---
title: "Introduction to corncob, no phyloseq"
author: "Bryan D Martin"
date: "`r Sys.Date()`"
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# Vignette Information

This is a version of the `corncob-intro.Rmd` vignette that does not rely on the package `phyloseq`. To see how to use `corncob` with `phyloseq` objects, or to see an additional analysis with an IBD microbiome dataset, check out the original `corncob-intro.Rmd` vignette.

We thank Dr. Thea Whitman for kindly providing us with the example data set we use for this vignette. You can read more about this data in Whitman, Thea, et al. "Dynamics of microbial community composition and soil organic carbon mineralization in soil following addition of pyrogenic and fresh organic matter." *The ISME Journal* 10.12 (2016): 2918.

# Introduction

Effectively modeling microbial relative abundance poses a number of statistical challenges, including:
\begin{itemize}
\item different sequencing depth,
\item excessive zeros from unobserved taxa,
\item high variability of empirical relative abundances (overdispersion),
\item within-taxon correlation,
\item hypothesis testing with categorical and continuous covariates.
\end{itemize}

```{r, echo = FALSE}
knitr::opts_chunk$set(fig.width=8, fig.height=4) 
```

Here, we introduce `corncob`, an individual taxon regression model that uses abundance tables and sample data. `corncob` is able to model differential abundance and differential variability, and addresses each of the challenges presented above.


Install `corncob` using:
```{r, eval = FALSE}
remotes::install_github("statdivlab/corncob")
```


To begin, we load our example data set we load our example data set as three different data frames

```{r, message = FALSE }
library(corncob)
library(magrittr)
data(soil_phylo_sample)
data(soil_phylo_otu)
data(soil_phylo_taxa)
```

We can see that we have 5 sample variables. Let's look at the first few observations.
```{r }
head(soil_phylo_sample)
```
Our covariates are as follows:
\begin{itemize}
\item  `Plants`: Indicator of whether plants are in the soil for this sample.
\item `Amdmt`: Categorical variable representing one of three soil additives; none, biochar, and freshbiomass, respectively.
\item `ID`: Categorical variable representing different plots of soil.
\item `Day`: Categorical variable representing one of three days of measurement; day 1, day 21, and day 81, respectively.
\item `DayAmdmt`: Categorical variable combining the `Day` and `Amdmt` variables into a single variable.
\end{itemize}

Additionally, we have a table that gives us counts of the phyla by sample. Let's take a look at the first 5 phyla and the first 5 samples. 
```{r }
soil_phylo_otu[1:5, 1:5]
```

Finally, we have a taxonomy table with 7 taxonomic ranks. 
```{r }
soil_phylo_taxa[1:3, ]
```

# Fitting a Model

First, let's subset our samples to only include those with the `DayAmdmt` covariate equal to 11 or 21 and then collapse the samples to the phylum level. We have already done this and saved the resulting sample data frame and otu data frame. 

```{r }
data(soil_phylum_small_sample)
sample_data <- soil_phylum_small_sample
data(soil_phylum_small_otu)
data <- soil_phylum_small_otu
```

Note that collapsing the samples is not necessary, and this model can work at any taxonomic rank. However, we will later be fitting a model to every taxa. We can see that by agglomerating taxa to the phylum level, we have gone from from 7770 to 40 taxa. Thus we collapse in order to increase the speed for the purposes of this tutorial.

Now we fit our model. We will demonstrate with Proteobacteria. We will need to make a data frame that includes our sample data as well as the Proteobacteria counts and sequencing depths for each sample. 

```{r}
pro_data <- cbind(sample_data, 
                  W = unlist(data["Proteobacteria", ]),
                  M = colSums(data))
```

For now, we will not include any covariates, so we use `~ 1` as our model formula responses.

```{r }
corncob <- bbdml(formula = cbind(W, M - W) ~ 1,
             phi.formula = ~ 1,
             data = pro_data)
```

\newpage

# Interpreting a Model

First, let's plot the data with our model fit on the relative abundance scale. To do this, we simply type:

```{r }
plot(corncob, B = 50)
```

You can access the documentation for this plotting function by typing `?plot.bbdml` into the console.

The points represent the relative abundances. The bars represent the 95\% prediction intervals for the observed relative abundance by sample. 
The parameter `B` determines the number of bootstrap simulations used to approximate the prediction intervals. For purposes of this tutorial, we use a small value `B = 50` for computational purposes, but recommend a higher setting for more accurate intervals, such as the default `B = 1000`.

Now let's look at the same plot, but on the counts scale with 95\% prediction intervals (since counts is not a parameter). To do this, we add the option `total = TRUE` to our plotting code.

```{r }
plot(corncob, total = TRUE, B = 50)
```

Finally, let's color the plot by the `DayAmdmt` covariate. To do this, we add the option `color = "DayAmdmt"` to our plotting code. 

```{r }
plot(corncob, total = TRUE, color = "DayAmdmt", B = 50)
```

```{r }
plot(corncob, color = "DayAmdmt", B = 50)
```

Notice that this plot also reorders our samples so that groups appear together so that they are easier to compare.

We can observe on this plot that it might be of interest to distinguish between the two groups with covariates. The average empirical relative abundance for the samples with `DayAmdmt = 21` tends to be lower and less variable than the samples with `DayAmdmt = 11`.

# Adding covariates

Let's try modeling the expected relative abundance and the variability of the counts with `DayAmdmt` as a covariate. We do this by modifying `formula` and `phi.formula` as:

```{r }
corncob_da <- bbdml(formula = cbind(W, M - W) ~ DayAmdmt,
             phi.formula = ~ DayAmdmt,
             data = pro_data)
```

Let's also plot this data on both the total count and relative abundance scales.

```{r }
plot(corncob_da, color = "DayAmdmt", total = TRUE, B = 50)
```

```{r }
plot(corncob_da, color = "DayAmdmt", B = 50)
```

Visually, the model with covariates seems to provide a much better fit to the data, but how can we compare the two models statistically?

# Model Selection

Let's use a likelihood ratio test to select our final model for this taxon. We want to test the null hypothesis that the likelihood of the model with covariates is equal to the likelihood of the model without covariates. To do this test, we use:

```{r }
lrtest(mod_null = corncob, mod = corncob_da)
```

We obtain a p-value much smaller than a cut-off of 0.05. Therefore we conclude that there is a statistically significant difference in the likelihood of the two models. Thus, we probably want to use the model with covariates for this taxon.

# Parameter Interpretation

Now that we have chosen our model, let's interpret our model output. To see a summary of the model, type:

```{r }
summary(corncob_da)
```

This output will look familiar if you have done regression analysis in R in the past. Covariates associated with the expected relative abundance are presented separately from covariates associated with the variance of the counts are preceded by.

From this model summary, we can see that the `DayAmdmt21` abundance coefficient is negative and statistically significant. This suggests that this taxon is differentially-abundant across `DayAmdmt`, and that samples with `DayAmdmt = 21` are expected to have a lower relative abundance. This matches what we saw from the observed abundances.

We can also see that the `DayAmdmt21` dispersion coefficient is negative and statistically significant. This suggests that this taxon is differentially-variable across `DayAmdmt`, and that samples with `DayAmdmt = 21` are expected to have a lower variability. This matches what we saw from the observed abundances.

# Analysis for Multiple Taxa

What if we want to test all the taxa in our data to see if they are differentially-abundant or differentially-variable? We use the `differentialTest` function. It will perform the above tests on all taxa, and it will control the false discovery rate to account for multiple comparisons.

Next, we use the `differentialTest` command. We specify the covariates of our model using `formula` and `phi.formula` as before, except we no longer include the response term because we are testing multiple taxa. We also specify which covariates we want to test for by removing them in the `formula_null` and `phi.formula_null` arguments. 

The difference between the formulas and the null version of the formulas are the variables that we test. We will go through several examples, starting with a test for differential abundance across the `DayAmdmt` coefficient.

We set `fdr_cutoff` to be our controlled false discovery rate.

Here, we will pass in our entire count table for all phyla as the `data` argument (setting `taxa_are_rows` to `TRUE`), and the sample data frame as the `sample_data` argument. 

```{r }
set.seed(1)
da_analysis <- differentialTest(formula = ~ DayAmdmt,
                                 phi.formula = ~ DayAmdmt,
                                 formula_null = ~ 1,
                                 phi.formula_null = ~ DayAmdmt,
                                 test = "Wald", boot = FALSE,
                                 data = data,
                                 sample_data = sample_data,
                                 taxa_are_rows = TRUE, 
                                 fdr_cutoff = 0.05)
```

We can see the output of the function by calling it:

```{r }
da_analysis
```

We can see a list of differentially-abundant taxa using: 

```{r }
da_analysis$significant_taxa
```

In this case, we identified `r length(da_analysis$significant_taxa)` taxa that are differentially-abundant across `DayAmdmt` (out of the 39 taxa tested).

We can see a list of differentially-variable taxa using:

```{r }
set.seed(1)
dv_analysis <- differentialTest(formula = ~ DayAmdmt,
                                 phi.formula = ~ DayAmdmt,
                                 formula_null = ~ DayAmdmt,
                                 phi.formula_null = ~ 1,
                                 test = "LRT", boot = FALSE,
                                 data = data,
                                 sample_data = sample_data,
                                 taxa_are_rows = TRUE, 
                                 fdr_cutoff = 0.05)
dv_analysis$significant_taxa
```

In this case, we identified `r length(dv_analysis$significant_taxa)` taxa that are differentially-variable across `DayAmdmt` (out of the 40 taxa tested).

We can examine a subset of the p-values of our tests using:

```{r }
da_analysis$p[1:5]
```

We can examine a subset of the p-values after controlling for the false discovery rate using:

```{r }
da_analysis$p_fdr[1:5]
```

where the values are now adjusted to control the false discovery rate at 0.05.

Now we can use the built-in plotting function in corncob to view the the model coefficients used in testing differential abundance across `DayAmdmt` via the `plot()` function. You can access the documentation for this plotting function by typing `?plot.differentialTest` into the console.

```{r }
plot(da_analysis)
```

Here, we can see that for `Bacteria_Armatimonadetes`, the effect of `DayAmdmt21` is positive compared to the baseline (in this case, `DayAmdmt11`).

If you wish to instead make your own custom plots with these same coefficients, you can easily extract the data used to generate the plots above from the `plot` function by setting `data_only = TRUE`:

```{r}
df <- plot(da_analysis, data_only = TRUE)

# we can easily remove special characters used in our formatting steps
df <- df %>%
  dplyr::mutate(variable = gsub("\nDifferential Abundance", "",
                                variable, fixed = TRUE))

head(df)
```

Finally, we can see a list of any taxa for which we were not able to fit a model using:

```{r }
which(is.na(da_analysis$p)) %>% names
```

In this case, we weren't able to fit `GN04` automatically. It's worthwhile to investigate the OTU individually if this is the case. 

It may be that the model is overparameterized because there aren't enough observations, or it may just be that the initializations were invalid for that taxa and it needs to be re-evaluated with new initializations.

Let's first try examining the data. 

```{r }
data["GN04", ]
```

We see that the observed counts of OTU is zero in all samples except for `S102`, where we observed a single count. We should be skeptical of **any** statistical model fit on a single observed count!

`corncob` is stable, but if you notice any issues, please [log them on Github](https://github.com/statdivlab/corncob/issues) to help us help you!

# Examples of Answering Scientific Questions

We will now walk through several scientific questions of interest and show how they can be answered using hypothesis testing with `corncob`. Note that `Day` and `Amdmt` are both factor covariates with levels 0, 1, and 2.

Note that some of these are rather strange tests, and shown for demonstration of the flexibility of the model only. Normally, when testing for differential variability across a covariate, we recommend always controlling for the effect of that covariate on the abundance. We first demonstrate examples with the small version of the soil dataset (note, this is a slightly different subset of the data from that used in `corncob-intro.Rmd`).

Testing for differential abundance across `Day`, without controlling for anything else:

```{r }
ex1 <- differentialTest(formula = ~ Day,
                        phi.formula = ~ 1,
                        formula_null = ~ 1,
                        phi.formula_null = ~ 1,
                        data = data,
                        taxa_are_rows = TRUE,
                        sample_data = sample_data, 
                        test = "Wald", boot = FALSE,
                        fdr_cutoff = 0.05)
plot(ex1)
```

Testing for differential abundance across `Day`, controlling for the effect of `Day` on dispersion:

```{r }
ex2 <- differentialTest(formula = ~ Day,
                        phi.formula = ~ Day,
                        formula_null = ~ 1,
                        phi.formula_null = ~ Day,
                        data = data,
                        taxa_are_rows = TRUE,
                        sample_data = sample_data, 
                        test = "Wald", boot = FALSE,
                        fdr_cutoff = 0.05)
plot(ex2)
```

Jointly testing for differential abundance and differential variability across `Day`:

```{r }
ex3 <- differentialTest(formula = ~ Day,
                        phi.formula = ~ Day,
                        formula_null = ~ 1,
                        phi.formula_null = ~ 1,
                        data = data,
                        taxa_are_rows = TRUE,
                        sample_data = sample_data, 
                        test = "Wald", boot = FALSE,
                        fdr_cutoff = 0.05)
plot(ex3)
```

More examples of answering scientific questions with a larger subset of this soil data set and with an IBD data set can be seen in `corncob-intro.Rmd`. 
