# R Package MADSEQ

#### Yu Kong

#### 2017-04-24

The MADSEQ package is a group of hierarchical Bayesian model for the detection and quantification of potential mosaic aneuploidy in sample using massive parallel sequencing data.

The MADSEQ package takes two pieces of information for the detection and quantification of mosaic aneuploidy:

1. The distribution of the alternative allele frequencies (AAF) of the sites that are genotyped as heterozygous.
2. The average sequencing coverage for regions. (for targeted sequencing it’s each targeted region; for whole genome sequencing, bin the genome into bins. And because sequencing coverage are usually biased by GC content, normalization is necessary, the normalization function is provided in the package).

MADSEQ works on the whole chromosome resolution. It applies all of the five models (normal, monosomy, mitotic trisomy, meiotic trisomy, loss of heterozygosity) to fit the distribution of the AAF of all the heterozygous sites, and fit the distribution of the coverage from that chromosome. After fitting the same data using all models, it does model comparison using BIC (Bayesian Information Criteria) to select the best model. The model selected tells us whether the chromosome is aneuploid or not, and also the type of mosaic aneuploidy. Then, from the posterior distribution of the best model, we could get the estimation of the fraction of aneuploidy cells.

## Data You Need

- A bam file for your sample: sorted, indexed.
- A vcf.gz file for your sample:
  - The vcf should be bgzip compressed (bgzip is a part of tabix package, which can be download from here).
  - The FORMAT section of the vcf file should contain depth (DP) and allelic depth (AD) field.
- A bed file containing targeted regions: if it’s targeted sequencing data, then the bed file contatining targeted regions (tiled regions is recommended) . If it’s whole genome sequencing, bin the whole genome into equal size bins (i.e. 500kb).

Note: Currently our package only supports one bam and one vcf file per sample. If you have more than one sample, please prepare multiple bam and vcf files for each of them.

### Example Data

There are two sets of example data come with the package:

1. A bam file named aneuploidy.bam and a vcf file named aneuploidy.vcf.gz for a sample containing trisomy in chromosome 18.
2. A bam file named normal.bam and a vcf file names normal.vcf.gz for a normal sample.
3. A bed file named target.bed containing the targeted regions.

To access the data use

    system.file("extdata","aneuploidy.bam",package="MADSEQ")
    system.file("extdata","aneuploidy.vcf.gz",package="MADSEQ")

Note:This is just a set of example data, only contains a very little region of the genome.

## Use The Package

We will start with the bam file, vcf file and bed file in the example data to show you each step for the analysis.

### Prepare coverage information

Started with bam file and bed file, you can use prepareCoverageGC function to get the coverage and GC information for each targeted regions.

    ## load the package
    suppressMessages(library("MADSEQ"))

    ## get path to the location of example data
    aneuploidy_bam = system.file("extdata","aneuploidy.bam",package="MADSEQ")
    normal_bam = system.file("extdata","normal.bam",package="MADSEQ")
    target = system.file("extdata","target.bed",package="MADSEQ")

    ## Note: for your own data, just specify the path to the location
    ## of your file using character.

    ## prepare coverage and GC content for each targeted region
    # aneuploidy sample
    aneuploidy_cov = prepareCoverageGC(target_bed=target,
                                        bam=aneuploidy_bam,
                                        "hg19")
    #> Warning: In BioC 3.5, the 'force' argument was replaced by the more
    #>   flexible 'pruning.mode' argument, and is deprecated. See ?seqinfo
    #>   for the supported pruning modes. Note that 'force=TRUE' is
    #>   equivalent to 'pruning.mode="coarse"'.
    #> 384 regions from 24 chromosomes in the bed file.
    #> calculating depth from BAM...
    #> Warning in .Internal(unique(x, incomparables, fromLast, nmax)): closing
    #> unused connection 5 (C:/Users/biocbuild/bbs-3.5-bioc/tmpdir/RtmpUfPq8Z/
    #> Rinst21a813b470ab/MADSEQ/extdata/target.bed)
    #>
    #> Attaching package: 'BiocGenerics'
    #> The following objects are masked from 'package:parallel':
    #>
    #>     clusterApply, clusterApplyLB, clusterCall, clusterEvalQ,
    #>     clusterExport, clusterMap, parApply, parCapply, parLapply,
    #>     parLapplyLB, parRapply, parSapply, parSapplyLB
    #> The following objects are masked from 'package:stats':
    #>
    #>     IQR, mad, sd, var, xtabs
    #> The following objects are masked from 'package:base':
    #>
    #>     Filter, Find, Map, Position, Reduce, anyDuplicated, append,
    #>     as.data.frame, cbind, colMeans, colSums, colnames, do.call,
    #>     duplicated, eval, evalq, get, grep, grepl, intersect,
    #>     is.unsorted, lapply, lengths, mapply, match, mget, order,
    #>     paste, pmax, pmax.int, pmin, pmin.int, rank, rbind, rowMeans,
    #>     rowSums, rownames, sapply, setdiff, sort, table, tapply,
    #>     union, unique, unsplit, which, which.max, which.min
    #>
    #> Attaching package: 'S4Vectors'
    #> The following object is masked from 'package:base':
    #>
    #>     expand.grid
    #>
    #> Attaching package: 'Biostrings'
    #> The following object is masked from 'package:base':
    #>
    #>     strsplit
    #> calculating GC content...

    # normal sample
    normal_cov = prepareCoverageGC(target_bed=target,
                                    bam=normal_bam,
                                    "hg19")
    #> Warning: In BioC 3.5, the 'force' argument was replaced by the more
    #>   flexible 'pruning.mode' argument, and is deprecated. See ?seqinfo
    #>   for the supported pruning modes. Note that 'force=TRUE' is
    #>   equivalent to 'pruning.mode="coarse"'.
    #> 384 regions from 24 chromosomes in the bed file.
    #> calculating depth from BAM...
    #> Warning in getClassDef(Class, where): closing unused connection 5 (C:/
    #> Users/biocbuild/bbs-3.5-bioc/tmpdir/RtmpUfPq8Z/Rinst21a813b470ab/MADSEQ/
    #> extdata/target.bed)
    #> calculating GC content...

    ## view the first two rows of prepared coverage data (A GRanges Object)
    aneuploidy_cov[1:2]
    #> GRanges object with 2 ranges and 2 metadata columns:
    #>       seqnames               ranges strand |     depth                GC
    #>          <Rle>            <IRanges>  <Rle> | <numeric>         <numeric>
    #>   [1]    chr14 [20528156, 20528257]      * |         0 0.303921568627451
    #>   [2]    chr14 [21538016, 21538117]      * |        21 0.647058823529412
    #>   -------
    #>   seqinfo: 24 sequences from an unspecified genome; no seqlengths

    normal_cov[1:2]
    #> GRanges object with 2 ranges and 2 metadata columns:
    #>       seqnames               ranges strand |     depth                GC
    #>          <Rle>            <IRanges>  <Rle> | <numeric>         <numeric>
    #>   [1]    chr14 [20528156, 20528257]      * |         0 0.303921568627451
    #>   [2]    chr14 [21538016, 21538117]      * |         4 0.647058823529412
    #>   -------
    #>   seqinfo: 24 sequences from an unspecified genome; no seqlengths

### Normalize coverage information

The normalization function takes prepared coverage GRanges object from prepareCoverageGC function, normalize the coverage and calculate the expected coverage for the sample. If there is only one sample, the function will correct the coverage by GC content, and take the average coverage for the whole genome as expected coverage. If there are more than one samples given, the function will first quantile normalize coverage across samples, then correct the coverage by GC for each sample. If control sample is not specified, the expected coverage is the median coverage across all samples, if a normal control is specified, the average coverage for control sample is taken as expected coverage for further analysis.

Note:

1. If you have more than one samples, please make sure they are targeted by the same regions.
2. If you have more than one samples, please separate female samples and male samples, which means normalize them separately; because sex chromosomes can bias the normalization.
3. One sample is good to go, however multiple samples will help you get a better normalization.

#### If you only have one sample

If you choose to write the output to file (recommended)

    ## normalize coverage data
    ## set plot=FALSE here because similar plot will show in the following example
    normalizeCoverage(aneuploidy_cov,writeToFile=TRUE, destination=".",plot=FALSE)
    #> no control provided
    #> there are 1 samples
    #> correct GC bias in sample 'aneuploidy_cov' ...
    #> normalized depth for sample aneuploidy_cov is written to ./aneuploidy_cov_normed_depth.txt

If you don’t want to write output to file

    ## normalize coverage data
    aneuploidy_normed = normalizeCoverage(aneuploidy_cov,writeToFile=FALSE,
                                          plot=FALSE)
    #> no control provided
    #> there are 1 samples
    #> correct GC bias in sample 'aneuploidy_cov' ...

    ## a GRangesList object will be produced by the function, look at it by
    names(aneuploidy_normed)
    #> [1] "aneuploidy_cov"
    aneuploidy_normed[["aneuploidy_cov"]]
    #> GRanges object with 188 ranges and 4 metadata columns:
    #>         seqnames               ranges strand |     depth                GC
    #>            <Rle>            <IRanges>  <Rle> | <numeric>         <numeric>
    #>     [1]    chr14 [21538016, 21538117]      * |        21 0.647058823529412
    #>     [2]    chr14 [22377854, 22377955]      * |        24 0.362745098039216
    #>     [3]    chr14 [26201540, 26201641]      * |        45 0.362745098039216
    #>     [4]    chr14 [26475535, 26475636]      * |         8 0.264705882352941
    #>     [5]    chr14 [29080232, 29080333]      * |         6 0.323529411764706
    #>     ...      ...                  ...    ... .       ...               ...
    #>   [184]    chr18 [69135718, 69135819]      * |        14 0.323529411764706
    #>   [185]    chr18 [72988854, 72988955]      * |       118 0.519607843137255
    #>   [186]    chr18 [73100816, 73100917]      * |        12 0.254901960784314
    #>   [187]    chr18 [73278496, 73278597]      * |        91 0.441176470588235
    #>   [188]    chr18 [77147421, 77147522]      * |        75 0.588235294117647
    #>         normed_depth ref_depth
    #>            <numeric> <numeric>
    #>     [1]           30         0
    #>     [2]           28         0
    #>     [3]           49         0
    #>     [4]           30         0
    #>     [5]           25         0
    #>     ...          ...       ...
    #>   [184]           33         0
    #>   [185]          100         0
    #>   [186]           32         0
    #>   [187]           85         0
    #>   [188]           67         0
    #>   -------
    #>   seqinfo: 24 sequences from an unspecified genome; no seqlengths

#### If you have more than one samples, without a normal control

If you choose to write the output to file (recommended)

    ## normalize coverage data
    normalizeCoverage(aneuploidy_cov, normal_cov,
                      writeToFile =TRUE, destination = ".", plot=FALSE)
    #> no control provided
    #> there are 2 samples
    #> Quantile normalizing ...
    #> correct GC bias in sample' aneuploidy_cov '...
    #> correct GC bias in sample' normal_cov '...
    #> normalized depth for sample aneuploidy_cov is written to ./aneuploidy_cov_normed_depth.txt
    #> normalized depth for sample normal_cov is written to ./normal_cov_normed_depth.txt

If you don’t want to write output to file

    ## normalize coverage data
    normed_without_control = normalizeCoverage(aneuploidy_cov, normal_cov,
                                               writeToFile=FALSE, plot=TRUE)
    #> no control provided
    #> there are 2 samples
    #> Quantile normalizing ...
    #> correct GC bias in sample' aneuploidy_cov '...
    #> correct GC bias in sample' normal_cov '...

    ## a GRangesList object will be produced by the function
    length(normed_without_control)
    #> [1] 2
    names(normed_without_control)
    #> [1] "aneuploidy_cov" "normal_cov"

    ## subsetting
    normed_without_control[["aneuploidy_cov"]]
    #> GRanges object with 188 ranges and 5 metadata columns:
    #>         seqnames               ranges strand |     depth quantiled_depth
    #>            <Rle>            <IRanges>  <Rle> | <numeric>       <numeric>
    #>     [1]    chr14 [21538016, 21538117]      * |        21              17
    #>     [2]    chr14 [22377854, 22377955]      * |        24              19
    #>     [3]    chr14 [26201540, 26201641]      * |        45              36
    #>     [4]    chr14 [26475535, 26475636]      * |         8               6
    #>     [5]    chr14 [29080232, 29080333]      * |         6               5
    #>     ...      ...                  ...    ... .       ...             ...
    #>   [184]    chr18 [69135718, 69135819]      * |        14              12
    #>   [185]    chr18 [72988854, 72988955]      * |       118              93
    #>   [186]    chr18 [73100816, 73100917]      * |        12              10
    #>   [187]    chr18 [73278496, 73278597]      * |        91              70
    #>   [188]    chr18 [77147421, 77147522]      * |        75              58
    #>                        GC normed_depth        ref_depth
    #>                 <numeric>    <numeric>        <numeric>
    #>     [1] 0.647058823529412           24 37.2867772108844
    #>     [2] 0.362745098039216           22 37.2867772108844
    #>     [3] 0.362745098039216           39 37.2867772108844
    #>     [4] 0.264705882352941           23 37.2867772108844
    #>     [5] 0.323529411764706           20 37.2867772108844
    #>     ...               ...          ...              ...
    #>   [184] 0.323529411764706           27            35.76
    #>   [185] 0.519607843137255           80            35.76
    #>   [186] 0.254901960784314           25            35.76
    #>   [187] 0.441176470588235           65            35.76
    #>   [188] 0.588235294117647           51            35.76
    #>   -------
    #>   seqinfo: 24 sequences from an unspecified genome; no seqlengths
    normed_without_control[["normal_cov"]]
    #> GRanges object with 190 ranges and 5 metadata columns:
    #>         seqnames               ranges strand |     depth quantiled_depth
    #>            <Rle>            <IRanges>  <Rle> | <numeric>       <numeric>
    #>     [1]    chr14 [21538016, 21538117]      * |         4               5
    #>     [2]    chr14 [22377854, 22377955]      * |        32              43
    #>     [3]    chr14 [26201540, 26201641]      * |        24              32
    #>     [4]    chr14 [26475535, 26475636]      * |        12              15
    #>     [5]    chr14 [29080232, 29080333]      * |        11              14
    #>     ...      ...                  ...    ... .       ...             ...
    #>   [186]    chr18 [69135718, 69135819]      * |        13              18
    #>   [187]    chr18 [72988854, 72988955]      * |        57              78
    #>   [188]    chr18 [73100816, 73100917]      * |        10              12
    #>   [189]    chr18 [73278496, 73278597]      * |        29              37
    #>   [190]    chr18 [77147421, 77147522]      * |        30              38
    #>                        GC normed_depth        ref_depth
    #>                 <numeric>    <numeric>        <numeric>
    #>     [1] 0.647058823529412           17 37.2867772108844
    #>     [2] 0.362745098039216           43 37.2867772108844
    #>     [3] 0.362745098039216           32 37.2867772108844
    #>     [4] 0.264705882352941           31 37.2867772108844
    #>     [5] 0.323529411764706           24 37.2867772108844
    #>     ...               ...          ...              ...
    #>   [186] 0.323529411764706           28            35.76
    #>   [187] 0.519607843137255           69            35.76
    #>   [188] 0.254901960784314           28            35.76
    #>   [189] 0.441176470588235           32            35.76
    #>   [190] 0.588235294117647           40            35.76
    #>   -------
    #>   seqinfo: 24 sequences from an unspecified genome; no seqlengths

#### If you have more than one samples, with a normal control

If you choose to write the output to file (recommended)

    ## normalize coverage data, normal_cov is the control sample
    normalizeCoverage(aneuploidy_cov, control=normal_cov,
                      writeToFile=TRUE, destination = ".",plot=FALSE)
    #> control: normal_cov
    #> there are 2 samples
    #> Quantile normalizing ...
    #> correct GC bias in sample' normal_cov '...
    #> correct GC bias in sample' aneuploidy_cov '...
    #> normalized depth for sample normal_cov is written to ./normal_cov_normed_depth.txt
    #> normalized depth for sample aneuploidy_cov is written to ./aneuploidy_cov_normed_depth.txt

If you don’t want to write output to file

    normed_with_control = normalizeCoverage(aneuploidy_cov, control=normal_cov,
                                            writeToFile =FALSE, plot=FALSE)
    #> control: normal_cov
    #> there are 2 samples
    #> Quantile normalizing ...
    #> correct GC bias in sample' normal_cov '...
    #> correct GC bias in sample' aneuploidy_cov '...

    ## a GRangesList object will be produced by the function
    length(normed_without_control)
    #> [1] 2
    names(normed_with_control)
    #> [1] "normal_cov"     "aneuploidy_cov"

### Prepare heterozygous sites

Having vcf.gz file and target bed file ready, use prepareHetero function to process the heterozygous sites.

    ## specify the path to vcf.gz file
    aneuploidy_vcf = system.file("extdata","aneuploidy.vcf.gz",package="MADSEQ")

    ## target bed file specified before

    ## If you choose to write the output to file (recommended)
    prepareHetero(aneuploidy_vcf, target, genome="hg19",
                  writeToFile=TRUE, destination=".")
    #> Warning: In BioC 3.5, the 'force' argument was replaced by the more
    #>   flexible 'pruning.mode' argument, and is deprecated. See ?seqinfo
    #>   for the supported pruning modes. Note that 'force=TRUE' is
    #>   equivalent to 'pruning.mode="coarse"'.
    #> starting filter
    #> filtering 387 records
    #> completed filtering
    #> filtered heterozygous sites for sample aneuploidy.vcf.gz is written to ./aneuploidy.vcf.gz_filtered_heterozygous.txt

    ## If you don't want to write output to file
    aneuploidy_hetero = prepareHetero(aneuploidy_vcf, target,
                                      genome="hg19", writeToFile=FALSE)
    #> Warning: In BioC 3.5, the 'force' argument was replaced by the more
    #>   flexible 'pruning.mode' argument, and is deprecated. See ?seqinfo
    #>   for the supported pruning modes. Note that 'force=TRUE' is
    #>   equivalent to 'pruning.mode="coarse"'.
    #> starting filter
    #> filtering 387 records
    #> completed filtering

### Run MadSeq model to detect potential mosaic aneuploidy

The function runMadSeq will run the models and select the best model for the input data.

Note:

1. Among three normalized coverage sets listed above (one sample, two samples without control, two samples with a control), we will use the two samplew with a control case for the following analysis.
2. Because these models are based on MCMC sampling, the running process can be very long. Running different chromosomes parallel in background or High Performance Computer Cluster is highly recommended.

#### If the processed data have been written into files

    ## specify the path to processed files
    aneuploidy_hetero = "./aneuploidy.vcf.gz_filtered_heterozygous.txt"
    aneuploidy_normed_cov = "./aneuploidy_cov_normed_depth.txt"

    ## run the model
    aneuploidy_chr18 = runMadSeq(hetero=aneuploidy_hetero,
                                 coverage=aneuploidy_normed_cov,
                                 target_chr="chr18",
                                 nChain=1, nStep=1000, thinSteps=1,
                                 adapt=100,burnin=200)
    #> total number of heterozygous site: 46
    #> total number of coverage 50
    #> module mix loaded
    #> 1. running normal model
    #> Compiling model graph
    #>    Resolving undeclared variables
    #>    Allocating nodes
    #> Graph information:
    #>    Observed stochastic nodes: 96
    #>    Unobserved stochastic nodes: 48
    #>    Total graph size: 302
    #>
    #> Initializing model
    #> 2. running monosomy model
    #> Compiling model graph
    #>    Resolving undeclared variables
    #>    Allocating nodes
    #> Graph information:
    #>    Observed stochastic nodes: 98
    #>    Unobserved stochastic nodes: 47
    #>    Total graph size: 349
    #>
    #> Initializing model
    #> 3. running mitotic trisomy model
    #> Compiling model graph
    #>    Resolving undeclared variables
    #>    Allocating nodes
    #> Graph information:
    #>    Observed stochastic nodes: 98
    #>    Unobserved stochastic nodes: 47
    #>    Total graph size: 349
    #>
    #> Initializing model
    #> 4. running meiotic trisomy model
    #> Compiling model graph
    #>    Resolving undeclared variables
    #>    Allocating nodes
    #> Graph information:
    #>    Observed stochastic nodes: 98
    #>    Unobserved stochastic nodes: 48
    #>    Total graph size: 379
    #>
    #> Initializing model
    #> 5. running loss of heterozygosity model
    #> Compiling model graph
    #>    Resolving undeclared variables
    #>    Allocating nodes
    #> Graph information:
    #>    Observed stochastic nodes: 96
    #>    Unobserved stochastic nodes: 51
    #>    Total graph size: 2861
    #>
    #> Initializing model
    #> models done, comparing models

    #> Order and delta BIC of the preference of models
    #>          BIC_normal BIC_mitotic_trisomy BIC_meiotic_trisomy
    #>            0.000000            8.463279           12.978304
    #>        BIC_monosomy             BIC_LOH
    #>           20.902613           22.343734
    #> model selected: normal

    ## An MadSeq object will be returned
    aneuploidy_chr18
    #> MadSeq object with the posterior distribution from normal model
    #>         kappa m_cov        mu      p_cov    r_cov
    #> [1,] 5.335972 31.96 0.4428044 0.15422429 5.827796
    #> [2,] 3.267671 31.96 0.4428044 0.16375117 6.258290
    #> [3,] 5.158145 31.96 0.4428044 0.11219189 4.038770
    #> [4,] 3.918075 31.96 0.4428044 0.16415119 6.276580
    #> [5,] 3.792722 31.96 0.4428044 0.08652953 3.027447
    #> [6,] 3.799169 31.96 0.4428044 0.14211082 5.294229
    #> ------
    #>          BIC_normal BIC_mitotic_trisomy BIC_meiotic_trisomy
    #>            0.000000            8.463279           12.978304
    #>        BIC_monosomy             BIC_LOH
    #>           20.902613           22.343734

Note: In order to save time, we only run 1 chain with a much less steps compared with default settings. For real cases, the default settings are recommended.

#### If the processed data have NOT been written into files

    ## subset normalized coverage for aneuploidy sample from the GRangesList
    ## returned by normalizeCoverage function
    aneuploidy_normed_cov = normed_with_control[["aneuploidy_cov"]]

    ## run the model
    aneuploidy_chr18 = runMadSeq(hetero=aneuploidy_hetero,
                                 coverage=aneuploidy_normed_cov,
                                 target_chr="chr18")

    ## An MadSeq object will be returned
    aneuploidy_chr18

The MadSeq object from the runMadSeq function contains:

- The posterior distribution of the selected model
- The delta BIC value between selected model and other models

Note: The value of delta BIC suggests the strength of the confidence of the selected model against other models. It can be summarized as

deltaBIC | Evidence against higher BIC
--- | ---
[0,2] | Not worth more than a bare mention
(2,6] | Positive
(6,10] | Strong
>10 | Very Strong

### Visualize the selected model

There are a group of plot functions to plot the output MadSeq object from the runMadSeq.

    ## plot the posterior distribution for all the parameters in selected model
    plotMadSeq(aneuploidy_chr18)

    ## plot the histogram for the estimated fraction of aneuploidy
    plotFraction(aneuploidy_chr18, prob=0.95)
    #> selected model is normal, f is estimated to be 0%

    ## plot the distribution of AAF as estimated by the model
    plotMixture(aneuploidy_chr18)

## Description of All Parameters

parameters | description
--- | ---
f | Fraction of mosaic aneuploidy
m | The midpoint of the alternative allele frequency (AAF) for all heterozygous sites
mu[1] | Mean AAF of mixture 1: the AAFs of this mixture shifted from midpoint to some higher values
mu[2] | Mean AAF of mixture 2: the AAFs of this mixture shifted from midpoint to some lower values
mu[3] (LOH model) | Mean AAF of mixture 3: In LOH model, mu[3] indicates normal sites without loss of heterozygosity
mu[3] (meiotic trisomy model) | Mean AAF of mixture 3: In meiotic model, the AAFs of this mixture shifted from 0 to some higher value
mu[4] | Mean AAF of mixture 4: the AAFs of this mixture shifted from 1 to some lower value (only in meiotic model)
kappa | Indicate variance of the AAF mixtures: larger kappa means smaller variance
p[1] | Weight of mixture 1: indicate the proportion of heterozygous sites in the mixture 1
p[2] | Weight of mixture 2: indicate the proportion of heterozygous sites in the mixture 2
p[3] | Weight of mixture 3: indicate the proportion of heterozygous sites in the mixture 3 (only in LOH and meiotic model)
p[4] | Weight of mixture 4: indicate the proportion of heterozygous sites in the mixture 4 (only in meiotic model)
p[5] | Weight of outlier component: the AAF of 1% sites might not well behaved, so these sites are treated as noise.
m_cov | Mean coverage of all the sites from the chromosome, estimated from a negative binomial distribution
p_cov | Prob of the negative binomial distribution for the coverage
r_cov | Another parameter (r) for the negative binomial disbribution of the coverage, small r means large variance