Rapid computer analysis of linkage data

A well tested and useful computer program for screening large amounts of pedigree data for linkage is described. All types of pedigree and genetic systems with up to six alleles present in the pedigree are allowed. An original procedure fills in uncertain genotypes and searches for internal inconsistencies in the data and is extremely thoroughgoing. Population gene frequencies are not used, although the program does take account of rare alleles. The scoring method is based on the division of the pedigree into sibships and the application of z-scores for the sexes considered separately wherever possible. About 2000 pedigrees have been analysed and scores for about the same number of pairs of loci are on tape. A mathematical exercise showed that the method of z-scores with score corrections, which is very much quicker than calculating complete likelihood, is acceptably efficient and its efficiency approaches 100% when scoring rare recessive systems. It is shown that the maximum value of the score is a good criterion for linkage, values of 3.1 and 4.0 indicating roughly 95% and 99% probabilities of linkage. The calculation of 2-unit support limits for the estimate of the recombination fraction between linked loci is advocated, and tables of these are given for syntenic loci. No new linkages were confirmed.     

Dyskeratosis congenita: three additional families show linkage to a locus in Xq28.

Dyskeratosis congenita (DC) is a rare inherited disorder with most families being of the X linked recessive type. We describe three families which show linkage to the marker DXS52 on Xq28. The combined maximum lod score was 2.00 at zero recombination. This is further evidence that the X linked DC gene is located at Xq28 and brings the reported maximum lod score for DC and DXS52 to 5.33 at zero recombination fraction, with a supporting recombination fraction interval of 0.00-0.10.     

Reifenstein''s syndrome: Investigation of linkage to X-chromosomal loci

Xg^a linkage data on a large family with Reifenstein's syndrome is reported. The pedigree was analyzed directly and the data were combined with that from the other linkage study in the literature to allow a numerical estimate of the likelihood of the recombination fraction, based on all the linkage data available. It was concluded that there is not close linkage between Xg^a and Reifenstein's syndrome. A suggestion was, however, found of tight linkage between the genes for color blindness and Reifenstein's syndrome, but this result was not definite because of the small amount of data available.     

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.     

Linkage of Chido and HL-A

Of 156 families who were HL-A typed and Chido-typed, 15 were found to be suitable for linkage analysis. It was calculated that the odds in favour of linkage of Chido and HL-A are 1,450,000:1. The maximum likelihood estimate of the Chido : 1.HL-A recombination fraction is 2½% with 95% confidence limits, obtained graphically, of 0 and 12%; the estimate of the Chido : 2.HL-A recombination fraction is 1½%, with limits of 0 and 9%. Among Chido negative subjects, the antigens HL-A12 and W5 occurred more frequently than in a control population.     

Linkage and association studies with C8A and C8B RFLPs on chromosome 1

Linkage relations for the C8A and C8B Bam HI RFLPs have been investigated. A peak lod score of 4·52 at recombination fraction zero was obtained between the two C8 genes. Combined with our previously obtained linkage data (Rogde et al. 1986) the maximum lod score is 7·53 at recombination fraction zero. The compiled C8-PGM1 linkage data from this and the previous study gave a maximum lod score of 22·02 at recombination fraction 0·11 (0·07–0·16) with no sex difference. A chromosome 1p reference marker, D1S57 , has been applied in this linkage study. A maximum lod score of 5·06 between the C8 cluster and D1S57 at θ= 0·18 (0·11–0·28) was recorded. The linkage analyses and triply informative families gave evidence that the C8 loci are situated about halfway between PGM1 and D1S57 on the short arm of chromosome 1. There was no evidence of allelic association between the C8A and C8B Bam HI RFLPs in 62 unrelated haplotypes.     

Quantitative linkage: a statistical procedure for its detection and estimation

A new approach for detecting and estimating quantitative linkage which uses sibship data is presented. Using a nested analysis of variance design (with marker genotype nested within sibship), it is shown that under the null hypothesis of no linkage, the expected between marker genotype within sibship mean square (EMSbeta) is equal to the expected within marker genotype within sibship mean square (EMSe), while under the alternative hypothesis of linkage, the first is greater than the second. Thus the regular F-ratio, MSbeta/MSe, can be used to test for quantitative linkage. This is true for both backcross and intercross matings and whether or not there is dominance at the marker locus. A second test involving the comparison of the within marker genotype within sibship variances is available for intercross matings. A maximum likelihood procedure for the estimation for the recombination frequency is also presented.     

The consistency of the posterior probability of linkage

When searching for trait loci along the genome, properly incorporating prior genomic information into the analysis will almost certainly increase the chance of success. Recently, we devised a method that utilizes such prior information in the mapping of trait genes for complex disorders (Vieland, 1998; Wang et al . 1999; Vieland et al . 2000). This method uses the posterior probability of linkage (PPL) based on the admixture model as a measure of linkage information. In this paper, we study the consistency of the PPL. It is shown that, as the number of pedigrees increases, the PPL converges in probability to 1 when there is linkage between the marker and a trait locus, and converges to 0 otherwise. This conclusion is shown to be true for general pedigrees and trait models, even when the likelihood functions are based on misspecified trait models. As part of the effort to prove this conclusion, it is shown that when there is no linkage, the maximum likelihood estimator of the recombination fraction in the admixture model is asymptotically 0.5, even when the admixture model misrepresents the true model.     

A Note on the Asymptotic Properties of Affected-sib-pair Linkage Tests

This article concerns the asymptotic properties of linkage tests for affected‐sib‐pair data under the null hypothesis of no linkage. We consider a popular single‐locus analysis model where the unknown parameters are the disease allele frequency, the three penetrances for the three genotypes at the disease locus, and the recombination fraction between the marker locus and the disease locus. These parameters are completely confounded under the null hypothesis of no linkage. We show that 1) If the total variance of the trait (i.e., the additive variance plus the dominance variance) is “separated” from 0, then the likelihood ratio statistic has an asymptotic 0.5χ²₀+ 0.5χ²₁ distribution; 2) If the prevalence of the trait is “separated” from 0 and the recombination fraction is fixed at 0, then the likelihood ratio statistic has an asymptotic distribution which is a mixture of χ²₀, χ²₁ and χ²₂ . The implications of these results are discussed.     

The linkage detection problem

Linkage data in man are usually analysed on families lacking a full set of grandparents on the assumption that there is one locus relating to an abnormal phenotype, the test locus. The basic problems of establishing linkage of this locus to another locus, the marker locus, with a defined likelihood and of estimating the recombination fraction were resolved by Morton(1955). However, the likelihood ratio derived after estimating the most likely recombination fraction is widely misunderstood to be a direct measure of the likelihood of linkage while the bias of estimates conditional on some likelihood criterion derived from the same data is usually ignored. Since data are limited this tolerance of excessive confidence in detecting linkages and a consistent tendency to underestimate recombination fractions is justified but the expected errors are not small and they are readily magnified when multiple loci are involved.
Multiple markers provide further opportunities for detecting false linkages although the corrections required are known. The problems imposed by multiple loci at which mutations lead to a common phenotype have so far had little consideration. As most proteins are multimers whose units are not necessarily coded for by neighbouring loci, and as disturbances to any of several enzymes acting in series may lead to similar consequences, the assumption of a one-to-one relationship between a locus and a phenotype will usually be wrong.     

Linked polymorphic DNA markers in the prediction of X-linked muscular dystrophy

Ten polymorphic DNA markers, including gene specific markers of loci DXS164 and DXS206 , were tested for allele frequencies, degree of heterozygosity and linkage in 34 Finnish families with X-linked muscular dystrophy. With the exception of the Bam HI RFLP of DXS164 subclone pERT87-15, allele frequencies and the degree of heterozygosity failed to show any significant deviation from the data published elsewhere. We document a high degree of linkage disequilibrium between several RFLPs belonging to locus DXS164 . Our linkage data include one recombination between DMD and DXS164 enabling a tentative location of the mutation site distal to DXS164 . The maximum lod score for linkage between the disease locus and DX164 was 7.828 at a recombination fraction of 0.02. According to our data DXS28 and DXS43 may be located further away from the disease locus than previously thought. We use only gene specific markers for genetic counselling. Excluding deletions, 97.1 % of women were heterozygous for at least one such marker. A diagnostic procedure in which useful information can be obtained in over 90 % of all diagnostic situations, using only four filters, is proposed.     

Application of the lod score method to detection of linkage between HLA and juvenile insulin-dependent diabetes

The lod score method has been applied to 28 informative families with at least one child suffering from juvenile insulin-dependent diabetes (JIDD), assuming autosomal recessive inheritance, for detection of linkage between HLA and a susceptibility locus for this disease. These 28 families were pooled with 21 other families from the literature. The maximum lod scores were obtained for recombination fractions from 4 to 16 %, according to the level of penetrance (10 to 90 %). These high estimates of the recombination fraction are not in agreement with the hypothesis that the association between JIDD and specific HLA haplotypes is due to a simple linkage disequilibrium between the HLA region and a susceptibility locus for JIDD.     

Power of the affected-sib-pair method to detect disease susceptibility loci of small effect: An application to multiple sclerosis

The distribution of marker locus identity-by-descent scores in affected sib pairs provides a powerful tool for detecting the presence of a linked non-Mendelian disease susceptibility locus. This basic approach is here extended to include a trio of sibs. A special type of sib trio consisting of two affected and one unaffected sib is investigated. It is shown that compared to affected-sib-pairs, trios with the above configuration are less efficient in detecting the presence of a linked disease susceptibility locus. When the generalized two-allele single locus model is fitted to sib pairs affected with multiple sclerosis, an estimate of the recombination fraction of 0.21 between the putative disease susceptibility locus and the HLA complex is obtained. However, this transmission model is deemed inadequate since a recombination fraction this large is inconsistent with the variety of HLA associations observed at the population level.     

The generalized sib pair IBD distribution: its use in the detection of linkage

General expressions for the distribution of identity by descent (IBD) scores at a marker locus have been derived given neither, one or both sibs affected with a disorder determined by a linked trait locus with arbitrary gene frequency and penetrance vector. It is shown that the distribution of IBD scores depends only on the additive and dominance variances and the population prevalence of the disorder. A one-sided test is suggested as an appropriate means of statistically testing the hypothesis that the recombination fraction is significantly less than. This sib pair approach is designed primarily to detect the presence of a critical disease susceptibility locus but when the assumptions of the incompletely penetrant single locus model are correct the methodology proposed here results in consistent estimates of the recombination fraction. The affected sib pair methodology seems especially suited to traits determined by single loci with non-Mendelian transmission. We wish to express our gratitude to Mr P. Van Eerdewegh and to Drs P. Fishman and S. Hodge for their many helpful comments and criticisms. This work has been supported by HEW grants MH 25430, MH 22521, MH 14677, MH 13002 and AA 00209.     

Multipoint mapping of adult onset polycystic kidney disease (PKD1) on chromosome 16.

Analysis of genetic linkage data in 33 adult onset polycystic kidney (ADPKD) families was carried out using probes for the D16S85, D16S84, and D16S94 loci. The data set of 33 families shows no evidence of genetic heterogeneity since one unlinked family was previously excluded. Two point linkage analysis showed maximum likelihood values of the recombination fraction of 0.07 for ADPKD and D16S85 (lod score 18.78), 0.02 for ADPKD and D16S84 (lod score 7.55), and 0.00 for ADPKD and D16S94 (lod score 6.73). Multipoint analysis showed a maximum likelihood order of tel-D16S85-0.06-D16S84-0.02-(PKD1, D16S94)-cen with a multipoint lod score of 32.16. Analysis of rare recombinants lying close to PKD1 gave results consistent with this order.     

Affected sibling pair linkage analysis of qualitative and quantitative traits for schizophrenia on chromosome 22 in a Chinese population

We performed nonparametric linkage analysis on 136 families with two or more siblings with schizophrenia from Sichuan, southwestern China. In addition to categorical diagnosis, we used quantitative trait information from the Positive and Negative Symptom Scale and the modified Overt Aggression Scale. Categorical analysis using the diagnosis of schizophrenia and a maximum likelihood identity-by-descent method produced scores of close to 0 throughout the whole region tested. Multipoint analysis allowed exclusion of most markers with a relative risk of > 2, but did not exclude the possibility of a relative risk of < 1.5 for four of the markers. Our results provide no significant evidence for a locus for schizophrenia on chromosome 22. Quantitative linkage analysis using the PANSS-G scale score produced a maximum LOD score of approximately 1.2 with the marker D22S310, using either the Haseman-Elston method or maximum likelihood variance estimation with or without dominance. PANSS-N produced a maximum LOD score of 1.2 at the D22S283 locus. LOD score of about 1 are easily produced by chance. Thus, we conclude that under quantitative trait we also find no evidence of linkage between schizophrenia and markers on chromosome 22 in our Chinese sibling pair sample.     

Adult polycystic kidney disease and linked RFLPs at the alpha globin locus: a genetic study in the South Wales population.

Thirteen South Wales kindreds with adult polycystic kidney disease have been studied for genetic linkage using the alpha globin and 3'HVR probes. A maximum lod score of 24.187 was found at a recombination fraction of 0.03. This study supports the existence of a single locus on chromosome 16 responsible for the disorder.     

Adult onset spinocerebellar ataxia linked to HLA in a South African kindred of mixed ancestry

Hereditary spinocerebellar ataxia (SCA) is a relatively common disorder in the Western Cape region of South Africa. At present there are no genetic markers available for prenatal or presymptomatic diagnosis. A large kindred of mixed ancestry with late onset SCA was studied in which the disorder segregated in an autosomal dominant fashion. HLA typing was undertaken on 44 family members, and the HLA haplotypes were assigned on the basis of segregation. The LIPED computer program, with a correction factor allowing for the age of onset, was used to analyze the pedigree for linkage to HLA. Of 22 individuals in whom disease status could be definitely assessed, only one recombinant between HLA and the SCA locus occurred. The lod score reached a maximum of 4.13 at a recombination fraction of 0.05, indicating the odds to be approximately 13,500 to 1 in favor of linkage between HLA and the putative disease allele for SCA. A possible recombination within the HLA region suggested that the disease allele lies telomeric of the HLA region. In view of the recent demonstration of tight linkage between SCA1 and D6S89, however, HLA should not be used for presymptomatic diagnosis or genetic counselling.     

A likelihood-based extended admixture model of oligogenic inheritance in 'model-based' and 'model-free' analysis

The admixture test of linkage heterogeneity is the most often and most successfully applied oligogenic-model linkage and/or LD analysis method. Full two-locus model linkage analysis is possible, but can be computationally intensive and difficult to interpret because of the need to specify so many indeterminate parameters. A novel, computationally efficient method is proposed for combining single locus lod scores which can allow for varying degrees of epistatic interaction. This method can be applied to two-point or multipoint (using complex-valued recombination fractions) linkage and/or linkage disequilibrium analysis to jointly test for multiple unlinked disease loci. Unlike the traditional admixture test, this algorithm permits joint analysis of multiple disease loci with different modes of inheritance for each, and can be applied to 'model-free' analysis as well through the use of 'pseudomarkers'. Software is available for computation of the various likelihood ratio tests described, for comparison of a variety of possible hypotheses regarding locus homogeneity, locus heterogeneity, and epistasis.     

Studies of genetic linkage between adult polycystic kidney disease and three markers on chromosome 16.

Adult polycystic kidney disease (APKD) is a common genetic disorder that is inherited as an autosomal dominant trait. Recent reports show that, in some families, the APKD gene shows close genetic linkage to two chromosome 16 specific genetic markers. We have been conducting a genetic linkage study using 29 polymorphic isoenzyme and antigenic markers in 184 members of 12 APKD families. We present here the results of linkage analysis using three of these markers which have also been reported to be located on chromosome 16: phosphoglycolate phosphatase (PGP), glutamate pyruvate transaminase (GPT), and haptoglobin (HP). The results show that APKD is closely linked to the PGP locus on the short arm of chromosome 16 (16p13----p12), which is consistent with the previously reported linkage both to PGP and to the alpha globin locus. The genetic distance between PGP and APKD shows a maximum likelihood value of the recombination fraction at zero with a lod score of 5 X 5. There is no evidence of linkage between APKD and either GPT or HP. The PGP polymorphism potentially provides a useful predictive test to complement the use of alpha globin probes in genetic counselling. These tests should provide an efficient means of primary screening of family members at risk, as well as introducing the possibility of prenatal diagnosis.