Version:	0.72
Date:	2023-03-21
Depends:	R (≥ 2.10.1), qtl
Title:	Tools for Analyzing Crossover Interference
Author:	Karl W Broman [aut, cre], Il-Youp Kwak [ctb]
Maintainer:	Karl W Broman <broman@wisc.edu>
Description:	Analysis of crossover interference in experimental crosses, particularly regarding the gamma model. See, for example, Broman and Weber (2000) <doi:10.1086/302923>.
Imports:	stats, utils
Suggests:	devtools, roxygen2, testthat
License:	GPL-3
URL:	https://github.com/kbroman/xoi
BugReports:	https://github.com/kbroman/xoi/issues
RoxygenNote:	7.1.0
Encoding:	UTF-8
ByteCompile:	true
LazyData:	true
NeedsCompilation:	yes
Packaged:	2023-03-22 12:21:39 UTC; kbroman
Repository:	CRAN
Date/Publication:	2023-03-22 12:40:02 UTC

BSS/BSB backcross data

Description

Data from two densely genotyped backcrosses.

Format

An object of class cross. See qtl::read.cross() for details.

Details

There are 94 individuals from each of two interspecific backcross: (C57BL/6J \times M. spretus) \times C57BL/6J and (C57BL/6J \times SPRET/Ei) \times SPRET/Ei. They were typed on 1372 and 4913 genetic markers, respectively, with 904 markers in common.

These data are from September, 2000. Updated data are available.

Source

Lucy Rowe, Jackson Laboratory

References

Rowe, L. B., Nadeau, J. H., Turner, R., Frankel, W. N., Letts, V. A., Eppig, J. T., Ko, M. S., Thurston, S. J. and Birkenmeier, E. H. (1994) Maps from two interspecific backcross DNA panels available as a community genetic mapping resource. Mamm. Genome 5, 253–274.

Broman, K. W., Rowe, L. B., Churchill, G. A. and Paigen, K. (2002) Crossover interference in the mouse. Genetics 160, 1123–1131.

Examples

data(bssbsb)
summary(bssbsb)
## Not run: plot(bssbsb)

Estimate chiasma distribution from crossover counts

Description

Fit several models, with an assumption of no chromatid interference, to crossover count data to obtain fitted distributions of the number of chiasmata.

Usage

chiasma(
  xo,
  max.chiasma = max(xo) * 2 + 5,
  n.iter = 10000,
  tol = 0.000001,
  verbose = FALSE
)

Arguments

Details

See Broman and Weber (2000) for details of the method.

We use R's stats::integrate() function for numerical integrals, stats::optimize() for optimizing the likelihood, and stats::uniroot() for identifying the endpoints of the likelihood support interval.

Value

A list with three components.

First, a matrix containing the observed distribution of the numbers of crossovers, followed by the fitted distributions under the Poisson model, the truncated Poisson model (assuming an obligate chiasma), the obligate chiasma model, and the freely varying model. In all cases we assume no chromatid interference.

Second, a matrix containing the estimated distributions of the number of chiasmata on the four-strand bundle for the above four models.

Third, the estimated average number of crossovers under the Poisson and truncated Poisson models.

Author(s)

Karl W Broman, broman@wisc.edu

References

Broman, K. W. and Weber, J. L. (2000) Characterization of human crossover interference. Am. J. Hum. Genet. 66, 1911–1926.

Broman, K. W., Rowe, L. B., Churchill, G. A. and Paigen, K. (2002) Crossover interference in the mouse. Genetics 160, 1123–1131.

Yu, K. and Feinbold, E. (2001) Estimating the frequency distribution of crossovers during meiosis from recombination data. Biometrics 57, 427–434.

See Also

fitGamma(), qtl::fitstahl(), countxo()

Examples

data(bssbsb)

# estimated number of crossovers on chr 1
nxo <- countxo(bssbsb, chr=1)

# estimate chiasma distribution
## Not run: chiasma(nxo)

Estimate coincidence function

Description

Estimate coincidence function for a chromosome.

Usage

coincidence(cross, chr = NULL, window = 5, ncalc = 500)

Arguments

Value

Data frame with columns distance and coincidence. The input argument window is kept as an attribute.

Author(s)

Il youp Kwak

See Also

intensity(), est.coi()

Examples

map1 <- sim.map(103, n.mar=104, anchor=TRUE, include.x=FALSE, eq=TRUE)
x <- sim.cross(map1, n.ind=2000, m=6, type="bc")

out <- coincidence(x, ncalc=101)
plot(out, type="l", lwd=2, ylim=c(0, max(out[,2])))

Convert format of crossover locations data

Description

Convert the format of data on crossover locations to that needed for the function fitGamma.

Usage

convertxoloc(breaks)

Arguments

Value

A data frame with two columns: the inter-crossover and crossover-to chromosome end differences ("distance") and indicators of censoring type ("censor"), with 0 = distance between crossovers, 1=start of chromosome to first crossover, 2 = crossover to end of chromosome, and 3 = whole chromosome.

Author(s)

Karl W Broman, broman@wisc.edu

See Also

find.breaks(), fitGamma(), simStahl()

Examples

data(bssbsb)

# crossover locations on chromosome 1
xoloc1 <- convertxoloc(find.breaks(bssbsb, chr=1))

# crossover locations on all chromosomes
xoloc <- convertxoloc(find.breaks(bssbsb))

Estimate number of crossovers

Description

Estimate the number of crossovers in each meiosis in a backcross.

Usage

countxo(cross, chr = NULL)

Arguments

Details

This works only a backcross. We use the internal function (within R/qtl) locate.xo.

Value

A vector with the estimated number of crossovers for each individual.

Author(s)

Karl W Broman, broman@wisc.edu

See Also

find.breaks()

Examples

data(bssbsb)

# estimated number of crossovers on chr 1
nxo <- countxo(bssbsb, chr=1)

# estimated number of crossovers genome-wide
nxo <- countxo(bssbsb)

Distance between crossovers given there are two

Description

Calculates the density of the distance between the crossovers on a meiotic product, given that there are precisely two crossovers, for the gamma model.

Usage

distance.given.two(
  nu,
  L = 103,
  x = NULL,
  n = 400,
  max.conv = 25,
  integr.tol = 0.00000001,
  max.subd = 1000,
  min.subd = 10
)

Arguments

Details

Let f(x;\nu) denote the density of a gamma random variable with parameters shape=\nu and rate=2\nu, and let f_k(x;\nu) denote the density of a gamma random variable with parameters shape=k \nu and rate=2\nu.

The distribution of the distance from one crossover to the next is f^*(x;\nu) = \sum_{k=1}^{\infty} f_k(x;\nu)/2^k.

The distribution of the distance from the start of the chromosome to the first crossover is g^*(x;\nu) = 1 - F^*(x;\nu) where F^* is the cdf of f^*.

We calculate the distribution of the distance between crossovers on a product with two crossovers by first calculating the joint distribution of the location of the two crossovers, given that they both occur before L and the third occurs after L, and then integrating out the location of the first crossover.

Value

A data frame with two columns: x is the distance (between 0 and L, in cM) at which the density was calculated and f is the density.

Warning

We sometimes have difficulty with the numerical integrals. You may need to use large min.subd (e.g. 25) to get accurate results.

Author(s)

Karl W Broman, broman@wisc.edu

References

Broman, K. W. and Weber, J. L. (2000) Characterization of human crossover interference. Am. J. Hum. Genet. 66, 1911–1926.

Broman, K. W., Rowe, L. B., Churchill, G. A. and Paigen, K. (2002) Crossover interference in the mouse. Genetics 160, 1123–1131.

McPeek, M. S. and Speed, T. P. (1995) Modeling interference in genetic recombination. Genetics 139, 1031–1044.

See Also

location.given.one(), first.given.two(),joint.given.two(), ioden(), firstden(), xoprob(), gammacoi()

Examples

f1 <- distance.given.two(1, L=200, n=101)
plot(f1, type="l", lwd=2, las=1,
     ylim=c(0,0.0122), yaxs="i", xaxs="i", xlim=c(0,200))

f2 <- distance.given.two(2.6, L=200, n=101)
lines(f2, col="blue", lwd=2)

## Not run: 
f3 <- distance.given.two(4.3, L=200, n=101)
lines(f3, col="red", lwd=2)

f4 <- distance.given.two(7.6, L=200, n=101)
lines(f4, col="green", lwd=2)

## End(Not run)

Estimate the coincidence function

Description

Estimate the coincidence function from backcross data.

Usage

est.coi(
  cross,
  chr = NULL,
  pos = NULL,
  window = 0,
  fill.method = c("imp", "argmax"),
  error.prob = 0.0000000001,
  map.function = c("haldane", "kosambi", "c-f", "morgan")
)

Arguments

Details

The coincidence function is the probability of a recombination event in both of two intervals, divided by the product of the two recombination fractions. We estimate this as a function of the distance between the two intervals.

Note that we first call qtl::fill.geno() to impute any missing genotype data.

Value

A data.frame containing the distance between intervals and the corresponding estimate of the coincidence. There are actually two columns of estimates of the coincidence. In the first estimate, we take a running mean of each of the numerator and denominator and then divide. In the second estimate, we first take a ratio and then take a running mean.

Author(s)

Karl W Broman, broman@wisc.edu

References

McPeek, M. S. and Speed, T. P. (1995) Modeling interference in genetic recombination. Genetics 139, 1031–1044.

See Also

gammacoi(), stahlcoi(), kfunc()

Examples

map1 <- sim.map(103, n.mar=104, anchor=TRUE, include.x=FALSE, eq=TRUE)
x <- sim.cross(map1, n.ind=2000, m=6, type="bc")

out <- est.coi(x, window=5)
plot(coi1 ~ d, data=out, type="l", lwd=2, col="blue")
lines(coi2 ~ d, data=out, lwd=2, col="green")
lines(gammacoi(7), lwd=2, col="red", lty=2)

Estimate the coincidence as a function of micron distance

Description

Estimate the coincidence as a function of micron distance, with data on XO locations in microns plus SC length in microns.

Usage

est.coi.um(
  xoloc,
  sclength,
  centromeres = NULL,
  group = NULL,
  intwindow = 0.05,
  coiwindow = NULL,
  intloc = NULL,
  coiloc = NULL
)

Arguments

Details

The coincidence function is the probability of a recombination event in both of two intervals, divided by the product of the two intensity function for the two intervals.

We estimate this as a function of the distance between the two intervals in microns, taking account of varying SC lengths,.

Value

A list containing the estimated coincidence (as a matrix with two columns, micron distance and corresponding estimated coincidence) and the estimated intensity functions (as a matrix with length(group)+1 columns (the locations at which the intensity functions were estimated followed by the group-specific estimates).

Author(s)

Karl W Broman, broman@wisc.edu

See Also

gammacoi(), stahlcoi(), kfunc(), est.coi()

Examples

# simple example using data simulated with no crossover interference
ncells <- 1000
L <- 2                      # chr lengths in Morgans (constant here)
nchi <- rpois(ncells, 2*L)  # number of chiasmata
xoloc <- lapply(nchi, function(a) runif(a, 0, L)) # chi locations
coi <- est.coi.um(xoloc, rep(L, ncells))

# plot estimated coincidence and intensity
#    (intensity is after scaling chromosome to length 1)
par(mfrow=c(2,1), las=1)
plot(coi$coincidence, type="l", lwd=2, ylim=c(0, max(coi$coincidence[,2])))
plot(coi$intensity, type="l", lwd=2, ylim=c(0, max(coi$intensity[,2])))

Estimate recombination rate

Description

Obtain a smoothed estimate of the recombination rate along a chromosome, using the cM and Mbp position of markers.

Usage

est.recrate(genmap, phymap, pos = NULL, window = 5)

Arguments

Details

We assume constant recombination rate within each marker interval.

Value

A data.frame containing the positions and estimate recombination rates.

Author(s)

Karl W Broman, broman@wisc.edu

See Also

est.coi(), intensity()

Examples

# create equally-spaced map
pmap <- sim.map(100, n.mar=51, anchor=TRUE, include.x=FALSE, eq.spacing=TRUE)

# simulate cross
x <- sim.cross(pmap, type="bc", n.ind=501)

# estimate map for that cross
emap <- est.map(x)

# empirical estimate of recombination rate
rr <- est.recrate(emap[[1]], pmap[[1]], window=5)
plot(rr, type="l", lwd=2)

Estimate crossover locations

Description

Estimate the locations of crossovers in a backcross.

Usage

find.breaks(cross, chr = NULL)

Arguments

Details

This works only a backcross, RIL, or intercross. We use the function qtl::locateXO() in R/qtl. Crossovers are estimated to be at the midpoint of the interval between the nearest flanking typed markers.

Value

If only one chromosome is considered, this is a list with one component for each individual. If multiple chromosomes were considered, this is a list with one element for each chromosome, each of which is a list with one element for each individual, as above.

For backcrosses and RIL, the componenets for the individuals are numeric(0) if there were no crossovers or a vector giving the crossover locations. The length of the chromosome (in cM) is saved as an attribute. (Note that the format is the same as the output of simStahl().)

For an intercross, the components for the individuals are themselves lists with all possible allocations of the crossovers to the two meiotic products; each component of this list is itself a list with two components, corresponding to the two meiotic products.

Author(s)

Karl W Broman, broman@wisc.edu

See Also

convertxoloc(), fitGamma(), simStahl()

Examples

data(bssbsb)

# crossover locations on chromosome 1
xoloc1 <- find.breaks(bssbsb, chr=1)

# crossover locations on all chromosomes
xoloc <- find.breaks(bssbsb)

Location of first crossover given there are two

Description

Calculates the density of the location of the first crossover on a random meiotic product, given that there are precisely two crossovers, for the gamma model.

Usage

first.given.two(
  nu,
  L = 103,
  x = NULL,
  n = 400,
  max.conv = 25,
  integr.tol = 0.00000001,
  max.subd = 1000,
  min.subd = 10
)

Arguments

Details

Let f(x;\nu) denote the density of a gamma random variable with parameters shape=\nu and rate=2\nu, and let f_k(x;\nu) denote the density of a gamma random variable with parameters shape=k \nu and rate=2\nu.

The distribution of the distance from one crossover to the next is f^*(x;\nu) = \sum_{k=1}^{\infty} f_k(x;\nu)/2^k.

The distribution of the distance from the start of the chromosome to the first crossover is g^*(x;\nu) = 1 - F^*(x;\nu) where F^* is the cdf of f^*.

We calculate the distribution of the location of the first crossover in a product with two crossovers by calculating the joint distribution of the location of the two crossovers, given that they both occur before L and the third occurs after L, and then integrating out the location of the second crossover.

Value

A data frame with two columns: x is the location (between 0 and L, in cM) at which the density was calculated and f is the density.

Warning

We sometimes have difficulty with the numerical integrals. You may need to use large min.subd (e.g. 25) to get accurate results.

Author(s)

Karl W Broman, broman@wisc.edu

References

Broman, K. W. and Weber, J. L. (2000) Characterization of human crossover interference. Am. J. Hum. Genet. 66, 1911–1926.

Broman, K. W., Rowe, L. B., Churchill, G. A. and Paigen, K. (2002) Crossover interference in the mouse. Genetics 160, 1123–1131.

McPeek, M. S. and Speed, T. P. (1995) Modeling interference in genetic recombination. Genetics 139, 1031–1044.

See Also

location.given.one(), distance.given.two(), joint.given.two(), ioden(), firstden(), xoprob(), gammacoi()

Examples

f1 <- first.given.two(1, L=200, n=101)
plot(f1, type="l", lwd=2, las=1,
     ylim=c(0,0.011), yaxs="i", xaxs="i", xlim=c(0,200))

f2 <- first.given.two(2.6, L=200, n=101)
lines(f2, col="blue", lwd=2)

## Not run: 
f3 <- first.given.two(4.3, L=200, n=101)
lines(f3, col="red", lwd=2)

f4 <- first.given.two(7.6, L=200, n=101)
lines(f4, col="green", lwd=2)

## End(Not run)

Distance to a crossover

Description

Calculates the density of the distance from an arbitrary point to the next crossover, for the gamma model.

Usage

firstden(nu, L = 103, x = NULL, n = 400, max.conv = 25)

Arguments

Details

Let f(x;\nu) denote the density of a gamma random variable with parameters shape=\nu and rate=2\nu, and let f_k(x;\nu) denote the density of a gamma random variable with parameters shape=k \nu and rate=2\nu.

The distribution of the distance from one crossover to the next is f^*(x;\nu) = \sum_{k=1}^{\infty} f_k(x;\nu)/2^k.

The distribution of the distance from an arbitrary point to the first crossover is g^*(x;\nu) = 1 - F^*(x;\nu) where F^* is the cdf of f^*.

Value

A data frame with two columns: x is the distance (between 0 and L, in cM) at which the density was calculated and f is the density.

Author(s)

Karl W Broman, broman@wisc.edu

References

Broman, K. W. and Weber, J. L. (2000) Characterization of human crossover interference. Am. J. Hum. Genet. 66, 1911–1926.

Broman, K. W., Rowe, L. B., Churchill, G. A. and Paigen, K. (2002) Crossover interference in the mouse. Genetics 160, 1123–1131.

McPeek, M. S. and Speed, T. P. (1995) Modeling interference in genetic recombination. Genetics 139, 1031–1044.

See Also

location.given.one(), first.given.two(), distance.given.two(), joint.given.two(), ioden(), xoprob(), gammacoi()

Examples

f1 <- firstden(1, L=200, n=201)
plot(f1, type="l", lwd=2, las=1,
     ylim=c(0,0.012), yaxs="i", xaxs="i", xlim=c(0,200))

f2 <- firstden(2.6, L=200, n=201)
lines(f2, col="blue", lwd=2)

f3 <- firstden(4.3, L=200, n=201)
lines(f3, col="red", lwd=2)

f4 <- firstden(7.6, L=200, n=201)
lines(f4, col="green", lwd=2)

Fit Gamma model

Description

Fit the gamma model for crossover interference to data on crossover locations.

Usage

fitGamma(
  d,
  censor = NULL,
  nu = NULL,
  lo = NULL,
  hi = NULL,
  se = FALSE,
  supint = FALSE,
  rescale = FALSE,
  drop = 1.5,
  tol = 0.00001,
  maxit = 1000,
  max.conv = 25,
  integr.tol = 0.00000001,
  max.subd = 1000,
  min.subd = 10,
  h = 0.1,
  hstep = 1.5
)

Arguments

Details

See Broman and Weber (2000) for details of the method.

We use R's stats::integrate() function for numerical integrals, stats::optimize() for optimizing the likelihood, and stats::uniroot() for identifying the endpoints of the likelihood support interval.

Value

If nu is specified, we return a data frame with two columns: nu and the corresponding log (base e) likelihood. If rescale=TRUE, the maximum log likelihood is subtracted off, so that its maximum is at 0.

If lo and hi is specified, the output contains a single row with the MLE of \nu and the corresponding log likelihood. If se=TRUE, we also include the estimated SE. If supint=TRUE, we include two additional rows with the lower and upper limits of the likelihood support interval.

Author(s)

Karl W Broman, broman@wisc.edu

References

Broman, K. W. and Weber, J. L. (2000) Characterization of human crossover interference. Am. J. Hum. Genet. 66, 1911–1926.

Broman, K. W., Rowe, L. B., Churchill, G. A. and Paigen, K. (2002) Crossover interference in the mouse. Genetics 160, 1123–1131.

McPeek, M. S. and Speed, T. P. (1995) Modeling interference in genetic recombination. Genetics 139, 1031–1044.

See Also

qtl::fitstahl()

Examples

data(bssbsb)


xodist <- convertxoloc(find.breaks(bssbsb, chr=1))

# plot a rough log likelihood curve
## Not run: out <- fitGamma(xodist, nu=seq(1, 19, by=2))

plot(out, type="l", lwd=2)

# get MLE
## Not run: mle <- fitGamma(xodist, lo=8, hi=12)

mle

abline(v=mle[1], h=mle[2], col="blue", lty=2)

# get MLE and SE
## Not run: mle <- fitGamma(xodist, lo=9.5, hi=10.5, se=TRUE)

mle

# get MLE and 10^1.5 support interval
## Not run: int <- fitGamma(xodist, lo=1, hi=20, supint=TRUE)

int
abline(v=mle[2:3,1], h=mle[2:3,2], col="red", lty=2)

Fit Stahl model

Description

Fit the Stahl model for crossover interference to data on crossover locations.

Usage

fitStahl(
  xoloc,
  chrlen = NULL,
  nu = c(1, 20),
  p = 0.02,
  max.conv = 25,
  integr.tol = 0.00000001,
  max.subd = 1000,
  min.subd = 10,
  verbose = TRUE,
  ...
)

Arguments

Details

See Housworth and Stahl (2003) and Broman and Weber (2000) for details of the method.

We first use stats::optimize() to find the MLE with the contraint p=0, followed by use of stats::optim() to do a 2-dimensional optimization for the MLEs of the pair.

Value

A vector with the estimates of \nu (interference parameter) and p (proportion of crossovers coming from the no interference pathway), the maximized log likelihood, the estimate of nu with p constrained to be 0, the maximized log likelihood in this case, and the log likelihood ratio for comparing the model with p allowed to vary freely versus contrained to be 0. (Note that it's the natural log of the likelihood ratio, and not twice that.)

Author(s)

Karl W Broman, broman@wisc.edu

References

Housworth, E. A. and Stahl, F. W. (2003) Crossover interference in humans. Am. J. Hum. Genet. 73, 188–197.

Broman, K. W. and Weber, J. L. (2000) Characterization of human crossover interference. Am. J. Hum. Genet. 66, 1911–1926.

See Also

fitGamma(), stahlLoglik(), simStahl()

Examples

data(bssbsb)


xoloc <- find.breaks(bssbsb, chr=1)
L <- attr(xoloc, "L")

# get MLE (limiting maximum iterations to 10, just for speed in this example)
## Not run: mle <- fitStahl(xoloc, L, nu=c(9, 12), control=list(maxit=10))

Coincidence function for the gamma model

Description

Calculates the coincidence function for the gamma model.

Usage

gammacoi(nu, L = 103, x = NULL, n = 400, max.conv = 25)

Arguments

Details

Let f(x;\nu) denote the density of a gamma random variable with parameters shape=\nu and rate=2\nu, and let f_k(x;\nu) denote the density of a gamma random variable with parameters shape=k \nu and rate=2\nu.

The coincidence function for the gamma model is C(x;\nu) = \sum_{k=1}^{\infty} f_k(x;\nu)/2.

Value

A data frame with two columns: x is the distance (between 0 and L, in cM) at which the coicidence was calculated and coincidence.

Author(s)

Karl W Broman, broman@wisc.edu

References

Broman, K. W. and Weber, J. L. (2000) Characterization of human crossover interference. Am. J. Hum. Genet. 66, 1911–1926.

Broman, K. W., Rowe, L. B., Churchill, G. A. and Paigen, K. (2002) Crossover interference in the mouse. Genetics 160, 1123–1131.

McPeek, M. S. and Speed, T. P. (1995) Modeling interference in genetic recombination. Genetics 139, 1031–1044.

See Also

stahlcoi(), location.given.one(), first.given.two(), distance.given.two(), joint.given.two(), ioden(), firstden(), xoprob()

Examples

f1 <- gammacoi(1, L=200)
plot(f1, type="l", lwd=2, las=1,
     ylim=c(0,1.25), yaxs="i", xaxs="i", xlim=c(0,200))

f2 <- gammacoi(2.6, L=200)
lines(f2, col="blue", lwd=2)

f3 <- gammacoi(4.3, L=200)
lines(f3, col="red", lwd=2)

f4 <- gammacoi(7.6, L=200)
lines(f4, col="green", lwd=2)

Estimate intensity function

Description

Estimate intensity function for a chromosome.

Usage

intensity(cross, chr = NULL, window = 2.5, ncalc = 500)

Arguments

Value

Data frame with columns position and intensity. The input argument window is kept as an attribute.

Author(s)

Il youp Kwak

See Also

coincidence()

Examples

map1 <- sim.map(103, n.mar=104, anchor=TRUE, include.x=FALSE, eq=TRUE)
x <- sim.cross(map1, n.ind=2000, m=6, type="bc")

out <- intensity(x)
plot(out, type="l", lwd=2, ylim=c(0, max(out[,2])))

Distance between crossovers

Description

Calculates the density of the distance from a given crossover to the next crossover, for the gamma model.

Usage

ioden(nu, L = 103, x = NULL, n = 400, max.conv = 25)

Arguments

Details

Let f(x;\nu) denote the density of a gamma random variable with parameters shape=\nu and rate=2\nu, and let f_k(x;\nu) denote the density of a gamma random variable with parameters shape=k \nu and rate=2\nu.

The distribution of the distance from one crossover to the next is f^*(x;\nu) = \sum_{k=1}^{\infty} f_k(x;\nu)/2^k.

Value

A data frame with two columns: x is the distance (between 0 and L, in cM) at which the density was calculated and f is the density.

Author(s)

Karl W Broman, broman@wisc.edu

References

Broman, K. W. and Weber, J. L. (2000) Characterization of human crossover interference. Am. J. Hum. Genet. 66, 1911–1926.

Broman, K. W., Rowe, L. B., Churchill, G. A. and Paigen, K. (2002) Crossover interference in the mouse. Genetics 160, 1123–1131.

McPeek, M. S. and Speed, T. P. (1995) Modeling interference in genetic recombination. Genetics 139, 1031–1044.

See Also

location.given.one(), first.given.two(), distance.given.two(), joint.given.two(), firstden(), xoprob(), gammacoi()

Examples

f1 <- ioden(1, L=200, n=201)
plot(f1, type="l", lwd=2, las=1,
     ylim=c(0,0.014), yaxs="i", xaxs="i", xlim=c(0,200))

f2 <- ioden(2.6, L=200, n=201)
lines(f2, col="blue", lwd=2)

f3 <- ioden(4.3, L=200, n=201)
lines(f3, col="red", lwd=2)

f4 <- ioden(7.6, L=200, n=201)
lines(f4, col="green", lwd=2)

Crossover locations given there are two

Description

Calculates the joint density of the crossover locations on a random meiotic product, given that there are precisely two crossovers, for the gamma model.

Usage

joint.given.two(
  nu,
  L = 103,
  x = NULL,
  y = NULL,
  n = 20,
  max.conv = 25,
  integr.tol = 0.00000001,
  max.subd = 1000,
  min.subd = 10
)

Arguments

Details

Let f(x;\nu) denote the density of a gamma random variable with parameters shape=\nu and rate=2\nu, and let f_k(x;\nu) denote the density of a gamma random variable with parameters shape=k \nu and rate=2\nu.

The distribution of the distance from one crossover to the next is f^*(x;\nu) = \sum_{k=1}^{\infty} f_k(x;\nu)/2^k.

The distribution of the distance from the start of the chromosome to the first crossover is g^*(x;\nu) = 1 - F^*(x;\nu) where F^* is the cdf of f^*.

Value

A data frame with three columns: x and y are the locations (between 0 and L, in cM) at which the density was calculated and f is the density.

Warning

We sometimes have difficulty with the numerical integrals. You may need to use large min.subd (e.g. 25) to get accurate results.

Author(s)

Karl W Broman, broman@wisc.edu

References

Broman, K. W. and Weber, J. L. (2000) Characterization of human crossover interference. Am. J. Hum. Genet. 66, 1911–1926.

Broman, K. W., Rowe, L. B., Churchill, G. A. and Paigen, K. (2002) Crossover interference in the mouse. Genetics 160, 1123–1131.

McPeek, M. S. and Speed, T. P. (1995) Modeling interference in genetic recombination. Genetics 139, 1031–1044.

See Also

location.given.one(), distance.given.two(), first.given.two(), ioden(), firstden(), xoprob(), gammacoi()

Examples

# Calculate the distribution of the average of the crossover locations,
# given that there are two and that they are separated by 20 cM
# (for a chromosome of length 200 cM)
L <- 200
d <- 20
x <- seq(0, L-d, by=0.5)
y <- x+d

f <- joint.given.two(4.3, L=L, x, y)
f$f <- f$f / distance.given.two(4.3, L, d)$f
plot((f$x+f$y)/2, f$f, type="l", xlim=c(0, L), ylim=c(0,max(f$f)),
     lwd=2, xlab="Average location", ylab="Density")
abline(v=c(d/2,L-d/2), h=1/(L-d), lty=2, lwd=2)

estimate Ripley's K function

Description

estimate the 1-d version of Ripley's K function

Usage

kfunc(
  x,
  d = seq(0, 100, by = 0.1),
  lengths = NULL,
  exclude = 0,
  tol = 0.000001
)

Arguments

Value

data frame with d, k, and se

See Also

gammacoi(), stahlcoi(), coincidence()

Examples

L <- 103
n <- 2000
map1 <- sim.map(L, n.mar=104, anchor=TRUE, include.x=FALSE, eq=TRUE)
x <- sim.cross(map1, n.ind=n, m=6, type="bc")

xoloc <- find.breaks(x)

d <- seq(0, 100, by=0.1)[-1]
kf <- kfunc(xoloc, d=d, lengths=rep(L, n))

plot(k ~ d, data=kf, type="n", yaxs="i", xaxs="i", las=1,
     ylim=c(0, max(kf$k + kf$se)))
polygon(c(kf$d, rev(kf$d)), c(kf$k + kf$se, rev(kf$k-kf$se)),
        border=NA, col="#add8e650")
lines(k ~ d, data=kf)

Location of crossover given there is one

Description

Calculates the density of the location of the crossover on a random meiotic product, given that there is precisely one crossover, for the gamma model.

Usage

location.given.one(
  nu,
  L = 103,
  x = NULL,
  n = 400,
  max.conv = 25,
  integr.tol = 0.00000001,
  max.subd = 1000,
  min.subd = 10
)

Arguments

Details

Let f(x;\nu) denote the density of a gamma random variable with parameters shape=\nu and rate=2\nu, and let f_k(x;\nu) denote the density of a gamma random variable with parameters shape=k \nu and rate=2\nu.

The distribution of the distance from one crossover to the next is f^*(x;\nu) = \sum_{k=1}^{\infty} f_k(x;\nu)/2^k.

The distribution of the distance from the start of the chromosome to the first crossover is g^*(x;\nu) = 1 - F^*(x;\nu) where F^* is the cdf of f^*.

We calculate the distribution of the location of the crossover on a product with a single crossover as the convolution of g^* with itself, and then rescaled to integrate to 1 on the interval (0,L).

Value

A data frame with two columns: x is the location (between 0 and L, in cM) at which the density was calculated and f is the density.

Author(s)

Karl W Broman, broman@wisc.edu

References

Broman, K. W. and Weber, J. L. (2000) Characterization of human crossover interference. Am. J. Hum. Genet. 66, 1911–1926.

Broman, K. W., Rowe, L. B., Churchill, G. A. and Paigen, K. (2002) Crossover interference in the mouse. Genetics 160, 1123–1131.

McPeek, M. S. and Speed, T. P. (1995) Modeling interference in genetic recombination. Genetics 139, 1031–1044.

See Also

first.given.two(), distance.given.two(), joint.given.two(), ioden(), firstden(), xoprob(), gammacoi()

Examples

f1 <- location.given.one(1, L=200, n=201)
plot(f1, type="l", lwd=2, las=1,
     ylim=c(0,0.006), yaxs="i", xaxs="i", xlim=c(0,200))

f2 <- location.given.one(2.6, L=200, n=201)
lines(f2, col="blue", lwd=2)

f3 <- location.given.one(4.3, L=200, n=201)
lines(f3, col="red", lwd=2)

f4 <- location.given.one(7.6, L=200, n=201)
lines(f4, col="green", lwd=2)

Convert recrate to scanone format

Description

Convert the result of est.recrate() to the format output by R/qtl's qtl::scanone() function.

Usage

recrate2scanone(recrate, phymap = NULL)

Arguments

Value

A data frame with class "scanone", in the format output by qtl::scanone().

Author(s)

Karl W Broman, broman@wisc.edu

See Also

est.recrate()

Examples

pmap <- sim.map(100, n.mar=51, anchor=TRUE, include.x=FALSE, eq.spacing=TRUE)

# simulate cross
x <- sim.cross(pmap, type="bc", n.ind=501)

# estimate map for that cross
emap <- est.map(x)

# empirical estimate of recombination rate
rr <- est.recrate(emap[[1]], pmap[[1]], window=5)

# make it a list (one component per chromosome, but here just the one chromosome)
rr <- list("1"=rr)

# convert to scanone output and plot
rr_scanone <- recrate2scanone(rr)
plot(rr_scanone)

Simulate crossover locations under the Stahl model

Description

Simulate crossover locations under the Stahl model.

Usage

simStahl(
  n.sim,
  nu = 1,
  p = 0,
  L = 100,
  obligate_chiasma = FALSE,
  n.bins4start = 10000
)

Arguments

Details

The Stahl model is an extension to the gamma model, in which chiasmata occur according to two independent mechanisms. A proportion p come from a mechanism exhibiting no interference, and a proportion 1-p come from a mechanism in which chiasma locations follow a gamma model with interference parameter \nu.

Value

A vector of length n.sim, each element being empty (for products with no crossovers) or a vector of crossover locations, in cM. An attribute, L, contains the chromosome length in cM.

Author(s)

Karl W Broman, broman@wisc.edu

References

Copenhaver, G. P., Housworth, E. A. and Stahl, F. W. (2002) Crossover interference in Arabidopsis. Genetics 160, 1631–1639.

Housworth, E. A. and Stahl, F. W. (2003) Crossover interference in humans. Am J Hum Genet 73, 188–197.

See Also

fitGamma(), qtl::sim.cross()

Examples

# simulations with no interference, chromosome of length 80 cM
xoNI <- simStahl(100, nu=1, p=0, L=80)

# simulations under gamma model with nu=7.6
xogamma <- simStahl(100, nu=7.6, p=0, L=80)

# simulations under Stahl model with nu=7.6, p=0.1
xostahl <- simStahl(100, nu=7.6, p=0.1, L=80)

# simulations under chi-square model with nu=11 (m=10) and obligate chiasma
xo_oblchi <- simStahl(100, nu=11, p=0, L=80, obligate_chiasma=TRUE)

# simulations under Stahl model with nu=11, p=0.1, and obligate chiasma
xo_oblchi_stahl <- simStahl(100, nu=11, p=0.1, L=80, obligate_chiasma=TRUE)

Calculate log likelihood for Stahl model

Description

Calculate the log likelihood for the Stahl model for varying parameters, with data on crossover locations.

Usage

stahlLoglik(
  xoloc,
  chrlen = NULL,
  nu,
  p,
  max.conv = 25,
  integr.tol = 0.00000001,
  max.subd = 1000,
  min.subd = 10
)

Arguments

Details

See Housworth and Stahl (2003) and Broman and Weber (2000) for details of the method.

If neither nu nor p has length 1, they both must have the same length. If one has length 1 and the other does not, the one with length 1 is repeated so that they both have the same length.

Value

A vector of log likelihoods.

The corresponding values of nu and p are saved as attributes.

Author(s)

Karl W Broman, broman@wisc.edu

References

Housworth, E. A. and Stahl, F. W. (2003) Crossover interference in humans. Am. J. Hum. Genet. 73, 188–197.

Broman, K. W. and Weber, J. L. (2000) Characterization of human crossover interference. Am. J. Hum. Genet. 66, 1911–1926.

See Also

qtl::fitstahl()

Examples

data(bssbsb)
xoloc <- find.breaks(bssbsb, chr=1)

loglik <- stahlLoglik(xoloc, nu=4, p=c(0.05, 0.1, 0.15))

Coincidence function for the Stahl model

Description

Calculates the coincidence function for the Stahl model.

Usage

stahlcoi(nu, p = 0, L = 103, x = NULL, n = 400, max.conv = 25)

Arguments

Details

The Stahl model is an extension to the gamma model, in which chiasmata occur according to two independent mechanisms. A proportion p come from a mechanism exhibiting no interference, and a proportion 1-p come from a mechanism in which chiasma locations follow a gamma model with interference parameter \nu.

Let f(x;\nu,\lambda) denote the density of a gamma random variable with parameters shape=\nu and rate=\lambda.

The coincidence function for the Stahl model is C(x;\nu,p) = [p + \sum_{k=1}^{\infty} f(x;k\nu, 2(1-p)\nu)]/2.

Value

A data frame with two columns: x is the distance (between 0 and L, in cM) at which the coicidence was calculated and coincidence.

Author(s)

Karl W Broman, broman@wisc.edu

References

Copenhaver, G. P., Housworth, E. A. and Stahl, F. W. (2002) Crossover interference in Arabidopsis. Genetics 160, 1631–1639.

Housworth, E. A. and Stahl, F. W. (2003) Crossover interference in humans. Am J Hum Genet 73, 188–197.

See Also

gammacoi(), location.given.one(), first.given.two(), distance.given.two(), ioden(), firstden(), xoprob()

Examples

f1 <- stahlcoi(1, p=0, L=200)
plot(f1, type="l", lwd=2, las=1,
     ylim=c(0,1.25), yaxs="i", xaxs="i", xlim=c(0,200))

f2 <- stahlcoi(2.6, p=0, L=200)
lines(f2, col="blue", lwd=2)

f2s <- stahlcoi(2.6, p=0.1, L=200)
lines(f2s, col="blue", lwd=2, lty=2)

f3 <- stahlcoi(4.3, p=0, L=200)
lines(f3, col="red", lwd=2)

f3s <- stahlcoi(4.3, p=0.1, L=200)
lines(f3s, col="red", lwd=2, lty=2)

f4 <- stahlcoi(7.6, p=0, L=200)
lines(f4, col="green", lwd=2)

f4s <- stahlcoi(7.6, p=0.1, L=200)
lines(f4s, col="green", lwd=2, lty=2)

Installed version of R/xoi

Description

Print the version number of the currently installed version of R/xoi.

Usage

xoiversion()

Value

A character string with the version number of the currently installed version of R/xoi.

Author(s)

Karl W Broman, broman@wisc.edu

Examples

xoiversion()

Distribution of number of crossovers

Description

Calculates the probability of 0, 1, 2, or >2 crossovers for a chromosome of a given length, for the gamma model.

Usage

xoprob(
  nu,
  L = 103,
  max.conv = 25,
  integr.tol = 0.00000001,
  max.subd = 1000,
  min.subd = 10
)

Arguments

Details

Let f(x;\nu) denote the density of a gamma random variable with parameters shape=\nu and rate=2\nu, and let f_k(x;\nu) denote the density of a gamma random variable with parameters shape=k \nu and rate=2\nu.

The distribution of the distance from one crossover to the next is f^*(x;\nu) = \sum_{k=1}^{\infty} f_k(x;\nu)/2^k.

The distribution of the distance from the start of the chromosome to the first crossover is g^*(x;\nu) = 1 - F^*(x;\nu) where F^* is the cdf of f^*.

We calculate the desired probabilities by numerical integration.

Value

A vector of length 4, giving the probabilities of 0, 1, 2, or >2 crossovers, respectively, on a chromosome of length L cM.

Author(s)

Karl W Broman, broman@wisc.edu

References

Broman, K. W. and Weber, J. L. (2000) Characterization of human crossover interference. Am. J. Hum. Genet. 66, 1911–1926.

Broman, K. W., Rowe, L. B., Churchill, G. A. and Paigen, K. (2002) Crossover interference in the mouse. Genetics 160, 1123–1131.

McPeek, M. S. and Speed, T. P. (1995) Modeling interference in genetic recombination. Genetics 139, 1031–1044.

See Also

location.given.one(), first.given.two(), distance.given.two(), joint.given.two(), ioden(), firstden(), gammacoi()

Examples

xoprob(1, L=103)
xoprob(4.3, L=103)