Analysis of genetic linkage and somatic loss of heterozygosity in affected pairs of first-degree relatives.

Recently, data on loss of constitutional heterozygosity (LOH) have been used by several groups to increase the power to detect linkage in pedigrees with an inherited cancer predisposition. This approach assumes that the predisposition is due to the inheritance of the defective copy of a tumor suppressor. In order to assess the gain of power expected from the inclusion of LOH data, we simulated segregation and somatic loss of alleles in pedigrees consisting of an affected pair of first-degree relatives. We explored the effects of pedigree structure, frequency of loss, penetrance, and recombination rate on the expected LOD score. The results indicate that, for establishment of genetic linkage, isolated parent-offspring pairs can be as informative as sib pairs and that they could represent an additional source of information in linkage studies.     

Intracranial aneurysms in Finnish families: confirmation of linkage and refinement of the interval to chromosome 19q13.3

We recently reported a two-stage genomewide screen of 48 sib pairs affected with intracranial aneurysms (IAs) that revealed suggestive linkage to chromosome 19q13, with a LOD score of 2.58. The region supporting linkage spanned ~22 cM. Here, we report a follow-up study of the locus at 19q13, with a sample size expanded to 139 affected sib pairs, along with 83 other affected relative pairs (222 affected relative pairs in total). Suggestive linkage was observed in both independent sample sets, and linkage was significant in the combined set at 70 cM (LOD score 3.50; P=.00006) and at 80 cM (LOD score 3.93; P=.00002). Linkage was highly significant at 70 cM (LOD score 5.70; P=.000001) and at 80 cM (LOD score 3.99; P=.00005) when a covariate measuring the number of affected individuals in the nuclear family was included. To evaluate further the contribution to the linkage signal from families with more than two affected relatives, we performed model-based linkage analysis with a recessive model and a range of penetrances, and we obtained maximum linkage at 70 cM (LOD score 3.16; P=.00007) with a penetrance of 0.3. We then estimated location by using GENEFINDER. The most likely location for a gene predisposing to IAs in the Finnish population is in a region with a 95% confidence interval of 11.6 cM (P=.00007) centered 2.0 cM proximal to D19S246.     

MOD-score analysis with simple pedigrees: an overview of likelihood-based linkage methods

A MOD-score analysis, in which the parametric LOD score is maximized with respect to the trait-model parameters, can be a powerful method for the mapping of complex traits. With affected sib pairs, it has been shown before that MOD scores asymptotically follow a mixture of chi(2) distributions with 2, 1 and 0 degrees of freedom under the null hypothesis of no linkage. In that context, a MOD-score analysis yields some (albeit limited) information regarding the trait-model parameters, and there is a chance for an increased power compared to a simple LOD-score analysis. Here, it is shown that with unilineal affected relative pairs, MOD scores asymptotically follow a mixture of chi(2) distributions with 1 and 0 degrees of freedom under the null hypothesis, that is, the same distribution as followed by simple LOD scores. No information regarding the trait model can be obtained in this setting, and no power is gained when compared to a LOD-score analysis. An outlook to larger pedigrees is given. The number of degrees of freedom underlying the null distribution of MOD scores, that depends on the type of pedigrees studied, corresponds to the number of explored dimensions related to power and to the number of parameters that can jointly be estimated.     

Genetics of Type I diabetes mellitus: a single, recessive predisposition gene mapping between HLA-B and GLO. With an appendix on the estimation of selection bias.

Three different published sets of HLA-typed families of juvenile diabetes mellitus (JDM) patients have been analyzed. There was no significant genetic heterogeneity between them according to the criterion of Morton, and the total material was analyzed on the assumption of a single recessive (JDM-P) gene with incomplete penetrance. The analysis, carried out with the NYLIP program modified to account for penetrance less than 1 and for selection bias, yields highly significant lod scores for linkage between HLA and JDM-P, with a maximum value of 7.40 at theta = .05 +/- .03. The segregation of HLA and GLO in five affected sib pairs, in which one of the sibs carries an HLA/GLO recombinant, places JDM-P closer to HLA than the GLO locus: four of these five pairs are HLA-identical and GLO-different, in agreement with the conclusions of the formal linkage analysis. The data from these three independent sets of families are therefore consistent with our earlier claim that JDM is inherited as a recessive trait closely linked to HLA with reduced penetrance, and its analysis does not require more complicated genetic models.     

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.     

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 power of identity-by-state methods for linkage analysis.

The affected-sib-pair method has been widely utilized for mapping. This methodology is aimed at mapping complex traits which have been observed to be familial but for which Mendelian segregation, even after allowing for partial penetrance, is not apparent. Indications of linkage are based on the observation of nonrandom segregation at a marker locus in two affected siblings. We extend this methodology to more distant genetic relationships and examine the power of identity-by-state methods for mapping when marker information is only available on pairs of affected relatives. The power depends on the polymorphism of the marker, the probability of identity by descent at the trait locus, and the recombination fraction between the trait and the marker loci.     

A linkage strategy for detection of human quantitative-trait loci. I. Generalized relative risk ratios and power of sib pairs with extreme trait values.

We generalize the concept of the relative risk ratio (lambda) to the case of quantitative traits, to take into account the various trait outcomes of a relative pair. Formulas are derived to express the expected proportions of genes shared identical by descent by a sib pair, in terms of the generalized lambda's for sib pairs (lambda S), parent-offspring pairs (lambda O), and monozygotic twins (lambda M) and in terms of the recombination fraction, with the assumption of no residual correlations. If residual correlations are nonzero among relative pairs, we assume that they are the same among sib pairs, parent-offspring pairs, and monozygotic twins, and we employ a slightly different definition for the generalized lambda so that the same set of formulas still hold. The power (or, the sample size necessary) to detect quantitative-trait loci (QTLs) by use of extreme sib pairs (ESPs) is shown to be a function of the three generalized lambda's. Since lambda M can be derived by use of values of lambda S and lambda O, estimates of the latter two lambda's will suffice for the analysis of power and the necessary sample sizes of ESPs, for a QTL linkage study.     

Evidence for a new Graves disease susceptibility locus at chromosome 18q21

Graves disease (GD) is a common autoimmune thyroid disorder that is inherited as a complex multigenic trait. By using a single microsatellite marker at each locus, we screened the type 1 diabetes loci IDDM4, IDDM5, IDDM6, IDDM8, and IDDM10 and the fucosyltransferase-2 locus for linkage in sib pairs with GD. This showed a two-point nonparametric linkage (NPL) score of 1.57 (P=.06) at the IDDM6 marker D18S41, but NPL scores were <1.0 at the other five loci. Thus, the investigation of the IDDM6 locus was extended by genotyping 11 microsatellite markers spanning 48 cM across chromosome 18q12-q22 in 81 sib pairs affected with autoimmune thyroid disease (AITD). Multipoint analysis, designating all AITD sib pairs as affected, showed a peak NPL score of 3.46 (P=.0003), at the marker D18S487. Designation of only GD cases as affected (74 sib pairs) showed a peak NPL score of 3.09 (P=.001). Linkage to this region has been demonstrated in type 1 diabetes (IDDM6), rheumatoid arthritis, and systemic lupus erythematosus, which suggests that this locus may have a role in several forms of autoimmunity.     

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.     

High-resolution genetic mapping of complex traits.

Positional cloning requires high-resolution genetic mapping. To plan a positional cloning project, one needs to know how many informative meioses will be required to narrow the search for a disease gene to an acceptably small region. For a simple Mendelian trait studied with linkage analysis, the answer is straightforward. In this paper, we address the situation of a complex trait studied with affected-relative-pair methods. We derive mathematical formulas for the size of an appropriate confidence region, as a function of the relative risk attributable to the gene. Using these results, we provide graphs showing the number of relative pairs required to narrow the gene hunt to an interval of a given size. For example, we show that localizing a gene to 1 cM requires a median of 200 sib pairs for a locus causing a fivefold increased risk to an offspring and 700 sib pairs for a locus causing a twofold increased risk. We discuss the implications of these results for the positional cloning of genes underlying complex traits.     

Linkage analysis of quantitative trait loci: sib pairs or sibships?

Sib pair linkage studies are now widely used to investigate the genetic factors implicated in complex quantitative traits. To increase the power of these approaches, it has been proposed to select extremely discordant (ED) sib pairs which are expected to contain the highest linkage information. However, it is known that sibships of larger size contain more linkage information than independent sib pairs. In this paper we compare, in terms of power and cost considerations, the ED strategy, which uses information on sib pairs only, to the recently developed 'Maximum Likelihood Binomial' sibship-oriented method performed on the whole sibships from which the ED sib pairs have been extracted. We show that the use of these whole sibships is an efficient alternative to approaches focusing on ED sib pairs only.     

Selection strategies for linkage studies using twins.

Genetic linkage analysis for complex diseases offers a major challenge to geneticists. In these complex diseases multiple genetic loci are responsible for the disease and they may vary in the size of their contribution; the effect of any single one of them is likely to be small. In many situations, like in extensive twin registries, trait values have been recorded for a large number of individuals, and preliminary studies have revealed summary measures for those traits, like mean, variance and components of variance, including heritability. Given the small effect size, a random sample of twins will require a prohibitively large sample size. It is well known that selective sampling is far more efficient in terms of genotyping effort. In this paper we derive easy expressions for the information contributed by sib pairs for the detection of linkage to a quantitative trait locus (QTL). We consider random samples as well as samples of sib pairs selected on the basis of their trait values. These expressions can be rapidly computed and do not involve simulation. We extend our results for quantitative traits to dichotomous traits using the concept of a liability threshold model. We present tables with required sample sizes for height, insulin levels and migraine, three of the traits studied in the GenomEUtwin project.     

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.     

The linkage information content value of polymorphism genetic markers in model-free linkage analysis

Guo and Elston [Hum Hered 1999;49:112-118] developed a linkage information content (LIC) value to measure the informativeness of a marker for identity-by-descent (IBD) sharing status of relative pairs. LIC values were derived for five types of relative pairs: full sib, half sib, grandparent-grandchild, first cousin and avuncular. In this paper, we give corrected LIC values for full sib, grandparent-grandchild, first cousin and avuncular pairs, and indicate the availability of a computer program to calculate them.     

Model-free linkage analysis with covariates confirms linkage of prostate cancer to chromosomes 1 and 4

As with many complex genetic diseases, genome scans for prostate cancer have given conflicting results, often failing to provide replication of previous findings. One factor contributing to the lack of consistency across studies is locus heterogeneity, which can weaken or even eliminate evidence for linkage that is present only in a subset of families. Currently, most analyses either fail to account for locus heterogeneity or attempt to account for it only by partitioning data sets into smaller and smaller portions. In the present study, we model locus heterogeneity among affected sib pairs with prostate cancer by including covariates in the linkage analysis that serve as surrogate measures of between-family linkage differences. The model is a modification of the Olson conditional logistic model for affected relative pairs. By including Gleason score, age at onset, male-to-male transmission, and/or number of affected first-degree family members as covariates, we detected linkage near three locations that were previously identified by linkage (1q24-25 [HPC1; LOD score 3.25, P=.00012], 1q42.2-43 [PCAP; LOD score 2.84, P=.0030], and 4q [LOD score 2.80, P=.00038]), near the androgen-receptor locus on Xq12-13 (AR; LOD score 3.06, P=.00053), and at five new locations (LOD score > 2.5). Without covariates, only a few weak-to-moderate linkage signals were found, none of which replicate findings of previous genome scans. We conclude that covariate-based linkage analysis greatly improves the likelihood that linked regions will be found by incorporation of information about heterogeneity within the sample.     

The power to detect linkage disequilibrium with quantitative traits in selected samples

Results from power studies for linkage detection have led to many ongoing and planned collections of phenotypically extreme nuclear families. Given the great expense of collecting these families and the imminent availability of a dense diallelic marker map, the families are likely to be used in allelic-association as well as linkage studies. However, optimal selection strategies for linkage may not be equally powerful for association. We examine the power to detect linkage disequilibrium for quantitative traits after phenotypic selection. The results encompass six selection strategies that are in widespread use, including single selection (two designs), affected sib pairs, concordant and discordant pairs, and the extreme-concordant and -discordant design. Selection of sibships on the basis of one extreme proband with high or low trait scores provides as much power as discordant sib pairs but requires the screening and phenotyping of substantially fewer initial families from which to select. Analysis of the role of allele frequencies within each selection design indicates that common trait alleles generally offer the most power, but similarities between the marker- and trait-allele frequencies are much more important than the trait-locus frequency alone. Some of the most widespread selection designs, such as single selection, yield power gains only when both the marker and quantitative trait loci (QTL) are relatively rare in the population. In contrast, discordant pairs and the extreme-proband design provide power for the broadest range of QTL-marker-allele frequency differences. Overall, proband selection from either tail provides the best balance of power, robustness, and simplicity of ascertainment for family-based association analysis.     

Interval Estimation of Simple Difference under Independent Negative Binomial Sampling

When the sample size is not large or when the underlying disease is rare, to assure collection of an appropriate number of cases and to control the relative error of estimation, one may employ inverse sampling, in which one continues sampling subjects until one obtains exactly the desired number of cases. This paper focuses discussion on interval estimation of the simple difference between two proportions under independent inverse sampling. This paper develops three asymptotic interval estimators on the basis of the maximum likelihood estimator (MLE), the uniformly minimum variance unbiased estimator (UMVUE), and the asymptotic likelihood ratio test (ALRT). To compare the performance of these three estimators, this paper calculates the coverage probability and the expected length of the resulting confidence intervals on the basis of the exact distribution. This paper finds that when the underlying proportions of cases in both two comparison populations are small or moderate (≤0.20), all three asymptotic interval estimators developed here perform reasonably well even for the pre-determined number of cases as small as 5. When the pre-determined number of cases is moderate or large (≥50), all three estimators are essentially equivalent in all the situations considered here. Because application of the two interval estimators derived from the MLE and the UMVUE does not involve any numerical iterative procedure needed in the ALRT, for simplicity we may use these two estimators without losing efficiency.     

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.     

Equivalence between Haseman-Elston and variance-components linkage analyses for sib pairs

The Haseman-Elston regression method offers a simpler alternative to variance-components (VC) models, for the linkage analysis of quantitative traits. However, even the "revisited" method, which uses the cross-product--rather than the squared difference--in sib trait values, is, in general, less powerful than VC models. In this report, we clarify the relative efficiencies of existing Haseman-Elston methods and show how a new Haseman-Elston method can be constructed to have power equivalent to that of VC models. This method uses as the dependent variable a linear combination of squared sums and squared differences, in which the weights are determined by the overall trait correlation between sibs in a population. We show how this method can be used for both the selection of maximally informative sib pairs for genotyping and the subsequent analysis of such selected samples.