc2 = ifelse(passed_ss_bmilean == 1 & diff_bmilean == 1, 
                                "aquamarine3", "black"),
                  lfc3 = ifelse(passed_ss_bmilean_bmioverweight == 1 & diff_bmilean_bmioverweight == 1, 
                                "aquamarine3", "black")) %>%
    tidyr::pivot_longer(cols = lfc1:lfc3, 
                        names_to = "group", values_to = "color") %>%
    dplyr::arrange(taxon)

df_fig_pair = df_fig_pair1 %>%
    dplyr::left_join(df_fig_pair2, by = c("taxon", "group"))

df_fig_pair$group = recode(df_fig_pair$group, 
                          `lfc1` = "Overweight - Obese",
                          `lfc2` = "Lean - Obese",
                          `lfc3` = "Lean - Overweight")
df_fig_pair$group = factor(df_fig_pair$group, 
                          levels = c("Overweight - Obese",
                                     "Lean - Obese", 
                                     "Lean - Overweight"))

lo = floor(min(df_fig_pair$value))
up = ceiling(max(df_fig_pair$value))
mid = (lo + up)/2
fig_pair = df_fig_pair %>%
    ggplot(aes(x = group, y = taxon, fill = value)) + 
    geom_tile(color = "black") +
    scale_fill_gradient2(low = "blue", high = "red", mid = "white", 
                         na.value = "white", midpoint = mid, limit = c(lo, up),
                         name = NULL) +
    geom_text(aes(group, taxon, label = value, color = color), size = 4) +
    scale_color_identity(guide = FALSE) +
    labs(x = NULL, y = NULL, title = "Log fold changes as compared to obese subjects") +
    theme_minimal() +
    theme(plot.title = element_text(hjust = 0.5))
fig_pair
```

## 5.7 ANCOM-BC2 multiple pairwise comparisons against a pre-specified group (Dunnett's type of test)

The Dunnett’s test [@dunnett1955multiple; @dunnett1991step; @dunnett1992step] 
is tailored for comparing multiple experimental groups against a control or 
reference group. ANCOM-BC2 Dunnett's type of test applies this framework but 
also controls the mdFDR. It's essential to highlight that ANCOM-BC2's primary 
results control for multiple testing across taxa but not for multiple 
comparisons between groups. As such, unlike the ANCOM-BC2 Dunnett's test, the 
primary output doesn't control the mdFDR.

In this illustration, our objective is to pinpoint taxa with differential 
abundance between "lean", "overweight", and the reference group "obese". The 
results encompass: 1) log fold changes, 2) standard errors, 3) test 
statistics, 4) p-values, 5) adjusted p-values, 6) indicators denoting 
whether the taxon is differentially abundant (TRUE) or not (FALSE), and 
7) indicators denoting whether the taxon passed the sensitivity analysis (TRUE) 
or not (FALSE).

In the subsequent heatmap, each cell represents a log fold-change 
(in natural log) value. Entries highlighted in green have successfully passed 
the sensitivity analysis for pseudo-count addition.

```{r}
res_dunn = output$res_dunn

df_fig_dunn1 = res_dunn %>%
    dplyr::filter(diff_bmioverweight == 1 | 
                      diff_bmilean == 1) %>%
    dplyr::mutate(lfc1 = ifelse(diff_bmioverweight == 1, 
                                round(lfc_bmioverweight, 2), 0),
                  lfc2 = ifelse(diff_bmilean == 1, 
                                round(lfc_bmilean, 2), 0)) %>%
    tidyr::pivot_longer(cols = lfc1:lfc2, 
                        names_to = "group", values_to = "value") %>%
    dplyr::arrange(taxon)

df_fig_dunn2 = res_dunn %>%
    dplyr::filter(diff_bmioverweight == 1 | 
                      diff_bmilean == 1) %>%
    dplyr::mutate(lfc1 = ifelse(passed_ss_bmioverweight == 1 & diff_bmioverweight == 1, 
                                "aquamarine3", "black"),
                  lfc2 = ifelse(passed_ss_bmilean == 1 & diff_bmilean == 1, 
                                "aquamarine3", "black")) %>%
    tidyr::pivot_longer(cols = lfc1:lfc2, 
                        names_to = "group", values_to = "color") %>%
    dplyr::arrange(taxon)

df_fig_dunn = df_fig_dunn1 %>%
    dplyr::left_join(df_fig_dunn2, by = c("taxon", "group"))

df_fig_dunn$group = recode(df_fig_dunn$group, 
                          `lfc1` = "Overweight - Obese",
                          `lfc2` = "Lean - Obese")
df_fig_dunn$group = factor(df_fig_dunn$group, 
                          levels = c("Overweight - Obese",
                                     "Lean - Obese"))

lo = floor(min(df_fig_dunn$value))
up = ceiling(max(df_fig_dunn$value))
mid = (lo + up)/2
fig_dunn = df_fig_dunn %>%
  ggplot(aes(x = group, y = taxon, fill = value)) + 
  geom_tile(color = "black") +
  scale_fill_gradient2(low = "blue", high = "red", mid = "white", 
                       na.value = "white", midpoint = mid, limit = c(lo, up),
                       name = NULL) +
  geom_text(aes(group, taxon, label = value, color = color), size = 4) +
    scale_color_identity(guide = FALSE) +
  labs(x = NULL, y = NULL, title = "Log fold changes as compared to obese subjects") +
  theme_minimal() +
  theme(plot.title = element_text(hjust = 0.5))
fig_dunn
```

## 5.8 ANCOM-BC2 pattern analysis {.tabset}

In certain experiments, groups inherently follow an order, such as in 
dose-response studies. In these cases, we're interested in discerning if 
microbial abundances align with specific patterns. Such patterns might manifest 
as monotonically increasing, monotonically decreasing, or an umbrella shape.

ANCOM-BC2 pattern analysis is able to identify potential patterns 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 are essentially testing:
$$H_0: 0 = \text{overweight - obese} = \text{lean - obese} \\
H_1: 0 \le \text{overweight - obese} \le \text{lean - obese} \quad \text{with at least one strict inequality}$$

Thus, we shall specify the contrast matrix $A$ as
$$A = \begin{bmatrix} 1 & 0 \\ -1 & 1 \end{bmatrix}$$
The first row of $A$ matrix, $(1, 0)$, indicates that $\text{"overweight - obese"} \ge 0$, and the second row, $(-1, 1)$ represents $\text{"lean - obese"} - \text{"overweight - obese"} \ge 0$.

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
$$A = \begin{bmatrix} -1 & 0 \\ 1 & -1 \end{bmatrix}$$
Lastly, to test for an umbrella trend:
$$H_0: \text{obese} = \text{overweight} = \text{lean} \\
H_1: \text{obese} \le \text{overweight} \ge \text{lean} \quad \text{with at least one strict inequality}$$
We shall specify the contrast matrix $A$ as
$$A = \begin{bmatrix} 1 & 0 \\ 1 & -1 \end{bmatrix}$$

For testing 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, such as in the above umbrella shape 
example, 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 the monotonically increasing and decreasing trends, as well as the 
umbrella trend 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 denoting 
whether the taxon is differentially abundant (TRUE) or not (FALSE), and 
7) indicators denoting whether the taxon passed the sensitivity analysis (TRUE) 
or not (FALSE).

In the subsequent heatmap, each cell represents a log fold-change (in natural 
log) value. Taxa marked in green have successfully passed the sensitivity 
analysis for pseudo-count addition.

```{r}
res_trend = output$res_trend

df_fig_trend = res_trend %>%
    dplyr::filter(diff_abn == 1) %>%
    dplyr::mutate(lfc1 = round(lfc_bmioverweight, 2),
                  lfc2 = round(lfc_bmilean, 2),
                  color = ifelse(passed_ss == 1, "aquamarine3", "black")) %>%
    tidyr::pivot_longer(cols = lfc1:lfc2, 
                        names_to = "group", values_to = "value") %>%
    dplyr::arrange(taxon)

df_fig_trend$group = recode(df_fig_trend$group, 
                          `lfc1` = "Overweight - Obese",
                          `lfc2` = "Lean - Obese")
df_fig_trend$group = factor(df_fig_trend$group, 
                          levels = c("Overweight - Obese", 
                                     "Lean - Obese"))

lo = floor(min(df_fig_trend$value))
up = ceiling(max(df_fig_trend$value))
mid = (lo + up)/2
fig_trend = df_fig_trend %>%
    ggplot(aes(x = group, y = taxon, fill = value)) + 
    geom_tile(color = "black") +
    scale_fill_gradient2(low = "blue", high = "red", mid = "white", 
                         na.value = "white", midpoint = mid, limit = c(lo, up),
                         name = NULL) +
    geom_text(aes(group, taxon, label = value), color = "black", size = 4) +
    labs(x = NULL, y = NULL, title = "Log fold changes as compared to obese subjects") +
    theme_minimal() +
    theme(plot.title = element_text(hjust = 0.5),
          axis.text.y = element_text(color = df_fig_trend %>%
                                         dplyr::distinct(taxon, color) %>%
                                         .$color))
fig_trend
```

## 5.9 Run `ancombc2` function using the `tse` object

One can also run the `ancombc2` function using the `TreeSummarizedExperiment` object.

```{r, eval=FALSE}
tse = mia::makeTreeSummarizedExperimentFromPhyloseq(atlas1006)
tse = tse[, tse$time == 0]
tse$bmi = recode(tse$bmi_group,
                 obese = "obese",
                 severeobese = "obese",
                 morbidobese = "obese")
tse = tse[, tse$bmi %in% c("lean", "overweight", "obese")]
tse$bmi = factor(tse$bmi, levels = c("obese", "overweight", "lean"))
tse$region = recode(as.character(tse$nationality),
                    Scandinavia = "NE", UKIE = "NE", SouthEurope = "SE", 
                    CentralEurope = "CE", EasternEurope = "EE",
                    .missing = "unknown")
tse = tse[, ! tse$region %in% c("EE", "unknown")]

set.seed(123)
output = ancombc2(data = tse, assay_name = "counts", tax_level = "Family",
                  fix_formula = "age + region + bmi", rand_formula = NULL,
                  p_adj_method = "holm", pseudo_sens = TRUE,
                  prv_cut = 0.10, lib_cut = 1000, s0_perc = 0.05,
                  group = "bmi", struc_zero = TRUE, neg_lb = TRUE,
                  alpha = 0.05, n_cl = 2, verbose = TRUE,
                  global = TRUE, pairwise = TRUE, dunnet = TRUE, trend = TRUE,
                  iter_control = list(tol = 1e-2, max_iter = 20, 
                                      verbose = TRUE),
                  em_control = list(tol = 1e-5, max_iter = 100),
                  lme_control = lme4::lmerControl(),
                  mdfdr_control = list(fwer_ctrl_method = "holm", B = 100),
                  trend_control = list(contrast = list(matrix(c(1, 0, -1, 1),
                                                              nrow = 2, 
                                                              byrow = TRUE),
                                                       matrix(c(-1, 0, 1, -1),
                                                              nrow = 2, 
                                                              byrow = TRUE),
                                                       matrix(c(1, 0, 1, -1),
                                                              nrow = 2, 
                                                              byrow = TRUE)),
                                       node = list(2, 2, 1),
                                       solver = "ECOS",
                                       B = 10))
```

## 5.10 Run `ancombc2` function by directly providing the abundance and metadata

Please ensure that you provide the following: 1) The abundance data at its lowest 
possible taxonomic level. 2) The aggregated data at the desired taxonomic level; 
if no aggregation is performed, it can be the same as the original abundance data.
3) The sample metadata.

```{r, eval=FALSE}
abundance_data = microbiome::abundances(pseq)
aggregate_data = microbiome::abundances(microbiome::aggregate_taxa(pseq, "Family"))
meta_data = microbiome::meta(pseq)

set.seed(123)
output = ancombc2(data = abundance_data, 
                  aggregate_data = aggregate_data,
                  meta_data = meta_data,
                  fix_formula = "age + region + bmi", rand_formula = NULL,
                  p_adj_method = "holm", pseudo_sens = TRUE,
                  prv_cut = 0.10, lib_cut = 1000, s0_perc = 0.05,
                  group = "bmi", struc_zero = TRUE, neg_lb = TRUE,
                  alpha = 0.05, n_cl = 2, verbose = TRUE,
                  global = TRUE, pairwise = TRUE, dunnet = TRUE, trend = TRUE,
                  iter_control = list(tol = 1e-2, max_iter = 20, 
                                      verbose = TRUE),
                  em_control = list(tol = 1e-5, max_iter = 100),
                  lme_control = lme4::lmerControl(),
                  mdfdr_control = list(fwer_ctrl_method = "holm", B = 100),
                  trend_control = list(contrast = list(matrix(c(1, 0, -1, 1),
                                                              nrow = 2, 
                                                              byrow = TRUE),
                                                       matrix(c(-1, 0, 1, -1),
                                                              nrow = 2, 
                                                              byrow = TRUE),
                                                       matrix(c(1, 0, 1, -1),
                                                              nrow = 2, 
                                                              byrow = TRUE)),
                                       node = list(2, 2, 1),
                                       solver = "ECOS",
                                       B = 10))
```

# 6. Run ANCOM-BC2 on a real longitudinal dataset {.tabset}

## 6.1 Import example data

A two-week diet swap study between western (USA) and traditional (rural Africa) 
diets [@lahti2014tipping]. The dataset is available via the 
microbiome R package [@lahti2017tools] in phyloseq [@mcmurdie2013phyloseq] 
format.

```{r}
data(dietswap, package = "microbiome")
```

## 6.2 Run `ancombc2` function using the `phyloseq` object

In this tutorial, we consider the following fixed effects:

* Continuous covariates: "timepoint"

* Categorical covariates: "nationality"

* The group variable of interest: "group"

    + Three groups: "DI", "ED", "HE"
    
    + The reference group: "DI"
    
and the following random effects:

* A random intercept

* A random slope: "timepoint"

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.

```{r}
set.seed(123)
# It should be noted that we have set the number of bootstrap samples (B) equal 
# to 10 in the 'trend_control' function for computational expediency. 
# However, it is recommended that users utilize the default value of B, 
# which is 100, or larger values for optimal performance.
output = ancombc2(data = dietswap, tax_level = "Family",
                  fix_formula = "nationality + timepoint + group",
                  rand_formula = "(timepoint | subject)",
                  p_adj_method = "holm", pseudo_sens = TRUE,
                  prv_cut = 0.10, lib_cut = 1000, s0_perc = 0.05,
                  group = "group", struc_zero = TRUE, neg_lb = TRUE,
                  alpha = 0.05, n_cl = 2, verbose = TRUE,
                  global = TRUE, pairwise = TRUE, dunnet = TRUE, trend = TRUE,
                  iter_control = list(tol = 1e-2, max_iter = 20, 
                                      verbose = TRUE),
                  em_control = list(tol = 1e-5, max_iter = 100),
                  lme_control = lme4::lmerControl(),
                  mdfdr_control = list(fwer_ctrl_method = "holm", B = 100),
                  trend_control = list(contrast = list(matrix(c(1, 0, -1, 1),
                                                              nrow = 2, 
                                                              byrow = TRUE)),
                                       node = list(2),
                                       solver = "ECOS",
                                       B = 10))
```

Again, tests for interactions are supported by specifying the interactions in the `fix_formula`. 
Please ensure that interactions between variables are included using the `*` operator, 
as the `:` operator is not recognized by the `ancombc2` function and will result in an error.

Additionally, when the `group` variable contains interaction terms, only the main effect 
will be considered in multi-group comparisons.

```{r, eval=FALSE}
set.seed(123)
output = ancombc2(data = dietswap, tax_level = "Family",
                  fix_formula = "nationality + timepoint * group",
                  rand_formula = "(timepoint | subject)",
                  p_adj_method = "holm", pseudo_sens = TRUE,
                  prv_cut = 0.10, lib_cut = 1000, s0_perc = 0.05,
                  group = "group", struc_zero = TRUE, neg_lb = TRUE,
                  alpha = 0.05, n_cl = 2, verbose = TRUE,
                  global = TRUE, pairwise = TRUE, dunnet = TRUE, trend = TRUE,
                  iter_control = list(tol = 1e-2, max_iter = 20, 
                                      verbose = TRUE),
                  em_control = list(tol = 1e-5, max_iter = 100),
                  lme_control = lme4::lmerControl(),
                  mdfdr_control = list(fwer_ctrl_method = "holm", B = 100),
                  trend_control = list(contrast = list(matrix(c(1, 0, -1, 1),
                                                              nrow = 2, 
                                                              byrow = TRUE)),
                                       node = list(2),
                                       solver = "ECOS",
                                       B = 10))
```

## 6.3 ANCOM-BC2 primary analysis

```{r}
res_prim = output$res %>%
    mutate_if(is.numeric, function(x) round(x, 2))
res_prim[1:6, ] %>%
    datatable(caption = "ANCOM-BC2 Primary Analysis")
```

## 6.4 ANCOM-BC2 global test

```{r}
res_global = output$res_global %>%
    mutate_if(is.numeric, function(x) round(x, 2))
res_global[1:6, ] %>%
    datatable(caption = "ANCOM-BC2 Global Test")
```

## 6.5 ANCOM-BC2 multiple pairwise comparisons

```{r}
res_pair = output$res_pair %>%
    mutate_if(is.numeric, function(x) round(x, 2))
res_pair[1:6, ] %>%
    datatable(caption = "ANCOM-BC2 Multiple Pairwise Comparisons")
```

## 6.6 ANCOM-BC2 Dunnett's type of test

```{r}
res_dunn = output$res_dunn %>%
    mutate_if(is.numeric, function(x) round(x, 2))
res_dunn[1:6, ] %>%
    datatable(caption = "ANCOM-BC2 Dunnett's Type of Test")
```

## 6.7 ANCOM-BC2 pattern analysis

```{r}
res_trend = output$res_trend %>%
    mutate_if(is.numeric, function(x) round(x, 2))
res_trend[1:6, ] %>%
    datatable(caption = "ANCOM-BC2 Pattern Analysis")
```

## 6.8 Run `ancombc2` function using the `tse` object

```{r, eval=FALSE}
tse = mia::makeTreeSummarizedExperimentFromPhyloseq(dietswap)

set.seed(123)
output = ancombc2(data = tse, assay_name = "counts", tax_level = "Family",
                  fix_formula = "nationality + timepoint + group",
                  rand_formula = "(timepoint | subject)",
                  p_adj_method = "holm", pseudo_sens = TRUE,
                  prv_cut = 0.10, lib_cut = 1000, s0_perc = 0.05,
                  group = "group", struc_zero = TRUE, neg_lb = TRUE,
                  alpha = 0.05, n_cl = 2, verbose = TRUE,
                  global = TRUE, pairwise = TRUE, dunnet = TRUE, trend = TRUE,
                  iter_control = list(tol = 1e-2, max_iter = 20, 
                                      verbose = TRUE),
                  em_control = list(tol = 1e-5, max_iter = 100),
                  lme_control = lme4::lmerControl(),
                  mdfdr_control = list(fwer_ctrl_method = "holm", B = 100),
                  trend_control = list(contrast = list(matrix(c(1, 0, -1, 1),
                                                              nrow = 2, 
                                                              byrow = TRUE)),
                                       node = list(2),
                                       solver = "ECOS",
                                       B = 10))
```

## 6.9 Run `ancombc2` function by directly providing the abundance and metadata

```{r, eval=FALSE}
abundance_data = microbiome::abundances(dietswap)
aggregate_data = microbiome::abundances(microbiome::aggregate_taxa(dietswap, "Family"))
meta_data = microbiome::meta(dietswap)

set.seed(123)
output = ancombc2(data = abundance_data, 
                  aggregate_data = aggregate_data,
                  meta_data = meta_data,
                  fix_formula = "nationality + timepoint + group",
                  rand_formula = "(timepoint | subject)",
                  p_adj_method = "holm", pseudo_sens = TRUE,
                  prv_cut = 0.10, lib_cut = 1000, s0_perc = 0.05,
                  group = "group", struc_zero = TRUE, neg_lb = TRUE,
                  alpha = 0.05, n_cl = 2, verbose = TRUE,
                  global = TRUE, pairwise = TRUE, dunnet = TRUE, trend = TRUE,
                  iter_control = list(tol = 1e-2, max_iter = 20, 
                                      verbose = TRUE),
                  em_control = list(tol = 1e-5, max_iter = 100),
                  lme_control = lme4::lmerControl(),
                  mdfdr_control = list(fwer_ctrl_method = "holm", B = 100),
                  trend_control = list(contrast = list(matrix(c(1, 0, -1, 1),
                                                              nrow = 2, 
                                                              byrow = TRUE)),
                                       node = list(2),
                                       solver = "ECOS",
                                       B = 10))
```

# 7. Bias-corrected log abundances

It is important to acknowledge that the estimation of sampling fractions in 
ANCOM-BC2 is limited to an additive constant. This means that only the 
difference between bias-corrected log abundances is meaningful, rather than the 
absolute values themselves. 

Moreover, within each taxon, the bias-corrected log abundances are centered 
across samples. ANCOM-BC2 operates on the assumption that while these 
taxon-specific biases differ between taxa, they stay consistent within a given 
taxon across various samples. This assumption enables ANCOM-BC2 to accommodate 
intra-taxon variations through the centering of the bias-corrected log 
abundances.

```{r}
bias_correct_log_table = output$bias_correct_log_table
# By default, ANCOM-BC2 does not add pseudo-counts to zero counts, which can 
# result in NAs in the bias-corrected log abundances. Users have the option to 
# either leave the NAs as they are or replace them with zeros. 
# This replacement is equivalent to adding pseudo-counts of ones to the zero counts. 
bias_correct_log_table[is.na(bias_correct_log_table)] = 0
# Show the first 6 samples
round(bias_correct_log_table[, 1:6], 2) %>% 
  datatable(caption = "Bias-corrected log abundances")
```

# Session information

```{r sessionInfo, message = FALSE, warning = FALSE, comment = NA}
sessionInfo()
```

# References









