A nonparametric bootstrap method for testing close linkage vs. pleiotropy of coincident quantitative trait loci

A novel method using the nonparametric bootstrap is proposed for testing whether a quantitative trait locus (QTL) at one chromosomal position could explain effects on two separate traits. If the single-QTL hypothesis is accepted, pleiotropy could explain the effect on two traits. If it is rejected, then the effects on two traits are due to linked QTLs. The method can be used in conjunction with several QTL mapping methods as long as they provide a straightforward estimate of the number of QTLs detectable from the data set. A selection step was introduced in the bootstrap procedure to reduce the conservativeness of the test of close linkage vs. pleiotropy, so that the erroneous rejection of the null hypothesis of pleiotropy only happens at a frequency equal to the nominal type I error risk specified by the user. The approach was assessed using computer simulations and proved to be relatively unbiased and robust over the range of genetic situations tested. An example of its application on a real data set from a saline stress experiment performed on a recombinant population of wheat (Triticum aestivum L. ) doubled haploid lines is also provided.

Strategies to utilize marker-quantitative trait loci associations

Marker-assisted selection holds promise because genetic markers provide completely heritable traits than can be measured at any age in either sex and that are potentially correlated with traits of economic value. Theoretical and simulation studies show that the advantage of using marker-assisted selection can be substantial, particularly when marker information is used, because normal selection is less effective, for example, for sex-limited or carcass traits. Assessment of the available information and its most effective use is difficult, but approaches such as crossvalidation may help in this respect. Marker systems are now becoming available that allow the high density of markers required for close associations between marker loci and trait loci. Emerging technologies could allow large numbers of polymorphic sites to be identified, practically guaranteeing that markers will be available that are in complete association with any trait locus. Identifying which polymorphism out of many that is associated with any trait will remain problematic, but multiple-locus disequilibrium measures may allow performance to be associated with unique marker haplotypes. This type of approach, combined with cheap and high density markers, could allow a move from selection based on a combination of "infinitesimal" effects plus individual loci to effective total genomic selection. In such a unified model, each region of the genome would be given its appropriate weight in a breeding program. However, the collection of good quality trait information will remain central to the use of these technologies for the foreseeable future.

On the sampling variance of intraclass correlations and genetic correlations

Widely used standard expressions for the sampling variance of intraclass correlations and genetic correlation coefficients were reviewed for small and large sample sizes. For the sampling variance of the intraclass correlation, it was shown by simulation that the commonly used expression, derived using a first-order Taylor series performs better than alternative expressions found in the literature, when the between-sire degrees of freedom were small. The expressions for the sampling variance of the genetic correlation are significantly biased for small sample sizes, in particular when the population values, or their estimates, are close to zero. It was shown, both analytically and by simulation, that this is because the estimate of the sampling variance becomes very large in these cases due to very small values of the denominator of the expressions. It was concluded, therefore, that for small samples, estimates of the heritabilities and genetic correlations should not be used in the expressions for the sampling variance of the genetic correlation. It was shown analytically that in cases where the population values of the heritabilities are known, using the estimated heritabilities rather than their true values to estimate the genetic correlation results in a lower sampling variance for the genetic correlation. Therefore, for large samples, estimates of heritabilities, and not their true values, should be used.

Empirical nonparametric bootstrap strategies in quantitative trait loci mapping: conditioning on the genetic model

Several nonparametric bootstrap methods are tested to obtain better confidence intervals for the quantitative trait loci (QTL) positions, i.e., with minimal width and unbiased coverage probability. Two selective resampling schemes are proposed as a means of conditioning the bootstrap on the number of genetic factors in our model inferred from the original data. The selection is based on criteria related to the estimated number of genetic factors, and only the retained bootstrapped samples will contribute a value to the empirically estimated distribution of the QTL position estimate. These schemes are compared with a nonselective scheme across a range of simple configurations of one QTL on a one-chromosome genome. In particular, the effect of the chromosome length and the relative position of the QTL are examined for a given experimental power, which determines the confidence interval size. With the test protocol used, it appears that the selective resampling schemes are either unbiased or least biased when the QTL is situated near the middle of the chromosome. When the QTL is closer to one end, the likelihood curve of its position along the chromosome becomes truncated, and the nonselective scheme then performs better inasmuch as the percentage of estimated confidence intervals that actually contain the real QTL's position is closer to expectation. The nonselective method, however, produces larger confidence intervals. Hence, we advocate use of the selective methods, regardless of the QTL position along the chromosome (to reduce confidence interval sizes), but we leave the problem open as to how the method should be altered to take into account the bias of the original estimate of the QTL's position.

Marker-assisted introgression in backcross breeding programs

The efficiency of marker-assisted introgression in backcross populations derived from inbred lines was investigated by simulation. Background genotypes were simulated assuming that a genetic model of many genes of small effects in coupling phase explains the observed breed difference and variance in backcross populations. Markers were efficient in introgression backcross programs for simultaneously introgressing an allele and selecting for the desired genomic background. Using a marker spacing of 10-20 cM gave an advantage of one to two backcross generations selection relative to random or phenotypic selection. When the position of the gene to be introgressed is uncertain, for example because its position was estimated from a trait gene mapping experiment, a chromosome segment should be introgressed that is likely to include the allele of interest. Even for relatively precisely mapped quantitative trait loci, flanking markers or marker haplotypes should cover approximately 10-20 cM around the estimated position of the gene, to ensure that the allele frequency does not decline in later backcross generations.

Detection of putative quantitative trait loci in line crosses under infinitesimal genetic models

Quantitative trait locus (QTL) mapping studies often employ segregating generations derived from a cross between genetically divergent inbred lines. In the analysis of such data it is customary to fit a single QTL and use a null hypothesis which assumes that the genomic region under study contributes no genetic variance. To explore the situation in which multiple linked genes contribute to the genetic variance, we simulated an F2-mapping experiment in which the genetic difference between the two original inbred strains was caused by a large number of loci, each having equal effect on the quantitative trait. QTLs were either in coupling, dispersion or repulsion phase in the base population of inbred lines, with the expected F2 genetic variance explained by the QTLs being equivalent in the three models. Where QTLs were in coupling phase, one inbred line was fixed for all plus alleles, and the other line was fixed for minus alleles. Where QTLs were in dispersion phase, they were assumed to be randomly fixed for one or other allele (as if the inbred lines had evolved from a common ancestor by random drift). Where QTLs were in repulsion phase alleles within an inbred line were alternating plus and minus at adjacent loci, and alternative alleles were fixed in the two inbred lines. In all these genetic models a standard interval mapping test statistic used to determine whether there is a QTL of large effect segregating in the population was inflated on average. Furthermore, the use of a threshold for QTL detection derived under the assumption that no QTLs were segregating would often lead to spurious conclusions regards the presence of genes of large effects (i.e. type I errors). The employment of an alternative model for the analysis, including linked markers as cofactors in the analysis of a single interval, reduced the problem of type I error rate, although test statistics were still inflated relative to the case of no QTLs. It is argued that in practice one should take into account the difference between the strains or the genetic variance in the F2 population when setting significance thresholds. In addition, tests designed to probe the adequacy of a single-QTL model or of an alternative infinitesimal coupling model are described. Such tests should be applied in QTL mapping studies to help dissect the true nature of genetic variation.

Confidence intervals in QTL mapping by bootstrapping

The determination of empirical confidence intervals for the location of quantitative trait loci (QTLs) was investigated using simulation. Empirical confidence intervals were calculated using a bootstrap resampling method for a backcross population derived from inbred lines. Sample sizes were either 200 or 500 individuals, and the QTL explained 1, 5, or 10% of the phenotypic variance. The method worked well in that the proportion of empirical confidence intervals that contained the simulated QTL was close to expectation. In general, the confidence intervals were slightly conservatively biased. Correlations between the test statistic and the width of the confidence interval were strongly negative, so that the stronger the evidence for a QTL segregating, the smaller the empirical confidence interval for its location. The size of the average confidence interval depended heavily on the population size and the effect of the QTL. Marker spacing had only a small effect on the average empirical confidence interval. The LOD drop-off method to calculate empirical support intervals gave confidence intervals that generally were too small, in particular if confidence intervals were calculated only for samples above a certain significance threshold. The bootstrap method is easy to implement and is useful in the analysis of experimental data.

Genetic parameters for milk yield, survival, workability, and type traits for Australian dairy cattle

Genetic parameters, such as heritabilities and genetic correlations were estimated for milk yield, survival, workability, and type traits for Australian Holstein-Friesian and Jersey cattle. All analyses were performed using multivariate REML with a sire model. Heritabilities for lactation yield traits were moderate, ranging from .20 to .28, and heritabilities for mean test day deviations were approximately .40 higher. Heritabilities for survival (probability of surviving to the next lactation) were low, ranging from .02 to .08. Genetic correlations between survival scores were high, ranging from .37 to .98, in particular between adjacent survival scores (on average .91 and .97 for Holstein-Friesians and Jerseys). Heritabilities for stayabilities were larger, ranging from .03 to .22. On average, genetic correlations between stayabilities were very high, ranging from .66 to .99. For milking speed, temperament, and likeability, heritability estimates ranged from .18 (for likeability in Holstein-Friesians) to .29 (for milking speed in Jerseys). Undesirable scores for milking speed and temperament had negative genetic correlations with stayabilities (correlations approximately -.20). Heritabilities for type traits were all moderate (.11 to .42), and genetic correlations among type traits and between type traits and production traits were large. Phenotypic correlations between type traits and stayabilities were low. Generally, genetic correlations between type traits and stayabilities were low although the standard errors of those estimates were large.

Power of likelihood ratio tests for heterogeneity of intraclass correlation and variance in balanced half-sib designs

Statistical power of likelihood ratio tests was investigated for detection of heterogeneous variances and intraclass correlation in balanced half-sib designs. Powers of likelihood ratio tests were obtained from simulations. For half-sib designs of sires nested within herds, true intraclass correlations and phenotypic variances, and estimates thereof, were repeatedly sampled, and likelihood ratio tests were conducted. The power for detecting heterogeneity of intraclass correlations was low, but the power for detecting heterogeneous phenotypic variances was nearly always 100%. For balanced cross-classified designs, sires had progeny in all herds, and data were simulated by assuming that heterogeneity of between- and within-sire components was the result of a herd-dependent scale effect. Using this model, the power to detect heterogeneous between-sire components was substantially higher than the corresponding power to detect heterogeneous intraclass correlations in the nested design. It seems unlikely that reliable inference about heterogeneity of genetic variances or heritabilities between individual herds from daily cattle field data can be made.