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.     

Ignoring linkage disequilibrium among tightly linked markers induces false-positive evidence of linkage for affected sib pair analysis

Most multipoint linkage programs assume linkage equilibrium among the markers being studied. The assumption is appropriate for the study of sparsely spaced markers with intermarker distances exceeding a few centimorgans, because linkage equilibrium is expected over these intervals for almost all populations. However, with recent advancements in high-throughput genotyping technology, much denser markers are available, and linkage disequilibrium (LD) may exist among the markers. Applying linkage analyses that assume linkage equilibrium to dense markers may lead to bias. Here, we demonstrated that, when some or all of the parental genotypes are missing, assuming linkage equilibrium among tightly linked markers where strong LD exists can cause apparent oversharing of multipoint identity by descent (IBD) between sib pairs and false-positive evidence for multipoint model-free linkage analysis of affected sib pair data. LD can also mimic linkage between a disease locus and multiple tightly linked markers, thus causing false-positive evidence of linkage using parametric models, particularly when heterogeneity LOD score approaches are applied. Bias can be eliminated by inclusion of parental genotype data and can be reduced when additional unaffected siblings are included in the analysis.     

Mathematical assumptions versus biological reality: myths in affected sib pair linkage analysis

Affected sib pair (ASP) analysis has become common ever since it was shown that, under very specific assumptions, ASPs afford a powerful design for linkage analysis. In 2003, Vieland and Huang, on the basis of a "fundamental heterogeneity equation," proved that heterogeneity and epistasis are confounded in ASP linkage analysis. A much more serious limitation of ASP linkage analysis is the implicit assumption that randomly sampled sib pairs share half their alleles identical by descent at any locus, whereas a critical assumption underlying Vieland and Huang's proof is that of joint Hardy-Weinberg equilibrium proportions at two trait loci. These are considered as examples of mathematical assumptions that may not always reflect biological reality. More-robust sib-pair designs and appropriate methods for their analysis have long been available.     

Nonparametric trend statistic incorporating dispersion differences in sib pair linkage for quantitative traits

The Haseman-Elston (HE) regression method and its extensions are widely used in genetic studies for detecting linkage to quantitative trait loci (QTL) using sib pairs. The principle underlying the simple HE regression method is that the similarity in phenotypes between two siblings increases as they share an increasing number of alleles identical by descent (IBD) from their parents at a particular marker locus. In such a procedure, similarity was identified with the locations, that is, means of groups of sib pairs sharing 0, 1, and 2 alleles IBD. A more powerful, rank-based nonparametric test to detect increasing similarity in sib pairs is presented by combining univariate trend statistics not only of locations, but also of dispersions of the squared phenotypic differences of two siblings for three groups. This trend test does not rely on distributional assumptions, and is applicable to the skewed or leptokurtic phenotypic distributions, in addition to normal or near normal phenotypic distributions. The performances of nonparametric trend statistics, including nonparametric regression slope, are compared with the HE regression methods as genetic linkage strategies.     

A note on linkage analysis with affected sib triplets

Previously, it has been shown for affected sib pairs that the mean test is the uniformly (in theta) most powerful test in case of a multiplicative mode of inheritance and that the mean test is equivalent to parametric linkage analysis calculated under an assumed multiplicative mode of inheritance. Here, these two results are extended to samples consisting of affected sib triplets. For affected sib quadruplets, however, it is shown that these results are no longer valid.     

Multilocus linkage analysis of affected sib pairs

OBJECTIVE: The conventional affected sib pair methods evaluate the linkage information at a locus by considering only marginal information. We describe a multilocus linkage method that uses both the marginal information and information derived from the possible interactions among several disease loci, thereby increasing the significance of loci with modest effects. METHODS: Our method is based on a statistic that quantifies the linkage information contained in a set of markers. By a marker selection-reduction process, we screen a set of polymorphisms and select a few that seem linked to disease. RESULTS: We test our approach on genome scan data for inflammatory bowel disease (InfBD) and on simulated data. On real data we detect 6 of the 8 known InfBD loci; on simulated data we obtain improvements in power of up to 40% compared to a conventional single-locus method. CONCLUSION: Our extensive simulations and the results on real data show that our method is in general more powerful than single-locus methods in detecting disease loci responsible for complex traits. A further advantage of our approach is that it can be extended to make use of both the linkage and the linkage disequilibrium between disease loci and nearby markers.     

p-Value simulation for affected sib pair multiple testing

A standard approach to calculation of critical values for affected sib pair multiple testing is based on: (a) fully informative markers, (b) Haldane map function assumptions leading to a Markov chain model for inheritance vectors, (c) central limit approximation to averages of sampled inheritance vectors leading to an Ornstein-Uhlenbeck process approximation, and (d) simple approximations to the maximum of such a process. Under these assumptions, assuming equispaced or close to equispaced markers, if the sample size is large, an approximation is available that is easy to calculate and performs well. However, for small sample sizes, a large number of markers, and for small p-values, there is good reason to be cautious about the use of the Gaussian approximation. We develop an algorithm for calculation of multiple testing p-values based on the standard Markov chain model, avoiding the use of Gaussian (large sample) approximation. We illustrate the use of this algorithm by demonstrating some inadequacies of the Gaussian approximation.     

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.     

Detecting gene-gene interactions using affected sib pair analysis with covariates

Interest has recently focussed on allowing for interactions between loci as a way to increase power to detect linkage. In this paper, a simplified logistic regression method was used to perform affected sib pair analyses allowing for the inclusion of data from other loci. A systematic search of two-locus disease models was carried out to determine the situations in which this was advantageous. If IBD information is available (e.g. from a genome scan), it is unlikely that allowing for interactions will give a large lod score in the absence of linkage evidence from sinlge-locus analysis. Furthermore, allowing for interactions rarely gave a significant increase in power to detect linkage over a single-locus analysis, except for heterogeneity models with low K(P). Conversely, the availability of disease-associated genotypes may greatly increase the power both to detect linkage to a second locus and interaction between the loci. These results indicate that when only IBD information is available, two-locus analysis of genome scan data should be restricted to regions giving peaks under single-locus analysis. If disease-associated genotypes are available, it may be worth re-analysing the whole genome.     

A general framework for statistical linkage analysis in multivalent tetraploids

In multivalent polyploids, simultaneous pairings among homologous chromosomes at meiosis result in a unique cytological phenomenon-double reduction. Double reduction casts an impact on chromosome evolution in higher plants, but because of its confounded effect on the pattern of gene cosegregation, it complicates linkage analysis and map construction with polymorphic molecular markers. In this article, we have proposed a general statistical model for simultaneously estimating the frequencies of double reduction, the recombination fraction, and optimal parental linkage phases between any types of markers, both fully and partially informative, or dominant and codominant, for a tetraploid species that undergoes only multivalent pairing. This model provides an in-depth extension of our earlier linkage model that was built upon Fisher's classifications for different gamete formation modes during the polysomic inheritance of a multivalent polyploid. By implementing a two-stage hierarchical EM algorithm, we derived a closed-form solution for estimating the frequencies of double reduction through the estimation of gamete mode frequencies and the recombination fraction. We performed different settings of simulation studies to demonstrate the statistical properties of our model for estimating and testing double reduction and the linkage in multivalent tetraploids. As shown by a comparative analysis, our model provides a general framework that covers existing statistical approaches for linkage mapping in polyploids that are predominantly multivalent. The model will have great implications for understanding the genome structure and organization of polyploid species.     

X-linked extension of the revised Haseman-Elston algorithm for linkage analysis in sib pairs

Haseman and Elston (H-E) proposed a regression-based robust test of linkage between a marker and an autosomal quantitative trait locus, using the squared sib pair trait difference as a dependent variable and the proportion of alleles shared identical by descent by the sib pair as an independent variable. Several authors have proposed improvement of the original H-E's seminal work by using an optimal linear combination of squared sum and squared difference as the dependent variable. In this paper, we extend Haseman and Elston's sib pair method to an X-linked locus. We give a general formulation of the complete regression model and details of the regression coefficients in terms of variance components. Simulation results are presented to describe the power of this technique for a theoretical best case scenario.     

Log-linear models for gene mapping with affected sib pair data

In genome-wide screens of genetic marker loci, non-mendelian inheritance of a marker is taken to indicate its vicinity to a disease locus. Heritable complex traits are thought to be under the influence of multiple possibly interacting susceptibility loci yet the most frequently used methods of linkage and association analysis focus on one susceptibility locus at a time. Here we introduce log-linear models for the joint analysis of multiple marker loci and interaction effects between them. Our approach focuses on affected sib pair data and identical by descent (IBD) allele sharing values observed on them. For each heterozygous parent, the IBD values at linked markers represent a sequence of dependent binary variables. We develop log-linear models for the joint distribution of these IBD values. An independence log-linear model is proposed to model the marginal means and the neighboring interaction model is advocated to account for associations between adjacent markers. Under the assumption of conditional independence, likelihood methods are applied to simulated data containing one or two susceptibility loci. It is shown that the neighboring interaction log-linear model is more efficient than the independence model, and incorporating interaction in the two-locus analysis provides increased power and accuracy for mapping of the trait loci.     

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.     

Selective genotyping for QTL detection using sib pair analysis in outbred populations with hierarchical structures

A simulation study illustrates the effects of the inclusion of half-sib pairs as well as the effects of selective genotyping on the power of detection and the parameter estimates in a sib pair analysis of data from an outbred population. The power of QTL detection obtained from samples of sib pairs selected according to their within family variance or according to the mean within family variance within half sib family was compared and contrasted with the power obtained when only full sib pair analysis was used. There was an increase in power (4–16%) and decrease in the bias of parameter estimates with the use of half-sib information. These improvements in power and parameter estimates depended on the number of the half sib pairs (half sib family size). Almost the same power as that obtained using all the available sib pairs could be achieved by selecting only 50–60% the animals. The most effective method was to select both full and half sib pairs on the basis of high within full sib family variance for the trait in question. The QTL position estimates were in general slightly biased towards the center of the chromosome and the QTL variance estimates were biased upwards, there being quite large differences in bias depending on the selection method.     

A systematic genomewide linkage study in 353 sib pairs with schizophrenia

We undertook a genomewide linkage study in a total of 353 affected sib pairs (ASPs) with schizophrenia. Our sample consisted of 179 ASPs from the United Kingdom, 134 from Sweden, and 40 from the United States. We typed 372 microsatellite markers at approximately 10-cM intervals. Our strongest finding was a LOD score of 3.87 on chromosome 10q25.3-q26.3, with positive results being contributed by all three samples and a LOD-1 interval of 15 cM. This finding achieved genomewide significance (P<.05), on the basis of simulation studies. We also found two regions, 17p11.2-q25.1 (maximum LOD score [MLS] = 3.35) and 22q11 (MLS = 2.29), in which the evidence for linkage was highly suggestive. Linkage to all of these regions has been supported by other studies. Moreover, we found strong evidence for linkage (genomewide P<.02) to 17p11.2-q25.1 in a single pedigree with schizophrenia. In our view, the evidence is now sufficiently compelling to undertake detailed mapping studies of these three regions.     

Multipoint linkage analysis using sib pairs: an interval mapping approach for dichotomous outcomes.

I propose an interval mapping approach suitable for a dichotomous outcome, with emphasis on samples of affected sib pairs. The method computes a lod score for each of a set of locations in the interval between two flanking markers and takes as its estimate of trait-locus location the maximum lod score in the interval, provided it exceeds the prespecified critical value. Use of the method depends on prior knowledge of the genetic model for the disease only through available estimates of recurrence risk to relatives of affected individuals. The method gives an unbiased estimate of location, provided the recurrence risk are correctly specified and provided the marker identity-by-descent probabilities are jointly, rather than individually, estimated. I also discuss use of the method for traits determined by two loci and give an approximation that has good power for a wide range of two-locus models.     

Power to detect linkage based on multiple sets of data in the presence of locus heterogeneity: comparative evaluation of model-based linkage methods for affected sib pair data

The development of rigorous methods for evaluating the overall strength of evidence for genetic linkage based on multiple sets of data is becoming increasingly important in connection with genomic screens for complex disorders. We consider here what happens when we attempt to increase power to detect linkage by pooling multiple independently collected sets of families under conditions of variable levels of locus heterogeneity across samples. We show that power can be substantially reduced in pooled samples when compared to the most informative constituent subsamples considered alone, in spite of the increased sample size afforded by pooling. We demonstrate that for affected sib pair data, a simple adaptation of the lod score (which we call the compound lod), which allows for intersample admixture differences can afford appreciably higher power than the ordinary heterogeneity lod; and also, that a statistic we have proposed elsewhere, the posterior probability of linkage, performs at least as well as the compound lod while having considerable computational advantages. The companion paper (this issue, pp 217-225) shows further that in application to multiple data sets, familiar model-free methods are in some sense equivalent to ordinary lod scores based on data pooling, and that they therefore will also suffer dramatic losses in power for pooled data in the presence of locus heterogeneity and other complicating factors.