Once a genetic region involved in a complex disease has been localized through linkage or association studies, we need methods to help us identify the actual disease predisposing genetic variant(s) in the region. A large number of single nucleotide polymorphic (SNP) sites may exist in this region. It is important to identify genetic variants directly involved in disease from those in linkage disequilibrium, and thus associated with, the disease predisposing variant(s). A question of great interest is to test whether a SNP, or a combination of SNPs, that influence the trait under investigation have been identified. For many complex HLA-associated diseases, patterns of amino acid site variability raise the possibility that HLA-variation association with a disease may not be due to a given allele but rather one or more variable amino acid sites occurring on several alleles. Here the question is whether an amino acid variant or a combination of amino acid variants involved in disease are identified. To address this question, this paper proposes a permutation procedure for the haplotype method, to test whether all the sites involved in the disease have been identified using the haplotypic data of patients and controls. The method is based on the theoretical result of Valdes and Thomson, that, for each haplotype combination containing all the amino acid sites involved in the disease process, the relative frequencies of amino acid variants at sites not involved in disease, but in linkage disequilibrium with the disease-predisposing sites, are expected to be the same in patients and controls. This procedure takes into account the non-independence of the sites sampled and is robust to mode of inheritance and penetrance of the disease, and can definitely specify when all the disease predisposing sites have not been identified. Application to both simulated data and real data sets on type 1 diabetes and alcoholism indicates that the proposed procedure works well in testing the important null hypothesis of whether all the predisposing sites are identified.

The dissection of complex traits frequently calls for multiple analyses to be performed, including the use of both multiple phenotypes and genetic models. These multiple phenotypes and models are often not independent, and hence the necessary correction for the multiple testing is not straightforward. In this paper we offer a new approach to address the problem of how to correct for non-independent multiple analyses in genomewide linkage studies. We describe one method of how to determine the number of ‘effectively independent’ tests performed in a linkage study using simple linear regression techniques. Further we describe how to use such information to establish genomewide significance thresholds for infinitely dense genomewide maps.

A common study design to map quantitative trait loci (QTL) is to compare the phenotypes and marker genotypes of two or more siblings in a sample of unrelated sib groups, and to test for linkage between chromosome location and quantitative trait values. The simplest case is sib pairs only, in particular dizygotic twin pairs, and a simple and elegant regression method was proposed by Haseman & Elston in 1972 to test for linkage. Since then, several other methods have been proposed to test for linkage. In this study, we derived the statistical power of linear regression and maximum likelihood methods to map QTL from sib pair data analytically, and determined which methods are superior under which set of population parameters. In particular, we considered four regression-based and three maximum likelihood-based approaches, and derived asymptotic approximations of the mean test statistic and statistical power for each method. It was found, both analytically and by computer simulation, that the revisited or new Haseman–Elston method (based upon the mean-corrected crossproduct of the observations on sib-pairs) is less powerful than a full maximum likelihood approach and is also inferior to the Haseman–Elston method under a realistic range of values for the population parameters. We found that a simple regression method, based upon both the squared difference and the mean-corrected squared sum of the observations on sib-pairs, is as powerful as a full maximum likelihood approach. Our derivations of statistical power for regression and maximum likelihood methods provide a simple way to compare alternative methods and obviate the need to perform elaborate computer simulations. DZ twin pairs are likely to be more powerful for linkage analysis than ordinary siblings because they may share more common environmental effects, thereby increasing the proportion of within-family variance that is explained by a QTL.

In case-control studies of complex disease genes, allele frequencies or allele positivities at candidate loci or markers are compared between cases and controls. Although 2 × 2 contingency tables based on allele frequency and allele positivity are generally used to perform simple statistical tests (e.g. a comparison of two proportions and a χ2 test), little is known about the difference in power between the two tables. In this study, we investigated the number of subjects required to obtain a power of 1 − β with a significance level of α for the allele frequency and allele positivity tables. A large difference in the required number of subjects was found between the two tables. Allele positivity tables were suitable for the detection of susceptibility alleles showing a dominant mode of inheritance (MOI). On the other hand, allele frequency tables were suitable for the identification of susceptibility alleles showing a recessive MOI or a multiplicative MOI. In the case of an additive MOI, a suitable table was determined by combining the frequency of the susceptibility allele and the penetrance. These results imply that there are cases in which true association is detected based on one contingency table and is not detected based on another. A simulation analysis revealed that the type I error rate was not much inflated under the null hypothesis of no association, even when a statistical test was performed twice using both allele frequency and allele positivity tables. In contrast, under the alternative hypothesis, the loss of power was marked when a test was performed once using an unsuitable table. In conclusion, statistical tests should be performed using both tables, without adjustment of multiplicity, in case-control studies of complex disease genes when the study objective is exploratory.

The development of the theory of estimation of gametic disequilibrium for multiallelic systems is particularly necessary, since a large number of the genetic markers available at present are highly polymorphic multiallelic systems. The D′ coefficient is one of the most commonly used measures of the extent of overall disequilibrium between all possible pairs of alleles at two multiallelic loci. Nevertheless, the sampling properties of this measure of overall disequilibrium, are to date, unknown. In this work, we have derived explicit expressions by large-sample theory to compute the approximate sampling variance of Dˆ′ between pairs of multiallelic loci, when samples of haplotypes are taken from populations. Formulae for calculating the asymptotic sampling variance were checked by Monte Carlo simulation. In addition, the magnitude of the sampling variance of Dˆ′ was investigated under different scenarios of disequilibrium between multiallelic loci. Extensive simulations were also carried out for describing the sampling distribution of Dˆ′, conditioned on the sample size, number of alleles and their frequencies, and disequilibrium components. It was found that the sampling distribution of Dˆ′ generally approaches well the theoretical normal distribution for experimental sample sizes, particularly when loci have many alleles. Disequilibrium data between microsatellite loci of human chromosome 11p are used for illustration. These investigations increase substantially our knowledge about this widely used measure of overall disequilibrium, which is relevant to evaluate disequilibrium between multiallelic loci in populations.

Transmission Disequilibrium Test (TDT)-based methods have been advocated by several authors for testing that a marker-phenotype association is actually due to linkage and not to uncontrolled stratification. As a pre-requisite of TDT-type methods is the presence of an association between marker and phenotype, one may wish to first investigate the association using a classical association study, and then to check by a TDT approach whether this association is actually due to linkage. We propose an estimating equation (EE) procedure, to compute analytically the minimum sample size of sibship data required to detect the association between a marker and a quantitative phenotype, and that required to confirm it by two TDT methods. We show that, when the marker allele frequency is low or high, the number of informative sibs needed in TDT-type methods can be lower than the number required in an association analysis, and even more so when the familial clustering is strong. However, in all cases, the number of sibs that need to be sampled to get the appropriate number of informative sibs for analysis is always larger for TDT methods than for an association study. In a phenotype-first strategy, this number may be critical when investigating costly phenotypes.

The gamete competition model is a likelihood version of the transmission disequilibrium test (TDT) that is inspired by conditional logistic regression and the Bradley–Terry ranking procedure. In family-based association studies, both the TDT and the gamete competition model apply directly to data on a single nucleotide polymorphism (SNP). Because any given SNP has limited polymorphism, it is tempting to collect several SNPs within a gene into a single super marker whose alleles are haplotypes. Unfortunately, this tactic wreaks havoc with the traditional TDT, which requires codominant markers (Spielman et al. 1993; Terwilliger & Ott, 1992). Eliminating phase ambiguities by assigning haplotypes to individuals before conducting the TDT may give misleading results because only the most probable haplotypes are then considered. Because pedigree implementations of the gamete competition model can accommodate dominant as well as codominant markers, they circumvent the phase problem by including all possible phases weighted by their estimated frequencies.

The publishers would like to apologise to the author of this paper and their readers for the serious oversight which occurred. Figure 1 illustrating the results of CGH on a single cell from one embryo revealing chromosome breakage and reciprocal loss/gain of chromosome fragments should have been printed in full colour. Please refer to the reprinted colour version below.

Hardy–Weinberg disequilibrium (HWD) among affected individuals has recently been proposed for fine-scale mapping of disease susceptibility genes. We investigate the statistical properties of several available HWD measures and develop a new HWD measure J for fine-scale mapping. It is shown both theoretically and through simulations that the available HWD measures depend not only on the genetic distance between the marker locus of interest and the disease susceptibility locus, but also on the allele frequencies at the marker locus. On the contrary, the new measure is not affected by the allele frequencies at the marker locus under the following assumptions: (a) there is initial complete linkage disequilibrium between the marker and the disease loci, (b) there are no new mutations at the marker and the disease loci, and (c) the population under study is large. We develop a novel method to estimate the location of the disease susceptibility gene based on the HWD measure J. The estimator is robust to low mutation rates at the marker and the disease loci. We compare the standard error of the estimated disease gene loci using Pexcess for case-control studies with the standard error using J for case-only studies under various disease models. The newly developed method is successfully applied to a data set on hereditary haemochromatosis (HH).

Typically, genome scans for complex disease have produced linkage peaks which have proved difficult to replicate in additional independent studies. Here we confirm that this may be due to the large variance in magnitude and position of the linkage statistics when maximized across a region. Simulations suggest that for genes of moderate effect (locus-specific sibling relative risks λs in the range 1.23–1.39), sample sizes of less than 500 affected sib pairs will give unacceptably large standard errors in the magnitudes and locations of significant linkage results. For genes of small effect (λs [les ] 1.13), sample sizes in the region of 1000–2000 pairs may be required to achieve consistency of results between different studies. These figures have important implications for our confidence in location estimates for disease genes obtained from linkage studies of modest size. In particular, collection of larger data sets and/or analysis strategies such as conditioning or narrowing the phenotype definition, in order to increase the relative effect size, may be required before embarking on positional cloning.

While extensive analyses have been conducted to test for, no formal analyses have been conducted to test against, the importance of candidate genes with random population samples. We develop a LOD score approach for exclusion analyses of candidate genes with random population samples. Under this approach, specific genetic effects and inheritance models at candidate genes can be analysed and if a LOD score is [les ] − 2.0, the locus can be excluded from having an effect larger than that specified. Computer simulations show that, with sample sizes often employed in association studies, this approach has high power to exclude a gene from having moderate genetic effects. In contrast to regular association analyses, population admixture will not affect the robustness of our analyses; in fact, it renders our analyses more conservative and thus any significant exclusion result is robust. Our exclusion analysis complements association analysis for candidate genes in random population samples and is parallel to the exclusion mapping analyses that may be conducted in linkage analyses with pedigrees or relative pairs. The usefulness of the approach is demonstrated by an application to test the importance of vitamin D receptor and estrogen receptor genes underlying the differential risk to osteoporotic fractures.

A year ago there was hope that a finished sequence of the human genome would soon be publicly available and would give a more reliable locus order than an unconstrained radiation hybrid or genetic map. Alas, there are now different draft orders for each region, none of which may be correct because of gaps, uncertain polarity of contigs, and errors in assembly. Shortly before these drafts became available, we analysed allelic association (also called linkage disequilibrium, LD) in the FRAX region in a large sample of haplotypes (Ennis et al. 2000). We demonstrate here that this material discriminates among alternative draft orders. To express support for discrimination between two values of χ2 = −2 ln L we use the Akaike criterion AIC = df[χ2/min χ2 −1]. Excluding premutations and full mutations at FMR1, all maps have 715 degrees of freedom (df) among 717 pairs of alleles after accepting L = 0 and estimating M, ∈ in the Malecot equation E(ρ) = Me −∈d, where ρ is the association between a pair of alleles at distance d. An AIC in excess of 2 provides evidence against a map with the larger χ2 .