A unified sampling approach for multipoint analysis of qualitative and quantitative traits in sib pairs

Recent advances in molecular biology have enhanced the opportunity to conduct multipoint mapping for complex diseases. Concurrently, one sees a growing interest in the use of quantitative traits in linkage studies. Here, we present a multipoint sib-pair approach to locate the map position (tau) of a trait locus that controls the observed phenotype (qualitative or quantitative), along with a measure of statistical uncertainty. This method builds on a parametric representation for the expected identical-by-descent statistic at an arbitrary locus, conditional on an event reflecting the sampling scheme, such as affected sib pairs, for qualitative traits, or extreme discordant (ED) sib pairs, for quantitative traits. Our results suggest that the variance about tau&d4;, the estimator of tau, can be reduced by as much as 60%-70% by reducing the length of intervals between markers by one half. For quantitative traits, we examine the precision gain (measured by the variance reduction in tau&d4;) by genotyping extremely concordant (EC) sib pairs and including them along with ED sib pairs in the statistical analysis. The precision gain depends heavily on the residual correlation of the quantitative trait for sib pairs but considerably less on the allele frequency and exact genetic mechanism. Since complex traits involve multiple loci and, hence, the residual correlation cannot be ignored, our finding strongly suggests that one should incorporate EC sib pairs along with ED sib pairs, in both design and analysis. Finally, we empirically establish a simple linear relationship between the magnitude of precision gain and the ratio of the number of ED pairs to the number of EC pairs. This relationship allows investigators to address issues of cost effectiveness that are due to the need for phenotyping and genotyping subjects.     

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.     

Mapping quantitative trait loci with extreme discordant sib pairs: sampling considerations.

Elsewhere we have proposed the use of extreme discordant sib pairs (EDSPs) for mapping quantitative trait loci in humans. Here we present sample sizes necessary to achieve a given level of power with this study design, as well as the number of sibs that need to be screened to obtain the required sample. Further, we present simple formulas for adjusting sample sizes to account for variable significance levels and power, as well as the density and informativeness of linkage markers in a multipoint sib-pair analysis. We conclude that with EDSPs, the most powerful study design, the smallest genetic effect detectable with a realistic sample size is approximately 10% of the variance of the trait.     

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.     

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.     

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.     

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.     

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.     

A score test for the linkage analysis of qualitative and quantitative traits based on identity by descent data from sib-pairs

We propose a general likelihood-based approach to the linkage analysis of qualitative and quantitative traits using identity by descent (IBD) data from sib-pairs. We consider the likelihood of IBD data conditional on phenotypes and test the null hypothesis of no linkage between a marker locus and a gene influencing the trait using a score test in the recombination fraction theta between the two loci. This method unifies the linkage analysis of qualitative and quantitative traits into a single inferential framework, yielding a simple and intuitive test statistic. Conditioning on phenotypes avoids unrealistic random sampling assumptions and allows sib-pairs from differing ascertainment mechanisms to be incorporated into a single likelihood analysis. In particular, it allows the selection of sib-pairs based on their trait values and the analysis of only those pairs having the most informative phenotypes. The score test is based on the full likelihood, i.e. the likelihood based on all phenotype data rather than just differences of sib-pair phenotypes. Considering only phenotype differences, as in Haseman and Elston (1972) and Kruglyak and Lander (1995), may result in important losses in power. The linkage score test is derived under general genetic models for the trait, which may include multiple unlinked genes. Population genetic assumptions, such as random mating or linkage equilibrium at the trait loci, are not required. This score test is thus particularly promising for the analysis of complex human traits. The score statistic readily extends to accommodate incomplete IBD data at the test locus, by using the hidden Markov model implemented in the programs MAPMAKER/SIBS and GENEHUNTER (Kruglyak and Lander, 1995; Kruglyak et al., 1996). Preliminary simulation studies indicate that the linkage score test generally matches or outperforms the Haseman-Elston test, the largest gains in power being for selected samples of sib-pairs with extreme phenotypes.     

The quantitative LOD score: test statistic and sample size for exclusion and linkage of quantitative traits in human sibships.

We present a test statistic, the quantitative LOD (QLOD) score, for the testing of both linkage and exclusion of quantitative-trait loci in randomly selected human sibships. As with the traditional LOD score, the boundary values of 3, for linkage, and -2, for exclusion, can be used for the QLOD score. We investigated the sample sizes required for inferring exclusion and linkage, for various combinations of linked genetic variance, total heritability, recombination distance, and sibship size, using fixed-size sampling. The sample sizes required for both linkage and exclusion were not qualitatively different and depended on the percentage of variance being linked or excluded and on the total genetic variance. Information regarding linkage and exclusion in sibships larger than size 2 increased as approximately all possible pairs n(n-1)/2 up to sibships of size 6. Increasing the recombination (theta) distance between the marker and the trait loci reduced empirically the power for both linkage and exclusion, as a function of approximately (1-2theta)4.     

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.     

A novel framework for sib pair linkage analysis

Sib pair linkage analysis of a dichotomous trait is a popular method for narrowing the search for genes that influence complex diseases. Although the pedigree structures are uncomplicated and the underlying genetic principles straightforward, a surprising degree of complexity is involved in implementing a sib pair study and interpreting the results. Ascertainment may be based on affected, discordant, or unaffected sib pairs, as well as on pairs defined by threshold values for quantitative traits, such as extreme discordant sib pairs. To optimize power, various domain restrictions and null hypotheses have been proposed for each of these designs, yielding a wide array of choices for the analyst. To begin, we systematically classify the major sources of discretion in sib pair linkage analysis. Then, we extend the work of Kruglyak and Lander (1995), to bring the various forms into a unified framework and to facilitate a more general approach to the analysis. Finally, we describe a new, freely available computer program, Splat (Sib Pair Linkage Analysis Testing), that can perform any sib pair statistical test currently in use, as well as any user-defined test yet to be proposed. Splat uses the expectation maximization algorithm to calculate maximum-likelihood estimates of sharing (subject to user-specified conditions) and then plots LOD scores versus chromosomal position. It includes a novel grid-scanning capability that enables simultaneous visualization of multiple test statistics. This can lead to further insight into the genetic basis of the disease process under consideration. In addition, phenotype definitions can be modified without the recalculation of inheritance vectors, thereby providing considerable flexibility for exploratory analysis. The application of Splat will be illustrated with data from studies on the genetics of diabetic nephropathy.     

A linkage strategy for detection of human quantitative-trait loci. II. Optimization of study designs based on extreme sib pairs and generalized relative risk ratios.

We are concerned here with practical issues in the application of extreme sib-pair (ESP) methods to quantitative traits. Two important factors-namely, the way extreme trait values are defined and the proportions in which different types of ESPs are pooled, in the analysis-are shown to determine the power and the cost effectiveness of a study design. We found that, in general, combining reasonable numbers of both extremely discordant and extremely concordant sib pairs that were available in the sample is more powerful and more cost effective than pursuing only a single type of ESP. We also found that dividing trait values with a less extreme threshold at one end or at both ends of the trait distribution leads to more cost-effective designs. The notion of generalized relative risk ratios (the lambda methods, as described in the first part of this series of two articles) is used to calculate the power and sample size for various choices of polychotomization of trait values and for the combination of different types of ESPs. A balance then can be struck among these choices, to attain an optimum design.     

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.     

Pleiotropic effects of a chromosome 3 locus on speech-sound disorder and reading

Speech-sound disorder (SSD) is a complex behavioral disorder characterized by speech-sound production errors associated with deficits in articulation, phonological processes, and cognitive linguistic processes. SSD is prevalent in childhood and is comorbid with disorders of language, spelling, and reading disability, or dyslexia. Previous research suggests that developmental problems in domains associated with speech and language acquisition place a child at risk for dyslexia. Recent genetic studies have identified several candidate regions for dyslexia, including one on chromosome 3 segregating in a large Finnish pedigree. To explore common genetic influences on SSD and reading, we examined linkage for several quantitative traits to markers in the pericentrometric region of chromosome 3 in 77 families ascertained through a child with SSD. The quantitative scores measured several processes underlying speech-sound production, including phonological memory, phonological representation, articulation, receptive and expressive vocabulary, and reading decoding and comprehension skills. Model-free linkage analysis was followed by identification of sib pairs with linkage and construction of core shared haplotypes. In our multipoint analyses, measures of phonological memory demonstrated the strongest linkage (marker D3S2465, P=5.6 x 10(-5), and marker D3S3716, P=6.8 x 10(-4)). Tests for single-word decoding also demonstrated linkage (real word reading: marker D3S2465, P=.004; nonsense word reading: marker D3S1595, P=.005). The minimum shared haplotype in sib pairs with similar trait values spans 4.9 cM and is bounded by markers D3S3049 and D3S3045. Our results suggest that domains common to SSD and dyslexia are pleiotropically influenced by a putative quantitative trait locus on chromosome 3.     

The relative efficiency of penetrance estimators for sib pairs

Here we present analytical studies to evaluate the relative efficiency of commonly used penetrance estimators using linkage designs. We investigated three different methods of estimating penetrance using sib pairs: Maximum likehood estimation (MLE) with trait information alone, MLE with both trait and marker information and the MOD score approach. Modeling sib pairs with unknown phase, we evaluated the asymptotic relative efficiency between estimators under either random sampling or single ascertainment for an autosomal dominant or recessive disease. We then provide plots of the asymptotic relative efficiency, enabling researchers to easily determine regions where the MOD score or segregation alone performs with comparable efficiency relative to joint segregation and linkage.     

Increasing the power and efficiency of disease-marker case-control association studies through use of allele-sharing information

Case-control disease-marker association studies are often used in the search for variants that predispose to complex diseases. One approach to increasing the power of these studies is to enrich the case sample for individuals likely to be affected because of genetic factors. In this article, we compare three case-selection strategies that use allele-sharing information with the standard strategy that selects a single individual from each family at random. In affected sibship samples, we show that, by carefully selecting sibships and/or individuals on the basis of allele sharing, we can increase the frequency of disease-associated alleles in the case sample. When these cases are compared with unrelated controls, the difference in the frequency of the disease-associated allele is therefore also increased. We find that, by choosing the affected sib who shows the most evidence for pairwise allele sharing with the other affected sibs in families, the test statistic is increased by >20%, on average, for additive models with modest genotype relative risks. In addition, we find that the per-genotype information associated with the allele sharing-based strategies is increased compared with that associated with random selection of a sib for genotyping. Even though we select sibs on the basis of a nonparametric statistic, the additional gain for selection based on the unknown underlying mode of inheritance is minimal. We show that these properties hold even when the power to detect linkage to a region in the entire sample is negligible. This approach can be extended to more-general pedigree structures and quantitative traits.     

Mapping quantitative trait loci in oligogenic models

We discuss strategies for mapping quantitative trait loci with emphasis on certain issues of study design that have recently received attention: e.g. genotyping only selected pedigrees and the comparative value of large pedigrees versus sib pairs. We use a standard variance components model and a parametrization of the genetic effects in which the 'segregation' parameters are locally orthogonal to the 'linkage' parameters. This permits simple explicit expressions for the expectation of the score statistic, which we use to compare the power of different strategies. We also discuss robustness of the score statistic.     

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.     

Powerful allele sharing statistics for nonparametric linkage analysis

Nonparametric linkage analysis is widely used to map susceptibility genes for complex diseases. This paper introduces six nonparametric statistics for measuring marker allele sharing among the affected members of a pedigree. We compare the power of these new statistics and three previous statistics to detect linkage with Mendelian diseases having recessive, additive, and dominant modes of inheritance. The nine statistics represent all possible combinations of three different IBD scoring functions and three different schemes for sampling genes among affecteds. Our results strongly suggest that the statistic T(rec)(blocks) is best for recessive traits, while the two statistics T(kin)(pairs) and T(all)(kin) vie for best for an additive trait. The best statistic for a dominant trait is less clear. The statistics T(kin)(pairs) and T(all)(kin) are equally promising for small sibships, but in extended pedigrees the statistics T(dom)(blocks) and T(dom)(pairs) appear best. For a complex trait, we advocate computing several of these statistics.