Genome scan with a single QTL model

Description

Genome scan with a single QTL model, with possible allowance for covariates, using any of several possible models for the phenotype and any of several possible numerical methods.

Usage

scanone(cross, chr, pheno.col=1, model=c("normal","binary","2part","np"),
        method=c("em","imp","hk","ehk","mr","mr-imp","mr-argmax"),
        addcovar=NULL, intcovar=NULL, weights=NULL,
        use=c("all.obs", "complete.obs"), upper=FALSE,
        ties.random=FALSE, start=NULL, maxit=4000,
        tol=1e-4, n.perm, perm.Xsp=FALSE, perm.strata=NULL, verbose,
        batchsize=250)

Arguments

Details

Use of the EM algorithm, Haley-Knott regression, and the extended Haley-Knott method require that multipoint genotype probabilities are first calculated using calc.genoprob. The imputation method uses the results of sim.geno.

Individuals with missing phenotypes are dropped.

In the case that n.perm>0, so that a permutation test is performed, the R function scanone is called repeatedly. If perm.Xsp=TRUE, separate permutations are performed for the autosomes and the X chromosome, so that an X-chromosome-specific threshold may be calculated. In this case, n.perm specifies the number of permutations used for the autosomes; for the X chromosome, n.perm * L_A/L_X permutations will be run, where L_A and L_X are the total genetic lengths of the autosomes and X chromosome, respectively. More permutations are needed for the X chromosome in order to obtain thresholds of similar accuracy.

For further details on the models, the methods and the use of covariates, see below.

Value

If n.perm is missing, the function returns a data.frame whose first two columns contain the chromosome IDs and cM positions. Subsequent columns contain the LOD scores for each phenotype. In the case of the two-part model, there are three LOD score columns for each phenotype: LOD(p,mu), LOD(p) and LOD(mu). The result is given class "scanone" and has attributes "model", "method", "df" and "type" (the latter is the type of cross analyzed).
If n.perm is specified, the function returns the results of a permutation test and the output has class "scanoneperm". If perm.Xsp=FALSE, the function returns a matrix with n.perm rows, each row containing the genome-wide maximum LOD score for each of the phenotypes. In the case of the two-part model, there are three columns for each phenotype, corresponding to the three different LOD scores. If perm.Xsp=TRUE, the result contains separate permutation results for the autosomes and the X chromosome respectively, and an attribute indicates the lengths of the chromosomes and an indicator of which chromosome is X.

Models

The normal model is the standard model for QTL mapping (see Lander and Botstein 1989). The residual phenotypic variation is assumed to follow a normal distribution, and analysis is analogous to analysis of variance.

The binary model is for the case of a binary phenotype, which must have values 0 and 1. The proportions of 1's in the different genotype groups are compared. Currently only methods em and mr are available for this model. See Xu and Atchley (1996) and Broman (2003).

The two-part model is appropriate for the case of a spike in the phenotype distribution (for example, metastatic density when many individuals show no metastasis, or survival time following an infection when individuals may recover from the infection and fail to die). The two-part model was described by Boyartchuk et al. (2001) and Broman (2003). Individuals with QTL genotype g have probability p[g] of having an undefined phenotype (the spike), while if their phenotype is defined, it comes from a normal distribution with mean mu[g] and common standard deviation s. Three LOD scores are calculated: LOD(p,mu) is for the test of the hypothesis that p[g] = p and mu[g] = mu. LOD(p) is for the test that p[g] = p while the mu[g] may vary. LOD(mu) is for the test that mu[g] = mu while the p[g] may vary.

With the non-parametric "model", an extension of the Kruskal-Wallis test is used; this is similar to the method described by Kruglyak and Lander (1995). In the case of incomplete genotype information (such as at locations between genetic markers), the Kruskal-Wallis statistic is modified so that the rank for each individual is weighted by the genotype probabilities, analogous to Haley-Knott regression. For this method, if the argument ties.random is TRUE, ties in the phenotypes are assigned random ranks; if it is FALSE, average ranks are used and a corrected LOD score is calculate. Currently the method argument is ignored for this model.

Methods

em: maximum likelihood is performed via the EM algorithm (Dempster et al. 1977), first used in this context by Lander and Botstein (1989).

imp: multiple imputation is used, as described by Sen and Churchill (2001).

hk: Haley-Knott regression is used (regression of the phenotypes on the multipoint QTL genotype probabilities), as described by Haley and Knott (1992).

ehk: the extended Haley-Knott method is used (like H-K, but taking account of the variances), as described in Feenstra et al. (2006).

mr: Marker regression is used. Analysis is performed only at the genetic markers, and individuals with missing genotypes are discarded. See Soller et al. (1976).

Covariates

Covariates are allowed only for the normal and binary models. The normal model is y = b[q] + A g + Z d[q] + e where q is the unknown QTL genotype, A is a matrix of additive covariates, and Z is a matrix of covariates that interact with the QTL genotype. The columns of Z are forced to be contained in the matrix A. The binary model is the logistic regression analog.

The LOD score is calculated comparing the likelihood of the above model to that of the null model y = m + A g + e.

Covariates must be numeric matrices. Individuals with any missing covariates are discarded.

X chromosome

The X chromosome must be treated specially in QTL mapping. See Broman et al. (in press).

If both males and females are included, male hemizygotes are allowed to be different from female homozygotes. Thus, in a backcross, we will fit separate means for the genotype classes AA, AB, AY, and BY. In such cases, sex differences in the phenotype could cause spurious linkage to the X chromosome, and so the null hypothesis must be changed to allow for a sex difference in the phenotype.

Numerous special cases must be considered, as detailed in the following table.

In the case that the number of degrees of freedom for the linkage test for the X chromosome is different from that for autosomes, a separate X-chromosome LOD threshold is recommended. Autosome- and X-chromosome-specific LOD thresholds may be estimated by permutation tests with scanone by setting n.perm>0 and using perm.Xsp=TRUE.

Author(s)

Karl W Broman, kbroman@biostat.wisc.edu; Hao Wu

References

Boyartchuk V. L., Broman, K. W., Mosher, R. E., D'Orazio S. E. F., Starnbach, M. N. and Dietrich, W. F. (2001) Multigenic control of Listeria monocytogenes susceptibility in mice. Nature Genetics 27, 259–260.

Broman, K. W. (2003) Mapping quantitative trait loci in the case of a spike in the phenotype distribution. Genetics 163, 1169–1175.

Broman, K. W., Sen, 'S, Owens, S. E., Manichaikul, A., Southard-Smith, E. M. and Churchill G. A. The X chromosome in quantitative trait locus mapping. Genetics, to appear.

Churchill, G. A. and Doerge, R. W. (1994) Empirical threshold values for quantitative trait mapping. Genetics 138, 963–971.

Dempster, A. P., Laird, N. M. and Rubin, D. B. (1977) Maximum likelihood from incomplete data via the EM algorithm. J. Roy. Statist. Soc. B, 39, 1–38.

Feenstra, B., Skovgaard, I. M. and Broman, K. W. (2006) Mapping quantitative trait loci by an extension of the Haley-Knott regression method using estimating equations. Genetics, 173, 2111–2119.

Haley, C. S. and Knott, S. A. (1992) A simple regression method for mapping quantitative trait loci in line crosses using flanking markers. Heredity 69, 315–324.

Kruglyak, L. and Lander, E. S. (1995) A nonparametric approach for mapping quantitative trait loci. Genetics 139, 1421–1428.

Lander, E. S. and Botstein, D. (1989) Mapping Mendelian factors underlying quantitative traits using RFLP linkage maps. Genetics 121, 185–199.

Sen, 'S. and Churchill, G. A. (2001) A statistical framework for quantitative trait mapping. Genetics 159, 371–387.

Soller, M., Brody, T. and Genizi, A. (1976) On the power of experimental designs for the detection of linkage between marker loci and quantitative loci in crosses between inbred lines. Theor. Appl. Genet. 47, 35–39.

Xu, S., and Atchley, W.R. (1996) Mapping quantitative trait loci for complex binary diseases using line crosses. Genetics 143, 1417–1424.

See Also

plot.scanone, summary.scanone, scantwo, calc.genoprob, sim.geno, max.scanone, summary.scanoneperm, -.scanone, +.scanone

Examples

###################
# Normal Model
###################
data(hyper)

# Genotype probabilities for EM and H-K
hyper <- calc.genoprob(hyper, step=2.5)
out.em <- scanone(hyper, method="em")
out.hk <- scanone(hyper, method="hk")

# Summarize results: peaks above 3
summary(out.em, thr=3)
summary(out.hk, thr=3)

# An alternate method of summarizing:
#     patch them together and then summarize
out <- c(out.em, out.hk)
summary(out, thr=3, format="allpeaks")

# Plot the results
plot(out.hk, out.em)
plot(out.hk, out.em, chr=c(1,4), lty=1, col=c("blue","black"))

# Imputation; first need to run sim.geno
# Do just chromosomes 1 and 4, to save time
hyper.c1n4 <- sim.geno(subset(hyper, chr=c(1,4)),
                       step=2.5, n.draws=8)
out.imp <- scanone(hyper.c1n4, method="imp")
summary(out.imp, thr=3)

# Plot all three results
plot(out.imp, out.hk, out.em, chr=c(1,4), lty=1,
     col=c("red","blue","black"))

# extended Haley-Knott
out.ehk <- scanone(hyper, method="ehk")
plot(out.hk, out.em, out.ehk, chr=c(1,4))

# Permutation tests
## Not run: 
permo <- scanone(hyper, method="hk", n.perm=1000)
## End(Not run)

# Threshold from the permutation test
summary(permo, alpha=c(0.05, 0.10))

# Results above the 0.05 threshold
summary(out.hk, perms=permo, alpha=0.05)

####################
# scan with square-root of phenotype
#   (Note that pheno.col can be a vector of phenotype values)
####################
out.sqrt <- scanone(hyper, pheno.col=sqrt(pull.pheno(hyper, 1)))
plot(out.em - out.sqrt, ylim=c(-0.1,0.1),
     ylab="Difference in LOD")
abline(h=0, lty=2, col="gray")

####################
# Stratified permutations
####################
extremes <- (nmissing(hyper)/totmar(hyper) < 0.5)

## Not run: 
operm.strat <- scanone(hyper, method="hk", n.perm=1000,
                       perm.strata=extremes)
## End(Not run)

summary(operm.strat)


####################
# X-specific permutations
####################
data(fake.f2)

fake.f2 <- calc.genoprob(fake.f2, step=2.5)

# genome scan
out <- scanone(fake.f2, method="hk")

# X-chr-specific permutations
## Not run: 
operm <- scanone(fake.f2, method="hk", n.perm=1000, perm.Xsp=TRUE)
## End(Not run)

# thresholds
summary(operm)

# scanone summary with p-values
summary(out, perms=operm, alpha=0.05, pvalues=TRUE)


###################
# Non-parametric
###################
out.np <- scanone(hyper, model="np")
summary(out.np, thr=3)

# Plot with previous results
plot(out.np, chr=c(1,4), lty=1, col="green")
plot(out.imp, out.hk, out.em, chr=c(1,4), lty=1,
     col=c("red","blue","black"), add=TRUE)

###################
# Two-part Model
###################
data(listeria)

listeria <- calc.genoprob(listeria,step=2.5)
out.2p <- scanone(listeria, model="2part", upper=TRUE)
summary(out.2p, thr=c(5,3,3), format="allpeaks")

# Plot all three LOD scores together
plot(out.2p, out.2p, out.2p, lodcolumn=c(2,3,1), lty=1, chr=c(1,5,13),
     col=c("red","blue","black"))

# Permutation test
## Not run: 
permo <- scanone(listeria, model="2part", upper=TRUE,
                 n.perm=1000)
## End(Not run)

# Thresholds
summary(permo)

###################
# Binary model
###################
binphe <- as.numeric(pull.pheno(listeria,1)==264)
out.bin <- scanone(listeria, pheno.col=binphe, model="binary")
summary(out.bin, thr=3)

# Plot LOD for binary model with LOD(p) from 2-part model
plot(out.bin, out.2p, lodcolumn=c(1,2), lty=1, col=c("black", "red"),
     chr=c(1,5,13))

# Permutation test
## Not run: 
permo <- scanone(listeria, pheno.col=binphe, model="binary",
                 n.perm=1000)
## End(Not run)

# Thresholds
summary(permo)

###################
# Covariates
###################
data(fake.bc)

plot(fake.bc)
fake.bc <- calc.genoprob(fake.bc, step=2.5)

# genome scans without covariates
out.nocovar <- scanone(fake.bc)

# genome scans with covariates
ac <- pull.pheno(fake.bc, c("sex","age"))
ic <- pull.pheno(fake.bc, "sex")

out.covar <- scanone(fake.bc, pheno.col=1,
                     addcovar=ac, intcovar=ic)
summary(out.nocovar, thr=3)
summary(out.covar, thr=3)
plot(out.covar, out.nocovar, chr=c(2,5,10))