| Title: | Linkage Disequilibrium Shrinkage Estimation for Polyploids |
|---|---|
| Description: | Estimate haplotypic or composite pairwise linkage disequilibrium (LD) in polyploids, using either genotypes or genotype likelihoods. Support is provided to estimate the popular measures of LD: the LD coefficient D, the standardized LD coefficient D', and the Pearson correlation coefficient r. All estimates are returned with corresponding standard errors. These estimates and standard errors can then be used for shrinkage estimation. The main functions are ldfast(), ldest(), mldest(), sldest(), plot.lddf(), format_lddf(), and ldshrink(). Details of the methods are available in Gerard (2021a) <doi:10.1111/1755-0998.13349> and Gerard (2021b) <doi:10.1038/s41437-021-00462-5>. |
| Authors: | David Gerard [aut, cre] |
| Maintainer: | David Gerard <[email protected]> |
| License: | GPL-3 |
| Version: | 2.1.5 |
| Built: | 2024-10-12 04:43:04 UTC |
| Source: | https://github.com/dcgerard/ldsep |
Estimate haplotypic or composite pairwise linkage disequilibrium (LD) in polyploids, using either genotypes or genotype likelihoods. Support is provided to estimate the popular measures of LD: the LD coefficient D, the standardized LD coefficient D', and the Pearson correlation coefficient r. All estimates are returned with corresponding standard errors. These estimates and standard errors can then be used for shrinkage estimation.
The main functions are:
ldfast()Fast, moment-based, bias-corrected LD LD estimates from marginal posterior distributions.
ldest()Estimates pairwise LD.
mldest()Iteratively apply ldest()
across many pairs of SNPs.
sldest()Iteratively apply ldest()
along a sliding window of fixed length.
plot.lddf()format_lddf()ldshrink()Shrink correlation estimates using adaptive shrinkage (Stephens, 2017; Dey and Stephens, 2018).
If you find the methods in this package useful, please run the following
in R for citation information: citation("ldsep")
David Gerard
This function will either standardize by the maximum covariance conditional on the marginal genotype distribution, or by the maximum covariance conditional on the marginal allele frequencies.
Dprime(qmat, type = c("allele", "geno"), constrain = FALSE)Dprime(qmat, type = c("allele", "geno"), constrain = FALSE)
qmat |
The observed joint genotype distribution. |
type |
Should we condition on the marginal genotype distribution
( |
constrain |
A logical. This option is only applicable when
|
Note that when type = "allele" and constrain = FALSE,
the resulting D' is constrained to fall between -K and K, where
K is the ploidy of the species. However, under HWE, this measure is
equal to haplotypic D'. Using constrain = TRUE will result
in a measure that is constrained to lie between -1 and 1, but
it will not equal haplotypic D' under HWE.
Using type = "geno" is its own thing and will not equal
D' generally under HWE. When type = "geno", then the
the constrain parameter has no effect.
A vector of length 2. The first element is the estimated D'. The second element is the normalization used.
David Gerard
K <- 6 qmat <- matrix(stats::runif((K+1)^2), nrow = K+1) qmat <- qmat / sum(qmat) Dprime(qmat, type = "geno") Dprime(qmat, type = "allele")K <- 6 qmat <- matrix(stats::runif((K+1)^2), nrow = K+1) qmat <- qmat / sum(qmat) Dprime(qmat, type = "geno") Dprime(qmat, type = "allele")
mldest() or
sldest() into an
upper-triangular matrix.Formats the LD estimates and standard errors output
from running mldest() or sldest()
into a more conventional upper-triangular matrix.
format_lddf(obj, element = "r2")format_lddf(obj, element = "r2")
obj |
An object of class |
element |
Which element in |
A matrix of the selected elements. Only the upper-triangle of the
matrix is filled. The lower-triangle and the diagonal are NA's.
David Gerard
set.seed(1) ## Simulate genotypes when true correlation is 0 nloci <- 5 nind <- 100 K <- 6 nc <- 1 genomat <- matrix(sample(0:K, nind * nloci, TRUE), nrow = nloci) ## Haplotypic LD estimates lddf <- mldest(geno = genomat, K = K, nc = nc, type = "hap") ## Obtain the D estimates in matrix form Dmat <- format_lddf(obj = lddf, element = "D") Dmatset.seed(1) ## Simulate genotypes when true correlation is 0 nloci <- 5 nind <- 100 K <- 6 nc <- 1 genomat <- matrix(sample(0:K, nind * nloci, TRUE), nrow = nloci) ## Haplotypic LD estimates lddf <- mldest(geno = genomat, K = K, nc = nc, type = "hap") ## Obtain the D estimates in matrix form Dmat <- format_lddf(obj = lddf, element = "D") Dmat
This function will calculate the (log) probabilities for all genotype combinations at two loci given just the haplotype frequencies. This is under the assumptions of HWE.
get_prob_array(K, prob, log_p = TRUE)get_prob_array(K, prob, log_p = TRUE)
K |
The ploidy of the species. |
prob |
Haplotype frequencies in the order of (ab, Ab, aB, AB). |
log_p |
A logical. Should we return the log-probabilities ( |
Element (i, j) is the (log) probability of genotype i-1 at locus 1 and genotype j-1 at locus 2.
David Gerard
get_prob_array(K = 6, prob = c(0.1, 0.2, 0.3, 0.4), log_p = FALSE)get_prob_array(K = 6, prob = c(0.1, 0.2, 0.3, 0.4), log_p = FALSE)
This will take genotype log-likelihoods and normalize them to sum to one. This corresponds to using a naive discrete uniform prior over the genotypes. It is not generally recommended that you use this function.
gl_to_gp(gl)gl_to_gp(gl)
gl |
A three dimensional array of genotype log-likelihoods.
Element |
A three-dimensional array, of the same dimensions as gl,
containing the posterior probabilities of each dosage.
David Gerard
data("glike") class(glike) dim(glike) gl_to_gp(glike)data("glike") class(glike) dim(glike) gl_to_gp(glike)
uit
Contains an array of genotype log-likelihoods from
the uit dataset. Element gp[i, j, k] is the
log-likelihood of dosage k-1 for individual j
at SNP i.
glikeglike
A three-dimensional array object.
doi:10.1371/journal.pone.0062355
Uitdewilligen, Jan GAML, Anne-Marie A. Wolters, B. Bjorn, Theo JA Borm, Richard GF Visser, and Herman J. Van Eck. "A next-generation sequencing method for genotyping-by-sequencing of highly heterozygous autotetraploid potato." PloS one 8, no. 5 (2013): e62355. doi:10.1371/journal.pone.0062355
uit for the full multidog() fit.
uit
Contains an array of posterior probabilities of the genotypes from
the uit dataset. Element gp[i, j, k] is the
posterior probability of dosage k-1 for individual j
at SNP i.
gpgp
A three-dimensional array object.
doi:10.1371/journal.pone.0062355
Uitdewilligen, Jan GAML, Anne-Marie A. Wolters, B. Bjorn, Theo JA Borm, Richard GF Visser, and Herman J. Van Eck. "A next-generation sequencing method for genotyping-by-sequencing of highly heterozygous autotetraploid potato." PloS one 8, no. 5 (2013): e62355. doi:10.1371/journal.pone.0062355
uit for the full multidog() fit.
lddf object.Tests if an argument is a lddf object.
is.lddf(x)is.lddf(x)
x |
Anything. |
A logical. TRUE if x is a lddf object,
and FALSE otherwise.
David Gerard
is.lddf("anything") # FALSEis.lddf("anything") # FALSE
Estimates either haplotypic or composite measures of LD using either genotypes are genotype likelihoods via maximum likelihood. The usual measures of LD are estimated (D, D', and r) along with the Fisher-z transformation of r (called "z"). All estimates are returned with standard errors. See Gerard (2021) for details.
ldest( ga, gb, K, se = TRUE, type = c("hap", "comp"), model = c("norm", "flex"), pen = ifelse(type == "hap", 2, 1) )ldest( ga, gb, K, se = TRUE, type = c("hap", "comp"), model = c("norm", "flex"), pen = ifelse(type == "hap", 2, 1) )
ga |
One of two possible inputs:
|
gb |
One of two possible inputs:
|
K |
The ploidy of the species. Assumed to be the same for all individuals. |
se |
A logical. Should we calculate standard errors ( |
type |
The type of LD to calculate. The available types are
haplotypic LD ( |
model |
When |
pen |
The penalty to be applied to the likelihood. You can think about
this as the prior sample size. Should be greater than 1. Does not
apply if |
A vector with some or all of the following elements:
DThe estimate of the LD coefficient.
D_seThe standard error of the estimate of the LD coefficient.
r2The estimate of the squared Pearson correlation.
r2_seThe standard error of the estimate of the squared Pearson correlation.
rThe estimate of the Pearson correlation.
r_seThe standard error of the estimate of the Pearson correlation.
DprimeThe estimate of the standardized LD
coefficient. When type = "comp", this corresponds
to the standardization where we fix allele frequencies.
Dprime_seThe standard error of Dprime.
DprimegThe estimate of the standardized LD coefficient. This corresponds to the standardization where we fix genotype frequencies.
Dprimeg_seThe standard error of Dprimeg.
zThe Fisher-z transformation of r.
z_seThe standard error of the Fisher-z
transformation of r.
p_abThe estimated haplotype frequency of ab. Only returned if estimating the haplotypic LD.
p_AbThe estimated haplotype frequency of Ab. Only returned if estimating the haplotypic LD.
p_aBThe estimated haplotype frequency of aB. Only returned if estimating the haplotypic LD.
p_ABThe estimated haplotype frequency of AB. Only returned if estimating the haplotypic LD.
q_ijThe estimated frequency of genotype i at locus 1 and genotype j at locus 2. Only returned if estimating the composite LD.
nThe number of individuals used to estimate pairwise LD.
This section describes the methods used when type = "hap" is
selected.
Haplotypic LD measures the association between two loci on the same haplotype. When haplotypes are known, estimating haplotypic LD is simple using just the haplotypic frequencies.
When haplotypes are not known, we can still estimate haplotypic frequencies using the genotypes or genotype likelihoods in autopolyploids as long as Hardy-Weinberg equilibrium (HWE) is satisfied. We do this via maximum likelihood using gradient ascent. Gradient ascent is performed over the unconstrained parameterization of the 3-simplex from Betancourt (2012). The estimated haplotype frequencies are then used to estimate haplotypic LD.
Standard errors are provided using standard maximum likelihood theory. In brief, the Hessian matrix of the log-likelihood is calculated at the MLE's of the haplotype frequencies. The negative inverse of this Hessian matrix is approximately the covariance matrix of the MLE's of the haplotype frequencies. Since all haplotypic LD measures are functions of the haplotype frequencies, we use the delta-method to obtain the standard errors for each LD estimate.
A Dirichlet(2,2,2,2) prior is placed over the frequencies of
haplotypes 00, 01, 10, and 11. This corresponds to the "add two" rule
of Agresti (1998). You can change this prior via the pen argument.
When you either do not have autopolyploids or when HWE is not
satisfied, then the estimates using type = "hap"
are nonsensical. However, the composite measures of LD are still
applicable (see below).
This section describes the methods used when type = "comp" is
selected.
When HWE is not satisfied, haplotype frequencies are not estimable. However, measures of association between two loci are still estimable. These associations may be caused by LD either on the same haplotype or between different haplotypes. Cockerham and Weir (1977) thus called such measures "composite" measures of LD.
When the genotypes are known, these composite measures have simple correspondences to well-known statistical measures of association. D is the covariance of genotypes between loci divided by the ploidy. r is the Pearson correlation of genotypes. D' is D divided by a term that involves only mean genotypes.
When genotypes are not known, we estimate the joint genotype frequencies and use these to estimate the composite measures of LD using genotype likelihoods. The distribution of genotypes is assumed to either follow a proportional bivariate normal model (by default) or a general categorical model.
These estimates of composite measures of LD estimate the haplotypic measures of LD when HWE is fulfilled, but are still applicable when HWE is not fulfilled.
When genotypes are known, standard errors are calculated using standard moment-based approaches. When genotypes are not known, standard errors are calculated using standard maximum likelihood theory, same as for the haplotypic LD estimates (see above), or using a bootstrap.
David Gerard
Agresti, Alan, and Brent A. Coull. "Approximate is better than "exact" for interval estimation of binomial proportions." The American Statistician 52, no. 2 (1998): 119-126. doi:10.1080/00031305.1998.10480550
Betancourt, Michael. "Cruising the simplex: Hamiltonian Monte Carlo and the Dirichlet distribution." In AIP Conference Proceedings 31st, vol. 1443, no. 1, pp. 157-164. American Institute of Physics, 2012. doi:10.1063/1.3703631
Cockerham, C. Clark, and B. S. Weir. "Digenic descent measures for finite populations." Genetics Research 30, no. 2 (1977): 121-147. doi:10.1017/S0016672300017547
Gerard, David. "Pairwise Linkage Disequilibrium Estimation for Polyploids." Molecular Ecology Resources 21, no. 4 (2021): 1230-1242. doi:10.1111/1755-0998.13349
ldfast()Fast, moment-based approach to LD estimation that also accounts for genotype uncertainty.
mldest()For calculating pairwise LD among all pairs of a collection of SNPs.
sldest()For calculating pairwise LD along a sliding window of SNPs.
ldest_hap()For the function that directly estimates haplotypic LD when HWE is fulfilled.
ldest_comp()For the function that directly estimates composite LD.
set.seed(1) n <- 100 # sample size K <- 6 # ploidy ## generate some fake genotypes when LD = 0. ga <- stats::rbinom(n = n, size = K, prob = 0.5) gb <- stats::rbinom(n = n, size = K, prob = 0.5) head(ga) head(gb) ## generate some fake genotype likelihoods when LD = 0. gamat <- t(sapply(ga, stats::dnorm, x = 0:K, sd = 1, log = TRUE)) gbmat <- t(sapply(gb, stats::dnorm, x = 0:K, sd = 1, log = TRUE)) head(gamat) head(gbmat) ## Haplotypic LD with genotypes ldout1 <- ldest(ga = ga, gb = gb, K = K, type = "hap") head(ldout1) ## Haplotypic LD with genotype likelihoods ldout2 <- ldest(ga = gamat, gb = gbmat, K = K, type = "hap") head(ldout2) ## Composite LD with genotypes ldout3 <- ldest(ga = ga, gb = gb, K = K, type = "comp") head(ldout3) ## Composite LD with genotype likelihoods and normal model ldout4 <- ldest(ga = gamat, gb = gbmat, K = K, type = "comp", model = "norm") head(ldout4) ## Composite LD with genotype likelihoods and general categorical model ldout5 <- ldest(ga = gamat, gb = gbmat, K = K, type = "comp", model = "flex", se = FALSE) head(ldout5) ldout1[["D"]] ldout2[["D"]] ldout3[["D"]] ldout4[["D"]] ldout5[["D"]]set.seed(1) n <- 100 # sample size K <- 6 # ploidy ## generate some fake genotypes when LD = 0. ga <- stats::rbinom(n = n, size = K, prob = 0.5) gb <- stats::rbinom(n = n, size = K, prob = 0.5) head(ga) head(gb) ## generate some fake genotype likelihoods when LD = 0. gamat <- t(sapply(ga, stats::dnorm, x = 0:K, sd = 1, log = TRUE)) gbmat <- t(sapply(gb, stats::dnorm, x = 0:K, sd = 1, log = TRUE)) head(gamat) head(gbmat) ## Haplotypic LD with genotypes ldout1 <- ldest(ga = ga, gb = gb, K = K, type = "hap") head(ldout1) ## Haplotypic LD with genotype likelihoods ldout2 <- ldest(ga = gamat, gb = gbmat, K = K, type = "hap") head(ldout2) ## Composite LD with genotypes ldout3 <- ldest(ga = ga, gb = gb, K = K, type = "comp") head(ldout3) ## Composite LD with genotype likelihoods and normal model ldout4 <- ldest(ga = gamat, gb = gbmat, K = K, type = "comp", model = "norm") head(ldout4) ## Composite LD with genotype likelihoods and general categorical model ldout5 <- ldest(ga = gamat, gb = gbmat, K = K, type = "comp", model = "flex", se = FALSE) head(ldout5) ldout1[["D"]] ldout2[["D"]] ldout3[["D"]] ldout4[["D"]] ldout5[["D"]]
This function will estimate the composite LD between two loci, either using genotype estimates or using genotype likelihoods. The resulting measures of LD are generalizations of Burrow's "composite" LD measure.
ldest_comp( ga, gb, K, pen = 1, useboot = TRUE, nboot = 50, se = TRUE, model = c("norm", "flex") )ldest_comp( ga, gb, K, pen = 1, useboot = TRUE, nboot = 50, se = TRUE, model = c("norm", "flex") )
ga |
One of two possible inputs:
|
gb |
One of two possible inputs:
|
K |
The ploidy of the species. Assumed to be the same for all individuals. |
pen |
The penalty to be applied to the likelihood. You can think about
this as the prior sample size. Should be greater than 1. Does not
apply if |
useboot |
Should we use bootstrap standard errors |
nboot |
The number of bootstrap iterations to use is
|
se |
A logical. Should we calculate standard errors ( |
model |
Should we assume the class of joint genotype distributions
is from the proportional bivariate normal ( |
A vector with some or all of the following elements:
DThe estimate of the LD coefficient.
D_seThe standard error of the estimate of the LD coefficient.
r2The estimate of the squared Pearson correlation.
r2_seThe standard error of the estimate of the squared Pearson correlation.
rThe estimate of the Pearson correlation.
r_seThe standard error of the estimate of the Pearson correlation.
DprimeThe estimate of the standardized LD
coefficient. When type = "comp", this corresponds
to the standardization where we fix allele frequencies.
Dprime_seThe standard error of Dprime.
DprimegThe estimate of the standardized LD coefficient. This corresponds to the standardization where we fix genotype frequencies.
Dprimeg_seThe standard error of Dprimeg.
zThe Fisher-z transformation of r.
z_seThe standard error of the Fisher-z
transformation of r.
p_abThe estimated haplotype frequency of ab. Only returned if estimating the haplotypic LD.
p_AbThe estimated haplotype frequency of Ab. Only returned if estimating the haplotypic LD.
p_aBThe estimated haplotype frequency of aB. Only returned if estimating the haplotypic LD.
p_ABThe estimated haplotype frequency of AB. Only returned if estimating the haplotypic LD.
q_ijThe estimated frequency of genotype i at locus 1 and genotype j at locus 2. Only returned if estimating the composite LD.
nThe number of individuals used to estimate pairwise LD.
David Gerard
set.seed(1) n <- 100 # sample size K <- 6 # ploidy ## generate some fake genotypes when LD = 0. ga <- stats::rbinom(n = n, size = K, prob = 0.5) gb <- stats::rbinom(n = n, size = K, prob = 0.5) head(ga) head(gb) ## generate some fake genotype likelihoods when LD = 0. gamat <- t(sapply(ga, stats::dnorm, x = 0:K, sd = 1, log = TRUE)) gbmat <- t(sapply(gb, stats::dnorm, x = 0:K, sd = 1, log = TRUE)) head(gamat) head(gbmat) ## Composite LD with genotypes ldout1 <- ldest_comp(ga = ga, gb = gb, K = K) head(ldout1) ## Composite LD with genotype likelihoods ldout2 <- ldest_comp(ga = gamat, gb = gbmat, K = K, se = FALSE, model = "flex") head(ldout2) ## Composite LD with genotype likelihoods and proportional bivariate normal ldout3 <- ldest_comp(ga = gamat, gb = gbmat, K = K, model = "norm") head(ldout3)set.seed(1) n <- 100 # sample size K <- 6 # ploidy ## generate some fake genotypes when LD = 0. ga <- stats::rbinom(n = n, size = K, prob = 0.5) gb <- stats::rbinom(n = n, size = K, prob = 0.5) head(ga) head(gb) ## generate some fake genotype likelihoods when LD = 0. gamat <- t(sapply(ga, stats::dnorm, x = 0:K, sd = 1, log = TRUE)) gbmat <- t(sapply(gb, stats::dnorm, x = 0:K, sd = 1, log = TRUE)) head(gamat) head(gbmat) ## Composite LD with genotypes ldout1 <- ldest_comp(ga = ga, gb = gb, K = K) head(ldout1) ## Composite LD with genotype likelihoods ldout2 <- ldest_comp(ga = gamat, gb = gbmat, K = K, se = FALSE, model = "flex") head(ldout2) ## Composite LD with genotype likelihoods and proportional bivariate normal ldout3 <- ldest_comp(ga = gamat, gb = gbmat, K = K, model = "norm") head(ldout3)
Given genotype (allele dosage) or genotype likelihood data for each individual at a pair of loci, this function will calculate the maximum likelihood estimates and their corresponding asymptotic standard errors of some measures of linkage disequilibrium (LD): D, D', the Pearson correlation, the squared Pearson correlation, and the Fisher-z transformation of the Pearson correlation. This function can be used for both diploids and polyploids.
ldest_hap( ga, gb, K, reltol = 10^-8, nboot = 100, useboot = FALSE, pen = 2, grid_init = FALSE, se = TRUE )ldest_hap( ga, gb, K, reltol = 10^-8, nboot = 100, useboot = FALSE, pen = 2, grid_init = FALSE, se = TRUE )
ga |
One of two possible inputs:
|
gb |
One of two possible inputs:
|
K |
The ploidy of the species. Assumed to be the same for all individuals. |
reltol |
The relative tolerance for the stopping criterion. |
nboot |
Sometimes, the MLE standard errors don't exist. So we use
the bootstrap as a backup. |
useboot |
A logical. Optionally, you may always use the bootstrap
to estimate the standard errors ( |
pen |
The penalty to be applied to the likelihood. You can think about
this as the prior sample size. Should be greater than 1. Does not
apply if |
grid_init |
A logical. Should we initialize the gradient ascent
at a grid of initial values ( |
se |
A logical. Should we calculate standard errors ( |
Let A and a be the reference and alternative alleles, respectively, at
locus 1. Let B and b be the reference and alternative alleles,
respectively, at locus 2. Let paa, pAb, paB, and pAB be the
frequencies of haplotypes ab, Ab, aB, and AB, respectively.
Let pA = pAb + pAB and let pB = paB + pAB
The ldest returns estimates of the following measures
of LD.
D: pAB - pA pB
D': D / Dmax, where Dmax = min(pA pB, (1 - pA) (1 - pB)) if D < 0 and Dmax = min(pA (1 - pB), pA (1 - pB)) if D > 0
r-squared: The squared Pearson correlation, r^2 = D^2 / (pA (1 - pA) pB (1 - pB))
r: The Pearson correlation, r = D / sqrt(pA (1 - pA) pB (1 - pB))
Estimates are obtained via maximum likelihood under the assumption of Hardy-Weinberg equilibrium. The likelihood is calculated by integrating over the possible haplotypes for each pair of genotypes.
The resulting standard errors are based on the square roots of the inverse of the negative Fisher-information. This is from standard maximum likelihood theory. The Fisher-information is known to be biased low, so the actual standard errors are probably a little bigger for small n (n < 20). In some cases the Fisher-information matrix is singular, and so we in these cases we return a bootstrap estimate of the standard error.
The standard error estimate of the squared Pearson correlation is not valid when r^2 = 0.
In cases where either SNP is estimated to be monoallelic
(pA %in% c(0, 1) or pB %in% c(0, 1)), this function
will return LD estimates of NA.
A vector with some or all of the following elements:
DThe estimate of the LD coefficient.
D_seThe standard error of the estimate of the LD coefficient.
r2The estimate of the squared Pearson correlation.
r2_seThe standard error of the estimate of the squared Pearson correlation.
rThe estimate of the Pearson correlation.
r_seThe standard error of the estimate of the Pearson correlation.
DprimeThe estimate of the standardized LD
coefficient. When type = "comp", this corresponds
to the standardization where we fix allele frequencies.
Dprime_seThe standard error of Dprime.
DprimegThe estimate of the standardized LD coefficient. This corresponds to the standardization where we fix genotype frequencies.
Dprimeg_seThe standard error of Dprimeg.
zThe Fisher-z transformation of r.
z_seThe standard error of the Fisher-z
transformation of r.
p_abThe estimated haplotype frequency of ab. Only returned if estimating the haplotypic LD.
p_AbThe estimated haplotype frequency of Ab. Only returned if estimating the haplotypic LD.
p_aBThe estimated haplotype frequency of aB. Only returned if estimating the haplotypic LD.
p_ABThe estimated haplotype frequency of AB. Only returned if estimating the haplotypic LD.
q_ijThe estimated frequency of genotype i at locus 1 and genotype j at locus 2. Only returned if estimating the composite LD.
nThe number of individuals used to estimate pairwise LD.
David Gerard
set.seed(1) n <- 100 # sample size K <- 6 # ploidy ## generate some fake genotypes when LD = 0. ga <- stats::rbinom(n = n, size = K, prob = 0.5) gb <- stats::rbinom(n = n, size = K, prob = 0.5) head(ga) head(gb) ## generate some fake genotype likelihoods when LD = 0. gamat <- t(sapply(ga, stats::dnorm, x = 0:K, sd = 1, log = TRUE)) gbmat <- t(sapply(gb, stats::dnorm, x = 0:K, sd = 1, log = TRUE)) head(gamat) head(gbmat) ## Haplotypic LD with genotypes ldout1 <- ldest_hap(ga = ga, gb = gb, K = K) head(ldout1) ## Haplotypic LD with genotype likelihoods ldout2 <- ldest_hap(ga = gamat, gb = gbmat, K = K) head(ldout2)set.seed(1) n <- 100 # sample size K <- 6 # ploidy ## generate some fake genotypes when LD = 0. ga <- stats::rbinom(n = n, size = K, prob = 0.5) gb <- stats::rbinom(n = n, size = K, prob = 0.5) head(ga) head(gb) ## generate some fake genotype likelihoods when LD = 0. gamat <- t(sapply(ga, stats::dnorm, x = 0:K, sd = 1, log = TRUE)) gbmat <- t(sapply(gb, stats::dnorm, x = 0:K, sd = 1, log = TRUE)) head(gamat) head(gbmat) ## Haplotypic LD with genotypes ldout1 <- ldest_hap(ga = ga, gb = gb, K = K) head(ldout1) ## Haplotypic LD with genotype likelihoods ldout2 <- ldest_hap(ga = gamat, gb = gbmat, K = K) head(ldout2)
Estimates the reliability ratios from posterior marginal moments and uses these to correct the biases in linkage disequilibrium estimation caused by genotype uncertainty. These methods are described in Gerard (2021).
ldfast( gp, type = c("r", "r2", "z", "D", "Dprime"), shrinkrr = TRUE, se = TRUE, thresh = TRUE, upper = 10, mode = c("zero", "estimate"), win = NULL )ldfast( gp, type = c("r", "r2", "z", "D", "Dprime"), shrinkrr = TRUE, se = TRUE, thresh = TRUE, upper = 10, mode = c("zero", "estimate"), win = NULL )
gp |
A three-way array with dimensions SNPs by individuals by dosage.
That is, |
type |
What LD measure should we estimate?
Note that these are all composite measures of LD (see
the description in |
shrinkrr |
A logical. Should we use adaptive shrinkage
(Stephens, 2016) to shrink the reliability ratios ( |
se |
Should we also return a matrix of standard errors ( |
thresh |
A logical. Should we apply an upper bound on the reliability
ratios ( |
upper |
The upper bound on the reliability ratios if
|
mode |
A character. Only applies if |
win |
A positive integer. The window size. This will constrain the correlations calculated to those +/- the window size. This will only improve speed if the window size is much less than the number of SNPs. |
A list with some or all of the following elements:
ldmatThe bias-corrected LD matrix.
rrThe estimated reliability ratio for each SNP. This is the multiplicative factor applied to the naive LD estimate for each SNP.
rr_rawThe raw reliability ratios (for the covariance,
not the correlation). Only returned if shrinkrr = TRUE.
rr_seThe standard errors for the log-raw
reliability ratios for each SNP. That is, we have
sd(log(rr_raw)) ~ rr_se. Only returned if shrinkrr = TRUE.
sematA matrix of standard errors of the corresponding estimators of LD.
Returns consistent and bias-corrected estimates of linkage disequilibrium.
The usual measures of LD are implemented: D, D', r, r2, and z
(Fisher-z of r). These are all composite measures of LD, not
haplotypic measures of LD (see the description in ldest()).
They are always appropriate measures of association
between loci, but only correspond to haplotypic measures of LD when
Hardy-Weinberg equilibrium is fulfilled in autopolyploids.
In order for these estimates to perform well, you need to use
posterior genotype probabilities that have been calculated using
adaptive priors, i.e. empirical/hierarchical Bayes approaches. There
are many approaches that do this, such as
updog,
polyRAD,
fitPoly, or
SuperMASSA.
Note that GATK uses a uniform prior, so would be inappropriate for
use in ldfast().
Calculating standard errors and performing hierarchical shrinkage of the
reliability ratios are both rather slow operations compared to just
raw method-of-moments based estimation for LD. If you don't need
standard errors, you can double your speed by setting
se = FALSE. It is not recommended that you disable the
hierarchical shrinkage.
Due to sampling variability, the estimates sometime lie outside of the
theoretical boundaries of the parameters being estimated. In such cases,
we truncate the estimates at the boundary and return NA for the
standard errors.
Let
be the sample correlation of posterior mean genotypes
between loci 1 and 2,
be the sample variance of posterior means at locus 1,
be the sample variance of posterior means at locus 2,
be the sample mean of posterior variances at locus 1, and
be the sample mean of posterior variances at locus 2.
Then the estimated Pearson correlation between the genotypes at loci 1 and 2 is
All other LD calculations are based on this equation. In particular,
the estimated genotype variances at loci 1 and 2 are
and , respectively, which can be
used to calculate D and D'.
The reliability ratio for SNP i is defined by .
By default, we apply ash() (Stephens, 2016)
to the log of these reliability ratios before adjusting the
Pearson correlation. Standard errors are required before using
ash(), but these are easily obtained
using the central limit theorem and the delta-method.
David Gerard
Gerard, David. Scalable Bias-corrected Linkage Disequilibrium Estimation Under Genotype Uncertainty. Heredity, 127(4), 357–362, 2021. doi:10.1038/s41437-021-00462-5.
T. Robertson and J. D. Cryer. An iterative procedure for estimating the mode. Journal of the American Statistical Association, 69(348):1012–1016, 1974. doi:10.1080/01621459.1974.10480246.
M. Stephens. False discovery rates: a new deal. Biostatistics, 18(2):275–294, 10 2016. doi:10.1093/biostatistics/kxw041.
ash()Function used to perform hierarchical shrinkage on the log of the reliability ratios.
ldest(), mldest(), sldest()
Maximum likelihood estimation of linkage disequilibrium.
data("gp") ldout <- ldfast(gp, "r") ldout$ldmat ldout$rr ldout$semat ldout <- ldfast(gp, "D") ldout$ldmat ldout$rr ldout$semat ldout <- ldfast(gp, "Dprime") ldout$ldmat ldout$rr ldout$sematdata("gp") ldout <- ldfast(gp, "r") ldout$ldmat ldout$rr ldout$semat ldout <- ldfast(gp, "D") ldout$ldmat ldout$rr ldout$semat ldout <- ldfast(gp, "Dprime") ldout$ldmat ldout$rr ldout$semat
mldest() or sldest().This will take the output of either mldest() or
sldest(), shrink the Fisher-z transformed
correlation estimates using ash()
(Stephens, 2017; Dey and Stephens, 2018), then return
the corresponding correlation estimates. You can obtain estimates of
r^2 by just squaring these estimates.
ldshrink(obj, ...)ldshrink(obj, ...)
obj |
An object of class |
... |
Additional arguments to pass to |
A correlation matrix.
David Gerard
Stephens, Matthew. "False discovery rates: a new deal." Biostatistics 18, no. 2 (2017): 275-294.
Dey, Kushal K., and Matthew Stephens. "CorShrink: Empirical Bayes shrinkage estimation of correlations, with applications." bioRxiv (2018): 368316.
This function is a wrapper to run ldest() for many pairs of
SNPs. This takes a maximum likelihood approach to LD estimation. See
ldfast() for a method-of-moments approach to LD estimation.
Support is provided for parallelization through the foreach and doParallel
packages. See Gerard (2021) for details.
mldest( geno, K, nc = 1, type = c("hap", "comp"), model = c("norm", "flex"), pen = ifelse(type == "hap", 2, 1), se = TRUE )mldest( geno, K, nc = 1, type = c("hap", "comp"), model = c("norm", "flex"), pen = ifelse(type == "hap", 2, 1), se = TRUE )
geno |
One of two possible inputs:
|
K |
The ploidy of the species. Assumed to be the same for all individuals. |
nc |
The number of computing cores to use. This should never be
more than the number of cores available in your computing environment.
You can determine the maximum number of available cores by running
|
type |
The type of LD to calculate. The available types are
haplotypic LD ( |
model |
When |
pen |
The penalty to be applied to the likelihood. You can think about
this as the prior sample size. Should be greater than 1. Does not
apply if |
se |
A logical. Should we calculate standard errors ( |
See ldest() for details on the different types of LD
estimators supported.
A data frame of class c("lddf", "data.frame")
with some or all of the following elements:
iThe index of the first SNP.
jThe index of the second SNP.
snpiThe row name corresponding to SNP i, if
row names are provided.
snpjThe row name corresponding to SNP j, if
row names are provided.
DThe estimate of the LD coefficient.
D_seThe standard error of the estimate of the LD coefficient.
r2The estimate of the squared Pearson correlation.
r2_seThe standard error of the estimate of the squared Pearson correlation.
rThe estimate of the Pearson correlation.
r_seThe standard error of the estimate of the Pearson correlation.
DprimeThe estimate of the standardized LD
coefficient. When type = "comp", this corresponds
to the standardization where we fix allele frequencies.
Dprime_seThe standard error of Dprime.
DprimegThe estimate of the standardized LD coefficient. This corresponds to the standardization where we fix genotype frequencies.
Dprimeg_seThe standard error of Dprimeg.
zThe Fisher-z transformation of r.
z_seThe standard error of the Fisher-z
transformation of r.
p_abThe estimated haplotype frequency of ab. Only returned if estimating the haplotypic LD.
p_AbThe estimated haplotype frequency of Ab. Only returned if estimating the haplotypic LD.
p_aBThe estimated haplotype frequency of aB. Only returned if estimating the haplotypic LD.
p_ABThe estimated haplotype frequency of AB. Only returned if estimating the haplotypic LD.
q_ijThe estimated frequency of genotype i at locus 1 and genotype j at locus 2. Only returned if estimating the composite LD.
nThe number of individuals used to estimate pairwise LD.
David Gerard
Gerard, David. "Pairwise Linkage Disequilibrium Estimation for Polyploids." Molecular Ecology Resources 21, no. 4 (2021): 1230-1242. doi:10.1111/1755-0998.13349
ldfast()Fast, moment-based approach to LD estimation that also accounts for genotype uncertainty.
ldest()For the base function that estimates pairwise LD.
sldest()For estimating pairwise LD along a sliding window.
format_lddf()For formatting the output of
mldest() as a matrix.
plot.lddf()For plotting the output of
mldest().
set.seed(1) ## Simulate genotypes when true correlation is 0 nloci <- 5 nind <- 100 K <- 6 nc <- 1 genomat <- matrix(sample(0:K, nind * nloci, TRUE), nrow = nloci) ## Composite LD estimates lddf <- mldest(geno = genomat, K = K, nc = nc, type = "comp") lddf[1:6, 1:6]set.seed(1) ## Simulate genotypes when true correlation is 0 nloci <- 5 nind <- 100 K <- 6 nc <- 1 genomat <- matrix(sample(0:K, nind * nloci, TRUE), nrow = nloci) ## Composite LD estimates lddf <- mldest(geno = genomat, K = K, nc = nc, type = "comp") lddf[1:6, 1:6]
Returns distribution of proportional bivariate normal.
pbnorm_dist(mu, sigma, K, log = FALSE)pbnorm_dist(mu, sigma, K, log = FALSE)
mu |
A vector of length 2 containing the mean. |
sigma |
A 2-by-2 positive definite covariance matrix |
K |
The ploidy of the individual. |
log |
A logical. If |
A matrix. Element (i,j) is the (log) probability of genotype i-1 at locus 1 and j-1 at locus 2.
David Gerard
mldest() or
sldest() using corrplot()
Formats the LD estimates in the form of a matrix and creates a heatmap of
these estimates. This heatmap is created using the
corrplot R package. I've adjusted a lot of the defaults
to suit my visualization preferences.
## S3 method for class 'lddf' plot( x, element = "r2", type = c("upper", "full", "lower"), method = c("color", "circle", "square", "ellipse", "number", "shade", "pie"), diag = FALSE, is.corr = NULL, tl.pos = "n", title = NULL, na.label = "square", ... )## S3 method for class 'lddf' plot( x, element = "r2", type = c("upper", "full", "lower"), method = c("color", "circle", "square", "ellipse", "number", "shade", "pie"), diag = FALSE, is.corr = NULL, tl.pos = "n", title = NULL, na.label = "square", ... )
x |
An object of class |
element |
Which element of |
type |
Character, |
method |
See |
diag |
Logical, whether display the correlation coefficients on the principal diagonal. |
is.corr |
See |
tl.pos |
See |
title |
What should the title be? Defaults to the element name. |
na.label |
See |
... |
Additional arguments to pass to
|
For greater plotting flexibility, see corrplot()
for the parameter options.
(Invisibly) returns a matrix of the selected elements.
David Gerard
set.seed(1) ## Simulate genotypes when true correlation is 0 nloci <- 5 nind <- 100 K <- 6 nc <- 1 genomat <- matrix(sample(0:K, nind * nloci, TRUE), nrow = nloci) ## Haplotypic LD estimates lddf <- mldest(geno = genomat, K = K, nc = nc, type = "hap") ## Plot estimates of z plot(lddf, element = "z")set.seed(1) ## Simulate genotypes when true correlation is 0 nloci <- 5 nind <- 100 K <- 6 nc <- 1 genomat <- matrix(sample(0:K, nind * nloci, TRUE), nrow = nloci) ## Haplotypic LD estimates lddf <- mldest(geno = genomat, K = K, nc = nc, type = "hap") ## Plot estimates of z plot(lddf, element = "z")
Given a matrix of prior probabilities for the genotypes at each SNP, this function will calculate the prior variance of genotypes.
pvcalc(priormat)pvcalc(priormat)
priormat |
A matrix of prior genotype probabilities. Element
|
A vector of prior variances.
David Gerard
data("uit") priormat <- uit$snpdf[, paste0("Pr_", 0:4)] pvcalc(priormat)data("uit") priormat <- uit$snpdf[, paste0("Pr_", 0:4)] pvcalc(priormat)
Calculates the pairwise Pearson correlation between all columns
within a fixed window size (win)
using the use = "pairwise.complete.obs" option
from cor(). That is, the correlation
between each pair of variables is computed using all complete pairs
of observations on those variables.
slcor(x, win = 1L)slcor(x, win = 1L)
x |
A numeric matrix. The variables index the columns. |
win |
The size of the window. Defaults to 1. |
A correlation matrix with only the observations within a window containing calculated correlations.
David Gerard
set.seed(1) n <- 10 p <- 100 xmat <- matrix(rnorm(n * p), ncol = n) xmat[sample(n * p, size = 30)] <- NA_real_ slcor(xmat, win = 2)set.seed(1) n <- 10 p <- 100 xmat <- matrix(rnorm(n * p), ncol = n) xmat[sample(n * p, size = 30)] <- NA_real_ slcor(xmat, win = 2)
This function is a wrapper for ldest() for estimating LD
along a sliding window of a fixed size. Support is provided for parallelization through the
foreach and doParallel packages.
sldest( geno, K, win = 50, nc = 1, type = c("hap", "comp"), model = c("norm", "flex"), pen = ifelse(type == "hap", 2, 1), se = TRUE )sldest( geno, K, win = 50, nc = 1, type = c("hap", "comp"), model = c("norm", "flex"), pen = ifelse(type == "hap", 2, 1), se = TRUE )
geno |
One of two possible inputs:
|
K |
The ploidy of the species. Assumed to be the same for all individuals. |
win |
The window size. Pairwise LD will be estimated plus or minus these many positions. Larger sizes significantly increase the computational load. |
nc |
The number of computing cores to use. This should never be
more than the number of cores available in your computing environment.
You can determine the maximum number of available cores by running
|
type |
The type of LD to calculate. The available types are
haplotypic LD ( |
model |
When |
pen |
The penalty to be applied to the likelihood. You can think about
this as the prior sample size. Should be greater than 1. Does not
apply if |
se |
A logical. Should we calculate standard errors ( |
See ldest() for details on the different types of LD
estimators supported.
A data frame of class c("lddf", "data.frame")
with some or all of the following elements:
iThe index of the first SNP.
jThe index of the second SNP.
snpiThe row name corresponding to SNP i, if
row names are provided.
snpjThe row name corresponding to SNP j, if
row names are provided.
DThe estimate of the LD coefficient.
D_seThe standard error of the estimate of the LD coefficient.
r2The estimate of the squared Pearson correlation.
r2_seThe standard error of the estimate of the squared Pearson correlation.
rThe estimate of the Pearson correlation.
r_seThe standard error of the estimate of the Pearson correlation.
DprimeThe estimate of the standardized LD
coefficient. When type = "comp", this corresponds
to the standardization where we fix allele frequencies.
Dprime_seThe standard error of Dprime.
DprimegThe estimate of the standardized LD coefficient. This corresponds to the standardization where we fix genotype frequencies.
Dprimeg_seThe standard error of Dprimeg.
zThe Fisher-z transformation of r.
z_seThe standard error of the Fisher-z
transformation of r.
p_abThe estimated haplotype frequency of ab. Only returned if estimating the haplotypic LD.
p_AbThe estimated haplotype frequency of Ab. Only returned if estimating the haplotypic LD.
p_aBThe estimated haplotype frequency of aB. Only returned if estimating the haplotypic LD.
p_ABThe estimated haplotype frequency of AB. Only returned if estimating the haplotypic LD.
q_ijThe estimated frequency of genotype i at locus 1 and genotype j at locus 2. Only returned if estimating the composite LD.
nThe number of individuals used to estimate pairwise LD.
David Gerard
ldest()For the base function that estimates pairwise LD.
mldest()For estimating pairwise LD between all provided SNPs.
ldfast()Fast, moment-based approach to LD estimation that also accounts for genotype uncertainty.
format_lddf()For formatting the output of
sldest() as a matrix.
plot.lddf()For plotting the output of
sldest().
set.seed(1) ## Simulate genotypes when true correlation is 0 nloci <- 100 nind <- 100 win <- 5 K <- 6 nc <- 1 genomat <- matrix(sample(0:K, nind * nloci, TRUE), nrow = nloci) ## Composite LD estimates lddf <- sldest(geno = genomat, K = K, win = win, nc = nc, type = "comp") plot(lddf, element = "z")set.seed(1) ## Simulate genotypes when true correlation is 0 nloci <- 100 nind <- 100 win <- 5 K <- 6 nc <- 1 genomat <- matrix(sample(0:K, nind * nloci, TRUE), nrow = nloci) ## Composite LD estimates lddf <- sldest(geno = genomat, K = K, win = win, nc = nc, type = "comp") plot(lddf, element = "z")
10 SNPs from the "PGSC0003DMB000000062" super scaffold were genotyped
using the multidog() function from the updog R package.
These data are the resulting output.
uituit
An object of class multidog().
See the documentation from the updog R package.
doi:10.1371/journal.pone.0062355
Uitdewilligen, Jan GAML, Anne-Marie A. Wolters, B. Bjorn, Theo JA Borm, Richard GF Visser, and Herman J. Van Eck. "A next-generation sequencing method for genotyping-by-sequencing of highly heterozygous autotetraploid potato." PloS one 8, no. 5 (2013): e62355. doi:10.1371/journal.pone.0062355
This function is a wrapper for adaptive shrinkage (Stephens, 2017) on the Fisher-z transformed estimates of the Pearson correlation. This approach was proposed in Dey and Stephens (2018) but is re-implemented here for now since the CorShrink package is not available on CRAN.
zshrink(zmat, smat, ...)zshrink(zmat, smat, ...)
zmat |
The matrix of Fisher-z transformed correlation estimates. |
smat |
The matrix of standard errors of the Fisher-z transformed correlation estimates. |
... |
Additional arguments to pass to |
A matrix of correlation estimates. These are posterior means of the correlation estimates after applying the CorShrink method (Dey and Stephens, 2018).
David Gerard
Stephens, Matthew. "False discovery rates: a new deal." Biostatistics 18, no. 2 (2017): 275-294.
Dey, Kushal K., and Matthew Stephens. "CorShrink: Empirical Bayes shrinkage estimation of correlations, with applications." bioRxiv (2018): 368316.