# Data importation in PLNmodels

#### PLN team

#### 2025-03-21

## Preliminaries

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.

## Format description

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

    library(PLNmodels)
    data(trichoptera)

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).

## Computing offsets

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

## Building data frame using `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

## Importing data from biom and phyloseq objects using `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

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)

### Reading from a phyloseq-class object

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)

## Mathematical details about the offsets

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.

### Total Sum Scaling

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

### Cumulative Sum Scaling

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`.

### Relative Log Expression

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"`)

### Geometric Mean of Pairwise Ratio

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.

### Wrench normalisation

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: \[ O_i = \frac{\hat{\Lambda}_i}{(\prod_{i = 1}^n \hat{\Lambda}_i)^{1/n}} \times \frac{d_i}{(\prod_{i = 1}^n d_i)^{1/n}} \]

## References

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.

1. although a `data.frame` is technically a `list`↩︎