On computation of p-values in parametric linkage analysis

Parametric linkage analysis is usually used to find chromosomal regions linked to a disease (phenotype) that is described with a specific genetic model. This is done by investigating the relations between the disease and genetic markers, that is, well-characterized loci of known position with a clear Mendelian mode of inheritance. Assume we have found an interesting region on a chromosome that we suspect is linked to the disease. Then we want to test the hypothesis of no linkage versus the alternative one of linkage. As a measure we use the maximal lod score Z(max). It is well known that the maximal lod score has asymptotically a (2 ln 10)(-1) x (1/2 chi2(0) + 1/2 chi2(1)) distribution under the null hypothesis of no linkage when only one point (one marker) on the chromosome is studied. In this paper, we show, both by simulations and theoretical arguments, that the null hypothesis distribution of Zmax has no simple form when more than one marker is used (multipoint analysis). In fact, the distribution of Zmax depends on the number of families, their structure, the assumed genetic model, marker denseness, and marker informativity. This means that a constant critical limit of Zmax leads to tests associated with different significance levels. Because of the above-mentioned problems, from the statistical point of view the maximal lod score should be supplemented by a p-value when results are reported.     

Mapping of facioscapulohumeral muscular dystrophy gene to chromosome 4q35-qter by multipoint linkage analysis and in situ hybridization

We have recently assigned the facioscapulohumeral muscular dystrophy (FSHD) gene to chromome 4 by linkage to the microsatellite marker Mfd 22 (locus D4S171). We now report that D4S139, a VNTR locus, is much more closely linked to FSHD. Two-point linkage analysis between FSHD and D4S139 in nine informative families showed a maximum combined lod score (Zmax) of 17.28 at a recombination fraction theta of 0.027. Multipoint linkage analysis between FSHD and the loci D4S139 and D4S171 resulted in a peak lod score of 20.21 at 2.7 cM from D4S139. Due to the small number of recombinants found with D4S139, the position of the FSHD gene relative to that of D4S139 could not be established with certainty. D4S139 was mapped to chromosome 4q35-qter by in situ hybridization, thus firmly establishing the location of the FSHD gene in the subtelomeric region of chromosome 4q. One small family yielded a negative lod score for D4S139. In the other families no significant evidence for genetic heterogeneity was obtained. Studies of additional markers and new families will improve the map of the FSHD region, reveal possible genetic heterogeneity, and allow better diagnostic reliability.     

Detecting linkage for genetically heterogeneous diseases and detecting heterogeneity with linkage data.

Interest in searching for genetic linkage between diseases and marker loci has been greatly increased by the recent introduction of DNA polymorphisms. However, even for the most well-behaved Mendelian disorders, those with clear-cut mode of inheritance, complete penetrance, and no phenocopies, genetic heterogeneity may exist; that is, in the population there may be more than one locus that can determine the disease, and these loci may not be linked. In such cases, two questions arise: (1) What sample size is necessary to detect linkage for a genetically heterogeneous disease? (2) What sample size is necessary to detect heterogeneity given linkage between a disease and a marker locus? We have answered these questions for the most important types of matings under specified conditions: linkage phase known or unknown, number of alleles involved in the cross at the marker locus, and different numbers of affected and unaffected children. In general, the presence of heterogeneity increases the recombination value at which lod scores peak, by an amount that increases with the degree of heterogeneity. There is a corresponding increase in the number of families necessary to establish linkage. For the specific case of backcrosses between disease and marker loci with two alleles, linkage can be detected at recombination fractions up to 20% with reasonable numbers of families, even if only half the families carry the disease locus linked to the marker. The task is easier if more than two informative children are available or if phase is known. For recessive diseases, highly polymorphic markers with four different alleles in the parents greatly reduce the number of families required.     

Model-free linkage analysis using likelihoods.

Misspecification of transmission model parameters can produce artifactually negative lod scores at small recombination fractions and in multipoint analysis. To avoid this problem, we have tried to devise a test that aims to detect a genetic effect at a particular locus, rather than attempting to estimate the map position of a locus with specified effect. Maximizing likelihoods over transmission model parameters, as well as linkage parameters, can produce seriously biased parameter estimates and so yield tests that lack power for the detection of linkage. However, constraining the transmission model parameters to produce the correct population prevalence largely avoids this problem. For computational convenience, we recommend that the likelihoods under linkage and non-linkage are independently maximized over a limited set of transmission models, ranging from Mendelian dominant to null effect and from null effect to Mendelian recessive. In order to test for a genetic effect at a given map position, the likelihood under linkage is maximized over admixture, the proportion of families linked. Application to simulated data for a wide range of transmission models in both affected sib pairs and pedigrees demonstrates that the new method is well behaved under the null hypothesis and provides a powerful test for linkage when it is present. This test requires no specification of transmission model parameters, apart from an approximate estimate of the population prevalence. It can be applied equally to sib pairs and pedigrees, and, since it does not diminish the lod score at test positions very close to a marker, it is suitable for application to multipoint data.     

Inter- and intrafamilial heterogeneity: effective sampling strategies and comparison of analysis methods.

Heterogeneity, both inter- and intrafamilial, represents a serious problem in linkage studies of common complex diseases. In this study we simulated different scenarios with families who phenotypically have identical diseases but who genotypically have two different forms of the disease (both forms genetic). We examined the proportion of families displaying intrafamilial heterogeneity, as a function of mode of inheritance, gene frequency, penetrance, and sampling strategies. Furthermore, we compared two different ways of analyzing linkage in these data sets: a two-locus (2L) analysis versus a one-locus (SL) analysis combined with an admixture test. Data were simulated with tight linkage between one disease locus and a marker locus; the other disease locus was not linked to a marker. Our findings are as follows: (1) In contrast to what has been proposed elsewhere to minimize heterogeneity, sampling only "high-density" pedigrees will increase the proportion of families with intrafamilial heterogeneity, especially when the two forms are relatively close in frequency. (2) When one form is dominant and one is recessive, this sampling strategy will greatly decrease the proportions of families with a recessive form and may therefore make it more difficult to detect linkage to the recessive form. (3) An SL analysis combined with an admixture test achieves about the same lod scores and estimate of the recombination fraction as does a 2L analysis. Also, a 2L analysis of a sample of families with intrafamilial heterogeneity does not perform significantly better than an SL analysis. (4) Bilineal pedigrees have little effect on the mean maximum lod score and mean maximum recombination fraction, and therefore there is little danger that including these families will lead to a false exclusion of linkage.     

Mapping of facioscapulohumeral muscular dystrophy gene to chromosome 4q35-qter by multipoint linkage analysis and in situ hybridization

We have recently assigned the facioscapulohumeral muscular dystrophy (FSHD) gene to chromosome 4 by linkage to the microsatellite marker Mfd 22 (locus D4S171). We now report that D4S139, a VNTR locus, is much more closely linked to FSHD. Two-point linkage analysis between FSHD and D4S139 in nine informative families showed a maximum combined lod score (Z_max) of 17.28 at a recombination fraction θ of 0.027. Multipoint linkage analysis between FSHD and the loci D4S139 and D4S171 resulted in a peak lod score of 20.21 at 2.7 cM from D4S139. Due to the small number of recombinants found with D4S139, the position of the FSHD gene relative to that of D4S139 could not be established with certainty. D4S139 was mapped to chromosome 4q35-qter by in situ hybridization, thus firmly establishing the location of the FSHD gene in the subtelomeric region of chromosome 4q. One small family yielded a negative lod score for D4S139. In the other families no significant evidence for genetic heterogeneity was obtained. Studies of additional markers and new families will improve the map of the FSHD region, reveal possible genetic heterogeneity, and allow better diagnostic reliability.     

Criteria to optimize designs for detection and estimation of linkage between marker loci from segregating populations containing several families

Summary Construction of a genome map of highly polymorphic markers has become possible in the past decade. Establishing a complete marker map is an enormous task. Therefore, designs to map molecular markers should be optimal. Designs to detect and estimate linkage between markers from segregating populations were studied. Two measures of design quality were used. The expectation of the maximum lod score indicates the possibility of designs to detect linkage. The accuracy of estimating recombination rate was measured as the probability that the true recombination rate is in a specified internal given the estimate. Accurate approximate methods were developed for rapid evaluation of designs. Seven family types (e.g., double backcross) can be distinguished that describe all families in a segregating population. The family type influences the expected maximum lod score and the accuracy of estimation. The frequency of favorable family types increased with increasing marker polymorphism. At a true recombination rate of 0.20,27 observations on offspring when five alleles were segregating, and 55 observations on offspring when two alleles were segregating, were necessary to obtain an expected maximum lod score of 3. The probability that the true recombination rate was between 0.15 and 0.25, given an estimate of 0.20, was about 0.85 for a design with 40 families with ten offspring and two alleles segregating and for a design with ten families with ten offspring and six alleles segregating. For smaller designs, accuracies were less, approximate evaluation of accuracy was not justified and, on average, true recombination rates were much greater than estimated given a specified value for the estimated recombination rate.     

Refining the localization of the PKD2 locus on chromosome 4q by linkage analysis in Spanish families with autosomal dominant polycystic kidney disease type 2.

Autosomal dominant polycystic kidney disease (ADPKD) is a genetically heterogeneous disorder. At least two distinct forms of ADPKD are now well defined. In approximately 86% of affected European families, a gene defect localized to 16p13.3 was responsible for ADPKD, while a second locus has been recently localized to 4q13-q23 as candidate for the disease in the remaining families. We present confirmation of linkage to microsatellite markers on chromosome 4q in eight Spanish families with ADPKD, in which the disease was not linked to 16p13.3. By linkage analysis with marker D4S423, a maximum lod score of 9.03 at a recombination fraction of .00 was obtained. Multipoint linkage analysis, as well as a study of recombinant haplotypes, placed the PKD2 locus between D4S1542 and D4S1563, thereby defining a genetic interval of approximately 1 cM. The refined map will serve as a genetic framework for additional genetic and physical mapping of the region and will improve the accuracy of presymptomatic diagnosis of PKD2.     

Evaluation of designs for reference families for livestock linkage mapping experiments

Abstract

The development of dense linkage maps consisting of highly polymorphic loci for livestock species is technically feasible. However, linkage mapping experiments are expensive as they involve many animals and marker typings per animal. To minimize costs of developing linkage maps for livestock species, optimizing designs for mapping studies is necessary. This study provides a general framework for evaluating the efficiency of designs for reference families consisting of two‐ or three‐ generation full‐sib or half‐sib families selected from a segregating population. The influence of number of families, number of offspring per family, family structure (either half‐sib or full‐sib) and marker polymorphism is determined. Evaluation is done for two markers with a recombination rate of .20 and for a marker and a dominant single gene with a recombination rate of .20. Two evaluation criteria are used: expected maximum lod score for detection of linkage and accuracy of an estimated recombination rate defined as probability that the true recombination rate is in an interval around the estimated recombination rate. First, for several designs the contribution of reference families to expected maximum lod score and accuracy is given. Second, the required number of families in a design to obtain a certain value for the evaluation criteria is calculated when number of offspring per family, family structure and marker polymorphism are specified. The required numbers increase when designs are optimized not only for expected maximum lod score but also for accuracy. The required number of animals to map a dominant single gene is very large. Therefore, a set of reference families should be designed for strictly mapping marker loci. Examples illustrate how tabulated results can be generalized to determine the values for a wide range of designs containing two‐ or three‐generation full‐sib or half‐sib families.     

Two-locus models of disease: comparison of likelihood and nonparametric linkage methods.

The power to detect linkage for likelihood and nonparametric (Haseman-Elston, affected-sib-pair, and affected-pedigree-member) methods is compared for the case of a common, dichotomous trait resulting from the segregation of two loci. Pedigree data for several two-locus epistatic and heterogeneity models have been simulated, with one of the loci linked to a marker locus. Replicate samples of 20 three-generation pedigrees (16 individuals/pedigree) were simulated and then ascertained for having at least 6 affected individuals. The power of linkage detection calculated under the correct two-locus model is only slightly higher than that under a single locus model with reduced penetrance. As expected, the nonparametric linkage methods have somewhat lower power than does the lod-score method, the difference depending on the mode of transmission of the linked locus. Thus, for many pedigree linkage studies, the lod-score method will have the best power. However, this conclusion depends on how many times the lod score will be calculated for a given marker. The Haseman-Elston method would likely be preferable to calculating lod scores under a large number of genetic models (i.e., varying both the mode of transmission and the penetrances), since such an analysis requires an increase in the critical value of the lod criterion. The power of the affected-pedigree-member method is lower than the other methods, which can be shown to be largely due to the fact that marker genotypes for unaffected individuals are not used.     

A genetic linkage map of microsatellites in the domestic cat (Felis catus)

Of the nonprimate mammalian species with developing comparative gene maps, the feline gene map (Felis catus, Order Carnivora, 2N = 38) displays the highest level of syntenic conservation with humans, with as few as 10 translocation exchanges discriminating the human and feline genome organization. To extend this model, a genetic linkage map of microsatellite loci in the feline genome has been constructed including 246 autosomal and 7 X-linked loci. Two hundred thirty-five dinucleotide (dC. dA)n. (dG. dT)n and 18 tetranucleotide repeat loci were identified and genotyped in a two-family, 108-member multigeneration interspecies backcross pedigree between the domestic cat (F. catus) and the Asian leopard cat (Prionailurus bengalensis). Two hundred twenty-nine loci were linked to at least one other marker with a lod score >/=3.0, identifying 34 linkage groups. Representative markers from each linkage group were assigned to specific cat chromosomes by somatic cell hybrid analysis, resulting in chromosomal assignments to 16 of the 19 feline chromosomes. Genome coverage spans approximately 2900 cM, and we estimate a genetic length for the sex-averaged map as 3300 cM. The map has an average intragroup intermarker spacing of 11 cM and provides a valuable resource for mapping phenotypic variation in the species and relating it to gene maps of other mammals, including human.     

The search for heterogeneity in insulin-dependent diabetes mellitus (IDDM): linkage studies, two-locus models, and genetic heterogeneity

One hundred families with insulin-dependent diabetes mellitus (IDDM) were analyzed for linkage with 27 genetic markers, including HLA, properdin factor B (BF), and glyoxalase 1(GLO) on chromosome 6, and Kidd blood group (Jk) on chromosome 2. The linkage analyses were performed under several different genetic models. An approximate correction for two-locus linkage analysis was developed and applied to four markers. Two different heterogeneity tests were implemented and applied to all the markers. One, the Predivided-Sample Test, utilizes various criteria thought to be relevant to genetic heterogeneity in IDDM. The other, the Admixture Test, looks for heterogeneity without specifying a prior how the sample should be divided. Results continued to support linkage of IDDM with three chromosome 6 markers: HLA, BF, and GLO. The total lod score for Kidd blood group, under the recessive model with 20% penetrance, is 1.63--down 1.2 from the 2.83 reported by us earlier. The only other marker whose lod score exceeded 1.0 under any model was pancreatic amylase (AMY2). The two-locus correction, which involved lowering the penetrance values used in the analysis, affected estimates of theta (recombination fraction) but did not markedly change the lod scores themselves. There was little evidence for heterogeneity within any of the lod scores, under either the Predivided-Sample Test or the Admixture Test.     

Estimating the power of a proposed linkage study for a complex genetic trait.

Many genetic traits have complex modes of inheritance; they may exhibit incomplete or age-dependent penetrance or fail to show any clear Mendelian inheritance pattern. As primary linkage maps for the human genome near completion, it is becoming increasingly possible to map these traits. Prior to undertaking a linkage study, it is important to consider whether the pedigrees available for the proposed study are likely to provide sufficient information to demonstrate linkage, assuming a linked marker is tested. In the current paper, we describe a computer simulation method to estimate the power of a proposed study to detect linkage for a complex genetic trait, given a hypothesized genetic model for the trait. Our method simulates trait locus genotypes consistent with observed trait phenotypes, in such a way that the probability to detect linkage can be estimated by sample statistics of the maximum lod score distribution. The method uses terms available when calculating the likelihood of the trait phenotypes for the pedigree and is applicable to any trait determined by one or a few genetic loci; individual-specific environmental effects can also be dealt with. Our method provides an objective answer to the question, Will these pedigrees provide sufficient information to map this complex genetic trait?     

Transmission test for linkage disequilibrium: the insulin gene region and insulin-dependent diabetes mellitus (IDDM).

A population association has consistently been observed between insulin-dependent diabetes mellitus (IDDM) and the "class 1" alleles of the region of tandem-repeat DNA (5'' flanking polymorphism [5''FP]) adjacent to the insulin gene on chromosome 11p. This finding suggests that the insulin gene region contains a gene or genes contributing to IDDM susceptibility. However, several studies that have sought to show linkage with IDDM by testing for cosegregation in affected sib pairs have failed to find evidence for linkage. As means for identifying genes for complex diseases, both the association and the affected-sib-pairs approaches have limitations. It is well known that population association between a disease and a genetic marker can arise as an artifact of population structure, even in the absence of linkage. On the other hand, linkage studies with modest numbers of affected sib pairs may fail to detect linkage, especially if there is linkage heterogeneity. We consider an alternative method to test for linkage with a genetic marker when population association has been found. Using data from families with at least one affected child, we evaluate the transmission of the associated marker allele from a heterozygous parent to an affected offspring. This approach has been used by several investigators, but the statistical properties of the method as a test for linkage have not been investigated. In the present paper we describe the statistical basis for this "transmission test for linkage disequilibrium" (transmission/disequilibrium test [TDT]). We then show the relationship of this test to tests of cosegregation that are based on the proportion of haplotypes or genes identical by descent in affected sibs. The TDT provides strong evidence for linkage between the 5''FP and susceptibility to IDDM. The conclusions from this analysis apply in general to the study of disease associations, where genetic markers are usually closely linked to candidate genes. When a disease is found to be associated with such a marker, the TDT may detect linkage even when haplotype-sharing tests do not.     

Replication of genetic linkage by follow-up of previously studied pedigrees.

Independent replication of linkage in previously studied pedigrees is desirable when genetic heterogeneity is suspected or when the illness is very rare. When the likelihood of the new data in this type of replication study is computed as conditional on the previously reported linkage results, it can be considered independent. We describe a simulation method using the SLINK program in which the initial data are fixed and newly genotyped individuals are simulated under theta = .01 and theta = .50. These give appropriate lod score criteria for rejection and acceptance of linkage in the follow-up study, which take into account the original marker genotypes in the data. An estimate of the power to detect linkage in the follow-up data is also generated.     

A Genetic Analysis of the Werner Syndrome Region on Human Chromosome 8p

Werner syndrome (WRN) is an inherited disorder that produces symptoms of premature aging. This disease is caused by a recessive mutation that has previously been mapped to chromosome 8p. We have now used genetic linkage analysis to map the WRN gene relative to chromosome 6 reference loci, to screen candidate genes, and to identify a novel dinucleotide repeat polymorphic marker closely linked to WRN. The WRN locus was mapped relative to the marker loci, PLAT, ANK1, D8S135, and D8S87 of the comprehensive chromosome 8 linkage map. The heregulin (HRG) and the fibroblast growth factor receptor 1 genes (FGFR1) have been mapped to chromosome 8p and are involved in cellular growth. Recombination events were detected between WRN and the HRG and FGFR1 genes, excluding them as candidates for the WRN gene. A polymorphic marker generated in this study, WT251, is linked to WRN at a recombination fraction of 0.006, with a lod score of 16.5.     

A linkage map of three anonymous human DNA fragments and SOD-1 on chromosome 21

Using DNA polymorphisms adjacent to single-copy genomic fragments derived from human chromosome 21, we initiated the construction of a linkage map of human chromosome 21. The probes were genomic EcoRI fragments pW228C, pW236B, pW231C and a portion of the superoxide dismutase gene (SOD-1). DNA polymorphisms adjacent to each of the probes were used as markers in informative families to perform classical linkage analysis. No crossing-over was observed between the polymorphic sites adjacent to genomic fragments pW228C and pW236B in 31 chances for recombination. Therefore, these fragments are closely linked to one another (theta = 0.00, lod score = 6.91, 95% confidence limits = 0-10 cM) and can be treated as one 'locus' with four high-frequency markers. There is a high degree of non-random association of markers adjacent to each of these two probes which suggests that they are physically very close to one another in the genome. The pW228C - pW236B 'locus' was also linked to the SOD-1 gene (theta = 0.07, lod score = 4.33, 95% confidence limits = 1-20 cM). On the other hand, no evidence for linkage was found between the pW228C-pW236B 'locus' and the genomic fragment pW231C (theta = 0.5, lod score = 0.00). Based on the fact that pW231C maps to 21q22.3 and SOD-1 to 21q22.1, we suggest that the pW228C-pW236B 'locus' lies in the proximal long arm of chromosome 21. These data provide the outline of a linkage map for the long arm of chromosome 21, and indicate that the pW228C-pW236B 'locus' is a useful marker system to differentiate various chromosome 21s in a population.     

Nance-Horan syndrome: localization within the region Xp21.1-Xp22.3 by linkage analysis.

Nance-Horan Syndrome (NHS) or X-linked cataract-dental syndrome (MIM 302350) is a disease of unknown pathogenesis characterized by congenital cataracts and dental anomalies. We performed linkage analysis in three kindreds with NHS by using six RFLP markers between Xp11.3 and Xp22.3. Close linkage was found between NHS and polymorphic loci DXS43 (theta = 0 with lod score 2.89), DXS41 (theta = 0 with lod score 3.44), and DXS67 (theta = 0 with lod score 2.74), defined by probes pD2, p99-6, and pB24, respectively. Recombinations were found with the marker loci DXS84 (theta = .04 with lod score 4.13), DXS143 (theta = .06 with lod score 3.11) and DXS7 (theta = .09 with lod score 1.68). Multipoint linkage analysis determined the NHS locus to be linked completely to DXS41 (lod score = 7.07). Our linkage results, combined with analysis of Xp interstitial deletions, suggest that the NHS locus is located within or close to the Xp22.1-Xp22.2 region.     

Mapping recessive ophthalmic diseases: linkage of the locus for Usher syndrome type II to a DNA marker on chromosome 1q

Usher syndrome is a heterogeneous group of autosomal recessive disorders that combines variably severe congenital neurosensory hearing impairment with progressive night-blindness and visual loss similar to that in retinitis pigmentosa. Usher syndrome type I is distinguished by profound congenital (preverbal) deafness and retinal disease with onset in the first decade of life. Usher syndrome type II is characterized by partial hearing impairment and retinal dystrophy that occurs in late adolescence or early adulthood. The chromosomal assignment and the regional localization of the genetic mutation(s) causing the Usher syndromes are unknown. We analyzed a panel of polymorphic genomic markers for linkage to the disease gene among six families with Usher syndrome type I and 22 families with Usher syndrome type II. Significant linkage was established between Usher syndrome type II and the DNA marker locus THH33 (D1S81), which maps to chromosome 1q. The most likely location of the disease gene is at a map distance of 9 cM from THH33 (lod score 6.5). The same marker failed to show linkage in families segregating an allele for Usher syndrome type I. These data confirm the provisional assignment of the locus for Usher syndrome type II to the distal end of chromosome 1q and demonstrate that the clinical heterogeneity between Usher types I and II is caused by mutational events at different genetic loci. Regional localization has the potential to improve carrier detection and to provide antenatal diagnosis in families at risk for the disease.     

Construction of a genetic linkage map in man using restriction fragment length polymorphisms.

We describe a new basis for the construction of a genetic linkage map of the human genome. The basic principle of the mapping scheme is to develop, by recombinant DNA techniques, random single-copy DNA probes capable of detecting DNA sequence polymorphisms, when hybridized to restriction digests of an individual's DNA. Each of these probes will define a locus. Loci can be expanded or contracted to include more or less polymorphism by further application of recombinant DNA technology. Suitably polymorphic loci can be tested for linkage relationships in human pedigrees by established methods; and loci can be arranged into linkage groups to form a true genetic map of "DNA marker loci." Pedigrees in which inherited traits are known to be segregating can then be analyzed, making possible the mapping of the gene(s) responsible for the trait with respect to the DNA marker loci, without requiring direct access to a specified gene's DNA. For inherited diseases mapped in this way, linked DNA marker loci can be used predictively for genetic counseling.