The effect of genotyping error in sib-pair genomewide linkage scans depends crucially upon the method of analysis

In genomewide linkage scans for complex diseases involving many loci with small genetic effects, it may be the case that no loci reach conventional statistical significance. A complementary method of evaluating linkage results, locus counting, may provide evidence for the existence of a number of genetic loci in these cases. Sib-pair study designs are often used in genomewide linkage scans, but because all genotype configurations are consistent with Mendelian inheritance, genotyping error will go largely undetected. Previous work on the effect of genotyping error has focused on a single disease locus. We considered the effect of two levels of genotyping error on genomewide evidence for linkage by using the simulated GAW 13 data. For affected sib-pair and non-parametric quantitative trait study designs, a 0.5% genotyping error rate reduced the number of independent linkage regions towards that expected under the null hypothesis of no linkage. A 2% genotyping error rate yielded less independent linkage regions than expected under the null hypothesis of no linkage. For a quantitative trait analysed using a parametric regression-based method, there was very little erosion of the linkage signal, even for error rates as high as 2%.     

The family based association test method: strategies for studying general genotype--phenotype associations

With possibly incomplete nuclear families, the family based association test (FBAT) method allows one to evaluate any test statistic that can be expressed as the sum of products (covariance) between an arbitrary function of an offspring's genotype with an arbitrary function of the offspring's phenotype. We derive expressions needed to calculate the mean and variance of these test statistics under the null hypothesis of no linkage. To give some guidance on using the FBAT method, we present three simple data analysis strategies for different phenotypes: dichotomous (affection status), quantitative and censored (eg, the age of onset). We illustrate the approach by applying it to candidate gene data of the NIMH Alzheimer Disease Initiative. We show that the RC-TDT is equivalent to a special case of the FBAT method. This result allows us to generalise the RC-TDT to dominant, recessive and multi-allelic marker codings. Simulations compare the resulting FBAT tests to the RC-TDT and the S-TDT. The FBAT software is freely available.     

Detection of linkage between a quantitative trait and a marker locus by the lod score method: sample size and sampling considerations

A simulation study is here conducted to measure the power of the lod score method to detect linkage between a quantitative trait and a marker locus in various situations. The number of families necessary to detect such linkage with 80% power is assessed for different sets of parameters at the trait locus and different values of the recombination fraction. The effects of varying the mode of sampling families and the sibship size are also evaluated.     

Affected Sib Pair Tests in Inbred Populations

The affected‐sib‐pair (ASP) method for detecting linkage between a disease locus and marker loci was first established 50 years ago, and since then numerous modifications have been made. We modify two identity‐by‐state (IBS) test statistics of Lange ( Lange, 1986a, 1986b ) to allow for inbreeding in the population. We evaluate the power and false positive rates of the modified tests under three disease models, using simulated data. Before estimating false positive rates, we demonstrate that IBS tests are tests of both linkage and linkage disequilibrium between marker and disease loci. Therefore, the null hypothesis of IBS tests should be no linkage and no LD. When the population inbreeding coefficient is large, the false positive rates of Lange's tests become much larger than the nominal value, while those of our modified tests remain close to the nominal value. To estimate power with a controlled false positive rate, we choose the cutoff values based on simulated datasets under the null hypothesis, so that both Lange's tests and the modified tests generate same false positive rate. The powers of Lange's z‐test and our modified z‐test are very close and do not change much with increasing inbreeding. The power of the modified chi‐square test also stays stable when the inbreeding coefficient increases. However, the power of Lange's chi‐square test increases with increasing inbreeding, and is larger than that of our modified chi‐square test for large inbreeding coefficients. The power is high under a recessive disease model for both Lange's tests and the modified tests, though the power is low for additive and dominant disease models. Allowing for inbreeding is therefore appropriate, at least for diseases known to be recessive.     

An extended transmission/disequilibrium test (TDT) for multi-allele marker loci

The transmission/disequilibrium test (TDT) was recently introduced by Spielman et al. (1993) as a test for linkage and linkage disequilibrium. The test is based on the unequal probability of transmission of two different marker alleles from parents to affected offspring, when the marker locus and the hypothetical disease locus are linked and are in linkage disequilibrium. The probabilities of marker allele transmission to affected offspring conditional on parental genotype have been derived by Ott (1989) for a biallelic marker and a recessive disorder with no phenocopies. Here, we derive the transmission probabilities for a multi-allele marker locus and a generalized single locus disease model in a random sample of affected individuals from a randomly mating population. The form of these transmission probabilities suggests an extension of the TDT to multi-allele marker loci, in which the alternative hypothesis is restricted to take account of the likely pattern of unequal transmission when the recombination fraction is near 0. We show how our extended TDT can be implemented by standard software for logistic regression, although we have also written our own program which is available on request. We have evaluated the approximate power of the test under a range of realistic assumptions, and it appears that the test will often have good power when linkage disequilibrium is strong and if the disease is recessive.     

Mapping Quantitative Trait Loci Using Linkage Disequilibrium: Marker- versus Trait-based Methods

Two approaches for mapping quantitative trait loci (QTL) using linkage disequilibrium at the population level were investigated. In the trait-based (TB) approach, the frequencies of marker alleles (or genotypes) are compared in individuals selected from the two tails of the trait distribution. The TB approach uses phenotypic information only in the selection step. In the marker-based (MB) approach, the quantitative trait values for the marker genotypes in the selected individuals are compared. The MB approach uses both the difference in marker allele (or genotype) frequencies and the phenotypic values of each marker genotype in the selected samples. We quantify the power of each approach and show that the power of the MB approach is greater than or equal to that of the TB approach. The advantage of the former is expected to increase with increasing number of traits phenotyped. Our accurate predictions obviate the need for elaborate simulation studies.     

Genetic contributions to LDL-C, Apo-B and LDL-C/Apo-B ratio in a sample of Israeli offspring with a parental history of myocardial infarction

One hundred and forty sibships consisting of 280 brothers and 256 sisters with a family history of myocardial infarction were investigated for the possible involvement of a major gene in the determination of LDL-C, Apo-B and LDL-C/Apo-B ratio (as a surrogate for LDL subclasses). The mean ages were 29.5 years (range 15–48) and 29.2 years (range 15–47), for brothers and sisters, respectively, and values were initially adjusted for age effects through multiple regression analysis. Results from commingling analysis indicated that for LDL-C a single normal distribution fitted the data as well as a mixture of two distributions. For Apo-B, a mixture of two normal distributions fitted the data significantly better than a single distribution (χ²= 7.8, df = 2, p = 0.02). For LDL-C/Apo-B ratio a mixture of three normal distributions fitted the data significantly better than two distributions (χ²= 9.2, df = 2, p = 0.01). A regression analysis applied to the logarithm of the sex-and age-adjusted mean and variance within sibship, showed no indication of a major gene involvement for LDL-C. For Apo-B and LDL-C/Apo-B ratio, mere existed, however, significant linear relationships between the logarithmically transformed means and wimin sibship variances which support the involvement of major genes. In addition, the Bartlett test applied to the data of within-sibship variances also rejected the null hypothesis of multifactorial transmission for Apo-B and LDL-C/Apo-B ratio (p < 0.0001). Lastly, the results from sib-pair linkage analyses provided significantly positive evidence for linkage between ApoB levels and the Apo-B Xbal restriction site.     

Tests of gene order from three-locus linkage data

Exact tests for gene order are derived and compared for three loci using linkage data from phase-known, completely informative marker loci (i.e. parents are heterozygotes with at most one allele identical at each locus), or from triple back-cross matings. A simulation method, based on resampling genotypes of children, is introduced to obtain approximations to the distribution of the test statistics for general mating types in families consisting of children and parents, with or without grandparents, as are used in many studies in human gene mapping. The method is illustrated by an application to linkage data on chromosome 13.     

Multipoint genomic scanning for quantitative loci: effects of map density, sibship size and computational approach

Multipoint interval mapping (MIM) and the MAPMAKER/SIBS program (M/S) are two methods of mapping quantitative loci by examining identity by descent (IBD) sharing in a region spanned by multiple microsatellite DNA markers. For the purpose of comparison, we simulated a quantitative trait controlled by a two-locus model, and evaluated the power and genome-wide false positive rate of both approaches. Based on our simulation, we examined the effects of marker density (5 cM, 10 cM and 20 cM) and sibship size (2, 3, 4 and 5) on the power to detect linkage. Our results indicate that a 10 cM map provides the optimal trade-off between power and type I error, and that the power of MIM increases with sibship size and, in general, performs better than MAPMAKER/SIBS. Furthermore, we conclude that using a reasonable sample of randomly ascertained sibships, it is possible to map a quantitative trait locus (QTL) which accounts for 25% of the phenotypic variance.     

European Mathematical Genetics Meeting, Heidelberg, Germany, 12th–13th April 2007

Saurabh Ghosh ¹
High correlations between two quantitative traits may be either due to common genetic factors or common environmental factors or a combination of both. In this study, we develop statistical methods to extract the contribution of a common QTL to the total correlation between the components of a bivariate phenotype. Using data on bivariate phenotypes and marker genotypes for sib-pairs, we propose a test for linkage between a common QTL and a marker locus based on the conditional cross-sib trait correlations (trait 1 of sib 1 – trait 2 of sib 2 and conversely) given the identity-by-descent sharing at the marker locus. The null hypothesis cannot be rejected unless there exists a common QTL. We use Monte-Carlo simulations to evaluate the performance of the proposed test under different trait parameters and quantitative trait distributions. An application of the method is illustrated using data on two alcohol-related phenotypes from the Collaborative Study On The Genetics Of Alcoholism project.     

Mating-type differentiation and mate selection in the homothallic Chlamydomonas monoica

By using combinations of phenotypically-distinct — but sexually-compatible — mutant strains of C. monoica (zym-1, zym-27, and nit-2) and assaying for zygote genotypes in genetically-mixed mating populations (where gametes of the two parental homothallic strains were present at similar frequencies), we have found that matings occur preferentially between cells of the same genotype. Additional support for an hypothesis of non-random mate selection was provided by using an easily-selectable genetic marker (sup-1) to verify the frequent occurrence of matings between cells of identical genotype in populations where the selectable genotype was present at very low relative frequency (10^-2 or 10^-3) in a mixed mating population. Direct evidence for non-random mate selection was obtained by presenting nitrogen-starved cells with limiting nitrate to synchronize gametic differentiation in wild-type strains. Under these conditions, the four, eight, or 16 mitotic daughters released from the same mother sporangium often immediately established mating pairs within the group. Thus successive mitotic divisions of a single mother cell yielded progeny of opposite expressed mating-type.     

Characteristics of simple sibship variance tests for the detection of major loci and application to height, weight and spatial performance

Simple methods have been proposed as screening tests for major loci. These methods rely primarily upon the detection of differences in within sibship variances expected for segregating and non-segregating sibships when a major locus is present. In the present study, computer simulation was used to investigate power and robustness of the test statistics. Power of the analyses depends upon the specific major locus model, but, in general, their application is quite practical for small samples. The test statistics were shown to be sensitive to deviations from normality, but robust under the conditions of either a polygenic or environmental model. Application of the test procedures to sibship data from the Boulder Family Study led to significant results for a three-dimensional spatial rotation test, but results for height and weight were non-significant. The simplest interpretation of the results for the spatial performance test wasin tcrms of a sex-linked major locus.     

An alternative model for quantitative trait loci (QTL) analysis in general pedigrees

Linkage detection of a trait involves detecting regions of the genome that influence the trait. A wide variety of statistical models are currently employed for linkage analysis of quantitative traits. Many of these models are developed under some assumptions of the trait distributions. Violation of the assumptions about the trait generally affects the type I error and power for linkage detection. In this paper, we have proposed a trait-model-free approach for linkage analysis of a quantitative trait in general pedigrees. The conditional segregation of marker alleles given the trait is modeled using a latent-variable logistic model. A likelihood-ratio test is used to test for linkage under our model. The main applicability of this approach lies in the fact that it always provides correct type I error no matter what the trait distribution is and thus can be used for nonnormal traits or for selected samples. By means of simulation studies, we have compared the power of our proposed model with existing approaches for nonnormal traits. The performance of these methods was also studied on a real dataset. We have demonstrated the usefulness of our approach in terms of power and robustness for linkage detection of quantitative traits in general pedigrees.     

Ankylosing Spondylitis in a Large Kindred: Clinical and Genetic Studies

In a kindred of 66 members spanning four generations, seven cases of ankylosing spondylitis (AS) have been found. Four of these were in a single sibship of 13. AS was associated with HL-A27 in three of the four involved siblings, but close linkage was shown to be unlikely. Knowledge of HL-A genotype has made possible informed counseling for younger members of the sibship of 13, some of whom, as teenagers, already have back pain.     

TDT statistics for mapping quantitative trait loci

The original transmission disequilibrium test (TDT), was introduced to test for linkage between a marker and a disease-susceptibility locus (Spielman et al . 1993). Allison (1997) extended the TDT procedure to quantitative traits. Allison's test, however, is restrictive in that it requires family trios consisting of one heterozygous parent, one homozygous parent and one child, and considers only the situation where the marker locus is analogous to the quantitative trait locus itself. In this paper, we propose, investigate and apply a general TDT for quantitative traits that permits more than one child per family, does not require only one parent to be heterozygous, and allows for the fact that the various alleles at the marker and trait loci may be at varying degree of linkage disequilibrium. We also show that this TDT for quantitative traits is still a valid test of linkage in the presence of population substructure. To provide guidelines for study design, we develop analytic formulae for calculation of the power of the TDT for mapping quantitative trait loci and investigate the impact of various factors on the power. Power calculations show that the proposed TDT for quantitative traits is more powerful than Allison's basic test statistic and the extreme discordant sib pair linkage method. The proposed TDT statistic for quantitative traits is applied to systolic blood pressure variation in the Rochester Family Heart Study using an extremely discordant sibling pair design.     

Power of regression and maximum likelihood methods to map QTL from sib-pair and DZ twin data

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.     

Sample size calculations for classical association and TDT-type methods using family data

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.     

Idiopathic haemochromatosis: An autosomal recessive disease

A genetic analysis of 96 pedigrees has confirmed our previously published report and demonstrated an autosomal recessive mode of inheritance for idiopathic haemochromatosis.
Three generations have been analyzed in each family: the sibship of the patient; the parents of the patient; the offspring of the patient. All the data are consistent with a recessive autosomal transmission: (1) An increased rate of consanguineous matings was found among the parents of the patients. (2) Not a single patient having the disease (latent or manifat) was found among either the parents or the children of the probands. (3) Three distinct levels of iron stores - normal, slightly increased and heavily overloaded - have been statistically separated in the haemochroma-totic families. (4) The result of a segregation analysis has shown a percentage of parsom with heavy iron overloads per sibship corresponding to that expected for a recessive autosomal transmission, taking into account a penetrance of 0.20 in the female (the estimate used in this study).     

Combined Linkage and Association Mapping of Quantitative Trait Loci with Missing Completely at Random Genotype Data

In genetics study, the genotypes or phenotypes can be missing due to various reasons. In this paper, the impact of missing genotypes is investigated for high resolution combined linkage and association mapping of quantitative trait loci (QTL). We assume that the genotype data are missing completely at random (MCAR). Two regression models, “genotype effect model” and “additive effect model”, are proposed to model the association between the markers and the trait locus. If the marker genotype is not missing, the model is exactly the same as those of our previous study, i.e., the number of genotype or allele is used as weight to model the effect of the genotype or allele in single marker case. If the marker genotype is missing, the expected number of genotype or allele is used as weight to model the effect of the genotype or allele. By analytical formulae, we show that the “genotype effect model” can be used to model the additive and dominance effects simultaneously, and the “additive effect model” can only be used to model the additive effect. Based on the two models, F-test statistics are proposed to test association between the QTL and markers. The non-centrality parameter approximations of F-test statistics are derived to calculate power and to compare power, which show that the power of the F-tests is reduced due to the missingness. By simulation study, we show that the two models have reasonable type I error rates for a dataset of moderate sample size. However, the type I error rates can be very slightly inflated if all individuals with missing genotypes are removed from analysis. Hence, the proposed method can help to get correct type I error rates although it does not improve power. As a practical example, the method is applied to analyze the angiotensin-1 converting enzyme (ACE) data. Edited by Pak Sham.