rehh
Searching for Footprints of Selection using 'Extended Haplotype Homozygosity' Based Tests

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Vignette for package *rehh*

Vignette for package *rehh*
In rehh: Searching for Footprints of Selection using 'Extended Haplotype Homozygosity' Based Tests 

knitr::opts_chunk$set(comment = ">",fig.height = 4.5, fig.width = 4.5, fig.show = "hold")

\clearpage

About the package

This vignette describes comprehensively how the R package rehh can be applied to perform whole genome scans for footprints of selection using statistics related to Extended Haplotype Homozygosity (EHH) [@Sabeti2002]. The vignette Examples in detail explains basic usage and methodology with the help of two tiny artificial data sets.

The package accepts multi-allelic genetic markers as input. Typically, albeit not necessarily, these will be bi-allelic SNPs.

The package is available for Linux, Windows and MacOS X from the CRAN repository https://cran.r-project.org/package=rehh and may be installed using a standard procedure. Once the package has been successfully installed, it can be loaded by:

library(rehh)

Background

The analysis of molecular population genetic data often comprises the search for genomic regions that might have experienced recent selection. Diverse approaches have been developed; for reviews on methodology see [@Sabeti2006], [@Oleksyk2010] and [@Vitti2013] and for practical advice [@Cadzow2014], [@Utsunomiya2015] and [@Weigand2018]. However only a few have found wide-spread application [@Haasl2016]. To the latter belong iHS [@Voight2006], Rsb [@Tang2007] and XP-EHH [@Sabeti2007], all of which are summary statistics aimed to distill a certain aspect of the genetic data into a single score and constructed in a way that extreme values are indicative of positive or "Darwinian" selection. iHS is intended for application on a single (presumably homogeneous) population, while XP-EHH and Rsb are targeted to differential selection between two populations. All three statistics are based on the concept of Extended Haplotype Homozygosity (EHH) as formulated by [@Sabeti2002].

iHS and XP-EHH can be calculated by the independent C++ command line tool Hapbin [@Maclean2015], which has been optimized for speed by exploiting bit-wise machine-level operations. The package rehh cannot compete on performance, but has the advantage of being able to work with multi-allelic markers and missing values. Moreover, it possesses a broader range of input and output options, including several graphical representations.

Changes between versions 2.X and 3.X

Due to changes in the API, although mostly small, the versions 3.X are not compatible with versions 2.X.

Data objects of class haplohh (see below) created by versions up to 2.0.4 must be updated via the command update_haplohh() (see its documentation by typing ?update_haplohh) in order to be accepted by the functions of the current version.

Example files

For illustration purposes, several example input files as well as R objects are provided.

Input files

All of the above files are copied into the current working directory via the command

make.example.files()

Throughout the vignette, this command is assumed to have been run!

R objects

Terminology

Depending on the context, we refer as chromosomes either to the different chromosomes of a species' reference sequence (e.g. "chr1", "chr2", "chrX", etc.) or to the homologous chromosomes from different individuals (e.g. 280 times "chr12" from 140 diploid individuals). The sequence of alleles on a chromosome is referred to as its haplotype and sometimes we use it interchangeably with chromosome. However, if we speak of a shared haplotype, the term is meant more abstractly for the unique sequence of alleles that a group of chromosomes share at each position within a certain region.

Overview

The package calculates three statistics which can be used to perform whole-genome scans for selection: iHS, XP-EHH and Rsb. iHS compares alleles within a single population while the other two compare sites between populations. The calculation proceeds technically in five steps which are performed by running two commands and combining tables:

Here is a minimal code example for calculating iHS on a single chromosome:

hh <-                  # data input
  data2haplohh(
    hap_file = "bta12_cgu.hap",
    map_file = "map.inp",
    chr.name = "12",
    allele_coding = "map"
  )
scan <- scan_hh(hh)    # calculation of EHH and integration
                       # (combine results from different chromosomes)
ihs <- ihh2ihs(scan)   # log ratio for alleles and standardization
manhattanplot(ihs)     # plot of the statistics

Data input {#input}

The package rehh requires as input:

and, if the haplotype data file is neither in variant call format nor in the format of ms output,

Since R can automatically recognize and read compressed files with standard name extensions like .gz or .bzip2, this holds as well for the input files that can be used with this package.

Haplotype data file {#hapfile}

Five haplotype input file formats are supported:

Alleles in standard or transposed haplotype format can be provided either coded (by integer numbers) or without coding (e.g. as nucleotides) (see section \@ref(LoadData)).

If alleles are not polarized, i.e. their ancestral/derived status is not known, some arbitrary assignation can be done either beforehand by the user or during input by the package; appropriate parameters must be set in subsequent functions (see section \@ref(polarizing)).

Marker information file {#mapfile}

For haplotype data files in standard, transposed or fastphase format a marker information file has to be provided. The file should contain columns without header corresponding to:

  1. marker name/identifier
  2. name of the chromosome (or scaffold)
  3. position on the chromosome (or scaffold), either physical (in bases) or genetic (e.g. in centiMorgans).

For a given chromosome, the markers must be ordered in the same way as in the haplotype data file.

If alleles in the haplotype file are not already provided in coded form (i.e. as positive integers), the package can (re)code them using two additional columns in the marker file:

  1. ancestral allele
  2. derived allele(s).

For instance, the provided file map.inp has five columns:

# show first 6 lines of file
cat(readLines("map.inp", n = 6), sep = "\n")

Loading data files: the function data2haplohh {#LoadData}

The data2haplohh() function converts data files into an R object of class haplohh, forming the basis for subsequent analysis. The following options are available to recode alleles and/or to filter markers and haplotypes:

1. Allele coding options:

The parameter allele_coding has four options: three of them expect the user to provide a coding, while the fourth option overwrites any user defined coding.

The package accepts two different integer codings within the haplotype file which can be specified as "12" or "01", respectively:

| | "12" | "01" | |-----------------|:----:|:----------:| |Missing value | 0 | NA or . | |Ancestral allele | 1 | 0 | |Derived allele 1 | 2 | 1 | |Derived allele 2 | 3 | 2 | |... | ... | ... |

The latter one is also the coding used internally by our package.

If the alleles are given in an un-coded form (e.g. as nucleotides), the alleles can be coded for each marker separately with help of the marker information file. This option is activated by allele_coding = "map". If an allele of a marker is found in the fourth column of the marker information file, it is re-coded as 0 (ancestral) and if it is found in the fifth column it is re-coded as 1 (derived). The fifth column may contain several alleles, separated by commas, like in the variant call format. In this case, the internal coding extends to higher integers. If an allele present in the haplotype file is not found neither in the fourth nor the fifth column of the marker information file, it is counted as NA (missing value).

If the alleles are not polarized, i.e. it is unknown which allele is ancestral and which derived, the option "none" should be selected. The alleles will then be coded in alpha-numeric order for each marker; the first becomes 0, the second 1, etc. Missing values must be given as NA or '.' while any other number, character or string will be counted as an allele. Options of subsequent analysis functions have to be set accordingly (see section \@ref(polarizing)).

2. Filtering options

2.1 Discard haplotypes with a high amount of missing data. If haplotypes contain missing data (presumably a rare case since most phasing programs allow imputing missing genotypes), it is possible to discard those with a fraction of less than min_perc_geno.hap of markers genotyped. Its default is NA, hence no restriction.

2.2. Discard markers with a high amount of missing data. It is possible to discard individual markers genotyped on a fraction of less than min_perc_geno.mrk of haplotypes. By default min_perc_geno.mrk = 100 meaning that only fully genotyped markers are retained. A value of NA or zero is not allowed since rehh cannot handle markers with no data.

2.3 Discard markers with a low Minor Allele Frequency. It is possible to discard markers with a Minor Allele Frequency (MAF) equal to or below min_maf. Its default is NA meaning no constraint. Setting this value to zero eliminates monomorphic markers. Note that since low-frequency alleles usually abound, any positive value may exclude a substantial fraction of the data.

The arguments min_perc_geno.hap, min_perc_geno.mrk and min_maf are evaluated in this order.

More details about input options can be found in the documentation using the command:

?data2haplohh

Example 1: reading haplotype file in standard format {#LoadDataEx1}

The following command converts the haplotype input file bta12_cgu.hap (in "standard" haplotype format) and marker information input file map.inp to an haplohh object named hh. Because the map file contains information for markers mapping to multiple chromosomes we have to specify by chr.name = 12 that the haplotype input file is about chromosome 12. Alleles are given as nucleotides in the haplotype file and coded with help of the map file by setting allele_coding = "map".

hh <- data2haplohh(hap_file = "bta12_cgu.hap",
                   map_file = "map.inp",
                   chr.name = 12,
                   allele_coding = "map")

If no value is specified for chr.name and more than one chromosome is detected in the map file, an error is produced:

hh <- data2haplohh(hap_file = "bta12_cgu.hap",
                   map_file = "map.inp",
                   allele_coding = "map")

Furthermore, the following message is prompted if the number of markers for the chromosome in the info file does not correspond to the one in the haplotype file (for instance when a wrong chromosome is specified):

hh <- data2haplohh(hap_file = "bta12_cgu.hap",
                   map_file = "map.inp",
                   chr.name = 18,
                   allele_coding = "map")

Example 2: reading haplotype file in transposed format (SHAPEIT2--like) {#LoadDataEx2}

If the haplotype input file is in "transposed" format (like bta12_cgu.thap), the option haplotype.in.columns has to be set to TRUE while all other parameters remain unaffected with respect to example 1. This is the only format which is not recognized automatically, but has to be explicitly declared by the user.

hh <- data2haplohh(hap_file = "bta12_cgu.thap",
                   map_file = "map.inp",
                   chr.name = 12,
                   allele_coding = "map",
                   haplotype.in.columns = TRUE)

Example 3: reading haplotype file in fastPHASE output format {#LoadDataEx3}

In this example, we use the fastPHASE output bta12_hapguess_switch.out as haplotype file. This format is recognized automatically. As before, we need to set chr.name = 12. Because haplotypes originated here from several populations (the fastPHASE option -u was used), it is necessary to select the population of interest (in our case "CGU") by its corresponding number in the file, hence popsel = 7.

hh <- data2haplohh(hap_file = "bta12_hapguess_switch.out",
                   map_file = "map.inp",
                   chr.name = 12,
                   popsel = 7,
                   allele_coding = "map")

If no value is specified for the popsel argument and more than one population is detected in the fastPHASE output file, an error is produced and the available population numbers printed:

hh <- data2haplohh(hap_file = "bta12_hapguess_switch.out",
                   map_file = "map.inp",
                   chr.name = 12,
                   allele_coding = "map")

Example 4: reading vcf files {#LoadDataEx4}

The function data2haplohh() checks automatically whether the specified haplotype input file is in variant call format (vcf). If this is the case, the parameters map_file and allele_coding are ignored. By default, the function tries to polarize (see section \@ref(polarizing)) the alleles of each marker using the ancestral allele, expected to be given by key AA of the INFO field.

Ancestral alleles are sometimes marked by upper case as "high confident" and by lower case as "low confident". The default setting capitalize_AA = TRUE lifts this distinction before polarization.

If the AA key is absent, the option polarize_vcf should be set to FALSE and the allele coding of the vcf file is directly used as internal coding.

If there is data for more than one chromosome in the file, the chromosome of interest has to be specified by chr.name. Since always the whole file is read in, it is advisable to split large data sets into chromosome-specific files.

In order to process vcf files, the package vcfR or the package data.table (which in turn needs R.utils to read compressed files) have to be installed. The parameter vcf_reader has to be set to either "vcfR" or "data.table".

In the file bta12_cgu.vcf.gz the ancestral allele was set as reference and hence no further polarizing is necessary.

hh <- data2haplohh(hap_file = "bta12_cgu.vcf.gz",
                   polarize_vcf = FALSE,
                   vcf_reader = "data.table")

Example 5: reading ms output {#LoadDataEx5}

The function data2haplohh() automatically checks whether the haplotype file is in the output format of the simulation program ms [@Hudson2002]. If this is the case, the parameters map_file and allele_coding are ignored. If the file contains several 'runs' (as referred to by the parameter nrep of ms), it is necessary to specify the number of the run in option chr.name. Note that always the whole file is read, so that it might be advisable to spread large simulations over separate files.

One argument of the data2haplohh function is specifically dedicated to ms output, although it works with other formats as well: ms gives chromosomal positions as fractions of the interval [0,1] and in order to obtain more realistic values, these positions can be multiplied by a factor, set by position_scaling_factor.

Note that ms output can contain multiple markers with the same (rounded) position, which rehh does not accept. In this case the numerical precision for chromosomal positions in the ms output should be increased (option -p of ms, option -oformat of msms).

Setting remove_multiple_markers to TRUE entails that from consecutive markers with the same position only the first one is retained and a warning containing the number of removed markers is printed. Note that this effectively transforms the "infinite sites model" used for simulations by ms into a "finite sites model".

In order to read the ms format, the package gap has to be installed. 

hh <- data2haplohh(hap_file = "ms.out",
                   chr.name = 2,
                   position_scaling_factor = 1000)

Subset data

Sometimes it might be of interest to restrict the computation of statistics to subsets of haplotypes or markers. This can be done with help of the function subset() and arguments select.hap and select.mrk. After subsetting, the same filters are applied as during data input from files in order to ensure that no marker has exclusively missing values and, optionally, to exclude markers which are monomorphic within the subset by setting min_maf to zero.

For example, to restrict the output of ms (see section \@ref(LoadDataEx5)) to the first five haplotypes and filter out monomorphic sites, we apply

hh_subset = subset(hh, select.hap = 1:5, min_maf = 0)

while the following command omits the first marker:

hh_subset = subset(hh, select.mrk = -1)

Computing EHH, EHHS and their "integrals" iHH and iES

Definition and computation

The (allele-specific) Extended Haplotype Homozygosity (EHH) {#EHH}

For an allele $a$ of a focal marker $s$, sometimes referred to as a core allele, the Extended Haplotype Homozygosity (EHH) is defined as the probability that two randomly chosen chromosomes, carrying the core allele, are homozygous over a given surrounding chromosomal region [@Sabeti2002]. It is estimated from a sample by calculating the homozygosity of the chromosomal chunk between the focal marker and another marker $t$ by the formula \begin{equation} \mathrm{EHH}{s,t}^a=\frac{1}{n{a}(n_a-1)}\sum\limits_{k=1}^{K^a_{s,t}}n_k(n_k-1) (#eq:ehh) \end{equation} where $n_a$ represents the number of chromosomes carrying the core allele $a$, $K^a_{s,t}$ represents the number of different shared haplotypes and $n_k$ refers to the number of chromosomes pertaining to the $k$-th such shared haplotype. If there is no missing data, it holds that $n_a=\sum\limits_{k=1}^{K^a_{s,t}}n_k$.

In the case of unphased chromosomes from diploid individuals (see section \@ref(phasing)) the extended haplotype homozygosity can be calculated as follows [@Tang2007]: we consider only chromosomes from individuals that are homozygous for the allele $a$ at the focal marker $s$ and estimate EHH at some marker $t$ by the fraction of individuals that are (still) homozygous over the entire chromosomal stretch between $s$ and $t$. Let $I^a_{s,t}$ denote the number of individuals that are homozygous from marker $s$ til marker $t$. \begin{equation} \mathrm{EHH}{s,t}^a=\frac{I{s,t}^a}{I_{s,s}^a}\;. (#eq:ehh2) \end{equation}

EHH is usually computed only for a region it surpasses a given threshold (e.g., $EHH > 0.05$).

The integrated EHH (iHH) {#iHH}

By definition, EHH starts at 1 and decays to 0 with increasing distance of t from the focal marker s. For a given core allele, the integrated EHH (iHH) is defined as the area under the EHH curve which, in turn, is defined by the EHH values and associated chromosomal positions [@Voight2006]. The integral is computed with a simple numerical method, called the trapezoidal rule.

Note that, technical details aside, the iHH value is nothing else than the average length of shared haplotypes.

The (site-specific) Extended Haplotype Homozygosity (EHHS) {#EHHS}

An extended haplotype homozygosity can be defined as well without regard to core alleles. In this case, the quantity is aimed to reflect the probability that any two randomly chosen chromosomes from a population are homozygous over a given surrounding chromosomal region of a focal marker. In contrast to the allele-specific EHH defined in the previous section, the chromosomes are not required to carry a specific allele at the focal marker. We adopt the naming by [@Tang2007] as site--specific EHH, abbreviated by EHHS. Note, however that this quantity is sometimes referred to as EHH, too, and there is no agreed notation in the literature.

EHHS was used in genome scans in two versions: un-normalized by [@Sabeti2007] and normalized by [@Tang2007].

In line with [@Sabeti2007] we define \begin{equation} (#eq:ehhssab) \mathrm{EHHS}{s,t}=\frac{1}{n_s(n_s-1)}\sum\limits{k=1}^{K_{s,t}}n_k(n_k-1) \end{equation} where we re-use notation from above and let $n_s$ refer to the number of chromosomes at marker $s$. If there are no missing values at that marker, this is simply the number of chromosomes in the sample.

[@Tang2007] proposed an apparently different estimator for the normalized EHHS, which we abbreviate to nEHHS, namely \begin{equation} (#eq:ehhstang) \mathrm{nEHHS}{s,t}=\frac{1-h{s,t}}{1-h_s} \end{equation} where:

  1. $h_s=\frac{n_s}{n_s-1}\left(1-\frac{1}{n_s^2}\left(\sum\limits_{k=1}^{K_{s,s}}n_k^2 \right )\right)$ is an estimator of the focal marker heterozygosity. In the bi-allelic case we can write this as $h_s=\frac{n_s}{n_s-1}\left(1-\frac{1}{n_s^2}\left(n_{a1}^2+n_{a2}^2\right) \right)$ with $a1$ and $a2$ referring to the numbers of the two alleles at marker $s$ ($n_{a1}+n_{a2}=n_s$).

  2. $h_{s,t}=\frac{n_s}{n_s-1}\left(1-\frac{1}{n_s^2}\left(\sum\limits_{k=1}^{K_{s,t}}n_k^2 \right )\right)$ is an estimator of haplotype heterozygosity in the chromosomal region extending from marker $s$ to marker $t$.

However both definitions are in fact equivalent, because it holds $\mathrm{EHHS}{s,t}=1-h{s,t}$ and hence \begin{equation} \mathrm{nEHHS}{s,t}=\frac{\mathrm{EHHS}{s,t}}{\mathrm{EHHS}_{s,s}}\;. \end{equation} Thus $\mathrm{nEHHS}_{s,t}$ is just normalized in order to yield 1 at the focal marker $s$. Note that the normalization factor depends on the frequency of the alleles at the focal marker and consequently is not necessarily the same for different focal markers.

Furthermore, we note that EHHS and EHH are related by \begin{equation} \mathrm{EHHS}{s,t}=\frac{n{a1}(n_{a1}-1)}{n_s(n_s-1)}\mathrm{EHH}{s,t}^{a1}+\frac{n{a2}(n_{a2}-1)}{n_s(n_s-1)}\mathrm{EHH}^{a2}_{s,t}\;, \end{equation} where for the sake of simplicity we assume that the focal marker has only two alleles $a1$ and $a2$. EHHS might hence be viewed as a linear combination of the EHH's of the focal alleles, weighted by roughly the square of the focal allele frequencies.

In the case of unphased chromosomes from diploid individuals (see section \@ref(phasing)) EHHS can be calculated like EHH in Equation \@ref(eq:ehh2), just without reference to core alleles: \begin{equation} \mathrm{EHHS}{s,t}=\frac{I{s,t}}{I_{s,s}}\;. (#eq:ehhs2) \end{equation} Note that defined this way, EHHS is always 1 at the focal marker. Hence there is no distinction between $\mathrm{EHHS}$ and $\mathrm{nEHHS}$.

Again, unphased EHHS can be related to unphased EHH by \begin{equation} \mathrm{EHHS}{s,t}=\frac{I{s,t}}{I_{s,s}}=\frac{I_{s,t}^{a1}+I_{s,t}^{a2}}{I_{s,s}^{a1}+I_{s,s}^{a2}}=\frac{I_{s,s}^{a1}}{I_{s,s}^{a1}+I_{s,s}^{a2}}\mathrm{EHH}{s,t}^{a1}+\frac{I{s,s}^{a2}}{I_{s,s}^{a1}+I_{s,s}^{a2}}\mathrm{EHH}_{s,t}^{a2}\; \end{equation} where for the sake of simplicity we assumed a bi-allelic focal marker with alleles $a1$ and $a2$.

As with EHH, the EHHS is usually computed only for the region where its value surpasses a given threshold (e.g., $EHHS > 0.05$).

The integrated EHHS (iES) {#iES}

Like EHH, EHHS has its maximum at the focal marker and decays to 0 with increasing distance from the focal marker. For a given focal marker, analogously to iHH, iES is defined as the integrated EHHS [@Tang2007]. Depending on whether EHHS or nEHHS is integrated, we yield iES and inES respectively. As with iHH, the numerical integration uses the trapezoidal rule.

Note that, technical details aside, the iES and inES values represent the average length of shared haplotypes. The length of shared haplotypes with different core alleles yields zero and these are included in the former but not the latter.

The function calc_ehh() {#calcehh}

The function calc_ehh() computes EHH for all alleles of a focal marker $s$ relative to markers $t$ upstream and downstream. For each allele the corresponding integral iHH of the EHH curve is calculated as well.

Three options can be specified to constrain the computation of EHH:

Several parameters influence the IHH values (the integral over EHH):

The option polarized, TRUE by default, in this function merely affects the order and labeling of alleles.

The parameter phased can be toggled to FALSE in order to calculate EHH by the formula for unphased data (see section \@ref(EHH) and \@ref(phasing)).

The option include_zero_values concerns only the output. If FALSE (by default), the output contains only EHH values for markers up to the specified cut-offs. If the parameter is toggled to TRUE, then values for all markers are returned, the remaining ones being zero.

Details are available in the R documentation by using the command:

?calc_ehh

In the following example, EHH is computed around the SNP F1205400. This SNP is associated with a strong signal of selection (using both iHS and Rsb statistics) and is located closely (<5kb) to a strong candidate gene involved in horn development [@Gautier2011].

Note that the R object haplohh_cgu_bta12 was generated using the data2haplohh() function with the example input files (see section \@ref(LoadDataEx1)). 

#example haplohh object (280 haplotypes, 1424 SNPs) see ?haplohh_cgu_bta12 for details
data(haplohh_cgu_bta12)
#computing EHH statistics for the focal SNP with name "F1205400"
#which displays a strong signal of selection
res <- calc_ehh(haplohh_cgu_bta12, 
                mrk = "F1205400", 
                include_nhaplo = TRUE)

The output contained in res is a list with four elements:

  1. mrk.name: the name/identifier of the focal marker.
  2. freq: a vector containing the frequencies of all alleles of the focal marker.
  3. ehh: a data frame containing the EHH values for each allele of the focal marker. Setting the parameter include_nhap to TRUE adds columns which show how many chromosomes contributed at each position to the calculation of EHH.
  4. ihh: a vector with iHH for each allele, i.e. the integrals over the EHH curves defined by the EHH values.
res$mrk.name

res$freq

res$ehh

res$ihh

The EHH values can be plotted by setting the output of calc_ehh() into the function plot() to yield Figure \@ref(fig:ehh):

plot(res)

For comparison, we might ignore that haplotypes have been phased and set the option phased to FALSE. Figure \@ref(fig:ehh2) shows that for this particular marker the differences are rather small: 

plot(calc_ehh(haplohh_cgu_bta12, 
              mrk = "F1205400", 
              phased = FALSE),
     xlim = c(2.55E7, 3.05E7))

The function calc_ehhs()

The calc_ehhs() function computes EHHS and normalized EHHS around the focal marker $s$ relative to another marker $t$. This function also computes the corresponding integrals iES and inES respectively. The options are identical to those of the function calc_ehh (see previous section), except that polarized is absent, because variant ancestry status does not figure in the formulas. Details are available by the command:

?calc_ehhs

In the following example, the EHHS statistics are computed around the SNP F1205400.

data(haplohh_cgu_bta12)
res <- calc_ehhs(haplohh_cgu_bta12, 
                 mrk = "F1205400", 
                 include_nhaplo = TRUE)

The output is similar to that of calc_ehh(), except that there are no alleles to be distinguished, but instead the wether EHHS is normalized or not. A list with four elements is obtained:

  1. mrk.name: the name/identifier of the focal marker.
  2. ehhs: a data frame with EHHS and nEHHS values along the chromosome around the focal marker. Optionally, the column NHAPLO can be included to show how many chromosomes were evaluated at each marker.
  3. IES: iES, i.e. the integral over the EHHS curve.
  4. INES: inES, i.e. the integral over the nEHHS curve.
res$mrk.name

res$ehhs 

res$IES

res$INES

Figure \@ref(fig:ehhs) can be obtained by setting the output of calc_ehhs into the function plot.ehhs():

plot(res)

By default, the un-normalized EHHS values are plotted. This can be toggled by setting nehhs = TRUE. As can be seen in Figure \@ref(fig:nehhs), the values are scaled to yield 1 at the focal marker. 

plot(res, nehhs = TRUE)

The parameter phased of the function calc_ehhs() can be toggled to FALSE yielding Figure \@ref(fig:ehhs2). As described above, for unphased data the EHHS values are always normalized. For this particular marker we get a similar picture as for phased nEHHS.

plot(calc_ehhs(haplohh_cgu_bta12,
               mrk = "F1205400",
               phased = FALSE))

The function scan_hh() {#scanhh}

While the functions calc_ehh() and calc_ehhs() return the integrated statistics iHH and iES for a particular marker, the function scan_hh() calculates these values for all markers in the haplohh object. However, in contrast to calc_ehh() which computes an iHH value for each allele of a possibly multi-allelic focal marker, scan_hh() calculates only two iHH values, namely (by default) for the ancestral allele and the derived allele with highest frequency. In case of unpolarized markers (see section \@ref(polarizing)), the option polarized should be set to FALSE, causing the function to use major and minor (most frequent and second most frequent) alleles instead.

The remainder of options are essentially the same as for the functions calc_ehh() and calc_ehhs() (see section \@ref(EHH)). Details are given in the documentation:

?scan_hh

As an example, in order to scan the haplohh_cgu_bta12 object (containing data on 1424 SNPs for 280 haplotypes), one may use the following command:

data(haplohh_cgu_bta12)
scan <- scan_hh(haplohh_cgu_bta12)

The resulting object scan is a data frame with r nmrk(haplohh_cgu_bta12) rows (the number of markers declared in the haplohh object) and 10 columns:

  1. chromosome name
  2. position in the chromosome
  3. sample frequency of the ancestral / major allele
  4. sample frequency of the second-most frequent remaining allele
  5. number of evaluated haplotypes at the focal marker for the ancestral / major allele
  6. number of evaluated haplotypes at the focal marker for the second-most frequent remaining allele
  7. iHH of the ancestral / major allele
  8. iHH of the second-most frequent remaining allele
  9. iES
  10. inES

Note that while for phased data the number of evaluated haplotypes of a core allele corresponds to its frequency in the sample, in case of unphased data the evaluation is restricted to haplotypes of homozygous individuals.

The following command prints a chunk of the scan containing the marker F1205400:

scan[453:459,]

Note that scan_hh() is more efficient than calc_ehh() and calc_ehhs() applied consecutively for each marker as can be seen by running the two code snippets below:

# perform scan using scan_hh
system.time(scan <- scan_hh(haplohh_cgu_bta12))

# perform scan applying calc_ehh and calc_ehhs to each marker
slow_scan_hh <- function(haplohh) {
  # create empty vectors of size nmrk
  IHH_A <- IHH_D <- IES <- INES <- vector("numeric", nmrk(haplohh))
  # invoke calc_ehh and calc_ehhs for each marker
  for (i in 1:nmrk(haplohh)) {
    res <- calc_ehh(haplohh, mrk = i)
    IHH_A[i] <- res$ihh["IHH_A"]
    IHH_D[i] <- res$ihh["IHH_D"]
    res <- calc_ehhs(haplohh, mrk = i)
    IES[i] <- res$IES
    INES[i] <- res$INES
  }
  # create data frame (the return value of this function)
  data.frame(IHH_A = IHH_A, 
             IHH_D = IHH_D,
             IES = IES,
             INES = INES)
}
system.time(slow_scan <- slow_scan_hh(haplohh_cgu_bta12))

Comparing columns shows that the computed values are identical:

identical(slow_scan[, "IHH_A"], scan[, "IHH_A"])
identical(slow_scan[, "IHH_D"], scan[, "IHH_D"])
identical(slow_scan[, "IES"], scan[, "IES"])
identical(slow_scan[, "INES"], scan[, "INES"])

Computing iHS, Rsb and XP-EHH

The iHS within-population statistic

Definition {#ihs}

The abbreviation iHS refers to "integrated haplotype homozygosity score". Let uniHS represent the un-standardized log-ratio of ancestral iHH$^A$ to derived iHH$^D$ of a certain focal marker $s$: $$\mathrm{uniHS}(s)=\ln\left(\frac{\mathrm{iHH}^A(s)}{\mathrm{iHH}^D(s)}\right)$$ Following [@Voight2006] we perform a standardization by setting: $$\mathrm{iHS}(s)=\frac{\mathrm{uniHS}(s) - \mathrm{mean}(\mathrm{uniHS}|p_s)}{\mathrm{sd}(\mathrm{uniHS}|p_s)}$$ where $\mathrm{mean}(\mathrm{uniHS}|p_s)$ and $\mathrm{sd}(\mathrm{uniHS}|p_s)$ represent the mean and standard deviation of uniHS restricted to those markers with a similar derived allele frequency as observed at the focal marker ($p_s$). In practice, the markers are binned so that each bin contains not a single frequency but a small range of frequencies to yield enough markers for reliable estimates of mean and standard deviation.

iHS is constructed to have an approximately standard Gaussian distribution and to be comparable across markers regardless of their underlying allele frequencies.

For a practical characterization of "outliers" we stretch statistical notation and associate a p-value to iHS [@Gautier2011] by: $$p_\mathrm{iHS}=-\log_{10}\left(2\Phi\left(-|\mathrm{iHS}|\right)\right)$$ where $\Phi\left(x\right)$ represents the Gaussian cumulative distribution function.

Assuming most of genotyped markers behave neutrally and the distribution of iHS values for neutral markers follows indeed a standard Gaussian distribution, we might interpret $p_\text{iHS}$ as a two-sided p-value (on a $-\log_{10}$ scale) of a test on the null hypothesis of no selection.

Alternatively, a one-sided p-value (in a $-\log_{10}$ scale) might be defined [@Gautier2011] as: \begin{equation} p^\text{left}\text{iHS}=-\log{10}\left(\Phi\left(\text{iHS}\right)\right) \end{equation} allowing the identification of those sites where the derived allele displays a significantly higher extended haplotype homozygosity than the ancestral allele or \begin{equation} p^\text{right}\text{iHS}=-\log{10}\left(1-\Phi\left(\text{iHS}\right)\right) \end{equation} for the opposite case.

Note that this procedure is controversial, because we identify the empirical distribution with the distribution under the null hypothesis of neutrality. This is an approximation at best and only warranted when it can be assumed that there are so few selected sites that their influence on the overall shape of the distribution can be neglected.

In case of unpolarized alleles, the uniHS is taken as the ratio of iHH from minor to major allele. Since derived allele frequency cannot be accounted for, no binning should be performed. The resulting standardized iHS cannot be expected to follow a normal distribution and p-values as defined above are not meaningful.

The function ihh2ihs() {#ihh2ihs}

The ihh2ihs() function computes iHS using a data frame containing the iHH values for ancestral and derived (resp. major and minor) alleles as obtained by the scan_hh() function (see section \@ref(scanhh)). The argument min_maf allows to exclude focal markers according to their minor allele frequency (by default min_maf=0.05). The argument freqbin controls the size (or number) of the allele frequency bins used to perform standardization (see section \@ref(ihs)). More precisely, allele frequency bins are built from min_maf to 1-min_maf in steps of size freqbin (by default freqbin=0.025). If an integer of 1 or greater is specified, a corresponding number of equally spaced bins is created. If freqbin is set to 0, standardization is performed considering each observed frequency as a discrete frequency class, which is useful when there are only a few distinct haplotypes.

For unphased data, iHH is calculated using only haplotypes of individuals which are homozygous at the focal marker. This number can be considerably lower than the absolute allele frequency. Hence, in addition to min_maf, the option min_nhaplo (default NA) should be used to reduce statistical noise arising from too few evaluated haplotypes.

Optionally, the allele frequencies of the input data frame can be included into the output by setting include_freq to TRUE.

A p-value is calculated for standardized iHS values. By default, it is two-sided, but a side can be chosen by setting argument p.side to "left" or "right".

As a typical workflow for performing a whole genome scan one might run scan_hh() on haplotype data from each chromosome and concatenate the resulting data frames before standardization. In the following example, we assume that the haplotype files are named as hap_chr_i.cgu with $i=1,...,29$ and the marker information file is named map.inp. The data frame wgscan contains the $iHH^A$ and $iHH^D$ values for the whole genome and serves as input for the ihh$ihs() function which calculates iHS, i.e. the standardized log ratio of the two $iHH$ values, for each marker.

## demo code - no data files for all chromosomes provided
for(i in 1:29) {
  # haplotype file name for each chromosome
  hap_file = paste("hap_chr_", i, ".cgu", sep = "")
  # create internal representation
  hh <- data2haplohh(hap_file = hap_file,
                     map_file = "map.inp", 
                     chr.name = i,
                     allele_coding = "map")
  # perform scan on a single chromosome (calculate iHH values)
  scan <- scan_hh(hh)
  # concatenate chromosome-wise data frames to
  # a data frame for the whole genome
  # (more efficient ways certainly exist...)
  if (i == 1) {
    wgscan <- scan
  } else {
    wgscan <- rbind(wgscan, scan)
  }
}
# calculate genome-wide iHS values
wgscan.ihs <- ihh2ihs(wgscan)

For illustration, the object wgscan.cgu contains the calculated and concatenated $iHH$ values of a whole genome scan on population CGU [@Gautier2011]:

library(rehh.data)
data(wgscan.cgu)
wgscan.ihs.cgu <- ihh2ihs(wgscan.cgu)

The resulting object wgscan.ihs.cgu is a list with two elements:

  1. ihs: a data frame with iHS and corresponding p-value $p_\mathrm{iHS}$ (p-value in a $-\log_{10}$ scale assuming the iHS are normally distributed under the neutral hypothesis). The first rows are displayed below:
head(wgscan.ihs.cgu$ihs)
  1. frequency.class: a data frame summarizing the derived allele frequency bins used for standardization and mean and standard deviation of the un-standardized values. The first rows are shown below:
head(wgscan.ihs.cgu$frequency.class)

The Rsb pairwise population statistic

Definition {#rsb}

The abbreviation Rsb stands for "ratio of EHHS between populations". For a given marker $s$, let $$\mathrm{unRsb}(s)=\ln\left(\frac{\mathrm{inES}\text{pop1}(s)}{\mathrm{inES}\text{pop2}(s)}\right)$$ represent the log-ratio of the inES values computed in two populations $pop1$ and $pop2$ (see section \@ref(iES)). The Rsb for marker $s$ is defined as the standardized unRsb [@Tang2007]:

\begin{equation} \mathrm{Rsb(s)}=\frac{\mathrm{unRsb}(s) - \text{median}({\mathrm{unRsb}})}{\mathrm {sd}({\mathrm{unRsb})}} \end{equation} where $\text{median}(\mathrm{unRsb})$ and $\mathrm{sd}(\mathrm{unRsb})$ represent the median and standard deviation of unRsb computed over all analyzed markers. Note that for the sake of uniformity, we included the logarithm into the definition of the quantity while in the original article it remained outside. However, we follow [@Tang2007] in using the median instead of the mean (hence unlike the definitions of iHS and XP-EHH). The authors argue that this might increase the robustness against different demographic scenarios.

Like XP-EHH, but unlike $iHS$, no binning with respect to allele frequencies is performed (see section \@ref(polarizing)).

As iHS (see section \@ref(ihs)), Rsb is constructed to have an approximately standard Gaussian distribution and may be transformed analogously into a p-value $p_\text{Rsb}$ (in a $-\log_{10}$ scale), either two-sided or one-sided. As for the latter, $p^\text{left}\text{Rsb}$ identifies strong extended homozygosity in population $pop2$ relative to population $pop1$ and vice versa for $p^\text{right}\text{Rsb}$.

The function ines2rsb() {#ines2rsb}

The ines2rsb() function computes Rsb using two data frames containing each the inES statistics of one population as obtained by the scan_hh() function (see section \@ref(scanhh)).

As with iHS the inES values have to be calculated chromosome-wise and concatenated afterwords (for each population separately) like in the example of section \@ref(ihh2ihs). The Rsb values are computed for all markers present in both data frames. They are identified by chromosome name and position, not their identifiers, thus within each population the chromosomal positions must be unique.

Optionally, the allele frequencies of both populations can be taken over from the input data frames (if present there) to the output data frame by setting include_freq to TRUE.

For illustration, we take calculated and concatenated $inES$ values from a genome scan [@Gautier2011] provided as example data and compute for each SNP the Rsb between the CGU and EUT populations:

data(wgscan.cgu) ; data(wgscan.eut)
## results from a genome scan (44,057 SNPs) see ?wgscan.eut and ?wgscan.cgu for details
rsb.cgu_eut <- ines2rsb(scan_pop1 = wgscan.cgu,
                        scan_pop2 = wgscan.eut,
                        popname1 = "CGU",
                        popname2 = "EUT")

The resulting object rsb.cgu_eut is a data frame which shows for each marker its Rsb and associated p-value. The latter are by default two-sided, but a side can be chosen by setting argument p.side to "left" or "right". The first rows of the resulting data frame are printed below:

head(rsb.cgu_eut)

The XP-EHH pairwise population statistic

Definition

The XP-EHH (cross-population EHH) statistic [@Sabeti2007] is similar to Rsb except that it is based on iES instead of inES (see section \@ref(iES)). Hence, for or a given marker $s$, let $$\text{unXP-EHH}(s)=\ln\left(\frac{\mathrm{iES}\text{pop1}(s)}{\mathrm{iES}\text{pop2}(s)}\right)$$ represent the log-ratio of the iES values computed by the function scan_hh() in the populations $pop1$ and $pop2$ (see section \@ref(iES)).

The XP-EHH for a given focal marker is defined as the standardized unXP-EHH [@Sabeti2007]:

\begin{equation} \text{XP-EHH}(s)=\frac{\text{unXP-EHH}(s) - \text{mean}(\text{unXP-EHH})}{\text{sd}(\text{unXP-EHH})} \end{equation} where $\text{mean}(\text{unXP-EHH})$ and $\text{sd}(\text{unXP-EHH})$ represent the mean and standard deviation of unXP-EHH computed over all analyzed markers.

As with Rsb, the information about the ancestral and derived status of alleles at the focal marker does not figure in the formula and no binning is performed.

As with the previous two scores iHS (section \@ref(ihs)) and Rsb (section \@ref(rsb)), XP-EHH is constructed to have an approximately standard Gaussian distribution and may be transformed analogously into a p-value $p_\text{XP-EHH}$ (in a $-\log_{10}$ scale), either two-sided or one-sided. Among the latter, $p^\text{left}\text{XP-EHH}$ identifies strong extended homozygosity in population $pop2$ relative to population $pop1$ and vice versa for $p^\text{right}\text{XP-EHH}$.

The function ies2xpehh() {#ies2xpehh}

The ies2xpehh() function computes XP-EHH using two data frames, one for each population, containing the iES statistics as obtained by the scan_hh() function (see section \@ref(scanhh)).

As with iHS and Rsb, the iES have to be calculated chromosome-wise and concatenated afterwords (for each population separately) as has been shown in section \@ref(ihh2ihs). The XP-EHH values are computed for all markers present in both data frames. They are identified by chromosome name and position, not their identifiers, thus within each population the chromosomal positions must be unique.

Optionally, the allele frequencies in both populations can be taken over from the input data frames (if present there) to the output data frame by setting include_freq to TRUE.

For illustration, we take calculated and concatenated $iES$ values from a genome scan [@Gautier2011] and compute for each SNP the XP-EHH between the CGU and EUT populations:

data(wgscan.cgu) ; data(wgscan.eut)
## results from a genome scan (44,057 SNPs) see ?wgscan.eut and ?wgscan.cgu for details
xpehh.cgu_eut <- ies2xpehh(scan_pop1 =  wgscan.cgu,
                           scan_pop2 =  wgscan.eut,
                           popname1 = "CGU",
                           popname2 = "EUT")

The resulting object xpehh.cgu_eut is a data frame containing for each SNP the XP-EHH and associated p-value. The latter are by default two-sided, but a side can be chosen by setting argument p.side to "left" or "right".

The first rows are displayed below:

head(xpehh.cgu_eut)

Delineating regions with "outliers": the function calc_candidate_regions()

In contrast to neutral evolution, selection tends to produce clusters of markers with outlier values [@Tang2007]. For this reason, typically not (only) the markers with outlier scores are reported, but intervals with a conspicuous number or percentage of such markers. Although often similar ad-hoc procedures can be found in the literature, their respective parameters vary widely. We offer the function calc_candidate_regions() which scans through the genome in (possibly overlapping) sliding windows and identifies those as "candidates" which fulfill three conditions:

Markers with extreme values are those with a score greater than specified by parameter threshold. Apart from the values of the above conditions, the average of all marker scores and the average of extremal scores is reported for each candidate window.

By default, neighboring and overlapping candidate windows are joined together and the associated values recalculated. This can be turned off by setting join_neighbors to FALSE.

If window_size is set to 1, only the set of extremal markers is returned.

B default, the score itself is tested and reported. Toggling pval to TRUE, uses the associated p-value instead.

cr.cgu <- calc_candidate_regions(wgscan.ihs.cgu,
                                 threshold = 4,
                                 pval = TRUE,
                                 window_size = 1E6,
                                 overlap = 1E5,
                                 min_n_extr_mrk = 2)
cr.cgu

\clearpage

Visualization of the statistics

Un-standardized iHS: the function freqbinplot() {#freqbinplot}

Since mutations and recombinations tend to shorten stretches of homozygosity between chromosomes, an ancestral (older) allele is expected to have lower extended haplotype homozygosity than a derived (younger) allele. Hence the distribution of unstandardized $iHS=\ln(\frac{iHH^A}{iHH^D})$ is biased towards negative values. More importantly, there is a additional dependency on both ancestral and derived allele frequencies (in the bi-allelic case one is determined by the other). This can be seen by setting the output of the function ihh2ihs into the function freqbinplot() yielding Figure \@ref(fig:freqbin).

freqbinplot(wgscan.ihs.cgu)

Note that this frequency dependence is expected under neutral evolution and not due to selection. In fact, an implicit assumption of the bin-wise standardization is that each bin is dominated by neutral markers. Thus, the bin-wise standardization reduces the dependency of iHS on the allele frequency; see section \@ref(polarizing) of this vignette and Figure 4 in [@Voight2006] [In their case the effect is even more pronounced, because they polarized alleles with help of an outgroup species while in the case of our data it was done using extant ancestral populations as a proxy.]

Rsb vs. XP-EHH comparison

Figure \@ref(fig:comp) shows a plot of Rsb against XP-EHH values across the CGU and EUT populations (see section \@ref(ines2rsb) and \@ref(ies2xpehh) respectively). Marked in red is the marker that was repeatedly used in the examples above. The plot was generated using the following R code:

plot(rsb.cgu_eut[, "RSB_CGU_EUT"],
     xpehh.cgu_eut[, "XPEHH_CGU_EUT"],
     xlab = "Rsb",
     ylab = "XP-EHH",
     pch = ".",
     xlim = c(-7.5, 7.5),
     ylim = c(-7.5, 7.5))
# add red circle for marker with name "F1205400"
points(rsb.cgu_eut["F1205400", "RSB_CGU_EUT"],
       xpehh.cgu_eut["F1205400", "XPEHH_CGU_EUT"],
       col = "red")
# add dashed diagonal
abline(a = 0, b = 1, lty = 2)

Distribution of standardized values: the function distribplot()

The distribplot() function allows to visualize the distributions of the standardized scores (either iHS, Rsb or XP-EHH) and compare them to the standard Gaussian distribution. As an example, in Figure \@ref(fig:distribplot) the function was applied on the iHS estimates obtained for the CGU population (see section \@ref(ihs)):

distribplot(wgscan.ihs.cgu$ihs$IHS, xlab = "iHS")

Setting the option qqplot to TRUE toggles the output to a qqplot, as in Figure \@ref(fig:qqplot). 

distribplot(wgscan.ihs.cgu$ihs$IHS, 
            xlab = "iHS", 
            qqplot = TRUE)

Genome wide score plots: the function manhattanplot()

The function manhattanplot() draws a Manhattan plot of the whole genome scores obtained by the functions ihh2ihs(), ines2rsb() or ies2xpehh(). Figure \@ref(fig:manhattanplot) was drawn using the command:

manhattanplot(wgscan.ihs.cgu,
              main = "iHS (CGU cattle breed)")

If the option pval is switched to TRUE the associated p-value instead of the score itself is plotted (Figure \@ref(fig:pval)).

manhattanplot(wgscan.ihs.cgu,
              pval = TRUE,
              threshold = 4,
              main = "p-value of iHS (CGU cattle breed)")

The colors of the chromosomes in Figures \@ref(fig:manhattanplot) and \@ref(fig:pval) are the default colors of R as obtained by the command palette(). It is possible to change them by the same command. Note that the colors are associated with the order of chromosomes in the scan and not their order in the plot.

Candidate regions as obtained by the function calc_candidate_regions() can be added to the plot as parameter cr. Individual markers can be highlighted by setting argument mrk to a vector of marker IDs or a data frame with positions (containing columns with name CHR and POSITION).

By default, chromosomes are separated by an inset of 5,000,000 bases. This value can be increased by the corresponding parameter in order to further reduce overlap between data points of neighboring chromosomes.

In order to reduce the number of plotted data points, the data set can be rasterized in both dimensions by parameter resolution. The data points are then rounded to the specified resolution and duplicate points removed.

Furthermore, it is possible to specify a subset or a re-ordering of chromosomes with help of parameter chr.name as in Figure \@ref(fig:manhattanplotsub). 

# re-define colors
palette(c("red", "green"))
manhattanplot(wgscan.ihs.cgu, 
              pval = TRUE,
              threshold = 4, 
              chr.name = c("1", "4", "5", "12"), 
              main = "iHS (CGU cattle breed)", 
              cr = cr.cgu,
              mrk = "F1205400",
              inset = 1E+7,
              resolution = c(200000, 0.05))
# set back to default colors
palette("default")

Genome wide score plots: the function manhattan() of package qqman

The package qqman contains a function manhattan() which is similar to the function manhattanplot() of this package. The input data frame is expected to have a slightly different format, though. Hence, before plotting we need to "translate" our data as in the following example with iHS values:

# extract data frame from result list
ihs <- wgscan.ihs.cgu$ihs
# create new data frame
wgscan.cgu.ihs.qqman <- data.frame(
    CHR = as.integer(factor(ihs$CHR, 
                            levels = unique(ihs$CHR))),
                               # chromosomes as integers
    BP = ihs$POSITION,         # base pairs
    P = 10**(-ihs$LOGPVALUE),  # transform back to p-values
    SNP = row.names(ihs)       # SNP names
    )

We can now use the new data frame as input for the function manhattan() to obtain Figure \@ref(fig:qqman):

library(qqman)
qqman::manhattan(wgscan.cgu.ihs.qqman,
                 col = c("red","green"),
                 chrlabs = unique(ihs$CHR),
                 suggestiveline = 4,
                 highlight = "F1205400",
                 annotatePval = 0.0001)

Visualization of the haplotype structure

The function plot.haplohh()

Chunks of the haplotype matrix can be plotted, provided they are not too large, with sequences in rows and markers in columns. The following plot shows a chunk of 200 SNPs in the vicinity of SNP F1205400 which is identified by the dashed line. The derived alleles are marked in red, the ancestral ones not shown. 

hh_subset <- subset(haplohh_cgu_bta12, select.mrk = 350:550)
oldpar <- par(mar = c(3, 2, 2, 2) + 0.1)
plot(
  hh_subset,
  mrk = "F1205400",
  group_by_allele = TRUE,
  ignore.distance = TRUE,
  col = c(NA, "red"),
  linecol = c("lightblue", "lightpink"),
  mrk.col = "black",
  cex = 0.1,
  pos.lab.hap = "none",
  pos.lab.mrk = "none"
)
par(oldpar)

The sequences are ordered with respect to the alleles of SNP F1205400: the lower two thirds of sequences contain the derived allele and are more homogenous around the SNP than the sequences carrying the ancestral allele.

\clearpage

The functions calc_furcation() and plot.furcation()

Furcation structures portray the decay of haplotype homozygosity around a focal marker [@Sabeti2002]. They represent the complete information contained in the concept of extended shared haplotypes of which EHH and EHHS can be regarded as summary statistics. For each allele of a focal marker and each side ("left" or "right"), furcations arise when the extension step reaches a marker whose alleles distinguish hitherto identical haplotypes. In case of bi-allelic markers without missing values these can be only bifurcations. The furcation structure for a specific allele and a specific side is hence a tree with its root at the focal marker. A stark contrast between furcation structures of different core alleles should correspond to outlier values of iHS.

The function calc_furcation() calculates such a furcation structure around a focal marker. 

data(haplohh_cgu_bta12)
furcation <- calc_furcation(haplohh_cgu_bta12,
                            mrk = "F1205400")

The result is an object of class furcation, comprising a set of tree structures with some additional information, see ?"furcation-class" for technical details.

The function plot.furcation() renders graphically the furcation object. Within the plot, the root (focal marker) is identified by a vertical dashed line. The diagram is bi-directional, portraying decay along both sides of the focal marker. The thickness of the lines corresponds to the number of chromosomes sharing a haplotype.

As an illustration, Figure \@ref(fig:furc) shows the bifurcation diagrams for derived and ancestral allele of the marker F1205400. 

plot(furcation)

It is possible to zoom into furcations using parameter xlim and label the leaves using the parameter hap.names, as shown in Figure \@ref(fig:furczoom), produced by the following command. Labels in bold indicate that further chromosomes are associated with the corresponding branch.

plot(furcation,
     xlim = c(2.8E+7, 3E+7),
     lwd = 0.05,
     hap.names = hap.names(haplohh_cgu_bta12),
     cex.lab = 0.3)

Individual trees can be exported into Newick format. The function as.newick produces a (possibly very long) string in this format. The following command extracts the furcation tree of the ancestral allele on the left side of the marker:

newick <- as.newick(furcation,
                    allele = 0,
                    side = "left",
                    hap.names = hap.names(haplohh_cgu_bta12))

Such a string can be rendered graphically e.g. by the R package ape yielding Figure \@ref(fig:newick):

library(ape)
tree <- ape::read.tree(text = newick)
plot(tree, 
     cex = 0.5, 
     direction = "leftwards", 
     edge.color = "blue",
     underscore = TRUE,
     no.margin = TRUE)

The functions calc_haplen() and plot.haplen()

To each haplotype in the sample the length of its longest shared haplotype is assigned, i.e. the range over which it is identical to at least one other haplotype (the latter might be different left and right to the focal marker). It corresponds to the maximal extension of the inner branches to both sides of the focal marker in a furcation diagram. The function calc_haplen() calculates this quantity:

haplen <- calc_haplen(furcation)

The haplen object is a list with four elements:

  1. mrk.name: the name/identifier of the focal marker
  2. position: the position of the focal marker
  3. xlim: the position of first and last marker in the data set
  4. haplen: a data frame with begin and end positions of the extended shared haplotypes.
haplen$mrk.name

haplen$position

haplen$xlim

head(haplen$haplen)

Haplotype length is visualized in Figure \@ref(fig:haplo) obtained by the command

plot(haplen)

As with a furcation plot, one can zoom into the picture using the parameter xlim and select alleles using parameter allele. The following command yields Figure \@ref(fig:haplozoom) showing only the region to the "left" of the focal marker and only the chromosomes of the ancestral core allele. 

plot(haplen,
     allele = 0,
     xlim = c(haplen$xlim[1], haplen$position),
     hap.names = hap.names(haplohh_cgu_bta12),
     cex.lab = 0.3,
     legend.xy.coords = "none")

We give a small example on how to check visual results by directly accessing the data of the haplohh-class object (see also its documentation by ?"haplohh-class"): from Figures \@ref(fig:newick) or \@ref(fig:haplozoom) it appears that two extended shared haplotypes reach the left border and we might identify them by their labels as CGU_MN147_2 and CGU_MN153_2. We can prove that there are indeed exactly two haplotypes in the sample which are identical within the complete region to the "left" of the focal marker:

# finding the index number of marker "F1205400"
mrk.nr = which(mrk.names(haplohh_cgu_bta12) == "F1205400")
# subset of all markers on the "left" of the focal one
hh_left = subset(haplohh_cgu_bta12, select.mrk = 1:mrk.nr)
# check for duplicated rows
which(duplicated(haplo(hh_left)))
# row 248 is identical to a previous row, but which one?
# get the other row by a search in opposite direction
which(duplicated(haplo(hh_left), fromLast = TRUE))
# extract the corresponding haplotype names
hap.names(hh_left)[c(236, 248)]

Finally, after working through the vignette, we remove the example files from the current working directory:

remove.example.files()

\clearpage

Data considerations

Multi-allelic markers {#multiallelic}

For many species, a low per-site mutation rate ensures that the vast majority of Single Nucleotide Polymorphisms (SNPs) is observed with only two alleles in a sample. Hence bi-allelic SNPs will constitute the foremost kind of data to apply our package onto. However, we think the ability to calculate the statistics for multi-allelic markers might be useful

Dealing with gaps {#gaps}

Certain genomic regions such as centromeres are difficult to sequence and can give rise to large gaps between consecutive markers. If not accounted for, these will cause spuriously inflated values of the "integrals" iHH and iES. [@Voight2006] applied two corrections for gaps. First, they introduced a penalty proportional to physical distance that resulted in any gap being greater than 20 kb to be re-scaled to this number. Second, they discarded integration, if two consecutive markers with a distance greater than 200 kb were encountered. Both methods are implemented in rehh, yet turned off by default, since the corresponding thresholds should be adapted manually to the specific data set.

Dealing with missing data {#missing}

Most phasing programs (such as fastPHASE or SHAPEIT2) interpolate ("impute") missing genotypes, hence at least for phased SNP markers, missing values should be rather the exception than the rule.

In cases where missing values cannot be avoided, there is no generally accepted way of handling them. Some authors replace them probabilistically by the "background" frequency of alleles observed on the remainder chromosomes (i.e. a simple kind of "imputing"). We implemented instead the following approach: every time when the extension of shared haplotypes reaches a new marker, all chromosomes with missing values are removed from further calculations; exempt are chromosomes which at that point do not share any more a haplotype with another chromosome.

The number of chromosomes evaluated at each position is reported in the results of calc_ehh() resp. calc_ehhs(), if the parameter include_nhaplo is set to TRUE. If there are missing values, this number will decrease monotonically with increasing distance to the focal marker. Without missing values, the number will be constant until calculation is stopped, e.g. by reaching the boundary condition limehh(s).

Our approach to handle missing values has two non-obvious consequences: first, under certain circumstances, EHH or EHHS can transiently increase, namely if the removal of chromosomes leads accidentally to a more homogeneous set. Second, the normalization factor for nEHHS has to be recalculated based on the reduced set and is no longer constant over the entire region. For realistic data, however, both effects should be negligible.

Dealing with multiple markers at the same position {#multiplemarkers}

Errors or typos aside, there are several possibilities how multiple markers with the same chromosomal position can arise:

Since it is unclear how rehh should handle such markers, they are not accepted as input. Ideally, multiple markers should be dealt with by a pre-processing of the data outside of the package. As a quick-and-dirty work-around, we offer the option remove_multiple_markers in function data2haplohh(), which, if set to TRUE, removes all but the first marker with identical positions.

Dealing with unphased data {#phasing}

Notwithstanding expensive experimental methods, current high-throughput genotyping / sequencing technologies cannot directly assign alleles to specific chromosomes of a heterozygous diploid (or multiploid) individual. Instead, this task of phasing is typically performed by specialized bioinformatic tools like SHAPEIT [@OConnell2014] and fastPHASE [@Scheet2006]. Although computationally demanding, the application of these tools is straight-forward and the results usually of sufficient quality for the calculation of EHH based statistics. Typically, the tools interpolate missing values away.

In the presumably rare cases where phasing is not feasible, EHH or EHHS can only be meaningfully estimated by reducing the set of compared chromosomes to those of homozygous (at the focal marker) individuals (assuming that the input data is ordered correspondingly). However, this reduction entails a substantial loss of power; even by an adapted parameter setting (see below) at the very minimum 10, but better up to 30 sequences are needed to obtain at least moderately accurate estimations.

For the within-population statistic iHS, the latter requirement concerns both major and minor alleles of a marker and scans should not be performed on samples comprising less than 100 sequences. Even for sample sizes of 200 sequences, meaningful estimation of iHS is hence restricted to variants of intermediate frequencies, i.e. high minor frequency.

For the cross-population statistics Rsb and XP-EHH a minimum number of 30 sequences from homozygous individuals is usually fulfilled if the sample size of each population exceeds 60 sequences. Hence, for unphased sequences the following parameters shoudl be set:

A few shared haplotypes of extreme length are usually encountered in neutrally evolving regions. In order to limit this "statistical noise", cut-off rules are for unphased sequences even more important than they are for phased ones

See [@Klassmann2020] for a study on estimating iHS, Rsb and XP-EHH using unphased sequences.

Dealing with unpolarized data {#polarizing}

The designation of alleles as 'ancestral' or 'derived' is referred to as polarization. Since sequences of ancient genomes are available only for a few species and restricted to a limited time back to the past, the 'ancestral' allele is usually inferred to be the one carried by one or more outgroup species such as chimpanzees or gorillas for humans. However, this presupposes the existence of a reference sequence of a suitable 'neighbor' species of sufficient quality as well as reliable genome-wide alignments. These requirements are not trivial and even if they are fulfilled, any alignment will not cover the whole genome and the covered part will contain mis-specified alleles due to invisible secondary or back-mutations [@Baudry2003], possibly causing spurious signals of selection [@Hernandez2007].

Note that the bin-wise standardization of iHS is the only calculation step within our package where the information about ancestry is exploited. The information of ancestry status is valuable since the expected values under neutral evolution depend on the respective allele frequencies at the focal marker (see Figure \@ref(fig:freqbin) of this vignette and Figure 4 of [@Voight2006]). The binning of markers by frequency before its standardization (see section \@ref(ihh2ihs)) is aimed to eliminate most of this dependence. For unpolarized alleles this correction cannot be done.

Hence two parameters are important when dealing with unpolarized data:

Simulations have shown that neglecting ancestry status reduces the strength of the signal, yet conspicuous values remain clustered; consequently, a delineation of candidate regions of selection should rely primarily on this latter feature. For a more detailed analysis see [@Klassmann2020].

\clearpage

Differences to the program hapbin {#hapbin}

The C++ program hapbin [@Maclean2015] is an alternative tool to calculate the statistics iHS and XP-EHH. The obtained values vary between hapbin and rehh. As far as we can tell, this is due to the differences in implementation listed below. The first 4 are hard-wired and cannot be remediated without changing the code while all further discrepancies can be largely eliminated by appropriate parameter settings. We refer to version 1.3.0 of hapbin.

  1. Hapbin disregards the markers directly at the border of chromosomes.

  2. If an allele of a focal marker is present only on a single chromosome, hapbin assigns a EHH value of 1 at the focal marker and zero otherwise. rehh assigns zero at the focal marker, too.

  3. Rehh stops computing EHH for each allele separately if a lower threshold set by limehh is reached. Hapbin continues the calculation until the values for both alleles fall below the threshold (set by -c or --cutoff).

  4. For the standardization of iHS resp. XP-EHH, hapbin uses the estimator $\sqrt{\frac{1}{n}\sum(x_i-\bar{x})^2}$ for the standard deviation while rehh uses $\sqrt{\frac{1}{n-1}\sum(x_i-\bar{x})^2}$.

  5. The bins used by hapbin for the standardization of iHS cover the whole interval ]0,1], while in rehh they span the interval [min_maf,1-min_maf[. Hapbin includes the upper endpoint into each bin, while rehh includes by default the lower endpoint. The latter can be changed by setting the parameter right of ihh2ihs() to TRUE.

  6. The default number of bins is 50 in hapbin, yielding a bin width of 0.02. The default width in rehh is 0.025 (yielding 36 bins, see point above!). Setting the number of bins in hapbin to 40 with option -b or --bin yields a bin width of 0.025.

  7. hapbin uses by default another estimator for homozygosity than rehh (see section \@ref(estimation)). If run with option -a or --binom, it uses the same as rehh.

  8. Integration over EHH resp. EHHS is performed by hapbin on the area between the curve spanned by these quantities and the x-axis (y=0) while rehh by default integrates only over the part of that area that is above the threshold set by the parameters limehh resp. limehhs, i.e. the area between the curve and the line y=threshold. This is not to be confused with the condition for truncation at left and right ends of the curve which is (for all practical purposes) handled identically by both programs. Setting in rehh the parameter lower_y_bound to zero makes the integration identical to that of hapbin. As mentioned above, limehh(s) of rehh corresponds to -c or --cutoff of hapbin.

  9. By default, the parameter discard_integration_at_border is TRUE in rehh. It has to be set to FALSE in order to conform to hapbin.

  10. Large differences can arise from different handling of gaps during the integration of EHH resp. EHHS yielding iHH resp. iES. Hapbin has a parameter -s or --scale to "down-weight" large gaps by capping them to the specified value. Its default value is 20000 while the corresponding option in rehh is turned off by default, but can be set by scalegap. The option maxgap within rehh leads to a stop of the integration and if the parameter discard_integration_at_border is set to TRUE, then no value is reported. This has no counterpart in hapbin. Instead, hapbin allows to specify a maximum length of Extended Haplotypes (disabled by default) which is available as option of the function scan_hh_full() in rehh.

\clearpage

About estimating homozygosity {#estimation}

The term homozygosity as component of the abbreviation EHH refers to the probability that two arbitrarily chosen chromosomes from a large population are identical at a given locus or in a given region. It does not make any statement about homozygosity of individuals or even presuppose that individuals are diploid. One might even argue, whether the term homogeneity would have been more appropriate.

If there are $K$ alleles in the population and each allele has a population frequency of $f_k$, then this probability is given by $$H=\sum_{k=1}^{K}f_k^2\;.$$ For each allele $k$, its population frequency can be estimated by its sample frequency $x_k$: if the sample contains $n$ chromosomes and allele $k$ is observed $n_k$ times, then $$\hat{f_k}=\frac{n_k}{n}=x_k\;.$$ It seems straightforward to estimate the population homozygosity from a sample by $$\hat{H_1}=\sum_{k=1}^Kx_k^2=\sum_{k=1}^K\left(\frac{n_k}{n}\right)^2\;.$$ However, it turns out that this estimator is biased (it yields values that tend to be slightly too high). The following estimator, instead, is unbiased [@Nei1974]: $$\hat{H_2}=\frac{n}{n-1}\sum_{k=1}^{K_{s,t}}x_k^2-\frac{1}{n-1}\;.$$ The latter is used by rehh. We can see this e.g. in Equation \@ref(eq:ehhssab), when we consider each (shared) haplotype in the region between markers $s$ and $t$ as an allele. We get

$$EHHS=\frac{1}{n(n-1)}\sum_{k=1}^{K_{s,t}}n_k(n_k-1)=\frac{n}{n-1}\frac{1}{n^2}\sum_{k=1}^{K_{s,t}}(n_k^2-n_k)=\frac{n}{n-1}\left(\sum_{k=1}^{K_{s,t}}\frac{n_k^2}{n^2}-\frac{n}{n^2}\right)=\hat{H_2}\;.$$ hapbin, in constrast, uses by default estimator $\hat{H_1}$ and refers to $\hat{H_2}$ as the "alternative" estimator. Evidently, for large $n$, the difference between the two becomes negligible. For small $n$ this is not necessarily so. If we consider a minimal sample of two non-identical chromosomes, hence $n=2$ and $K=2$, then we have $$\hat{H_1}=\left(\frac{1}{2}\right)^2+\left(\frac{1}{2}\right)^2=\frac{1}{2}$$ and $$\hat{H_2}=\frac{1}{2\cdot 1}(1\cdot 0+1\cdot 0)=0\;.$$ Interestingly, although $\hat{H_1}$ is biased, it yields values which are on average closer to the population value than $\hat{H_2}$, since the variance of $\hat{H_1}$ is smaller than that of $\hat{H_2}$ [@Nei1974].

It is unlikely, though, that the choice of the estimator has a major effect on detecting selection. \clearpage

References



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rehh documentation built on Sept. 15, 2021, 5:06 p.m.

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