# Estimating two-locus IBD coefficients Estimates by simulation the two-locus IBD coefficients of a non-inbred pairwise relationship. ```r twoLocusIBD(x, ind1, ind2, rho=NULL, cM=NULL, Nsim, Xchrom=FALSE, verbose=TRUE, ...) ``` ## Arguments - `x`: A pedigree in the form of a `linkdat` object. - `ind1, ind2`: Numeric ID labels of the two individuals. - `rho`: NULL, or a number in the interval [0, 0.5]: the recombination fraction between the two loci. If non-NULL, it is converted to centiMorgan using Haldanes map function: `cM = -50*log(1-2*rho)`. - `cM`: NULL, or a non-negative number: the distance in centiMorgan between the two loci. - `Nsim`: The number of simulations to be performed. - `Xchrom`: A logical indicating if the loci are X-linked (if TRUE) or autosomal (FALSE). - `verbose`: A logical. - ``...``: Further arguments to be passed on to `IBDsim`. ## Details For a given pair of autosomal loci, the two-locus IBD coefficients of a non-inbred pairwise relationship are defined by $$ \kappa[i,j] = Pr(sharing i alleles IBD at locus 1, and j alleles IBD at locus 2), $$ where $0 \le i,j \le 2$. For X-chromosomal loci, and if at least one of the individuals is male, $\kappa[i,j]$ is defined only for $0 \le i,j \le 1$. While the single-locus IBD coefficients $\kappa=(\kappa[0], \kappa[1], \kappa[2])$ (see `oneLocusIBD`) depend only on the genealogy relating the two individuals, the two-locus coefficients also depend on the genetic distance between the loci. In particular, if the loci are completely linked (rho=0; cM=0) we have $\kappa[i,j] = \kappa[i]$ if i=j and 0 otherwise. If the loci are unlinked (rho=0.5; cM=Inf) we have $\kappa[i,j] = \kappa[i]*\kappa[j]$. (See examples.) ## Returns A numerical matrix, where the entry in row $a$ and column $b$ is the estimate of $\kappa[a-1, b-1]$ defined above. ## Author(s) Magnus Dehli Vigeland ## See Also `oneLocusIBD`, `oneLocusJacquard`, `twoLocusJacquard` ## Examples ```r ### Example 1: Full siblings x <- nuclearPed(2) Nsim <- 100 # Should be increased substantially # One-locus kappa estimates (autosomal and X): k_est = oneLocusIBD(x, ind1=3, ind2=4, Nsim=Nsim, seed=123) k_est_X = oneLocusIBD(x, ind1=3, ind2=4, Nsim=Nsim, Xchrom=TRUE, seed=123) ### Two-locus IBD estimation # Completely linked, autosomal rho = 0 k2_linked = twoLocusIBD(x, ind1=3, ind2=4, rho=rho, Nsim=Nsim, seed=123) stopifnot(identical(diag(k2_linked), k_est)) # Completely unlinked, autosomal rho = 0.5 k2_unlinked = twoLocusIBD(x, ind1=3, ind2=4, rho=rho, Nsim=Nsim, seed=123) stopifnot(identical(k2_unlinked, outer(k_est, k_est))) # Recombination rate 10%, autosomal rho <- 0.1 cM <- -50*log(1-2*rho) r1 <- twoLocusIBD(x, ind1=3, ind2=4, rho=rho, Nsim=Nsim, seed=17) r2 <- twoLocusIBD(x, ind1=3, ind2=4, cM=cM, Nsim=Nsim, seed=17) stopifnot(identical(r1, r2)) ### Two-locus IBD on X # Completely linked, X chromosome rho = 0 k2_linked_X = twoLocusIBD(x, ind1=3, ind2=4, rho=rho, Nsim=Nsim, Xchrom=TRUE, seed=123) stopifnot(identical(diag(k2_linked_X), k_est_X)) # Completely unlinked, X chromosome rho = 0.5 k2_unlinked_X = twoLocusIBD(x, ind1=3, ind2=4, rho=rho, Nsim=Nsim, Xchrom=TRUE, seed=123) stopifnot(identical(k2_unlinked_X, outer(k_est_X, k_est_X))) # Recombination rate 10%, X chromosome rho <- 0.1 cM <- -50*log(1-2*rho) r1_X <- twoLocusIBD(x, ind1=3, ind2=4, rho=rho, Nsim=Nsim, Xchrom=TRUE, seed=123) r2_X <- twoLocusIBD(x, ind1=3, ind2=4, cM=cM, Nsim=Nsim, Xchrom=TRUE, seed=123) stopifnot(identical(r1_X, r2_X)) ### Example 2: Comparing half sib, grandparent and uncle # These are indistinguishable with unlinked loci, see e.g. # pages 182-183 in Egeland, Kling and Mostad (2016). # Each simulations followed by exact counterpart. x <- addSon(addSon(nuclearPed(2, sex=1:2), 4), 5) plot(x) rho <- 0.25 Nsim <- 10 # Should be increased to at least 10000 twoLocusIBD(x, 1, 6, rho=rho, Nsim=Nsim, verbose=FALSE)[2,2]; .5*(1-rho) twoLocusIBD(x, 8, 6, rho=rho, Nsim=Nsim, verbose=FALSE)[2,2]; .5*(rho^2+(1-rho)^2) twoLocusIBD(x, 3, 6, rho=rho, Nsim=Nsim, verbose=FALSE)[2,2]; .5*((1-rho)*(rho^2+(1-rho)^2) + rho/2) ### Example 3: X chromosome, granddaughter vs maternal grandfather. y <- addDaughter(nuclearPed(1, sex=2), 3) plot(y) rho <- 0.25 Nsim <-10 twoLocusIBD(y, 1, 5, rho=rho, Nsim=Nsim, Xchrom=TRUE) # Exact matrix(c(1-rho, rho, rho, 1-rho)/2, ncol=2) ```

twoLocusIBD function