The null distribution of the heterogeneity lod score does depend on the assumed genetic model for the trait.

It is well known that the asymptotic null distribution of the homogeneity lod score (LOD) does not depend on the genetic model specified in the analysis. When appropriately rescaled, the LOD is asymptotically distributed as 0.5 chi(2)(0) + 0.5 chi(2)(1), regardless of the assumed trait model. However, because locus heterogeneity is a common phenomenon, the heterogeneity lod score (HLOD), rather than the LOD itself, is often used in gene mapping studies. We show here that, in contrast with the LOD, the asymptotic null distribution of the HLOD does depend upon the genetic model assumed in the analysis. In affected sib pair (ASP) data, this distribution can be worked out explicitly as (0.5 - c)chi(2)(0) + 0.5chi(2)(1) + cchi(2)(2), where c depends on the assumed trait model. E.g., for a simple dominant model (HLOD/D), c is a function of the disease allele frequency p: for p = 0.01, c = 0.0006; while for p = 0.1, c = 0.059. For a simple recessive model (HLOD/R), c = 0.098 independently of p. This latter (recessive) distribution turns out to be the same as the asymptotic distribution of the MLS statistic under the possible triangle constraint, which is asymptotically equivalent to the HLOD/R. The null distribution of the HLOD/D is close to that of the LOD, because the weight c on the chi(2)(2) component is small. These results mean that the cutoff value for a test of size alpha will tend to be smaller for the HLOD/D than the HLOD/R. For example, the alpha = 0.0001 cutoff (on the lod scale) for the HLOD/D with p = 0.05 is 3.01, while for the LOD it is 3.00, and for the HLOD/R it is 3.27. For general pedigrees, explicit analytical expression of the null HLOD distribution does not appear possible, but it will still depend on the assumed genetic model.     

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.     

MOD-score analysis with simple pedigrees: an overview of likelihood-based linkage methods

A MOD-score analysis, in which the parametric LOD score is maximized with respect to the trait-model parameters, can be a powerful method for the mapping of complex traits. With affected sib pairs, it has been shown before that MOD scores asymptotically follow a mixture of chi(2) distributions with 2, 1 and 0 degrees of freedom under the null hypothesis of no linkage. In that context, a MOD-score analysis yields some (albeit limited) information regarding the trait-model parameters, and there is a chance for an increased power compared to a simple LOD-score analysis. Here, it is shown that with unilineal affected relative pairs, MOD scores asymptotically follow a mixture of chi(2) distributions with 1 and 0 degrees of freedom under the null hypothesis, that is, the same distribution as followed by simple LOD scores. No information regarding the trait model can be obtained in this setting, and no power is gained when compared to a LOD-score analysis. An outlook to larger pedigrees is given. The number of degrees of freedom underlying the null distribution of MOD scores, that depends on the type of pedigrees studied, corresponds to the number of explored dimensions related to power and to the number of parameters that can jointly be estimated.     

Genetic mapping of autosomal dominant Charcot-Marie-Tooth disease in a large French-Acadian kindred: identification of new linked markers on chromosome 17.

We have performed linkage analysis in a large French-Acadian kindred segregating one form of autosomal dominant Charcot-Marie-Tooth disease (CMTD) (type IA) using 17 polymorphic DNA markers spanning human chromosome 17 and demonstrate linkage to several markers in the pericentromeric region, including DNA probes pA10-41, EW301, S12-30, pTH17.19, c11-2B, and p11-2c11.5. Linkage of markers pA10-41 and EW301 to CMTD type IA has been reported elsewhere. Four new markers, 1516, 1517, 1541, and LL101, which map to chromosome 17 have been identified. The marker 1516 appears to be closely linked to the CMTD locus on chromosome 17 as demonstrated by a maximum lod score of 3.42 at theta (recombination fraction) = 0. This marker has been mapped to 17p11.2 using a somatic cell hybrid constructed from a patient with Smith-Magenis syndrome [46,XY, del(17)(p11.2p11.2)]. A lod score of 6.16 has been obtained by multipoint linkage analysis with 1516 and two markers from 17q11.2, pTH17.19, and c11-2B. The markers 1517 and 1541 have been mapped to 17p12-17q11.2 and demonstrate maximum lod scores of 2.35 and 0.63 at recombination values of .1 and .2, respectively. The marker LL101 has been mapped to 17p13.105-17p13.100 and demonstrates a maximum lod score of 1.56 at a recombination value of .1. Our study confirms the localization of CMTD type IA to the pericentromeric region of chromosome 17.     

Distribution of lod scores under uncertain mode of inheritance.

We consider probability distributions of alternative lod statistics, differing in their treatment of segregation parameters when mode of inheritance is uncertain. A particular pedigree structure and a dominant genetic system displaying incomplete penetrance are analyzed. Lod scores calculated assuming an incorrect segregation model appear to conform quite well to the chi 2 distribution in the absence of linkage. In the presence of linkage, some power is lost. However, if lod scores are calculated under several different segregation models and the best one is accepted, opportunity for chance occurrence of high lod scores is enhanced. The distribution is still chi 2, but with extra degrees of freedom. These results hold over a wide range of sample sizes and segregation models, including small samples and low levels of penetrance.     

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.     

Estimation and test for linkage between markers: a comparison of lod score and χ 2 test in a linkage study of maritime pine (Pinus pinaster Ait.)

The first step in the construction of a linkage map involves the estimation and test for linkage between all possible pairs of markers. The lod score method is used in many linkage studies for the latter purpose. In contrast with classical statistical tests, this method does not rely on the choice of a first-type error level. We thus provide a comparison between the lod score and a ² test on linkage data from a gymnosperm, the maritime pine. The lod score appears to be a very conservative test with the usual thresholds. Its severity depends on the type of data used.     

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.     

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.     

Approximate likelihood-ratio test for branches: A fast, accurate, and powerful alternative

We revisit statistical tests for branches of evolutionary trees reconstructed upon molecular data. A new, fast, approximate likelihood-ratio test (aLRT) for branches is presented here as a competitive alternative to nonparametric bootstrap and Bayesian estimation of branch support. The aLRT is based on the idea of the conventional LRT, with the null hypothesis corresponding to the assumption that the inferred branch has length 0. We show that the LRT statistic is asymptotically distributed as a maximum of three random variables drawn from the chi(0)2 + chi(1)2 distribution. The new aLRT of interior branch uses this distribution for significance testing, but the test statistic is approximated in a slightly conservative but practical way as 2(l1- l2), i.e., double the difference between the maximum log-likelihood values corresponding to the best tree and the second best topological arrangement around the branch of interest. Such a test is fast because the log-likelihood value l2 is computed by optimizing only over the branch of interest and the four adjacent branches, whereas other parameters are fixed at their optimal values corresponding to the best ML tree. The performance of the new test was studied on simulated 4-, 12-, and 100-taxon data sets with sequences of different lengths. The aLRT is shown to be accurate, powerful, and robust to certain violations of model assumptions. The aLRT is implemented within the algorithm used by the recent fast maximum likelihood tree estimation program PHYML (Guindon and Gascuel, 2003).     

Asymptotic distribution of the lod score for familial data

Using a linkage model with mixed parental mating types between a trait locus and a marker locus the asymptotic null distribution of the statistic U = 2ln(10)Z(theta) was stimulated and compared to the chi square type distribution function 1/2 + 1/2 Pr [chi 2(1) less than mu]. The stimulation results show that the chi square approximation fits the asymptotic null distribution well when both loci are predominated by either one of the two alleles at their loci, respectively, or the linkage phase tends to disequilibrium.     

Refinement of the locus for autosomal dominant hereditary gingival fibromatosis (GINGF) to a 3.8-cM region on 2p21

Hereditary gingival fibromatosis (HGF, MIM 135300; approved gene symbol GINGF) is an oral disease characterized by enlargement of gingiva. Recently, a locus for autosomal dominant HGF has been mapped to an 11-cM region on chromosome 2p21. In the current investigation, we genotyped four Chinese HGF families using polymorphic microsatellite markers on 2p21. The HOMOG test provided evidence for genetic homogeneity, with evidence for linkage in four families (heterogeneity versus homogeneity test HOMOG, chi(2) = 0. 00). A cumulative maximum two-point lod score of 5.04 was produced with marker D2S390 at a recombination frequency of θ = 0 in the four linked families. Haplotype analysis localized the hereditary gingival fibromatosis locus within the region defined by D2S352 and D2S2163. This region overlaps by 3.8 cM with the previously reported HGF region. Single-strand conformation polymorphism and sequence analysis of the coding region of cytochrome P450 1B1 (CYP1B1) excluded it as a likely candidate gene.     

Multipoint linkage analysis in Menkes disease.

Linkage analyses were performed in 11 families with X-linked Menkes disease. In each family more than one affected patient had been diagnosed. Forty informative meioses were tested using 11 polymorphic DNA markers. From two-point linkage analyses high lod scores are seen for DXS146 (pTAK-8; maximal lod score 3.16 at recombination fraction [theta] = .0), for DXS1 (p-8; maximal lod score 3.44 at theta = .0), for PGK1 (maximal lod score 2.48 at theta = .0), and for DXS3 (p19-2; maximal lod score 2.90 at theta = .0). This indicates linkage to the pericentromeric region. Multilocus linkage analyses of the same data revealed a peak for the location score between DXS146(pTAK-8) and DXYS1X(pDP34). The most likely location is between DXS159 (cpX289) and DXYS1X(pDP34). Odds for this location relative to the second-best-supported region, between DXS146(pTAK-8) and DXS159 (cpX289), are better than 74:1. Visualization of individual recombinant X chromosomes in two of the Menkes families showed the Menkes locus to be situated between DXS159(cpX289) and DXS94(pXG-12). Combination of the present results with the reported absence of Menkes symptoms in male patients with deletions in Xq21 leads to the conclusion that the Menkes locus is proximal to DXSY1X(pDP34) and located in the region Xq12 to Xq13.3.     

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.     

Genetic polymorphisms of Pe and Po salivary proteins with probable linkage of their genes to the salivary protein gene complex (SPC)

Two new genetic polymorphisms (Pe and Po) are found in human parotid saliva. Each polymorphism is determined by the autosomal inheritance of one expressed (dominant) and one unexpressed (recessive) allele. Autosomal inheritance is supported by studies of 63 families including 264 children for Pe and 57 families including 242 children for Po. For randomly collected salivas, gene frequencies in 317 whites are Pe+ = 0.76 and Pe- = 0.24; in 408 whites, Po+ = 0.75 and Po- = 0.25; in 51 blacks, Pe+ = 0.76 and Pe- = 0.24; and in 59 blacks, Po+ = 0.77 and Po- = 0.23. Both Pe and Po proteins react immunologically with polyclonal antisera prepared to proline-rich proteins PRPs. The Pe protein has an isoelectric point of approximately pH 6.1-6.3, and the Po protein has an isoelectric point greater than pH 8.0. In randomly collected salivas, the Pe and Po proteins are associated with other known salivary PRPs. The Pe protein is most strongly associated with the CON 1 and Ps proteins, is less strongly associated with the Pr and Pa proteins, and is not significantly associated with the PmF, PmS, PIF, Db, Con 2, or Gl proteins. If it is assumed that the strength of these associations (presumed linkage disequilibrium) may be related in part to map distance, then these data roughly fit the linear order of PRP genes as previously determined from recombination data derived from family linkage studies. The Po protein is associated with the PmS protein. There is evidence for probable linkage of Pe and Po to the SPC (salivary protein gene complex): Pe to Pa (nine families, lod score at theta = 0 is 2.67) and Po to CON 2 (three families, lod score at theta = 0 is 2.35).     

A genome scan for Loci associated with aerobic running capacity in rats

Aerobic capacity is a complex trait that defines the efficiency to use atmospheric oxygen as an electron acceptor in energy transfer. Copenhagen (COP) and DA inbred rat strains show a wide difference in a test for aerobic treadmill running and serve as contrasting genetic models for aerobic capacity. A genome scan was carried out on an F(2)(COP x DA) segregating population (n=224) to detect quantitative trait loci (QTLs) associated with aerobic running capacity. Linkage analysis revealed a significant QTL on chromosome 16 (lod score, 4.0). A suggestive linkage was found near the p-terminus of chromosome 3 (lod score, 2.2) with evidence of an interaction with another QTL on chromosome 16 (lod score, 2.9). All three QTLs showed a dominant mode of inheritance in which the presence of at least one DA allele was associated with a greater distance run. These results represent the first aerobic capacity QTLs identified in genetic models.     

Data on linkage relations between GLO and 21-hydroxylase

Summary Linkage between GLO and 21-hydroxylase was investigated in 11 families with 24 children. Positive lod score values with a maximum of +1.618 at =0.05 indicate close linkage between these marker loci.     

Statistical analysis of synaptic transmission: model discrimination and confidence limits.

Procedures for discriminating between competing statistical models of synaptic transmission, and for providing confidence limits on the parameters of these models, have been developed. These procedures were tested against simulated data and were used to analyze the fluctuations in synaptic currents evoked in hippocampal neurones. All models were fitted to data using the Expectation-Maximization algorithm and a maximum likelihood criterion. Competing models were evaluated using the log-likelihood ratio (Wilks statistic). When the competing models were not nested, Monte Carlo sampling of the model used as the null hypothesis (H0) provided density functions against which H0 and the alternate model (H1) were tested. The statistic for the log-likelihood ratio was determined from the fit of H0 and H1 to these probability densities. This statistic was used to determine the significance level at which H0 could be rejected for the original data. When the competing models were nested, log-likelihood ratios and the chi 2 statistic were used to determine the confidence level for rejection. Once the model that provided the best statistical fit to the data was identified, many estimates for the model parameters were calculated by resampling the original data. Bootstrap techniques were then used to obtain the confidence limits of these parameters.     

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.