Two-locus models of disease: comparison of likelihood and nonparametric linkage methods.

The power to detect linkage for likelihood and nonparametric (Haseman-Elston, affected-sib-pair, and affected-pedigree-member) methods is compared for the case of a common, dichotomous trait resulting from the segregation of two loci. Pedigree data for several two-locus epistatic and heterogeneity models have been simulated, with one of the loci linked to a marker locus. Replicate samples of 20 three-generation pedigrees (16 individuals/pedigree) were simulated and then ascertained for having at least 6 affected individuals. The power of linkage detection calculated under the correct two-locus model is only slightly higher than that under a single locus model with reduced penetrance. As expected, the nonparametric linkage methods have somewhat lower power than does the lod-score method, the difference depending on the mode of transmission of the linked locus. Thus, for many pedigree linkage studies, the lod-score method will have the best power. However, this conclusion depends on how many times the lod score will be calculated for a given marker. The Haseman-Elston method would likely be preferable to calculating lod scores under a large number of genetic models (i.e., varying both the mode of transmission and the penetrances), since such an analysis requires an increase in the critical value of the lod criterion. The power of the affected-pedigree-member method is lower than the other methods, which can be shown to be largely due to the fact that marker genotypes for unaffected individuals are not used.     

The power and statistical behaviour of allele-sharing statistics when applied to models with two disease loci

We have evaluated the power for detecting a common trait determined by two loci, using seven statistics, of which five are implemented in the computer program SimWalk2, and two are implemented in GENEHUNTER. Unlike most previous reports which involve evaluations of the power of allelesharing statistics for a single disease locus, we have used a simulated data set of general pedigrees in which a twolocus disease is segregating and evaluated several nonparametric linkage statistics implemented in the two programs. We found that the power for detecting linkage using the S_all statistic in GENEHUNTER (GH, version 2.1), implemented as statisticE in SimWalk2 (version 2.82), is different in the two. TheP values associated with statisticE output by SimWalk2 are consistently more conservative than those from GENEHUNTER except when the underlying model includes heterogeneity at a level of 50% where theP values output are very comparable. On the other hand, when the thresholds are determined empirically under the null hypothesis, S_all in GENEHUNTER and statisticE have similar power.     

Two-locus heterogeneity cannot be distinguished from two-locus epistasis on the basis of affected-sib-pair data

The observation of multiple linkage signals in the course of conducting genome screens for complex disorders raises the question of whether distinct genes represent independent causes of disease (heterogeneity) or whether they interact to produce the phenotype of interest (epistasis); and there has been a corresponding interest in statistical methods for detecting and/or exploiting the distinction between these two possibilities. At the same time, researchers are increasingly relying on affected-sib-pair (ASP) data. Here, we demonstrate an apparently unrecognized fact about two-locus (2L) models and ASP data, namely, 2L heterogeneity and 2L epistasis cannot, in general, be distinguished from one another on the basis of ASP marker data, as a matter of mathematical principle and therefore regardless of sample size. By the same token, correlations across ASPs in single-locus LOD scores or other measures also cannot be used to distinguish 2L heterogeneity from 2L epistasis. This raises questions about the measurement of gene-gene interactions in terms of patterns of correlation in marker data. Portions of our results carry over to larger pedigree structures as well, as long as only affected individuals are included in analyses; the extent to which our overall findings apply to general pedigrees (including unaffected individuals) remains to be investigated.     

Linkage between quantitative trait loci and marker loci: resolution power of three statistical approaches in single marker analysis

This paper presents a comparison of three methods of parameter estimation in analysis of linkage between a quantitative trait locus (QTL) and a marker locus: maximum likelihood, mean square for trait cumulative distribution function, and method of moments, employing simulated backcross data. The sensitivity of estimates to violation of assumptions of normality and equal variances were also studied. Some measures of discrepancy between the trait distributions in the QTL groups are considered to evaluate the potential dependence of the resolution capacity of the QTL substitution effect with respect to trait mean value and variance.     

Linkage between loci of quantitative traits and marker loci: multi-trait analysis with a single marker

An efficient approach to increase the resolution power of linkage analysis between a quantitative trait locus (QTL) and a marker is described in this paper. It is based on a counting of the correlations between the QTs of interest. Such correlations may be caused by the segregation of other genes, environmental effects and physiological limitations. Let a QT locus A/a affect two correlated traits, x and y. Then, within the framework of mixture models, the accuracy of the parameter estimates may be seriously increased, if bivariate densities f _aa(x, y), f _Aa(x, y) and f _AA(x, y) rather than the marginals are considered as the basis for mixture decomposition. The efficiency of the proposed method was demonstrated employing Monte-Carlo simulations. Several types of progeny were considered, including backcross, F₂ and recombinant inbred lines. It was shown that provided the correlation between the traits involved was high enough, a good resolution to the problem is possible even if the QTL groups are strongly overlapping for their marginal densities.     

Testing for association between disease and linked marker loci: a log-linear-model analysis.

One approach frequently used for identifying genetic factors involved in the process of a complex disease is the comparison of patients and controls for a number of genetic markers near a candidate gene. The analysis of such association studies raises some specific problems because of the fact that genotypic and not gametic data are generally available. We present a log-linear-model analysis providing a valid method for analyzing such studies. When studying the association of disease with one marker locus, the log-linear model allows one to test for the difference between allelic frequencies among affected and unaffected individuals, Hardy-Weinberg (H-W) equilibrium in both groups, and interaction between the association of alleles at the marker locus and disease. This interaction provides information about the dominance of the disease susceptibility locus, with dominance defined using the epidemiological notion of odds ratio. The degree of dominance measured at the marker locus depends on the strength of linkage disequilibrium between the marker locus and the disease locus. When studying the association of disease with several linked markers, the model becomes rapidly complex and uninterpretable unless it is assumed that affected and unaffected populations are in H-W equilibrium at each locus. This hypothesis must be tested before going ahead in the analysis. If it is not rejected, the log-linear model offers a stepwise method of identification of the parameters causing the difference between populations. This model can be extended to any number of loci, alleles, or populations.     

Some population genetic models combining artificial and natural selection pressures: II. Two-locus theory

Some population genetic systems of traits controlled principally at two or three main loci subject to the combined effects of artificial and natural selection are studied. A number of genotypic-phenotypic associations are formulated, and both random and selfing mating patterns are examined. The interaction of some forms of epistasis and linkage in these systems are evaluated.     

How to model a complex trait. 2. Analysis with two disease loci

Complex traits are often governed by more than one trait locus. The first step towards an adequate model for such diseases is a linkage analysis with two trait loci. Such an analysis can be expected to have higher power to detect linkage than a standard single-trait-locus linkage analysis. However, it is crucial to accurately specify the parameters of the two-locus model. Here, we recapitulate the general two-locus model with and without genomic imprinting. We relate heterogeneity, multiplicative, and additive two-locus models to biological or pathophysiological mechanisms, and give the corresponding averaged ("best-fitting") single-trait-locus models for each of the two loci. Furthermore, we derive the two-locus penetrances from the averaged single-locus models, under the assumption of one of the three model classes mentioned above. Using these formulae, if the best-fitting single-locus models are available, investigators may perform a two-trait-locus linkage analysis under a realistic model. This procedure will maximize the power to detect linkage for traits which are governed by two or more loci, and lead to more accurate estimates of the disease-locus positions.     

A two loci viability matrix with implications in current population genetics models

Summary Some equilibrium properties of a two loci deterministic system subject to a postulated viability matrix are given. Two conclusions are reached: i) linkage can not stabilize additive equally contributing two-loci systems under optimizing selection. ii) Hybrid protein superiority can not be invoked for the maintenance of polymorphisms shared by duplicated structural loci.     

Genetic response from marker assisted selection in an outbred population for differing marker bracket sizes and with two identified quantitative trait loci.

Effect of flanking quantitative trait loci (QTL)-marker bracket size on genetic response to marker assisted selection in an outbred population was studied by simulation of a nucleus breeding scheme. In addition, genetic response with marker assisted selection (MAS) from two quantitative trait loci on the same and different chromosome(s) was investigated. QTL that explained either 5% or 10% of phenotypic variance were simulated. A polygenic component was simulated in addition to the quantitative trait loci. In total, 35% of the phenotypic variance was due to genetic factors. The trait was measured on females only. Having smaller marker brackets flanking the QTL increased the genetic response from MAS selection. This was due to the greater ability to trace the QTL transmission from one generation to the next with the smaller flanking QTL-marker bracket, which increased the accuracy of estimation of the QTL allelic effects. Greater negative covariance between effects at both QTL was observed when two QTL were located on the same chromosome compared to different chromosomes. Genetic response with MAS was greater when the QTL were on the same chromosome in the early generations and greater when they were on different chromosomes in the later generations of MAS.     

Levels of genetic polymorphism: marker loci versus quantitative traits.

Species are the units used to measure ecological diversity and alleles are the units of genetic diversity. Genetic variation within and among species has been documented most extensively using allozyme electrophoresis. This reveals wide differences in genetic variability within, and genetic distances among, species, demonstrating that species are not equivalent units of diversity. The extent to which the pattern observed for allozymes can be used to infer patterns of genetic variation in quantitative traits depends on the forces generating and maintaining variability. Allozyme variation is probably not strictly neutral but, nevertheless, heterozygosity is expected to be influenced by population size and genetic distance will be affected by time since divergence. The same is true for quantitative traits influenced by many genes and under weak stabilizing selection. However, the limited data available suggest that allozyme variability is a poor predictor of genetic variation in quantitative traits within populations. It is a better predictor of general phenotypic divergence and of postzygotic isolation between populations or species, but is only weakly correlated with prezygotic isolation. Studies of grasshopper and planthopper mating signal variation and assortative mating illustrate how these characters evolve independently of general genetic and morphological variation. The role of such traits in prezygotic isolation, and hence speciation, means that they will contribute significantly to the diversity of levels of genetic variation within and among species.     

The number of heterozygous loci between two randomly chosen completely linked sequences of loci in two subdivided population models

The stationary probability distribution of the number of heterozygous loci in two randomly chosen sequences of completely linked infinite alleles loci, with mutation at each locus, is found in the island model for within and between islands. Results for an infinite site model are found as a limit. A single charge state locus is also studied in the island model and distributions found for the charge difference between two genes. Similar results are derived for a stepping stone model.     

The detection of linkage and heterogeneity in nuclear families for complex disorders: one versus two marker loci.

Using exact expected likelihoods, we have computed the average number of phase-unknown nuclear families needed to detect linkage and heterogeneity. We have examined the case of both dominant and recessive inheritance with reduced penetrance and phenocopies. Most of our calculations have been carried out under the assumption that 50% of families are linked to a marker locus. We have varied both the number of offspring per family and the sampling scheme. We have also investigated the increased power when the disease locus is midway between two marker loci 10 cM apart. For recessive inheritance, both linkage and heterogeneity can be detected in clinically feasible sample sizes. For dominant inheritance, linkage can be detected but heterogeneity cannot be detected unless larger sibships (four offspring) are sampled or two linked markers are available. As expected, if penetrance is reduced, sampling families with all sibs affected is most efficient. Our results provide a basis for estimating the amount of resources needed to find genes for complex disorders under conditions of heterogeneity.     

On the power of experimental designs for the detection of linkage between marker loci and quantitative loci in crosses between inbred lines

Summary The power of experiments aimed at detecting linkage between a quantitative locus and a marker locus, both segregating in the backross or F₂ generation of a cross between two inbred lines, is examined. Given that the two lines are close to fixation for alternative alleles of both marker locus and quantitative locus, it is concluded that experiments involving a few thousand offspring should be able to detect close linkages involving quantitative loci (or groups of loci) having rather modest effects (i.e., that contribute, say, 1% of the total phenotypic variance in the F₂).     

Detection of quantitative trait loci in outbred populations with incomplete marker data.

Augmentation of marker genotypes for ungenotyped individuals is implemented in a Bayesian approach via the use of Markov chain Monte Carlo techniques. Marker data on relatives and phenotypes are combined to compute conditional posterior probabilities for marker genotypes of ungenotyped individuals. The presented procedure allows the analysis of complex pedigrees with ungenotyped individuals to detect segregating quantitative trait loci (QTL). Allelic effects at the QTL were assumed to follow a normal distribution with a covariance matrix based on known QTL position and identity by descent probabilities derived from flanking markers. The Bayesian approach estimates variance due to the single QTL, together with polygenic and residual variance. The method was empirically tested through analyzing simulated data from a complex granddaughter design. Ungenotyped dams were related to one or more sons or grandsires in the design. Heterozygosity of the marker loci and size of QTL were varied. Simulation results indicated a significant increase in power when ungenotyped dams were included in the analysis.     

Fine mapping of quantitative trait loci using linkage disequilibria with closely linked marker loci

A multimarker linkage disequilibrium mapping method was developed for the fine mapping of quantitative trait loci (QTL) using a dense marker map. The method compares the expected covariances between haplotype effects given a postulated QTL position to the covariances that are found in the data. The expected covariances between the haplotype effects are proportional to the probability that the QTL position is identical by descent (IBD) given the marker haplotype information, which is calculated using the genedropping method. Simulation results showed that a QTL was correctly positioned within a region of 3, 1.5, or 0.75 cM in 70, 62, and 68%, respectively, of the replicates using markers spaced at intervals of 1, 0.5, and 0.25 cM, respectively. These results were rather insensitive to the number of generations since the QTL occurred and to the effective population size, except that 10 generations yielded rather poor estimates of the QTL position. The position estimates of this multimarker disequilibrium mapping method were more accurate than those from a single marker transmission disequilibrium test. A general approach for identifying QTL is suggested, where several stages of disequilibrium mapping are used with increasingly dense marker spacing.     

How many marker loci are necessary? Analysis of dominant marker data sets using two popular population genetic algorithms

The number of marker loci required to answer a given research question satisfactorily is especially important for dominant markers since they have a lower information content than co‐dominant marker systems. In this study, we used simulated dominant marker data sets to determine the number of dominant marker loci needed to obtain satisfactory results from two popular population genetic analyses: STRUCTURE and AMOVA (analysis of molecular variance). Factors such as migration, level of population differentiation, and unequal sampling were varied in the data sets to mirror a range of realistic research scenarios. AMOVA performed well under all scenarios with a modest quantity of markers while STRUCTURE required a greater number, especially when populations were closely related. The popular ΔK method of determining the number of genetically distinct groups worked well when sampling was balanced, but underestimated the true number of groups with unbalanced sampling. These results provide a window through which to interpret previous work with dominant markers and we provide a protocol for determining the number of markers needed for future dominant marker studies.     

General equations for Pt, Ps, and the power of the TDT and the affected-sib-pair test

Several equations are highlighted here, whose algebraic symmetries and generality make them very useful for understanding and comparing the properties of the transmission disequilibrium test (TDT) and affected sib-pair test. Methods using the equations are also presented that yield precise estimates of sample sizes needed for genome scans or for testing a single candidate gene, and these power methods are shown to compare favorably with alternative approaches recently described by Knapp (1999) and by Tu and Whittemore (1999). Simple relationships are also noted that summarize the relative sample sizes required for equivalent power to detect association by the TDT or case-control designs. As single-nucleotide polymorphism (SNP) maps revolutionize the search for disease-causing genes, the equations should prove useful for planning and evaluating studies of linkage and association across a broad range of possible disease models and relationships between markers and linked disease loci.