This vignette documents the data format used in
**PLNmodel** by `PLN`

and its variants. It also
shows how to create an object in the proper format for further analyses
from (i) tabular data, (ii) biom-class objects and (iii) phyloseq-class
objects.

We illustrate the format using trichoptera data set, a full description of which can be found in the corresponding vignette.

The trichoptera data set is a list made of two data frames:
`Abundance`

(hereafter referred to as the *counts*)
and `Covariate`

(hereafter the *covariates*).

`str(trichoptera, max.level = 1)`

```
## List of 2
## $ Abundance:'data.frame': 49 obs. of 17 variables:
## $ Covariate:'data.frame': 49 obs. of 7 variables:
```

The covariates include, among others, the wind, pressure and humidity.

`names(trichoptera$Covariate)`

```
## [1] "Temperature" "Wind" "Pressure" "Humidity"
## [5] "Cloudiness" "Precipitation" "Group"
```

In the PLN framework, we model the counts from the covariates, let’s
say wind and pressure, using a Poisson Log-Normal model. Most models in
R use the so-called *formula interface* and it would thus be
natural to write something like

`PLN(Abundance ~ Wind + Pressure, data = trichoptera)`

Unfortunately and unlike many generalized linear models, the response
in PLN is intrinsically **multivariate**: it has 17
dimensions in our example. The left hand side (LHS) must encode a
multivariate response across multiple samples, using a 2D-array (e.g. a
matrix or a data frame).

We must therefore prepare a data structure where
`Abundance`

refers to a count *matrix* whereas
`Wind`

and `Pressure`

refer to *vectors*
before feeding it to `PLN`

. That’s the purpose of
`prepare_data`

.

```
trichoptera2 <- prepare_data(counts = trichoptera$Abundance,
covariates = trichoptera$Covariate)
str(trichoptera2)
```

```
## 'data.frame': 49 obs. of 9 variables:
## $ Abundance : num [1:49, 1:17] 0 0 0 0 0 0 0 0 0 0 ...
## ..- attr(*, "dimnames")=List of 2
## .. ..$ : chr [1:49] "1" "2" "3" "4" ...
## .. ..$ : chr [1:17] "Che" "Hyc" "Hym" "Hys" ...
## $ Temperature : num 18.7 19.8 22 23 22.5 23.9 15 17.2 15.4 14.1 ...
## $ Wind : num -2.3 -2.7 -0.7 2.3 2.3 -2 -4.7 -1 -2.7 -3.7 ...
## $ Pressure : num 998 1000 997 991 990 ...
## $ Humidity : num 60 63 73 71 62 64 93 84 88 75 ...
## $ Cloudiness : num 19 0 6 81 50 50 100 19 69 6 ...
## $ Precipitation: num 0 0 0 0 0 0 1.6 0 1.6 0 ...
## $ Group : Factor w/ 12 levels "1","2","3","4",..: 1 1 1 1 1 1 1 1 1 1 ...
## $ Offset : num 29 13 38 192 79 18 8 34 12 4 ...
```

If you look carefully, you can notice a few difference between
`trichoptera`

and `trichoptera2`

:

- the first is a
`list`

whereas the second is a`data.frame`

^{1}; -
`Abundance`

is a matrix-column of`trichoptera2`

that you can extract using the usual functions`[`

and`[[`

to retrieve the count matrix; -
`trichoptera2`

has an additional`Offset`

column (more on that later).

It is common practice when modeling count data to introduce an offset
term to control for different sampling efforts, exposures, baselines,
etc. The *proper way* to compute sample-specific offsets in still
debated and may vary depending on the field. There are nevertheless a
few popular methods:

- Total Sum Scaling (TSS), where the offset of a sample is the total count in that sample
- Cumulative Sum Scaling (CSS), introduced in (Paulson et al. 2013), where the offset of a sample if the cumulative sum of counts in that sample, up to a quantile determined in a data driven way.
- Relative Log-Expression (RLE), implemented in (Anders and Huber 2010), where all samples are used to compute a reference sample, each sample is compared to the reference sample using log-ratios and the offset is the median log-ratio.
- Geometric Mean of Pairwise Ratio (GMPR), introduced in (Chen et al. 2018) where each sample is compared to each other to compute a median log-ratio and the offset of a sample is the geometric means of those pairwise ratios.
- Wrench, introduced in (Kumar et al. 2018)
and fully implemented in the Wrench
package, where all samples are used to compute reference proportions
and each sample is compared to the reference using ratios (and
**not log-ratios**) of proportions to compute compositional correction factors. In that case, the offset is the product of (geometrically centered) compositional factors and (geometrically centered) depths.

Each of these offset be computed from a counts matrix using the
`compute_offset`

function and changing its
`offset`

argument:

```
## same as compute_offset(trichoptera$Abundance, offset = "TSS")
compute_offset(trichoptera$Abundance)
```

```
## 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
## 29 13 38 192 79 18 8 34 12 4 4 3 49 33 600 172
## 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
## 58 51 56 127 35 13 17 3 27 40 44 8 9 1599 2980 88
## 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
## 135 327 66 90 63 15 14 20 70 53 95 43 62 149 16 31
## 49
## 86
```

In this particular example, the counts are too sparse and sophisticated offset methods all fail (numeric output hidden)

`compute_offset(trichoptera$Abundance, "CSS")`

```
## Warning in offset_function(counts, ...): Some samples only have 1 positive
## values. Can't compute quantiles and fall back to TSS normalization
```

`compute_offset(trichoptera$Abundance, "RLE")`

```
## Warning in offset_function(counts, ...): Because of high sparsity, some samples
## have null or infinite offset.
```

`compute_offset(trichoptera$Abundance, "GMPR")`

We can mitigate this problem for the RLE offset by adding pseudocounts to the counts although doing so has its own drawbacks.

`compute_offset(trichoptera$Abundance, "RLE", pseudocounts = 1)`

```
## 1 2 3 4 5 6 7 8
## 0.9186270 0.8349121 0.8570257 0.9186270 0.9186270 0.8349121 0.8192245 0.8570257
## 9 10 11 12 13 14 15 16
## 0.7322797 0.7322797 0.6321923 0.6321923 0.9240361 0.9186270 1.3788037 0.9240361
## 17 18 19 20 21 22 23 24
## 0.9186270 0.9240361 0.9240361 1.7140514 0.8908577 0.8570257 0.8349121 0.6321923
## 25 26 27 28 29 30 31 32
## 0.9240361 0.9240361 0.9186270 0.8570257 0.8349121 2.7721084 3.2934218 0.9584503
## 33 34 35 36 37 38 39 40
## 1.0406547 0.9584503 0.9584503 0.9186270 0.9584503 0.8908577 0.8349121 0.7322797
## 41 42 43 44 45 46 47 48
## 0.9240361 0.9240361 0.9584503 0.9584503 1.2643846 1.7140514 0.8233555 0.8908577
## 49
## 0.9186270
```

A better solution consists in using only positive counts to compute the offsets:

`compute_offset(trichoptera$Abundance, "RLE", type = "poscounts")`

```
## 1 2 3 4 5 6 7 8
## 0.5631099 0.9462046 0.8299806 0.9462046 0.6460415 0.5756774 0.6308031 0.4947789
## 9 10 11 12 13 14 15 16
## 0.7988730 0.3574190 0.2207270 0.1260525 0.8051707 0.7289732 1.5591914 1.1850090
## 17 18 19 20 21 22 23 24
## 0.6389431 1.0000000 1.4606879 1.8224330 0.8403498 0.3942114 0.4058586 0.1997183
## 25 26 27 28 29 30 31 32
## 0.6286560 0.5631099 0.8777257 0.3440303 0.2504662 2.7086423 2.5800932 1.4584997
## 33 34 35 36 37 38 39 40
## 2.6050843 4.5898981 1.2185072 1.2616062 0.8823673 0.6250713 0.3950030 0.5631099
## 41 42 43 44 45 46 47 48
## 1.4511384 1.0000000 0.9462046 1.0882448 1.0000000 1.0631099 0.4621924 0.7900060
## 49
## 1.0672361
```

Finally, we can use wrench to compute the offsets:

`compute_offset(trichoptera$Abundance, "Wrench")`

```
## [1] 0.41269451 0.17385897 0.31925785 0.70682391 0.44749382 0.20676142
## [7] 0.10226641 0.21204241 0.09228293 0.03919178 0.03946910 0.02295077
## [13] 24.18096281 1.54038084 3.95187144 0.80172207 10.52136472 4.55854738
## [19] 0.46968859 51.88528943 0.32803605 0.14764973 4.13635353 0.03152826
## [25] 1.63626578 0.53769290 0.38482355 0.62689594 0.08030685 83.87692476
## [31] 33.40810265 0.75508731 1.05809625 1.18385013 0.58657751 0.43005272
## [37] 4.79925224 0.18381986 0.16078594 0.18490773 4.25999137 5.46689591
## [43] 0.60139850 7.56594658 9.89766712 33.73727074 0.74264459 29.13747796
## [49] 47.12017975
```

**Note** TSS is the only methods that produces offset on
the same scale as the counts, all others produces offsets that are
(hopefully) *proportional* to library sizes but on a different
scale. To force the offsets to be on the same scale as the counts for
all methods, you can use the option `scale = "count"`

.

`compute_offset(trichoptera$Abundance, "Wrench", scale = "count")`

```
## 1 2 3 4 5 6
## 17.2788462 7.2791915 13.3668057 29.5935647 18.7358369 8.6567635
## 7 8 9 10 11 12
## 4.2817277 8.8778701 3.8637358 1.6408958 1.6525069 0.9609112
## 13 14 15 16 17 18
## 1012.4174950 64.4932346 165.4584151 33.5667961 440.5123899 190.8589478
## 19 20 21 22 23 24
## 19.6650957 2172.3524890 13.7343345 6.1818537 173.1823794 1.3200369
## 25 26 27 28 29 30
## 68.5077809 22.5123254 16.1119346 26.2471113 3.3623170 3511.7901098
## 31 32 33 34 35 36
## 1398.7427987 31.6142749 44.3007651 49.5658750 24.5590443 18.0056062
## 37 38 39 40 41 42
## 200.9368679 7.6962380 6.7318451 7.7417851 178.3588942 228.8900199
## 43 44 45 46 47 48
## 25.1795750 316.7738500 414.3991879 1412.5245302 31.0933184 1219.9387043
## 49
## 1972.8451146
```

`prepare_data`

We’ll already learned that `prepare_data`

can join counts
and covariates into a single data.frame. It can also compute offset
through `compute_offset`

and does so by default with
`offset = "TSS"`

, hence the `Offset`

column in
`trichoptera2`

. You can change the offset method and provide
additional arguments that will passed on to
`compute_offset`

.

```
str(prepare_data(trichoptera$Abundance,
trichoptera$Covariate,
offset = "RLE", pseudocounts = 1))
```

```
## 'data.frame': 49 obs. of 9 variables:
## $ Abundance : num [1:49, 1:17] 0 0 0 0 0 0 0 0 0 0 ...
## ..- attr(*, "dimnames")=List of 2
## .. ..$ : chr [1:49] "1" "2" "3" "4" ...
## .. ..$ : chr [1:17] "Che" "Hyc" "Hym" "Hys" ...
## $ Temperature : num 18.7 19.8 22 23 22.5 23.9 15 17.2 15.4 14.1 ...
## $ Wind : num -2.3 -2.7 -0.7 2.3 2.3 -2 -4.7 -1 -2.7 -3.7 ...
## $ Pressure : num 998 1000 997 991 990 ...
## $ Humidity : num 60 63 73 71 62 64 93 84 88 75 ...
## $ Cloudiness : num 19 0 6 81 50 50 100 19 69 6 ...
## $ Precipitation: num 0 0 0 0 0 0 1.6 0 1.6 0 ...
## $ Group : Factor w/ 12 levels "1","2","3","4",..: 1 1 1 1 1 1 1 1 1 1 ...
## $ Offset : num 0.919 0.835 0.857 0.919 0.919 ...
```

Different communities use different standard for the count data where
samples are either or columns of the counts matrix.
`prepare_data`

uses heuristics to guess the direction of the
counts matrix (or fail informatively doing so) and automatically
transpose it if needed.

Finally, `prepare_data`

enforces sample-consistency
between the counts and the covariates and automatically trims away: -
samples for which only covariates or only counts are available; -
samples with no positive counts

For example, if we remove the first sample from the counts and the last one from the covariates, we end up with 49 - 2 = 47 samples left, as expected.

```
nrow(prepare_data(trichoptera$Abundance[-1, ], ## remove first sample
trichoptera$Covariate[-49,] ## remove last sample
))
```

`## [1] 47`

`prepare_data_from_[phyloseq|biom]`

Community composition data are quite popular in microbial ecology and usually stored in flat files using the biom format and/or imported in R as phyloseq-class objects (McMurdie 2013) using the Bioconductor phyloseq package.

We show here how to import data from a biom file (or biom-class object) and form a phyloseq-class object.

Reading from a biom file requires the bioconductor package biomformat.
This package is **not** a standard dependency of PLNmodels
and needs to be installed separately.

You can easily prepare your data from a biom file using the following steps:

- read your biom file with
`biomformat::read_biom()`

- extract the count table with
`biomformat::biom_data()`

- extract the covariates with
`biomformat::sample_metadata()`

(or build your own) - feed them to
`prepare_data`

as illustrated below:

```
## If biomformat is not installed, uncomment the following lines
# if (!requireNamespace("BiocManager", quietly = TRUE)) {
# install.packages("BiocManager")
# }
# BiocManager::install("biomformat")
library(biomformat)
biomfile <- system.file("extdata", "rich_dense_otu_table.biom", package = "biomformat")
biom <- biomformat::read_biom(biomfile)
## extract counts
counts <- as(biomformat::biom_data(biom), "matrix")
## extract covariates (or prepare your own)
covariates <- biomformat::sample_metadata(biom)
## prepare data
my_data <- prepare_data(counts = counts, covariates = covariates)
str(my_data)
```

Likewise, preparing data from a phyloseq-class object requires the
bioconductor package phyloseq.
This package is **not** a standard dependency of PLNmodels
and needs to be installed separately.

You can easily prepare your data from a phyloseq object using the following steps:

- extract the count table with
`phyloseq::otu_table()`

- extract the covariates with
`phyloseq::sample_data()`

(or build your own) - feed them to
`prepare_data`

as illustrated below:

```
## If biomformat is not installed, uncomment the following lines
# if (!requireNamespace("BiocManager", quietly = TRUE)) {
# install.packages("BiocManager")
# }
# BiocManager::install("phyloseq")
library(phyloseq)
data("enterotype")
## extract counts
counts <- as(phyloseq::otu_table(enterotype), "matrix")
## extract covariates (or prepare your own)
covariates <- phyloseq::sample_data(enterotype)
## prepare data
my_data <- prepare_data(counts = counts, covariates = covariates)
str(my_data)
```

We detail here the mathematical background behind the various offsets and the way they are computed. Note \(\mathbf{Y} = (Y_{ij})\) the counts matrix where \(Y_{ij}\) is the count of species \(j\) in sample \(i\). Assume that there are \(p\) species and \(n\) samples in total. The offset of sample \(i\) is noted \(O_i\) and computed in the following way.

Offsets are simply the total counts of a sample (frequently called depths in the metabarcoding literature): \[ O_i = \sum_{j=1}^p Y_{ij} \]

Positive counts are used to compute sample-specific quantiles \(q_i^l\) and cumulative sums \(s_i^l\) defined as \[
q_i^l = \min \{q \text{ such that } \sum_j 1_{Y_{ij} \leq q} \geq l
\sum_j 1_{Y_{ij} > 0} \} \qquad s_i^l = \sum_{j: Y_{ij} \leq q_i^l}
Y_{ij}
\] The sample-specific quantiles are then used to compute
reference quantiles defined as \(q^l =
\text{median} \{q^i_l\}\) and median average deviation around the
quantile \(q^l\) as \(d^l = \text{median} |q_i^l - q^l|\). The
method then searches for the smallest quantile \(l\) for which it detects instability,
defined as large relative increase in the \(d^l\). Formally, \(\hat{l}\) is the smallest \(l\) satisfying \(\frac{d^{l+1} - d^l}{d^l} \geq 0.1\). The
scaling sample-specific offset are then chosen as: \[
O_i = s_i^{\hat{l}} / \text{median}_i \{ s_i^{\hat{l}} \}
\] Dividing by the median of the \(s_i^{\hat{l}}\) ensures that offsets are
centered around \(1\) and compare sizes
differences with respect to the reference sample. Note also that the
reference quantiles \(q^l\) can be
computed using either the median (default, as in the original Paulson et al. (2013) paper) or the mean, by specifying
`reference = mean`

, as implemented in
`metagenomeseq`

.

A reference sample \((q_j)_j\) is
first built by computing the geometric means of each species count:
\[
q_j = \exp \left( \frac{1}{n} \sum_{i} \log(Y_{ij})\right)
\] Each sample is then compared to the reference sample to
compute one ratio per species and the final offset \(O_i\) is the median of those ratios: \[
O_i = \text{median}_j \frac{Y_{ij}}{q_j}
\] The method fails when no species is shared across all sample
(as all \(q_j\) are then \(0\)) or when a sample shares less than 50%
of species with the reference (in which case the median of the ratios
may be null or infinite). The problem can be alleviated by adding
pseudocounts to the \(c_{ij}\) with
`pseudocounts = 1`

or using positive counts in the
computations (`type = "poscounts"`

)

This method is similar to RLE but does create a reference sample. Instead, each sample is compared to each other to compute a median ratio (similar to RLE) \[ r_{ii'} = {\text{median}}_{j: Y_{ij}.Y_{i'j} > 0} \frac{Y_{ij}}{Y_{i'j}} \] The offset is then taken as the median of all the \(r_{ii'}\): \[ O_i = \text{median}_{i' != i} r_{ii'} \] The method fails when there is only one sample in the data set or when a sample shares no species with any other.

This method is fully detailed in Kumar et al.
(2018) and we
only provide a barebone implementation corresponding to the defaults
parameters of `Wrench::wrench()`

. Assume that samples belong
to \(K\) discrete groups and note \(g_i\) the group of sample \(i\). Wrench is based on the following
(simplified) log-normal model for counts: \[
Y_{ij} \sim \pi_{ij} \delta_0 + (1 - \pi_{ij})\log\mathcal{N}(\mu_{ij},
\sigma^2_j)
\] where the \(Y_{ij}\) are
independent and the mean \(\mu_{ij}\)
is decomposed as: \[
\mu_{ij} = \underbrace{\log{p_{0j}}}_{\text{log-ref. prop.}}
+ \underbrace{\log{d_i}}_{\text{log-depth}} +
\underbrace{\log{\zeta_{0g_i}}}_{\text{log effect of group } g_i} +
\underbrace{a_{i}}_{\text{(f|m)ixed effect}} +
\underbrace{b_{ij}}_{\text{mixed effects}}
\] where the random effects are independents centered gaussian
and the depths is the total sum of counts: \[
\begin{align*}
d_i & = \sum_{j=1}^p c_{ij} \\
b_{ij} & \sim \mathcal{N}(0, \eta^2_{g_i}) \\
\end{align*}
\]

The **net** log fold change \(\theta_{ij}\) of the **proportion
ratio** \(r_{ij} = c_{ij} / d_i
p_{0j}\) of species \(j\)
relative to the reference is \(\log(\theta_{ij}) \overset{\Delta}{=}
\mathbb{E}[\log(r_{ij}) | a_i, b_{ij}] = \log{\zeta_{0g_i}} + a_i +
b_{ij}\). We can decompose it as \(\theta_{ij} = \Lambda_i^{-1} v_{ij}\) where
\(\Lambda_i^{-1}\) is the
*compositional correction factor* and \(v_{ij}\) is the fold change of **true
abundances**.

With the above notations, the net fold change compounds both the fold change of true abundances and the compositional correction factors. With the assumption that the \(b_{ij}\) are centered, \(\log(\hat{\Lambda}_i)\) can be estimated through a robust average of the \(\hat{\theta}_{ij}\), which can themselves be computed from the log-ratio of proportions.

We detail here how the different parameters and/or effects are estimated.

- The reference proportions \(p_{0j}\) are constructed as averages of the sample proportions \(p_{ij}\) and the ratio are derived from both quantities \[ p_{ij} = \frac{Y_{ij}}{\sum_{j=1}^p Y_{ij}} \qquad p_{0j} = \frac{1}{n} \sum_{i=1}^n p_{ij} \qquad r_{ij} = \frac{p_{ij}}{p_{0j}} \]
- The probabilities of absence \(\pi_{ij}\) are estimated by fitting the following Bernoulli models: \[ 1_{\{Y_{ij} = 0\}} \sim \mathcal{B}(\pi_{j}^{d_i}) \] and setting \(\hat{\pi}_{ij} = \hat{\pi}^{d_i}\)
- The species variances \(\sigma^2_j\) are estimated by fitting the
following linear model (with no zero-inflation component) \[
\log Y_{ij} \sim \log(d_i) + \mu_{g_i} + \mathcal{N}(0, \sigma^2_j)
\] Note that in the original
`Wrench::wrench()`

, the log depth \(\log(d_i)\) is used as predictor but I believe it makes more sense to use it an offset. - set the group proportions \(p_{gj}\) and group ratios \(r_{gj}\) to: \[ p_{gj} = \frac{\sum_{i : g_i = g} Y_{ij}}{\sum_{j, i : g_i = g} Y_{ij}} \qquad r_{gj} = \frac{p_{gj}}{p_{0j}} \]
- Estimate the location and dispersion parameters as: \[ \hat{\zeta}_{0g} = \frac{\sum_{j=1}^p r_{gj}}{p} \qquad \log{r_{g.}} = \frac{\sum_{j: r_{gj} \neq 1} \log{r_{gj}}}{\sum_{j: r_{gj} \neq 0} 1} \qquad \hat{\eta}_{g}^2 = \frac{\sum_{j: r_{gj} \neq 1} (\log{r_{gj}} - \log{r_{g.}})^2}{\sum_{j: r_{gj} \neq 0} 1} \]
- Estimate the mixed effects as shrunken (and scaled) averages of the ratios \[ \hat{a}_i = \frac{\sum_{j = 1}^p \frac{1}{\hat{\eta}^2_{g_i} + \hat{\sigma}^2_j} (\log{r_{ij}} - \log{\hat{\zeta}_{0g_i}})}{\sum_{j = 1}^p \frac{1}{\hat{\eta}^2_{g_i} + \hat{\sigma}^2_j}} \qquad \hat{b}_{ij} = \frac{\hat{\eta}^2_{g_i}}{\hat{\eta}^2_{g_i} + \hat{\sigma}^2_j} \left( \log{r_{ij}} - \log\hat{\zeta}_{0g_i} - \hat{a}_i\right) \]
- Estimate the regularized ratios as: \[ \hat{\theta}_{ij} = \exp\left( \log\hat{\zeta}_{0g_i} + \hat{a}_i + \hat{b}_{ij} \right) \]
- Estimate the compositional correction factors as (weighted) means of the regularized (and possibly corrected) ratios: \[ \hat{\Lambda}_i = \begin{cases} \sum_{j = 1}^p \hat{\theta}_{ij} \bigg/ p& \text{ if type = "simple"} \\ \sum_{j = 1}^p \hat{\theta}_{ij} e^{-\hat{\sigma}_j^2 / 2} / w_{ij} \bigg/ \sum_{j=1}^p 1/w_{ij} & \text{ if type = "wrench"} \\ \end{cases} \] where \(w_{ij} = (1 - \hat{\pi}_{ij})(\hat{\pi}_{ij} + e^{\hat{\sigma}_j^2 + \hat{\eta}_i^2} - 1)\). The correction term \(e^{\hat{\sigma}_j^2 / 2}\) arises from the relation \(\mathbb{E}[r_{ij} | r_{ij} > 0] = \theta_{ij} e^{\sigma_j^2/2}\) and the weight \(w_{ij}\) are marginal variances: \(\mathbb{V}[r_{ij}] = w_{ij}\).

The offsets are then the product of compositional correction factors and depths: \[ O_i = \frac{\hat{\Lambda}_i}{(\prod_{i = 1}^n \hat{\Lambda}_i)^{1/n}} \times \frac{d_i}{(\prod_{i = 1}^n d_i)^{1/n}} \]

Anders, Simon, and Wolfgang Huber. 2010. “Differential Expression
Analysis for Sequence Count Data.” *Genome Biology* 11
(10): R106. https://doi.org/10.1186/gb-2010-11-10-r106.

Chen, Li, James Reeve, Lujun Zhang, Shengbing Huang, Xuefeng Wang, and
Jun Chen. 2018. “GMPR: A Robust Normalization Method for
Zero-Inflated Count Data with Application to Microbiome Sequencing
Data.” *PeerJ* 6 (April): e4600. https://doi.org/10.7717/peerj.4600.

Kumar, M. Senthil, Eric V. Slud, Kwame Okrah, Stephanie C. Hicks,
Sridhar Hannenhalli, and Héctor Corrada Bravo. 2018. “Analysis and
Correction of Compositional Bias in Sparse Sequencing Count
Data.” *BMC Genomics* 19 (1). https://doi.org/10.1186/s12864-018-5160-5.

McMurdie, Paul J. AND Holmes. 2013. “Phyloseq: An r Package for
Reproducible Interactive Analysis and Graphics of Microbiome Census
Data.” *PLoS ONE* 8 (4): e61217. https://doi.org/10.1371/journal.pone.0061217.

Paulson, Joseph N, O. Colin Stine, Héctor Corrada Bravo, and Mihai Pop.
2013. “Differential Abundance Analysis for Microbial Marker-Gene
Surveys.” *Nat Methods* 10 (September): 1200–1202. https://doi.org/10.1038/nmeth.2658.

although a

`data.frame`

is technically a`list`

↩︎