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.     

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.     

Further evidence for the increased power of LOD scores compared with nonparametric methods.

In genetic analysis of diseases in which the underlying model is unknown, "model free" methods-such as affected sib pair (ASP) tests-are often preferred over LOD-score methods, although LOD-score methods under the correct or even approximately correct model are more powerful than ASP tests. However, there might be circumstances in which nonparametric methods will outperform LOD-score methods. Recently, Dizier et al. reported that, in some complex two-locus (2L) models, LOD-score methods with segregation analysis-derived parameters had less power to detect linkage than ASP tests. We investigated whether these particular models, in fact, represent a situation that ASP tests are more powerful than LOD scores. We simulated data according to the parameters specified by Dizier et al. and analyzed the data by using a (a) single locus (SL) LOD-score analysis performed twice, under a simple dominant and a recessive mode of inheritance (MOI), (b) ASP methods, and (c) nonparametric linkage (NPL) analysis. We show that SL analysis performed twice and corrected for the type I-error increase due to multiple testing yields almost as much linkage information as does an analysis under the correct 2L model and is more powerful than either the ASP method or the NPL method. We demonstrate that, even for complex genetic models, the most important condition for linkage analysis is that the assumed MOI at the disease locus being tested is approximately correct, not that the inheritance of the disease per se is correctly specified. In the analysis by Dizier et al., segregation analysis led to estimates of dominance parameters that were grossly misspecified for the locus tested in those models in which ASP tests appeared to be more powerful than LOD-score analyses.     

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.     

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.     

Effects of stratification in the analysis of affected-sib-pair data: benefits and costs

The benefits and costs of stratification of affected-sib-pair (ASP) data were examined in three situations: (1) when there is no difference in identity-by-descent (IBD) allele sharing between stratified and unstratified ASP data sets; (2) when there is an increase in IBD allele sharing in one of the stratified groups; and (3) when the data are stratified on the basis of IBD allele-sharing status at one locus, and the stratified ASPs are then analyzed for linkage at a second locus. When there is no difference in IBD sharing between strata, a penalty is always paid for stratifying the data. The loss of power to detect linkage in the stratified ASP data sets is the result of multiple testing and the smaller sample size within individual strata. In the case in which etiologic heterogeneity (i.e., severity of phenotype, age at onset) represents genetic heterogeneity, the power to detect linkage can be increased by stratifying the ASP data. This benefit is obtained when there is sufficient IBD allele sharing and sample sizes. Once linkage has been established for a given locus, data can be stratified on the basis of IBD status at this locus and can be tested for linkage at a second locus. When the relative risk is in the vicinity of 1, the power to detect linkage at the second locus is always greater for the unstratified ASP data set. Even for values of the relative risk that diverge sufficiently from 1, with adequate sample sizes and IBD allele sharing, the benefits of stratifying ASP data are minimal.     

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.     

The score statistic of the LD-lod analysis: detecting linkage adaptive to linkage disequilibrium

We study the properties of a modified lod score method for testing linkage that incorporates linkage disequilibrium (LD-lod). By examination of its score statistic, we show that the LD-lod score method adaptively combines two sources of information: (a) the IBD sharing score which is informative for linkage regardless of the existence of LD and (b) the contrast between allele-specific IBD sharing scores which is informative for linkage only in the presence of LD. We also consider the connection between the LD-lod score method and the transmission-disequilibrium test (TDT) for triad data and the mean test for affected sib pair (ASP) data. We show that, for triad data, the recessive LD-lod test is asymptotically equivalent to the TDT; and for ASP data, it is an adaptive combination of the TDT and the ASP mean test. We demonstrate that the LD-lod score method has relatively good statistical efficiency in comparison with the ASP mean test and the TDT for a broad range of LD and the genetic models considered in this report. Therefore, the LD-lod score method is an interesting approach for detecting linkage when the extent of LD is unknown, such as in a genome-wide screen with a dense set of genetic markers.     

The structure of genetic linkage data: from LIPED to 1M SNPs

There are three assumptions of independence or conditional independence that underlie linkage likelihood computations on sets of related individuals. The first is the independence of meioses, which gives rise to the conditional independence of haplotypes carried by offspring, given those of their parents. The second derives from the assumption of absence of genetic interference, which gives rise to the conditional independence of inheritance vectors, given the inheritance vector at an intermediate location. The third is the assumption of independence of allelic types, at the population level, both among haplotypes of unrelated individuals and also over the loci along a given haplotype. These three assumptions have been integral to likelihood computations since the first lod scores were computed, and remain key components in analysis of modern genetic data. In this paper we trace the role of these assumptions through the history of linkage likelihood computation, through to a new framework of genetic linkage analysis in the era of dense genomic marker data.     

Multicenter linkage study of schizophrenia candidate regions on chromosomes 5q, 6q, 10p, and 13q: schizophrenia linkage collaborative group III

Schizophrenia candidate regions 33-51 cM in length on chromosomes 5q, 6q, 10p, and 13q were investigated for genetic linkage with mapped markers with an average spacing of 5.64 cM. We studied 734 informative multiplex pedigrees (824 independent affected sibling pairs [ASPs], or 1,003 ASPs when all possible pairs are counted), which were collected in eight centers. Cases with diagnoses of schizophrenia or schizoaffective disorder (DSM-IIIR criteria) were considered affected (n=1,937). Data were analyzed with multipoint methods, including nonparametric linkage (NPL), ASP analysis using the possible-triangle method, and logistic-regression analysis of identity-by-descent (IBD) sharing in ASPs with sample as a covariate, in a test for intersample heterogeneity and for linkage with allowance for intersample heterogeneity. The data most supportive for linkage to schizophrenia were from chromosome 6q; logistic-regression analysis of linkage allowing for intersample heterogeneity produced an empirical P value <.0002 with, or P=.0004 without, inclusion of the sample that produced the first positive report in this region; the maximum NPL score in this region was 2.47 (P=.0046), the maximum LOD score (MLS) from ASP analysis was 3.10 (empirical P=.0036), and there was significant evidence for intersample heterogeneity (empirical P=.0038). More-modest support for linkage was observed for chromosome 10p, with logistic-regression analysis of linkage producing an empirical P=. 045 and with significant evidence for intersample heterogeneity (empirical P=.0096).     

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.     

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.     

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.     

Problems in the definition, interpretation, and evaluation of genetic heterogeneity

Suppose that we wish to classify families with multiple cases of disease into one of three categories: those that segregate mutations of a gene of interest, those which segregate mutations of other genes, and those whose disease is due to nonhereditary factors or chance. Among families in the first two categories (the hereditary families), we wish to estimate the proportion, p, of families that segregate mutations of the gene of interest. Although this proportion is a commonly accepted concept, it is well defined only with an unambiguous definition of "family." Even then, extraneous factors such as family sizes and structures can cause p to vary across different populations and, within a population, to be estimated differently by different studies. Restrictive assumptions about the disease are needed, in order to avoid this undesirable variation. The assumptions require that mutations of all disease-causing genes (i) have no effect on family size, (ii) have very low frequencies, and (iii) have penetrances that satisfy certain constraints. Despite the unverifiability of these assumptions, linkage studies often invoke them to estimate p, using the admixture likelihood introduced by Smith and discussed by Ott. We argue against this common practice, because (1) it also requires the stronger assumption of equal penetrances for all etiologically relevant genes; (2) even if all assumptions are met, estimates of p are sensitive to misspecification of the unknown phenocopy rate; (3) even if all the necessary assumptions are met and the phenocopy rate is correctly specified, estimates of p that are obtained by linkage programs such as HOMOG and GENEHUNTER are based on the wrong likelihood and therefore are biased in the presence of phenocopies. We show how to correct these estimates; but, nevertheless, we do not recommend the use of parametric heterogeneity models in linkage analysis, even merely as a tool for increasing the statistical power to detect linkage. This is because the assumptions required by these models cannot be verified, and their violation could actually decrease power. Instead, we suggest that estimation of p be postponed until the relevant genes have been identified. Then their frequencies and penetrances can be estimated on the basis of population-based samples and can be used to obtain more-robust estimates of p for specific populations.     

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.     

DNA fingerprinting for forensic identification: potential effects on data interpretation of subpopulation heterogeneity and band number variability

Some methods of statistical analysis of data on DNA fingerprinting suffer serious weaknesses. Unlinked Mendelizing loci that are at linkage equilibrium in subpopulations may be statistically associated, not statistically independent, in the population as a whole if there is heterogeneity in gene frequencies between subpopulations. In the populations where DNA fingerprinting is used for forensic applications, the assumption that DNA fragments occur statistically independently for different probes, different loci, or different fragment size classes lacks supporting data so far; there is some contrary evidence. Statistical association of alleles may cause estimates based on the assumption of statistical independence to understate the true matching probabilities by many orders of magnitude. The assumptions that DNA fragments occur independently and with constant frequency within a size class appear to be contradicted by the available data on the mean and variance of the number of fragments per person. The mistaken use of the geometric mean instead of the arithmetic mean to compute the probability that every DNA fragment of a randomly chosen person is present among the DNA fragments of a specimen may substantially understate the probability of a match between blots, even if other assumptions involved in the calculations are taken as correct. The conclusion is that some astronomically small probabilities of matching by chance, which have been claimed in forensic applications of DNA fingerprinting, presently lack substantial empirical and theoretical support.     

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.     

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 robust identity-by-descent procedure using affected sib pairs: multipoint mapping for complex diseases

Multipoint linkage analysis is a powerful tool to localize susceptibility genes for complex diseases. However, the conventional lod score method relies critically on the correct specification of mode of inheritance for accurate estimation of gene position. On the other hand, allele-sharing methods, as currently practiced, are designed to test the null hypothesis of no linkage rather than estimate the location of the susceptibility gene(s). In this paper, we propose an identity-by-descent (IBD)-based procedure to estimate the location of an unobserved susceptibility gene within a chromosomal region framed by multiple markers. Here we deal with the practical situation where some of the markers might not be fully informative. Rather the IBD statistic at an arbitrary within the region is imputed using the multipoint marker information. The method is robust in that no assumption about the genetic mechanism is required other than that the region contains no more than one susceptibility gene. In particular, this approach builds upon a simple representation for the expected IBD at any arbitrary locus within the region using data from affected sib pairs. With this representation, one can carry out a parametric inference procedure to locate an unobserved susceptibility gene. In addition, here we derive a sample size formula for the number of affected sib pairs needed to detect linkage with multiple markers. Throughout, the proposed method is illustrated through simulated data. We have implemented this method including exploratory and formal model-fitting procedures to locate susceptibility genes, plus sample size and power calculations in a program, GENEFINDER, which will be made available shortly.     

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.