Nonparametric simulation-based statistics for detecting linkage in general pedigrees.

We present here four nonparametric statistics for linkage analysis that test whether pairs of affected relatives share marker alleles more often than expected. These statistics are based on simulating the null distribution of a given statistic conditional on the unaffecteds' marker genotypes. Each statistic uses a different measure of marker sharing: the SimAPM statistic uses the simulation-based affected-pedigree-member measure based on identity-by-state (IBS) sharing. The SimKIN (kinship) measure is 1.0 for identity-by-descent (IBD) sharing, 0.0 for no IBD status sharing, and the kinship coefficient when the IBD status is ambiguous. The simulation-based IBD (SimIBD) statistic uses a recursive algorithm to determine the probability of two affecteds sharing a specific allele IBD. The SimISO statistic is identical to SimIBD, except that it also measures marker similarity between unaffected pairs. We evaluated our statistics on data simulated under different two-locus disease models, comparing our results to those obtained with several other nonparametric statistics. Use of IBD information produces dramatic increases in power over the SimAPM method, which uses only IBS information. The power of our best statistic in most cases meets or exceeds the power of the other nonparametric statistics. Furthermore, our statistics perform comparisons between all affected relative pairs within general pedigrees and are not restricted to sib pairs or nuclear families.     

Selection and subsequent analysis of sib pair data for QTL detection.

Haseman and Elston (1972) developed a robust regression method for the detection of linkage between a marker and a quantitative trait locus (QTL) using sib pair data. The principle underlying this method is that the difference in phenotypes between pairs of sibs becomes larger as they share a decreasing number of alleles at a particular QTL identical by descent (IBD) from their parents. In this case, phenotypically very different sibs will also on average share a proportion of alleles IBD at any marker linked to the QTL that is lower than the expected value of 0.5. Thus, the deviation of the proportion of marker alleles IBD from the expected value in pairs of sibs selected to be phenotypically different (i.e. discordant) can provide a test for the presence of a QTL. A simple regression method for QTL detection in sib pairs selected for high phenotypic differences is presented here. The power of the analytical method was found to be greater than the power obtained using the standard analysis when samples of sib pairs with high phenotypic differences were used. However, the use of discordant sib pairs was found to be less powerful for QTL detection than alternative selective genotyping schemes based on the phenotypic values of the sibs except with intense selection, when its advantage was only marginal. The most effective selection scheme overall was the use of sib pairs from entire families selected on the basis of high within-family variance for the trait in question. There is little effect of selection on QTL position estimates, which are in good agreement with the simulated values. However, QTL variance estimates are biased to a greater or lesser degree, depending on the selection method.     

Construction of a confidence set of markers for the location of a disease gene using affected sib pair data

We have previously proposed a confidence set approach for finding tightly linked genomic regions under the setting of parametric linkage analysis. In this article, we extend the confidence set approach to nonparametric linkage analysis of affected sib pair (ASP) data based on their identity-by-descent (IBD) information. Two well-known statistics in nonparametric linkage analysis, the Two-IBD test (proportion of ASPs sharing two alleles IBD), and the Mean test (average number of alleles shared IBD in the ASPs), are used for constructing confidence sets. Some numerical analyses as well as a simulation study were carried out to demonstrate the utility of the methods. Our results show that the fundamental advantages of the confidence set approach in parametric linkage analysis are retained when the method is generalized to nonparametric analysis. Our study on the accuracy of confidence sets, in terms of choice of tests, underlying disease incidence data, and amount of data available, leads us to conclude, among other things, that the Mean test outperforms the Two-IBD test in most situations, with the reverse being true only for traits with small additive variance. Although we describe how to construct confidence sets based on only two familiar tests, one can construct confidence sets similarly using other allele sharing statistics.     

Nonparametric longitudinal allele-sharing model

Basically no methods are available for the analysis of quantitative traits in longitudinal genetic epidemiological studies. We introduce a nonparametric factorial design for longitudinal data on independent sib pairs, modelling the phenotypic quadratic differences as the dependent variable. Factors are the number of alleles shared identically by descent (IBD) and the age categories at which the dependent variable is measured, allowing for dependence due to age. To identify a linked marker a rank statistic tests the influence of IBD group on phenotypic quadratic differences. No assumptions are made on normality or variances of the dependent variable. We apply our method to 71 sib pairs from the Framingham Heart Study data provided at the Genetic Analysis Workshop 13. For all 15 available markers on chromosome 17 we analyzed the influence on systolic blood pressure. In addition, different selection strategies to sample from the whole data are discussed.     

Consanguinity and the sib-pair method: an approach using identity by descent between and within individuals.

To test for linkage between a trait and a marker, one can consider identical marker alleles in related individuals, for instance, sibs. For recessive diseases, it has been shown that some information may be gained from the identity by descent (IBD) of the two alleles of an affected inbred individual at the marker locus. The aim of this paper is to extend the sib-pair method of linkage analysis to the situation of sib pairs sampled from consanguineous populations. This extension takes maximum advantage of the information provided by both the IBD pattern between sibs and allelic identity within each sib of the pair. This is possible through the use of the condensed identity coefficients. Here, we propose a new test of linkage based on a chi2. We compare the performance of this test with that of the classical chi2 test based on the distribution of sib pairs sharing 0, 1, or 2 alleles IBD. For sib pairs from first-cousin matings, the proposed test can better detect the role of a disease-susceptibility (DS) locus. Its power is shown to be greater than that of the classical test, especially for models where the DS allele may be common and incompletely penetrant; that is to say for situations that may be encountered in multifactorial diseases. A study of the impact of inbreeding on the expected proportions of sib pairs sharing 0, 1, or 2 alleles IBD is also performed here. Ignoring inbreeding, when in fact inbreeding exists, increases the rate of type I errors in tests of linkage.     

Power of sib-pair and sib-trio linkage analysis with assortative mating and multiple disease loci.

Sib-pair linkage analysis has been proposed for identifying genes that predispose to common diseases. We have shown that the presence of assortative mating and multiple disease-susceptibility loci (genetic heterogeneity) can increase the required sample size for affected-affected sib pairs several fold over the sample size required under random mating. We propose a new test statistic based on sib trios composed of either one unaffected and two affected siblings or one affected and two unaffected siblings. The sample-size requirements under assortative mating and multiple disease loci for these sib-trio statistics are much smaller, under most conditions, than the corresponding sample sizes for sib pairs. Study designs based on data from sib trios with one or two affected members are recommended whenever assortative mating and genetic heterogeneity are suspected.     

Efficiency robust tests for mapping quantitative trait loci using extremely discordant sib pairs

In 1972, Haseman and Elston proposed a pioneering regression method for mapping quantitative trait loci using randomly selected sib pairs. Recently, the statistical power of their method was shown to be increased when extremely discordant sib pairs are ascertained. While the precise genetic model may not be known, prior information that constrains IBD probabilities is often available. We investigate properties of tests that are robust against model uncertainty and show that the power gain from further constraining IBD probabilities is marginal. The additional linkage information contained in the trait values can be incorporated by combining the Haseman-Elston regression method and a robust allele sharing test.     

The affected sib-set method: revision based on the mixed model using a conditional probability approach]

One of the implicit assumptions of the single locus model, having been used so far in the analysis of linkage between the genetic marker locus and the disease predisposition locus, is the requirement of independent--from the rest of genotype--action of genotypes of the disease predisposition locus considered. In this communication, it is emphasized that the lack of this requirement makes problematical the theoretical substantiation of the affected sib-pair method in the linkage analysis. To remove this obstacle, explicit pointing out of independence of the action of the single locus genotypes on the rest of the genotype is necessary in formulating of the single locus model which, with due regard for this assumption, represents a special, perhaps, unique case of the gene action characterized by incomplete differential penetrances of the genotypes under conditions, when the genes of the rest of genotype involved to the disease, are fixed. In this connection, the mixed model of inheritance with the "major gene", proposed by Morton and MacLean (1974), is considered, on the basis of which the theoretical expectations of the proportions of the affected sib pairs, sharing the x = 2, 1, 0 haplotypes, identical by descent (IBD) in phenotypic matings with the h = 2, 1, 0 affected parents are derived. Based on the combinatorial analysis of IBD relationships in sib pairs and of the distribution of sibships of any size s greater than or equal to 2 by the numbers L = 2, 3, 4 haplotypes, inherited by s siblings, the empirical assessment of data on sibships of any size with r greater than or equal to 2 affected siblings is considered, which makes it possible to reduce the data observed on distribution of the numbers L in sibships, to that of the IBD relationships in the affected sib pairs. It is also pointed out that conditional probability approach, proposed by the author earlier, allows at the same time to obtain the empirical estimates of the recurrence risks, conditional both on phenotypes of siblings (r affected; s-r normal siblings), and on the number of L haplotypes inherited by sibships.     

Using identity-by-descent information in affected sib pairs to increase the efficiency of genetic association studies.

The aim of this study was to determine whether identity-by-descent (IBD) information for affected sib pairs (ASPs) can be used to select a sample of cases for a genetic case-control study which will provide more power for detecting association with loci in a known linkage region. By modeling the expected frequency of the disease allele in ASPs showing IBD sharing of 0, 1, or 2 alleles, and considering additive, recessive, and dominant disease models, we show that cases selected from IBD 2 families are best for this purpose, followed by those selected from IBD 1 families; least useful are cases selected from IBD 0 families.     

The bias introduced by population stratification in IBD based linkage analysis

The lack of replication of model-free linkage analyses performed on complex diseases raises questions about the robustness of these methods to various biases. The confounding effect of population stratification on a genetic association study has long been recognized in the genetic epidemiology community. Because the estimation of the number of alleles shared identical by descent (IBD) does not depend on the marker allele frequency when founders of families are observed, model-free linkage analysis is usually thought to be robust to population stratification. However, for common complex diseases, the genotypes of founders are often unobserved and therefore population stratification has the potential to impair model-free linkage analysis. Here, we demonstrate that, when some or all of the founder genotypes are missing, population stratification can introduce deleterious effects on various model-free linkage methods or designs. For an affected sib pair design, it can cause excess false-positive discoveries even when the trait distribution is homogeneous among subpopulations. After incorporating a control group of discordant sib pairs or for a quantitative trait, two circumstances must be met for population stratification to be a confounder: the distributions for both the marker and the trait must be heterogeneous among subpopulations. When this occurs, the bias can result in either a liberal, and hence invalid, test or a conservative test. Bias can be eliminated or alleviated by inclusion of founders' or other family members' genotype data. When this is not possible, new methods need to be developed to be robust to population stratification.     

A unified Haseman-Elston method for testing linkage with quantitative traits

The Haseman and Elston (H-E) method uses a simple linear regression to model the squared trait difference of sib pairs with the shared allele identical by descent (IBD) at marker locus for linkage testing. Under this setting, the squared mean-corrected trait sum is also linearly related to the IBD sharing. However, the resulting slope estimate for either model is not efficient. In this report, we propose a simple linkage test that optimally uses information from the estimates of both models. We also demonstrate that the new test is more powerful than both the traditional one and the recently revisited H-E methods.     

Genetic association mapping based on discordant sib pairs: the discordant-alleles test.

Family-based tests of association provide the opportunity to test for an association between a disease and a genetic marker. Such tests avoid false-positive results produced by population stratification, so that evidence for association may be interpreted as evidence for linkage or causation. Several methods that use family-based controls have been proposed, including the haplotype relative risk, the transmission-disequilibrium test, and affected family-based controls. However, because these methods require genotypes on affected individuals and their parents, they are not ideally suited to the study of late-onset diseases. In this paper, we develop several family-based tests of association that use discordant sib pairs (DSPs) in which one sib is affected with a disease and the other sib is not. These tests are based on statistics that compare counts of alleles or genotypes or that test for symmetry in tables of alleles or genotypes. We describe the use of a permutation framework to assess the significance of these statistics. These DSP-based tests provide the same general advantages as parent-offspring trio-based tests, while being applicable to essentially any disease; they may also be tailored to particular hypotheses regarding the genetic model. We compare the statistical properties of our DSP-based tests by computer simulation and illustrate their use with an application to Alzheimer disease and the apolipoprotein E polymorphism. Our results suggest that the discordant-alleles test, which compares the numbers of nonmatching alleles in DSPs, is the most powerful of the tests we considered, for a wide class of disease models and marker types. Finally, we discuss advantages and disadvantages of the DSP design for genetic association mapping.     

Matrix representation of the Haseman-Elston method.

The Hasemann-Elston method of linkage detection is based on the probabilities of a sib pair having 0, 1, or 2 alleles identical by descent (IBD) at a marker and a trait locus. These probabilities form a 3x3 matrix. Here, the characteristic values and characteristic vectors of this matrix were used to clarify the structure of the equations and to simplify calculations. As examples, the regression coefficients were derived for three genetic systems: a trait and a marker, two epistatic traits and two markers, and one trait locus and two markers. The last model was studied under the assumption of no crossover interference, the expression for allele IBD sharing at a trait locus was derived as a function of allele IBD sharing at two marker loci, and the regression is shown to be non-linear.     

A likelihood ratio approach for identifying three-quarter siblings in genetic databases

The detection of family relationships in genetic databases is of interest in various scientific disciplines such as genetic epidemiology, population and conservation genetics, forensic science, and genealogical research. Nowadays, screening genetic databases for related individuals forms an important aspect of standard quality control procedures. Relatedness research is usually based on an allele sharing analysis of identity by state (IBS) or identity by descent (IBD) alleles. Existing IBS/IBD methods mainly aim to identify first-degree relationships (parent–offspring or full siblings) and second degree (half-siblings, avuncular, or grandparent–grandchild) pairs. Little attention has been paid to the detection of in-between first and second-degree relationships such as three-quarter siblings (3/4S) who share fewer alleles than first-degree relationships but more alleles than second-degree relationships. With the progressively increasing sample sizes used in genetic research, it becomes more likely that such relationships are present in the database under study. In this paper, we extend existing likelihood ratio (LR) methodology to accurately infer the existence of 3/4S, distinguishing them from full siblings and second-degree relatives. We use bootstrap confidence intervals to express uncertainty in the LRs. Our proposal accounts for linkage disequilibrium (LD) by using marker pruning, and we validate our methodology with a pedigree-based simulation study accounting for both LD and recombination. An empirical genome-wide array data set from the GCAT Genomes for Life cohort project is used to illustrate the method.Subject terms: Genetic markers, Population genetics     

Adding further power to the Haseman and Elston method for detecting linkage in larger sibships: weighting sums and differences

Haseman and Elston (H-E) proposed a robust test to detect linkage between a quantitative trait and a genetic marker. In their method the squared sib-pair trait difference is regressed on the estimated proportion of alleles at a locus shared identical by descent by sib pairs. This method has recently been improved by changing the dependent variable from the squared difference to the mean-corrected product of the sib-pair trait values, a significantly positive regression indicating linkage. Because situations arise in which the original test is more powerful, a further improvement of the H-E method occurs when the dependent variable is changed to a weighted average of the squared sib-pair trait difference and the squared sib-pair mean-corrected trait sum. Here we propose an optimal method of performing this weighting for larger sibships, allowing for the correlation between pairs within a sibship. The optimal weights are inversely proportional to the residual variances obtained from the two different regressions based on the squared sib-pair trait differences and the squared sib-pair mean-corrected trait sums, respectively, allowing for correlations among sib pairs. The proposed method is compared with the existing extension of the H-E approach for larger sibships. Control of the type I error probabilities for sibships of any size can be improved by using a generalized estimating equation approach and the robust sandwich estimate of the variance, or a Monte-Carlo permutation test.     

Regression-based sib pair linkage analysis for binary traits

The Haseman-Elston (HE) regression method offers a mathematically and computationally simpler alternative to variance-components (VC) models for the linkage analysis of quantitative traits. However, current versions of HE regression and VC models are not optimised for binary traits. Here, we present a modified HE regression and a liability-threshold VC model for binary-traits. The new HE method is based on the regression of a linear combination of the trait squares and the trait cross-product on the proportion of alleles identical by descent (IBD) at the putative locus, for sibling pairs. We have implemented both the new HE regression-based method and have performed analytic and simulation studies to assess its type 1 error rate and power under a range of conditions. These studies showed that the new HE method is well-behaved under the null hypothesis in large samples, is more powerful than both the original and the revisited HE methods, and is approximately equivalent in power to the liability-threshold VC model.     

Gene-environment interaction and affected sib pair linkage analysis

OBJECTIVES: Gene-environment (GxE) interaction influences risk for many complex disease traits. However, genome screens using affected sib pair linkage techniques are typically conducted without regard for GxE interaction. We propose a simple extension of the commonly used mean test and evaluate its power for several forms of GxE interaction. METHODS: We compute expected IBD sharing by sibling exposure profile, that is by whether two sibs are exposed (EE), unexposed (UU), or are discordant for exposure (EU). We describe a simple extension of the mean test, the "mean-interaction" test that utilizes heterogeneity in IBD sharing across EE, EU, and UU sib pairs in a test for linkage. RESULTS: The mean-interaction test provides greater power than the mean test for detecting linkage in the presence of moderate or strong GxE interaction, typically when the interaction relative risk (R(ge)) exceeds 3 or is less than 1/3. In the presence of strong interaction (R(ge) = 10), the required number of affected sib pairs to achieve 80% power for detecting linkage is approximately 30% higher when the environmental factor is ignored in the mean test, than when it is utilized in the mean-interaction test. CONCLUSION: Linkage methods that incorporate environmental data and allow for interaction can lead to increased power for localizing a disease gene involved in a GxE interaction.     

A Discordant-Sibship Test for Disequilibrium and Linkage: No Need for Parental Data

The sibship disequilibrium test (SDT) is designed to detect both linkage in the presence of association and association in the presence of linkage (linkage disequilibrium). The test does not require parental data but requires discordant sibships with at least one affected and one unaffected sibling. The SDT has many desirable properties: it uses all the siblings in the sibship; it remains valid if there are misclassifications of the affectation status; it does not detect spurious associations due to population stratification; asymptotically it has a chi2 distribution under the null hypothesis; and exact P values can be easily computed for a biallelic marker. We show how to extend the SDT to markers with multiple alleles and how to combine families with parents and data from discordant sibships. We discuss the power of the test by presenting sample-size calculations involving a complex disease model, and we present formulas for the asymptotic relative efficiency (which is approximately the ratio of sample sizes) between SDT and the transmission/disequilibrium test (TDT) for special family structures. For sib pairs, we compare the SDT to a test proposed both by Curtis and, independently, by Spielman and Ewens. We show that, for discordant sib pairs, the SDT has good power for testing linkage disequilibrium relative both to Curtis''s tests and to the TDT using trios comprising an affected sib and its parents. With additional sibs, we show that the SDT can be more powerful than the TDT for testing linkage disequilibrium, especially for disease prevalence >.3.     

A simulation study of the effects of assignment of prior identity-by-descent probabilities to unselected sib pairs, in covariance-structure modeling of a quantitative-trait locus.

Sib pair-selection strategies, designed to identify the most informative sib pairs in order to detect a quantitative-trait locus (QTL), give rise to a missing-data problem in genetic covariance-structure modeling of QTL effects. After selection, phenotypic data are available for all sibs, but marker data-and, consequently, the identity-by-descent (IBD) probabilities-are available only in selected sib pairs. One possible solution to this missing-data problem is to assign prior IBD probabilities (i.e., expected values) to the unselected sib pairs. The effect of this assignment in genetic covariance-structure modeling is investigated in the present paper. Two maximum-likelihood approaches to estimation are considered, the pi-hat approach and the IBD-mixture approach. In the simulations, sample size, selection criteria, QTL-increaser allele frequency, and gene action are manipulated. The results indicate that the assignment of prior IBD probabilities results in serious estimation bias in the pi-hat approach. Bias is also present in the IBD-mixture approach, although here the bias is generally much smaller. The null distribution of the log-likelihood ratio (i.e., in absence of any QTL effect) does not follow the expected null distribution in the pi-hat approach after selection. In the IBD-mixture approach, the null distribution does agree with expectation.     

The affected-/discordant-sib-pair design can guarantee validity of multipoint model-free linkage analysis of incomplete pedigrees when there is marker-marker disequilibrium

Genomewide linkage studies are tending toward the use of single-nucleotide polymorphisms (SNPs) as the markers of choice. However, linkage disequilibrium (LD) between tightly linked SNPs violates the fundamental assumption of linkage equilibrium (LE) between markers that underlies most multipoint calculation algorithms currently available, and this leads to inflated affected-relative-pair allele-sharing statistics when founders' multilocus genotypes are unknown. In this study, we investigate the impact that the degree of LD, marker allele frequency, and association type have on estimating the probabilities of sharing alleles identical by descent in multipoint calculations and hence on type I error rates of different sib-pair linkage approaches that assume LE. We show that marker-marker LD does not inflate type I error rates of affected sib pair (ASP) statistics in the whole parameter space, and that, in any case, discordant sib pairs (DSPs) can be used to control for marker-marker LD in ASPs. We advocate the ASP/DSP design with appropriate sib-pair statistics that test the difference in allele sharing between ASPs and DSPs.