---
title: "ANCOM-BC2 Tutorial"
author: 
  - Huang Lin$^1$
  - $^1$NICHD, 6710B Rockledge Dr, Bethesda, MD 20892
date: '`r format(Sys.Date(), "%B %d, %Y")`'
output: rmarkdown::html_vignette
bibliography: bibliography.bib
vignette: >
  %\VignetteIndexEntry{ANCOMBC2}
  %\VignetteEngine{knitr::rmarkdown}
  \usepackage[utf8]{inputenc}
---

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

# 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 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 [@costea2014fair; @paulson2014reply],
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 [@guo2010controlling; @grandhi2016multiple].
    
    + 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 [@grandhi2016multiple] 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 [@jelsema2016clme] 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. 

```{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)
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 = S4Vectors::SimpleList(counts = otu_data)
smd = S4Vectors::DataFrame(smd)
tse = TreeSummarizedExperiment::TreeSummarizedExperiment(assays = assays, colData = smd)
```

## 3.2 Run ancombc2 function

```{r}
set.seed(123)
output = ancombc2(data = tse, assay_name = "counts", tax_level = NULL,
                  fix_formula = "cont_cov + cat_cov", rand_formula = NULL,
                  p_adj_method = "holm", pseudo = 0, 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 = FALSE, 
                  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 = NULL, 
                  trend_control = NULL)

res_prim = output$res
tab_sens = output$pseudo_sens_tab
```

## 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.

```{r, fig.width=10}
sens_cat = tab_sens %>%
    transmute(taxon, sens_cat = cat_cov2) %>%
    left_join(res_prim %>%
                  transmute(taxon, diff_cat = diff_cat_cov2), 
              by = "taxon") %>%
    mutate(group = "Cat2 vs. Cat1") %>%
    bind_rows(
        tab_sens %>%
            transmute(taxon, sens_cat = cat_cov3) %>%
            left_join(res_prim %>%
                          transmute(taxon, diff_cat = diff_cat_cov3), 
                      by = "taxon") %>%
            mutate(group = "Cat3 vs. Cat1")
    )
sens_cat$diff_cat = recode(sens_cat$diff_cat * 1, 
                           `1` = "Significant",
                           `0` = "Nonsignificant")

fig_sens_cat = sens_cat %>%
    ggplot(aes(x = taxon, y = sens_cat, color = diff_cat)) +
    geom_point() +
    scale_color_brewer(palette = "Dark2", name = NULL) +
    facet_grid(rows = vars(group), scales = "free") +
    labs(x = NULL, y = "Sensitivity Score") +
    theme_bw() +
    theme(axis.text.x = element_text(angle = 60, vjust = 0.5))
fig_sens_cat

sens_cut1 = 5
sens_cut2 = 5
```

## 3.4 Power and FDR

```{r}
# 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) %>%
    mutate(taxon = res_prim$taxon) %>%
    left_join(fmd %>%
                  rownames_to_column("taxon"), 
              by = "taxon") %>%
    transmute(
        taxon = taxon,
        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, by = "taxon")

# 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")
tab_summ2 %>%
    datatable(caption = "Cat 3 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 [@lahti2014tipping]. The dataset is also 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)

# Subset to baseline
tse = atlas1006[, atlas1006$time == 0]

# Re-code the bmi group
tse$bmi = recode(tse$bmi_group,
                 obese = "obese",
                 severeobese = "obese",
                 morbidobese = "obese")
# Subset to lean, overweight, and obese subjects
tse = tse[, tse$bmi %in% c("lean", "overweight", "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:
tse$bmi = factor(tse$bmi, levels = c("obese", "overweight", "lean"))
# You can verify the change by checking:
# levels(sample_data(tse)$bmi)

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

# Discard "EE" as it contains only 1 subject
# Discard subjects with missing values of region
tse = tse[, ! tse$region %in% c("EE", "unknown")]

print(tse)
```
    
## 4.2 Run ancombc2 function

```{r}
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 = 0, 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)),
                                       node = list(2, 2),
                                       solver = "ECOS",
                                       B = 100))
```

## 4.3 Structural zeros

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

## 4.4 Sensitivity scores

```{r}
tab_sens = output$pseudo_sens_tab
tab_sens %>%
    datatable(caption = "Sensitivity Scores") %>%
    formatRound(colnames(tab_sens)[-1], digits = 2)
```

## 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).

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

### Results for age {.tabset}

#### Original results

```{r}
df_age = res_prim %>%
    dplyr::select(taxon, ends_with("age")) 
df_fig_age = df_age %>%
    filter(diff_age == 1) %>% 
    arrange(desc(lfc_age)) %>%
    mutate(direct = ifelse(lfc_age > 0, "Positive LFC", "Negative LFC"))
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))
fig_age
```

#### Pseudo-count sensitivity analysis for age

For the covariate of age, no outlying sensitivity scores are observed. 
All significant taxa have low sensitivity scores.

```{r}
sens_age = tab_sens %>%
    transmute(taxon, sens_age = age) %>%
    left_join(df_age, by = "taxon")
sens_age$diff_age = recode(sens_age$diff_age * 1, 
                           `1` = "Significant",
                           `0` = "Nonsignificant")

fig_sens_age = sens_age %>%
    ggplot(aes(x = taxon, y = sens_age, color = diff_age)) +
    geom_point() +
    scale_color_brewer(palette = "Dark2", name = NULL) +
    labs(x = NULL, y = "Sensitivity Score") +
    theme_bw() +
    theme(axis.text.x = element_text(angle = 60, vjust = 0.5))
fig_sens_age
```

### Results for bmi {.tabset}

#### Original results

```{r}
df_bmi = res_prim %>%
    dplyr::select(taxon, contains("bmi")) 
df_fig_bmi = df_bmi %>%
    filter(diff_bmilean == 1 | diff_bmioverweight == 1) %>%
    mutate(lfc_overweight = ifelse(diff_bmioverweight == 1, 
                                   lfc_bmioverweight, 0),
           lfc_lean = ifelse(diff_bmilean == 1, 
                             lfc_bmilean, 0)) %>%
    transmute(taxon, 
              `Overweight vs. Obese` = round(lfc_overweight, 2),
              `Lean vs. Obese` = round(lfc_lean, 2)) %>%
    pivot_longer(cols = `Overweight vs. Obese`:`Lean vs. Obese`, 
                 names_to = "group", values_to = "value") %>%
    arrange(taxon)
  
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 = "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))
fig_bmi
```

#### Pseudo-count sensitivity analysis for bmi

For the covariate of bmi, no outlying sensitivity scores are observed. 
All significant taxa have low sensitivity scores.

```{r}
sens_bmi = tab_sens %>%
    transmute(taxon, sens_bmi = bmilean) %>%
    left_join(df_bmi %>%
                  transmute(taxon, diff_bmi = diff_bmilean), 
              by = "taxon") %>%
    mutate(group = "Lean vs. Obese") %>%
    bind_rows(
        tab_sens %>%
            transmute(taxon, sens_bmi = bmioverweight) %>%
            left_join(df_bmi %>%
                          transmute(taxon, diff_bmi = diff_bmioverweight), 
                      by = "taxon") %>%
            mutate(group = "Overweight vs. Obese")
    )
sens_bmi$diff_bmi = recode(sens_bmi$diff_bmi * 1, 
                           `1` = "Significant",
                           `0` = "Nonsignificant")

fig_sens_bmi = sens_bmi %>%
    ggplot(aes(x = taxon, y = sens_bmi, color = diff_bmi)) +
    geom_point() +
    scale_color_brewer(palette = "Dark2", name = NULL) +
    facet_grid(rows = vars(group), scales = "free") +
    labs(x = NULL, y = "Sensitivity Score") +
    theme_bw() +
    theme(axis.text.x = element_text(angle = 60, vjust = 0.5))
fig_sens_bmi
```

## 4.6 ANCOM-BC2 global test {.tabset}

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).

```{r}
res_global = output$res_global
```

### Results for BMI

```{r}
df_bmi = res_prim %>%
    dplyr::select(taxon, contains("bmi")) 
df_fig_global = df_bmi %>%
    left_join(res_global %>%
                  transmute(taxon, diff_bmi = diff_abn)) %>%
    dplyr::filter(diff_bmi == 1) %>%
    mutate(lfc_lean = ifelse(diff_bmilean == 1, 
                             lfc_bmilean, 0),
           lfc_overweight = ifelse(diff_bmioverweight == 1, 
                                   lfc_bmioverweight, 0)) %>%
    transmute(taxon, 
              `Lean vs. Obese` = round(lfc_lean, 2), 
              `Overweight vs. Obese` = round(lfc_overweight, 2)) %>%
    pivot_longer(cols = `Lean vs. Obese`:`Overweight vs. Obese`, 
                 names_to = "group", values_to = "value") %>%
    arrange(taxon)
  
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))
fig_global
```

### Pseudo-count sensitivity analysis for the global test

For the global test of bmi, no outlying sensitivity scores are observed. 
All significant taxa have low sensitivity scores.

```{r}
sens_global = tab_sens %>%
    transmute(taxon, sens_global = global) %>%
    left_join(res_global %>%
                  transmute(taxon, diff_global = diff_abn * 1), 
              by = "taxon") 
sens_global$diff_global = recode(sens_global$diff_global, 
                                 `1` = "Significant",
                                 `0` = "Nonsignificant")

fig_sens_global = sens_global %>%
    ggplot(aes(x = taxon, y = sens_global, color = diff_global)) +
    geom_point() +
    scale_color_brewer(palette = "Dark2", name = NULL) +
    labs(x = NULL, y = "Sensitivity Score") +
    theme_bw() +
    theme(axis.text.x = element_text(angle = 60, vjust = 0.5))
fig_sens_global
```

## 4.7 ANCOM-BC2 pairwise directional test {.tabset}

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).

```{r}
res_pair = output$res_pair
```

### Results for BMI

```{r}
df_fig_pair = res_pair %>%
    filter(diff_bmilean == 1 | diff_bmioverweight == 1 |
               diff_bmilean_bmioverweight == 1) %>%
    mutate(lfc_lean = ifelse(diff_bmilean == 1, 
                             lfc_bmilean, 0),
           lfc_overweight = ifelse(diff_bmioverweight == 1, 
                                   lfc_bmioverweight, 0),
           lfc_lean_overweight = ifelse(diff_bmilean_bmioverweight == 1, 
                                        lfc_bmilean_bmioverweight, 0)) %>%
    transmute(taxon, 
              `Lean vs. Obese` = round(lfc_lean, 2), 
              `Overweight vs. Obese` = round(lfc_overweight, 2),
              `Lean vs. Overweight` = round(lfc_lean_overweight, 2)
              ) %>%
    pivot_longer(cols = `Lean vs. Obese`:`Lean vs. Overweight`, 
                 names_to = "group", values_to = "value") %>%
    arrange(taxon)
df_fig_pair$group = factor(df_fig_pair$group, 
                           levels = c("Lean vs. Obese",
                                      "Overweight vs. Obese",
                                      "Lean vs. 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 = "black", size = 4) +
  labs(x = NULL, y = NULL, title = "Log fold change of pairwise comparisons") +
  theme_minimal() +
  theme(plot.title = element_text(hjust = 0.5))
fig_pair
```

### Pseudo-count sensitivity analysis for bmi

For the pairwise analysis of bmi, no outlying sensitivity scores are observed. 
All significant taxa have low sensitivity scores.

```{r, fig.height=8}
sens_pair = tab_sens %>%
    transmute(taxon, sens_pair = `obese - lean`) %>%
    left_join(res_pair %>%
                  transmute(taxon, 
                            diff_pair = diff_bmilean), 
              by = "taxon") %>%
    mutate(group = "Lean vs. Obese") %>%
    bind_rows(
        tab_sens %>%
            transmute(taxon, sens_pair = `obese - overweight`) %>%
            left_join(res_pair %>%
                          transmute(taxon, 
                                    diff_pair = diff_bmioverweight), 
                      by = "taxon") %>%
            mutate(group = "Overweight vs. Obese")
    ) %>%
    bind_rows(
        tab_sens %>%
            transmute(taxon, sens_pair = `overweight - lean`) %>%
            left_join(res_pair %>%
                          transmute(taxon, 
                                    diff_pair = diff_bmilean_bmioverweight), 
                      by = "taxon") %>%
            mutate(group = "Lean vs. Overweight")
    )
sens_pair$diff_pair = recode(sens_pair$diff_pair * 1, 
                             `1` = "Significant",
                             `0` = "Nonsignificant")
sens_pair$group = factor(sens_pair$group, 
                         levels = c("Lean vs. Obese",
                                    "Overweight vs. Obese",
                                    "Lean vs. Overweight"))

fig_sens_pair = sens_pair %>%
    ggplot(aes(x = taxon, y = sens_pair, color = diff_pair)) +
    geom_point() +
    scale_color_brewer(palette = "Dark2", name = NULL) +
    facet_grid(rows = vars(group), scales = "free") +
    labs(x = NULL, y = "Sensitivity Score") +
    theme_bw() +
    theme(axis.text.x = element_text(angle = 60, vjust = 0.5))
fig_sens_pair
```

## 4.8 ANCOM-BC2 Dunnett's type of test {.tabset}

The Dunnett’s test [@dunnett1955multiple; @dunnett1991step; @dunnett1992step] 
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).

```{r}
res_dunn = output$res_dunn
```

### Results for BMI

```{r}
df_fig_dunn = res_dunn %>%
    dplyr::filter(diff_bmilean == 1 | diff_bmioverweight == 1) %>%
    mutate(lfc_lean = ifelse(diff_bmilean == 1, 
                             lfc_bmilean, 0),
           lfc_overweight = ifelse(diff_bmioverweight == 1, 
                                   lfc_bmioverweight, 0)) %>%
    transmute(taxon, 
              `Lean vs. Obese` = round(lfc_lean, 2), 
              `Overweight vs. Obese` = round(lfc_overweight, 2)) %>%
    pivot_longer(cols = `Lean vs. Obese`:`Overweight vs. Obese`, 
                 names_to = "group", values_to = "value") %>%
    arrange(taxon)
  
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 = "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))
fig_dunn
```

### Pseudo-count sensitivity analysis for bmi

For the covariate of bmi, no outlying sensitivity scores are observed. 
All significant taxa have low sensitivity scores.

```{r}
sens_dunn = tab_sens %>%
    transmute(taxon, sens_bmi = bmilean) %>%
    left_join(res_dunn %>%
                  transmute(taxon, diff_bmi = diff_bmilean), 
              by = "taxon") %>%
    mutate(group = "Lean vs. Obese") %>%
    bind_rows(
        tab_sens %>%
            transmute(taxon, sens_bmi = bmioverweight) %>%
            left_join(res_dunn %>%
                          transmute(taxon, diff_bmi = diff_bmioverweight), 
                      by = "taxon") %>%
            mutate(group = "Overweight vs. Obese")
    )
sens_dunn$diff_bmi = recode(sens_dunn$diff_bmi * 1, 
                            `1` = "Significant",
                            `0` = "Nonsignificant")

fig_sens_dunn = sens_dunn %>%
    ggplot(aes(x = taxon, y = sens_bmi, color = diff_bmi)) +
    geom_point() +
    scale_color_brewer(palette = "Dark2", name = NULL) +
    facet_grid(rows = vars(group), scales = "free") +
    labs(x = NULL, y = "Sensitivity Score") +
    theme_bw() +
    theme(axis.text.x = element_text(angle = 60, vjust = 0.5))
fig_sens_dunn
```

## 4.9 ANCOM-BC2 trend test {.tabset}

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).

```{r}
res_trend = output$res_trend
```

### Results for BMI

Note that the LFC and SE for the reference group ("obese" in this example) will
set to be 0s.

```{r, fig.width=10}
df_fig_trend = res_trend %>%
    dplyr::filter(diff_abn) %>%
    transmute(taxon,
              lfc = lfc_bmioverweight,
              se = se_bmioverweight,
              q_val,
              group = "Overweight - Obese") %>%
    bind_rows(
        res_trend %>%
            dplyr::filter(diff_abn) %>%
            transmute(taxon,
                      lfc = lfc_bmilean,
                      se = se_bmilean,
                      q_val,
                      group = "Lean - Obese")
    )
    
df_fig_trend$group = factor(df_fig_trend$group, 
                           levels = c("Overweight - Obese", "Lean - Obese"))
  
fig_trend = df_fig_trend %>%
  ggplot(aes(x = group, y = lfc, fill = group)) + 
  geom_bar(stat = "identity", position = position_dodge(), color = "black") +
  geom_errorbar(aes(ymin = lfc - se, ymax = lfc + se), width = .2,
                position = position_dodge(.9)) +
  facet_wrap(vars(taxon), nrow = 2, scales = "free") +
  labs(x = NULL, y = NULL, title = "Log fold change as compared to obese subjects") +
  scale_fill_brewer(palette = "Set2", name = NULL) +
  theme_bw() +
  theme(plot.title = element_text(hjust = 0.5),
        axis.text.x = element_blank(),
        axis.ticks.x = element_blank(),
        legend.position = c(0.9, 0.2))
fig_trend
```

### Pseudo-count sensitivity analysis for bmi

For the covariate of bmi, no outlying sensitivity scores are observed. 
All significant taxa have low sensitivity scores.

```{r}
sens_trend = tab_sens %>%
    transmute(taxon, sens_bmi = bmilean) %>%
    left_join(res_trend %>%
                  transmute(taxon, diff_bmi = diff_abn), 
              by = "taxon") %>%
    mutate(group = "Lean vs. Obese") %>%
    bind_rows(
        tab_sens %>%
            transmute(taxon, sens_bmi = bmioverweight) %>%
            left_join(res_trend %>%
                          transmute(taxon, diff_bmi = diff_abn), 
                      by = "taxon") %>%
            mutate(group = "Overweight vs. Obese")
    )
sens_trend$diff_bmi = recode(sens_trend$diff_bmi * 1, 
                            `1` = "Significant",
                            `0` = "Nonsignificant")

fig_sens_trend = sens_trend %>%
    ggplot(aes(x = taxon, y = sens_bmi, color = diff_bmi)) +
    geom_point() +
    scale_color_brewer(palette = "Dark2", name = NULL) +
    facet_grid(rows = vars(group), scales = "free") +
    labs(x = NULL, y = "Sensitivity Score") +
    theme_bw() +
    theme(axis.text.x = element_text(angle = 60, vjust = 0.5))
fig_sens_trend
```

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

## 5. 1 Import example data

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

```{r}
data(dietswap)
tse = dietswap
print(tse)
```

## 5.2 Run ancombc2 function

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)
output = ancombc2(data = tse, assay_name = "counts", tax_level = "Family",
                  fix_formula = "nationality + timepoint + group",
                  rand_formula = "(timepoint | subject)",
                  p_adj_method = "holm", pseudo = 0, 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 = 100))

res_prim = output$res %>%
    mutate_if(is.numeric, function(x) round(x, 2))
res_prim %>%
    datatable(caption = "ANCOM-BC2 Primary Results")
```

# Session information

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

# References









