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.     

Using lod-score differences to determine mode of inheritance: a simple, robust method even in the presence of heterogeneity and reduced penetrance.

Determining the mode of inheritance is often difficult under the best of circumstances, but when segregation analysis is used, the problems of ambiguous ascertainment procedures, reduced penetrance, heterogeneity, and misdiagnosis make mode-of-inheritance determinations even more unreliable. The mode of inheritance can also be determined using a linkage-based method (maximized maximum lod score or mod score) and association-based methods, which can overcome many of these problems. In this work, we determined how much information is necessary to reliably determine the mode of inheritance from linkage data when heterogeneity and reduced penetrance are present in the data set. We generated data sets under both dominant and recessive inheritance with reduced penetrance and with varying fractions of linked and unlinked families. We then analyzed those data sets, assuming reduced penetrance, both dominant and recessive inheritance, and no heterogeneity. We investigated the reliability of two methods for determining the mode of inheritance from the linkage data. The first method examined the difference (delta) between the maximum lod scores calculated under the two mode-of-inheritance assumptions. We found that if delta was > 1.5, then the higher of the two maximum lod scores reflected the correct mode of inheritance with high reliability and that a delta of 2.5 appeared to practically guarantee a correct mode-of-inheritance inference. Furthermore, this reliability appeared to be virtually independent of alpha, the fraction of linked families in the data set, although the reliability decreased slightly as alpha fell below .50.(ABSTRACT TRUNCATED AT 250 WORDS)     

Genetic analysis of multiply-affected families of insulin-dependent diabetes mellitus (IDDM) probands

The present study combines segregation and linkage information on 30 families ascertained through a proband and a first degree relative affected with insulin-dependent diabetes mellitus (IDDM). An autosomal dominant model with incomplete penetrance was much more likely to fit the family data than a recessive model, whether or not linkage to HLA was assumed. The lod scores for linkage to HLA were 2.46 at theta M = theta F = 0.00 for dominant and 1.45 at theta M = theta F = 0.22 for a recessive model. The results are discussed in light of heterogeneity in likelihood and lod scores when the families are grouped by familial types, which indicate that the increase in likelihood of a dominant hypothesis can be attributed to the parent-child families and not the sib-sib families.     

Genetics of Type I diabetes mellitus: a single, recessive predisposition gene mapping between HLA-B and GLO. With an appendix on the estimation of selection bias.

Three different published sets of HLA-typed families of juvenile diabetes mellitus (JDM) patients have been analyzed. There was no significant genetic heterogeneity between them according to the criterion of Morton, and the total material was analyzed on the assumption of a single recessive (JDM-P) gene with incomplete penetrance. The analysis, carried out with the NYLIP program modified to account for penetrance less than 1 and for selection bias, yields highly significant lod scores for linkage between HLA and JDM-P, with a maximum value of 7.40 at theta = .05 +/- .03. The segregation of HLA and GLO in five affected sib pairs, in which one of the sibs carries an HLA/GLO recombinant, places JDM-P closer to HLA than the GLO locus: four of these five pairs are HLA-identical and GLO-different, in agreement with the conclusions of the formal linkage analysis. The data from these three independent sets of families are therefore consistent with our earlier claim that JDM is inherited as a recessive trait closely linked to HLA with reduced penetrance, and its analysis does not require more complicated genetic models.     

Percentiles of the null distribution of 2 maximum lod score tests

We here consider the null distribution of the maximum lod score (LOD-M) obtained upon maximizing over transmission model parameters (penetrance values, dominance, and allele frequency) as well as the recombination fraction. Also considered is the lod score maximized over a fixed choice of genetic model parameters and recombination-fraction values set prior to the analysis (MMLS) as proposed by Hodge et al. The objective is to fit parametric distributions to MMLS and LOD-M. Our results are based on 3,600 simulations of samples of n = 100 nuclear families ascertained for having one affected member and at least one other sibling available for linkage analysis. Each null distribution is approximately a mixture p(2)(0) + (1 - p)(2)(v). The values of MMLS appear to fit the mixture 0.20(2)(0) + 0.80chi(2)(1.6). The mixture distribution 0.13(2)(0) + 0.87chi(2)(2.8). appears to describe the null distribution of LOD-M. From these results we derive a simple method for obtaining critical values of LOD-M and MMLS.     

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.     

Magnitude of type I error when single-locus linkage analysis is maximized over models: a simulation study.

It is well known that maximizing the maximum LOD score over multiple parameter values or models (i.e., the method of mod scores, or MMLS), will inflate type I error, compared with assuming only one parameter value/model in the linkage analysis. On the other hand, a mod score often has greater power to detect linkage than does a LOD score (Z) calculated under a wrong genetic model. Therefore, it is of interest to determine the actual magnitude of type I error in realistic genetic situations. Simulated data sets with no linkage were generated under three dominant and three recessive single-locus models, with reduced penetrance (f = .8, .5, and .2). Data sets were analyzed for linkage by (1) maximizing over penetrance only, (2) maximizing over "dominance model" (i.e., dominant versus recessive), and (3) maximizing over both penetrance and dominance model simultaneously. In (1), the resultant significance levels were approximately doubled, compared with baseline values if one had not maximized over penetrances (i.e., compared with a one-sided chi2(1)). In (2), significance levels were increased somewhat less, and, in (3), they were increased by approximately two to three times (but not more than four times) over those of the one-sided chi2(1). This means that, for a given size of test alpha, an investigator would need to increase the Z used as a test criterion, by approximately 0.30 LOD units for analyses as in (1) or (2) and by 0.60 Z units for analyses as in (3). These guidelines, which are valid up to approximately Z = 3.0, are conservative for (1) and are very conservative for (2) and (3). By quantifying the increase in significance level (or, correspondingly, the increase in Z), our findings will enable users to rationally assess the advantages versus the disadvantages of mod scores.     

HLA and the inheritance of multiple sclerosis: linkage analysis of 72 pedigrees.

Linkage analysis of 72 pedigrees by the maximum-likelihood method provides evidence of linkage between HLA and the hypothesized multiple sclerosis susceptibility gene (MSSG) for both the dominant and recessive models of inheritance and for penetrance values ranging from 5%--65% (or higher). This MSSG, if it exists, is most likely located at 15%--20% recombination units from the HLA complex, probably on the B-D side. The analysis also shows that there is no heterogeneity in the estimates of linkage, and lod scores are not artifically inflated because of the association of multiple sclerosis (MS) with HLA-B7.     

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.     

Linkage to D3S47 (C17) in one large autosomal dominant retinitis pigmentosa family and exclusion in another: confirmation of genetic heterogeneity.

Recently Dryja and his co-workers observed a mutation in the 23d codon of the rhodopsin gene in a proportion of autosomal dominant retinitis pigmentosa (ADRP) patients. Linkage analysis with a rhodopsin-linked probe C17 (D3S47) was carried out in two large British ADRP families, one with diffuse-type (D-type) RP and the other with regional-type (R-type) RP. Significantly positive lod scores (lod score maximum [Zmax] = +5.58 at recombination fraction [theta] = .0) were obtained between C17 and our D-type ADRP family showing complete penetrance. Sequence and oligonucleotide analysis has, however, shown that no point mutation at the 23d codon exists in affected individuals in our complete-penetrance pedigree, indicating that another rhodopsin mutation is probably responsible for ADRP in this family. Significantly negative lod scores (Z less than -2 at theta = .045) were, however, obtained between C17 and our R-type family which showed incomplete penetrance. Previous results presented by this laboratory also showed no linkage between C17 and another large British R-type ADRP family with incomplete penetrance. This confirms genetic heterogeneity. Some types of ADRP are being caused by different mutations in the rhodopsin locus (3q21-24) or another tightly linked gene in this region, while other types of ADRP are the result of mutations elsewhere in the genome.     

Linkage analysis of the apolipoprotein C2 gene and myotonic dystrophy on human chromosome 19 reveals linkage disequilibrium in a French-Canadian population.

The gene for human apolipoprotein C2 (APOC2), situated on the proximal long arm of chromosome 19, is closely linked to the gene for the most common form of adult muscular dystrophy, myotonic dystrophy (DM). Six APOC2 RFLPs (TaqI, BglI, BanI, BamHI, NcoI, and AvaII) have been identified to date. We have conducted a comprehensive DM linkage study utilizing all six RFLPs and involving 50 families and 372 individuals. The most informative RFLPs are, in descending order, NcoI (lod = 6.64, theta = 0.05), BglI (lod = 6.12, theta = 0.05), AvaII (lod = 6.02, theta = 0.03), BanI (lod = 5.76, theta = 0.04), TaqI (lod = 4.29, theta = 0.06), and BamHI (lod = 1.75, theta = 0.01). A substantial increase in the lod scores over those seen with the individual RFLPs was obtained when the linkage of the entire APOC2 haplotype (composed of the six RFLPs) was studied (lod = 17.87, theta = 0.04). We have observed significant inter-APOC2 RFLP linkage disequilibrium. Consequently, the three most informative RFLPs have been found to be BanI, TaqI, and either BglI, AvaII, or NcoI polymorphisms. We also demonstrate linkage disequilibrium between DM and APOC2 in our French-Canadian population (standardized disequilibrium constant phi = .22, chi 2 = 5.12, df = 1, P less than 0.04). This represents the first evidence of linkage disequilibrium between APOC2 and the DM locus.     

Probable genetic linkage between a locus for human urinary pepsinogen and the HL-A loci.

The genetic basis of familial variation in the relative intensities of human urinary pepsinogen isozymes is not completely clear from family studies. An investigation of the linkage relationships of pepsinogen isozyme 5, considering only segregation for the presence or absence of Pg 5, yields a peak lod score of 4.1 at theta = .1 for linkage with HL-A1 or HL-A2. Added to data from segregation interpreted according to a scheme proposed for the inheritance of intensity differences in Pg 5, the peak lod score becomes 3.0 at theta = .2. Data derived from the segregation of pepsinogen isozyme 4, possibly determined by an allele to that controlling the presence or absence of Pg 5, further reduces the total lod score at theta = .2 to 2.9. The results indicate probable linkage between a locus for urinary pepsinogen and the HL-A loci, but are insufficient to permit any conclusion concerning possible heterogeneity in the linkage relationships of Pg 4 and Pg 5 to HL-A.     

Segregation and linkage analyses of dopamine-beta-hydroxylase activity.

Dopamine-beta-hydroxylase (DBH) activity in serum was measured by spectrophotometric methods in 95 persons of a large family (HGAR 2), along with 27 polymorphic markers from blood, urine and saliva. The distribution of DBH activity, after appropriate transformation and age adjustment, showed a significantly better fit to a mixture of two normal distributions than a single normal distribution. Pedigree segregation analyses showed evidence of a possible major gene governing low levels of DBH activity, segregating in this family in a recessive fashion. Linkage analyses between that major locus and the 27 polymorphic markers showed no significant lod scores favoring linkage. The highest lod score obtained was 0.81 with Lp at zero recombination fraction. In addition, published data on DBH activity measured by radiochemical assays on 22 families with 161 members were reanalyzed as a quantitative trait, with appropriate correction for ascertainment bias. The results were similar to that of HGAR 2, corroborating the existence of a major locus for DBH activity.     

Hereditary Hyperparathyroidism–Jaw Tumor Syndrome: The Endocrine Tumor Gene HRPT2 Maps to Chromosome 1q21-q31

The syndrome of hereditary hyperparathyroidism and jaw tumors (HPT-JT) is characterized by inheritance, in an autosomal dominant pattern, of recurrent parathyroid adenomas, fibro-osseous tumors of the mandible and/or maxilla, Wilms tumor, and parathyroid carcinoma. This syndrome is clinically and genetically distinct from other endocrine neoplasia syndromes and appears to result from mutation of an endocrine tumor gene designated “HRPT2.” We studied five HPT-JT families (59 persons, 20 affected); using PCR-based markers, we instituted a genomewide linkage search after excluding several candidate genes. Lod scores were calculated at various recombination fractions (θ), penetrance 90%. We mapped HRPT2 to the long arm of chromosome 1 (1q21-q31). The maximal lod score was 6.10 at θ = .0 with marker D1S212, or >10⁶ odds in favor of linkage. In six hereditary Wilms tumor families (96 persons, 29 affected), we found no linkage to 1q markers closely linked with HRPT2 (lod scores ?15.6 [D1S191] and ?17.8 [D1S196], θ = .001). Nine parathyroid adenomas and one Wilms tumor from nine members of three HPT-JT families were examined for loss of heterozygosity at linked loci. The parathyroid adenomas and Wilms tumor showed no loss of heterozygosity for these DNA markers. Our data establish that HRPT2, an endocrine tumor gene on the long arm of chromosome 1, is responsible for the HPT-JT syndrome but not for the classical hereditary Wilms tumor syndrome.     

Quasi-linkage: a confounding factor in linkage analysis of complex diseases?

Human linkage analysis is based on the assumption that unlinked genomic loci, particularly loci located on non-homologous chromosomes, segregate independently during meiosis. An exception to this rule is the phenomenon of quasi-linkage (QL) that describes the non-random segregation of non-homologous chromosomes, which can undermine the basic concept of linkage. Molecular mechanisms of QL are not clear; however, observations in mice and plants suggest a possible affinity between non-homologous chromosomal regions containing repetitive or like sequences. QL has not been investigated in humans. As QL may generate false linkages in genome scans of complex diseases, we sought to determine whether genomic loci detected in such genome scans exhibit QL. A number of individual markers showing linkage to schizophrenia, asthma, multiple sclerosis, inflammatory bowel disease and type-1 diabetes were tested for QL in a pairwise linkage analysis against all other markers exhibiting evidence for linkage in each specific study. The Marshfield genotype dataset of eight CEPH families was used for this purpose. The best QL lod scores generated from the analysis were within the range of the lukewarm lod scores reported in the majority of linkage studies for complex disorders. In addition, we performed a genome-wide QL analysis on the Marshfield family database which detected eight QL lod scores >6. The replication of the best Marshfield QL scores was performed using the deCODE families and although none of the eight pairs demonstrated independent evidence for QL, three pairs generated maximal lod scores of 0.11, 0.3, and 1.51. In conclusion, although complex disease relevant markers did not produce high QL lod scores, the general phenomenon of QL in humans cannot be excluded and potentially can be a confounding factor in genetic studies of complex traits.     

Model misspecification and multipoint linkage analysis.

Pairwise linkage analysis is robust to genetic model misspecification provided dominance is correctly specified, the primary effect being inflation of the recombination fraction. By contrast, we show that multipoint analysis under misspecified models is not robust when a putative disease locus is placed between close flanking markers, with potentially spuriously negative multipoint lod scores being produced. The problem is due to incorrect attribution of segregation of a disease allele and the consequent conclusion of (unlikely) double crossovers between flanking markers. As a possible solution, we propose the use of high disease allele frequencies, as this allows probabilistically for nonsegregation (through parental homozygosity or dual matings). We show analytically and through analysis of pedigree data simulated under a two-locus heterogeneity model that using a disease allele frequency of 0.05 in the dominant case and 0.25 in the recessive case is quite robust in producing positive multipoint lod scores with close flanking markers across a broad range of conditions including varying allele frequencies, epistasis, genetic heterogeneity and phenocopies.     

Mapping of a gene determining tuberous sclerosis to human chromosome 11q14-11q23

Tuberous sclerosis (TSC) is a dominantly inherited disorder characterized by hamartomas and hamartias in one or more organs, most often in skin, brain, and kidneys. Analysis of the basic genetic defect in tuberous sclerosis would be greatly expedited by definitive determination of the chromosomal location of the TSC gene or genes. We have carried out genetic linkage studies in 15 TSC families, using 34 polymorphic markers including protein markers and DNA markers. Pairwise lod scores were calculated using LIPED, and multipoint analyses were carried out using MENDEL. In the pairwise linkage analysis, using a penetrance value of 90%, a significant positive lod score was obtained with MCT128.1 (D11S144), 11q22-11q23, Zmax 3.26 at theta = 0.08. The tyrosinase probe TYR (11q14-11q22) gave a maximum lod score of 2.88 at theta = 0. In the multipoint analyses the most likely order is (TYR,TSC)-MCT128.1-HHH172. Homogeneity analysis was carried out using the USERM9 subprogram of MENDEL, which conducts the admixture test of C. Smith (1963, Ann. Hum. Genet. 27: 175-182). This test provided no evidence for genetic heterogeneity (that is, non-11-linked families) in this data set.     

Heterogeneity in X-linked recessive Charcot-Marie-Tooth neuropathy.

Three families presenting with X-linked recessive Charcot-Marie-Tooth neuropathies (CMT) were studied both clinically and genetically. The disease phenotype in family 1 was typical of CMT type 1, except for an infantile onset; two of five affected individuals were mentally retarded, and obligate-carrier females were unaffected. Families 2 and 3 showed distal atrophy with weakness, juvenile onset, and normal intelligence. Motor-nerve conduction velocities were significantly slowed, and electromyography data were consistent with denervation in affected CMT males in all three families. Thirty X-linked RFLPs were tested for linkage studies against the CMT disease loci. Family 1 showed tight linkage (recombination fraction [theta] = 0) to Xp22.2 markers DXS16, DXS143, and DXS43, with peak lod scores of 1.75, 1.78, and 2.04, respectively. A maximum lod score of 3.48 at DXS16 (theta = 0) was obtained by multipoint linkage analysis of the map DXS143-DXS16-DXS43. In families 2 and 3 there was suggestion of tight linkage (theta = 0) to Xq26 markers DXS86, DXS144, and DXS105, with peak lod scores of 2.29, 1.33, and 2.32, respectively. The combined maximum multipoint lod score of 1.81 at DXS144 (theta = 0) for these two families occurred in the map DXS10-DXS144-DXS51-DXS105-DXS15-DXS52++ +. A joint homogeneity analysis including both regions (Xp22.2 and Xq26-28) provided evidence against homogeneity (chi 2 = 9.12, P less than .005). No linkage to Xp11.12-q22 markers was observed, as was reported for X-linked dominant CMT and the Cowchock CMT variant. Also, the chromosomes 1 and 17 CMT loci were excluded by pairwise linkage analysis in all three families.     

Joint linkage of multiple loci for a complex disorder.

Many investigators who have been searching for linkage to complex diseases have by now accumulated a drawer full of negative results. If disease is actually caused by genes at several loci, these data might contain multiple-locus system (MLS) information that the investigator does not realize. Trying to obtain this information formally, through the MLS likelihood, leads to severe computational and statistical difficulties. Therefore, we propose a scheme of inference based on single-locus (SL) statistics, considered jointly. By simulation, we find that the MLS lod score is closely approximated by the sum of SL lod scores. However, we also find that for moderately large systems, say three of four loci, both MLS and SL lod scores are likely to be inconclusive. Nonetheless, MLS can often be detected through the correlation of individual pedigree SL lod scores. Significant correlation is itself evidence of an MLS, because, in the absence of linkage, false-positive lod scores are necessarily random. Under epistasis SL lod scores tend to be positively correlated among pedigrees, while under independent action SL lod scores from high-density samples tend to be negatively correlated.