Correlation-based inference for linkage disequilibrium with multiple alleles

The correlation between alleles at a pair of genetic loci is a measure of linkage disequilibrium. The square of the sample correlation multiplied by sample size provides the usual test statistic for the hypothesis of no disequilibrium for loci with two alleles and this relation has proved useful for study design and marker selection. Nevertheless, this relation holds only in a diallelic case, and an extension to multiple alleles has not been made. Here we introduce a similar statistic, R(2), which leads to a correlation-based test for loci with multiple alleles: for a pair of loci with k and m alleles, and a sample of n individuals, the approximate distribution of n(k - 1)(m - 1)/(km)R(2) under independence between loci is chi((k-1)(m-1))(2). One advantage of this statistic is that it can be interpreted as the total correlation between a pair of loci. When the phase of two-locus genotypes is known, the approach is equivalent to a test for the overall correlation between rows and columns in a contingency table. In the phase-known case, R(2) is the sum of the squared sample correlations for all km 2 x 2 subtables formed by collapsing to one allele vs. the rest at each locus. We examine the approximate distribution under the null of independence for R(2) and report its close agreement with the exact distribution obtained by permutation. The test for independence using R(2) is a strong competitor to approaches such as Pearson's chi square, Fisher's exact test, and a test based on Cressie and Read's power divergence statistic. We combine this approach with our previous composite-disequilibrium measures to address the case when the genotypic phase is unknown. Calculation of the new multiallele test statistic and its P-value is very simple and utilizes the approximate distribution of R(2). We provide a computer program that evaluates approximate as well as "exact" permutational P-values.     

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.     

Analysis of the spectra of mutations and polymorphic loci of cystic fibrosis transmembrane conductance regulator in the population of Bashkortostan

The spectra of mutations and polymorphic loci of the gene of cystic fibrosis transmembrane conductance regulator (CFTR) was studied in 60 cystic fibrosis (CF) families from Bashkortostan. Mutations delF508, 394delTT, CFTRdele2,3(21 kb), R334W, and S1196X (33.3, 3.3, 1.7, 0.8, and 0.8%, respectively) were identified. The frequencies of tandem tetranucleotide repeat (TTR) alleles were determined for locus IVS6a-GATT of intron 6 of the CFTR gene and two extragenic loci flanking the CFTR gene, D7S23 and MET (probes CS.7 and MetH) in mutant and normal chromosomes. Allelic and haplotypic associations of these loci with the mutations found were estimated. An absolute linkage between the 6TTR allele of locus IVS6a-GATT and the delF508 mutation was ascertained. A considerable linkage disequilibrium between the delF508 mutation and the C2 allele of locus D7S23 and between this mutation and the A1 allele of locus MET was found. Most of the other mutant chromosomes carried marker alleles 7TTR, C1, and A2. It was demonstrated that 67% of CF chromosomes carrying delF508 had haplotype 6-2-1 for loci IVS6a-GATT/D7S23/MET, respectively. The frequency distribution of haplotypes in CF chromosomes without delF508 had a high variance and did not differ significantly from the distribution in normal chromosomes (chi 2 = 9.415; p > 0.05).     

Haplotype distribution of and linkage disequilibrium between four polymorphic markers near the CFTR locus in Brazilian cystic fibrosis patients

To contribute to a better understanding of the origin and distribution of CFTR mutations in the Brazilian population, we have investigated the linkage between four polymorphic markers (XV2c, KM19, GATT, and TUB9) within or near the CFTR locus. The distribution of alleles for each polymorphism for both parental and cystic fibrosis (CF) chromosomes from Rio de Janeiro CF families were ascertained using a maximum-likelihood method. This same method was applied to study the distribution of the haplotypes defined by these markers. There was no significant association between the XV2c and KM19 loci on the parental and CF chromosomes. On the other hand, a strong association between GATT and TUB9 loci was observed on both CF and parental chromosomes, and striking linkage disequilibrium between the GATT-TUB9 pair and deltaF508 was observed (chi2 = 26.48, p < 0.0001). Remarkable linkage disequilibrium between the GATT-TUB9 marker pair and non-deltaF508 was also found (chi2 = 17.05, p < 0.0001). Our finding of a linkage disequilibrium between GATT-TUB9 and the CFTR locus could suggest that gene flow between different ethnic groups, mainly sub-Saharan and Mediterranean populations, with Brazilian populations could have resulted in some CF mutations originating on chromosomes that carried the GATT-TUB9 marker haplotype 7-2 (OR = 1.34 < 2.83 < 6.00; p = 0.0066).     

An RFLP linkage map of Upland cotton, Gossypium hirsutum L.

 Ninety-six F₂.F₃ bulked sampled plots of Upland cotton, Gossypium hirsutum L., from the cross of HS46×MARCABUCAG8US-1-88, were analyzed with 129 probe/enzyme combinations resulting in 138 RFLP loci. Of the 84 loci that segregated as co-dominant, 76 of these fit a normal 1 :  2 : 1 ratio (non-significant chi square at P=0.05). Of the 54 loci that segregated as dominant genotypes, 50 of these fit a normal 3: 1 ratio (non-significant chi square at P=0.05). These 138 loci were analyzed with the MAPMAKER∖ EXP program to determine linkage relationships among them. There were 120 loci arranged into 31 linkage groups. These covered 865 cM, or an estimated 18.6% of the cotton genome. The linkage groups ranged from two to ten loci each and ranged in size from 0.5 to 107 cM. Eighteen loci were not linked. Received: 31 March 1998 / Accepted: 29 April 1998     

The Sampling Distribution of Linkage Disequilibrium

The probabilities of obtaining particular samples of gametes with two completely linked loci are derived. It is assumed that the population consists of N diploid, randomly mating individuals, that each of the two loci mutate according to the infinite allele model at a rate µ and that the population is at equilibrium. When 4Nµ is small, the most probable samples of gametes are those that segregate only two alleles at either locus. The probabilities of various samples of gametes are discussed. The results show that most samples with completely linked loci have either a very small or a very large association between the alleles of each locus. This causes the distribution of linkage disequilibrium to be skewed and the distribution of the correlation coefficient to be bimodal. The correlation coefficient is commonly used as a test statistic with a chi square distribution and yet has a bimodal distribution when the loci are completely linked. Thus, such a test is not likely to be accurate unless the rate of recombination between the loci and/or the effective population size are sufficiently large enough so that the loci can be treated as unlinked.     

Estimation of linkage in the presence of multiplicative viability effects

Log-linear analysis of contingency tables is applied to trihybrid backcross data to estimate linkage and viability. Whereas nonadditive viability differences perturb recombination estimates in the classical analysis, this statistical procedure yields maximum likelihood crossover frequency estimates in the presence of multiplicative viability effects. Other advantages of this method include: (1) estimation of viability effects of gene substitution at each locus, (2) estimation of asymptotic confidence intervals on recombination frequencies and viabilities, and (3) it tests the null hypothesis of no interference and no viability interactions. Extensions to cover more loci and to allow certain kinds of epistasis are easily made. Relative merits of the proposed and classical methods are discussed.     

Linkage disequilibrium between the juvenile neuronal ceroid lipofuscinosis gene and marker loci on chromosome 16p 12.1.

The neuronal ceroid lipofuscinoses (NCL; Batten disease) are a collection of autosomal recessive disorders characterized by the accumulation of autofluorescent lipopigments in the neurons and other cell types. Clinically, these disorders are characterized by progressive encephalopathy, loss of vision, and seizures. CLN3, the gene responsible for juvenile NCL, has been mapped to a 15-cM region flanked by the marker loci D16S148 and D16S150 on human chromosome 16. CLN2, the gene causing the late-infantile form of NCL (LNCL), is not yet mapped. We have used highly informative dinucleotide repeat markers mapping between D16S148 and D16S150 to refine the localization of CLN3 and to test for linkage to CLN2. We find significant linkage disequilibrium between CLN3 and the dinucleotide repeat marker loci D16S288 (chi 2(7) = 46.5, P < .005), D16S298 (chi 2(6) = 36.6, P < .005), and D16S299 (chi 2(7) = 73.8, P < .005), and also a novel RFLP marker at the D16S272 locus (chi 2(1) = 5.7, P = .02). These markers all map to 16p12.1. The D16S298/D16S299 haplotype "5/4" is highly overrepresented, accounting for 54% of CLN3 chromosomes as compared with 8% of control chromosomes (chi 2 = 117, df = 1, P < .001). Examination of the haplotypes suggests that the CLN3 locus can be narrowed to the region immediately surrounding these markers in 16p12.1. Analysis of D16S299 in our LNCL pedigrees supports our previous finding that CLN3 and CLN2 are different genetic loci. This study also indicates that dinucleotide repeat markers play a valuable role in disequilibrium studies.     

Isolation and characterization of 11 polymorphic microsatellite loci in collared lizards (Crotaphytus collaris)

We identified 11 polymorphic microsatellite loci in collared lizards (Crotaphytus collaris). Polymorphism assessment in 512 individuals from 52 populations sampled across much of the species distribution revealed a fairly high degree of genetic diversity (six to 20 alleles per locus) and a wide range of average expected heterozygosity values (0.143–0.530). We found no evidence for linkage, very few deviations from HW expectation (two of 572 possible population/locus analyses) and thus no evidence for null alleles. There was a tendency for reduced polymorphism towards the northern periphery.     

Linkage analysis of quantitative trait loci in the presence of heterogeneity

Variance component modeling for linkage analysis of quantitative traits is a powerful tool for detecting and locating genes affecting a trait of interest, but the presence of genetic heterogeneity will decrease the power of a linkage study and may even give biased estimates of the location of the quantitative trait loci. Many complex diseases are believed to be influenced by multiple genes and therefore genetic heterogeneity is likely to be present for many real applications of linkage analysis. We consider a mixture of multivariate normals to model locus heterogeneity by allowing only a proportion of the sampled pedigrees to segregate trait-influencing allele(s) at a specific locus. However, for mixtures of normals the classical asymptotic distribution theory of the maximum likelihood estimates does not hold, so tests of linkage and/or heterogeneity are evaluated using resampling methods. It is shown that allowing for genetic heterogeneity leads to an increase in power to detect linkage. This increase is more prominent when the genetic effect of the locus is small or when the percentage of pedigrees not segregating trait-influencing allele(s) at the locus is high.     

Linkage of loci encoding a kidney endothelial antigen and fumarate hydratase (Fh-1) in the rat

In the rat a single locus, provisionally designated Eag-1, controls the expression of an antigen present on the endothelium of kidney peritubular capillaries and veins. We have examined the linkage relationship between Eag-1 and 10 polymorphic loci including hemoglobin b, fumarate hydratase, peptidase-3, urinary pepsinogen, seminal vesicle protein, glycerophosphate dehydrogenase, esterase-1, esterase-6, pinkeye, and hooded. Tissue samples from animals derived from (AUG X BN.1C)F1 X AUG and (AUG X BN.1C)F1 X BN.1C backcrosses were examined and a linkage association between Eag-1 and Fh-1 (EC 4.2.2.1) was detected. The linkage distance between Eag-1 and Fh-1 is 21 cM (chi 2 = 27.9; p = 0.00001) and this association defines the third locus in the tenth (X) linkage group of the rat.     

Testing for Homogeneity in Meta‐Analysis I. The One‐Parameter Case: Standardized Mean Difference

Summary Meta‐analysis seeks to combine the results of several experiments in order to improve the accuracy of decisions. It is common to use a test for homogeneity to determine if the results of the several experiments are sufficiently similar to warrant their combination into an overall result. Cochran’s Q statistic is frequently used for this homogeneity test. It is often assumed that Q follows a chi‐square distribution under the null hypothesis of homogeneity, but it has long been known that this asymptotic distribution for Q is not accurate for moderate sample sizes. Here, we present an expansion for the mean of Q under the null hypothesis that is valid when the effect and the weight for each study depend on a single parameter, but for which neither normality nor independence of the effect and weight estimators is needed. This expansion represents an order O(1/n) correction to the usual chi‐square moment in the one‐parameter case. We apply the result to the homogeneity test for meta‐analyses in which the effects are measured by the standardized mean difference (Cohen’s d‐statistic). In this situation, we recommend approximating the null distribution of Q by a chi‐square distribution with fractional degrees of freedom that are estimated from the data using our expansion for the mean of Q. The resulting homogeneity test is substantially more accurate than the currently used test. We provide a program available at the Paper Information link at the Biometrics website http://www.biometrics.tibs.org for making the necessary calculations.     

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.     

Genetics of rye phosphatases: evidence of a duplication

Summary Genetic analyses were conducted on alkaline phosphatases of the endosperm of dry kernels and leaf acid phosphatases in four open pollinated and one inbred line of cultivated rye (Secale cereale L.). A total of seven alkaline phosphatase isozymes were observed occurring at variable frequencies in the different cultivars analyzed. We propose that at least five loci control the alkaline phosphatases of rye endosperm — Alph-1, Alph-2, Alph-3, Alph-4 and Alph-5 — all of which have monomeric behaviour. The leaf acid phosphatases are controlled by one locus and have a dimeric quaternary structure. All loci coding for alkaline phosphatase isozymes showed one active, dominant allele and one null, recessive allele, except for the locus Alph-3 which showed two active, dominant alleles and one null, recessive one. The linkage analyses suggest the existence of two linkage groups for alkaline phosphatases: one of them would contain Alph-2, Alph-4, Alph-5 and the locus/loci coding isozymes 6 and 7. This linkage group is located in the 7RS chromosome arm. The other group would include Alph-1 and Alph-3 loci, being located in the 1RL chromosome arm. Leaf acid phosphatases have been previously located in the 7RL chromosome arm. Our data also support an independent relationship between loci controlling the endosperm alkaline phosphatases and leaf acid phosphatases.     

An autosomal genomic scan for loci linked to type II diabetes mellitus and body-mass index in Pima Indians.

Genetic factors influence the development of type II diabetes mellitus, but genetic loci for the most common forms of diabetes have not been identified. A genomic scan was conducted to identify loci linked to diabetes and body-mass index (BMI) in Pima Indians, a Native American population with a high prevalence of type II diabetes. Among 264 nuclear families containing 966 siblings, 516 autosomal markers with a median distance between adjacent markers of 6.4 cM were genotyped. Variance-components methods were used to test for linkage with an age-adjusted diabetes score and with BMI. In multipoint analyses, the strongest evidence for linkage with age-adjusted diabetes (LOD = 1.7) was on chromosome 11q, in the region that was also linked most strongly with BMI (LOD = 3.6). Bivariate linkage analyses strongly rejected both the null hypothesis of no linkage with either trait and the null hypothesis of no contribution of the locus to the covariation among the two traits. Sib-pair analyses suggest additional potential diabetes-susceptibility loci on chromosomes 1q and 7q.     

Theoretical and empirical power of regression and maximum-likelihood methods to map quantitative trait loci in general pedigrees

Both theoretical calculations and simulation studies have been used to compare and contrast the statistical power of methods for mapping quantitative trait loci (QTLs) in simple and complex pedigrees. A widely used approach in such studies is to derive or simulate the expected mean test statistic under the alternative hypothesis of a segregating QTL and to equate a larger mean test statistic with larger power. In the present study, we show that, even when the test statistic under the null hypothesis of no linkage follows a known asymptotic distribution (the standard being chi(2)), it cannot be assumed that the distribution under the alternative hypothesis is noncentral chi(2). Hence, mean test statistics cannot be used to indicate power differences, and a comparison between methods that are based on simulated average test statistics may lead to the wrong conclusion. We illustrate this important finding, through simulations and analytical derivations, for a recently proposed new regression method for the analysis of general pedigrees to map quantitative trait loci. We show that this regression method is not necessarily more powerful nor computationally more efficient than a maximum-likelihood variance-component approach. We advocate the use of empirical power to compare trait-mapping methods.     

Survival quantitative trait locus fine mapping by measuring and testing for Hardy-Weinberg and linkage disequilibrium

I show that fine-scale localization of a survival-related locus can be accomplished on the basis of deviations from Hardy-Weinberg equilibrium and linkage disequilibrium at closely linked marker loci. The method is based on chi(2)-tests and they can be performed for age-specific samples of alive (or dead) individuals, as for combined samples of alive and dead individuals.     

A logistic regression mixture model for interval mapping of genetic trait loci affecting binary phenotypes

Often in genetic research, presence or absence of a disease is affected by not only the trait locus genotypes but also some covariates. The finite logistic regression mixture models and the methods under the models are developed for detection of a binary trait locus (BTL) through an interval-mapping procedure. The maximum-likelihood estimates (MLEs) of the logistic regression parameters are asymptotically unbiased. The null asymptotic distributions of the likelihood-ratio test (LRT) statistics for detection of a BTL are found to be given by the supremum of a chi2-process. The limiting null distributions are free of the null model parameters and are determined explicitly through only four (backcross case) or nine (intercross case) independent standard normal random variables. Therefore a threshold for detecting a BTL in a flanking marker interval can be approximated easily by using a Monte Carlo method. It is pointed out that use of a threshold incorrectly determined by reading off a chi2-probability table can result in an excessive false BTL detection rate much more severely than many researchers might anticipate. Simulation results show that the BTL detection procedures based on the thresholds determined by the limiting distributions perform quite well when the sample sizes are moderately large.     

Tests for linkage and association in nuclear families.

The transmission/disequilibrium test (TDT) originally was introduced to test for linkage between a genetic marker and a disease-susceptibility locus, in the presence of association. Recently, the TDT has been used to test for association in the presence of linkage. The motivation for this is that linkage analysis typically identifies large candidate regions, and further refinement is necessary before a search for the disease gene is begun, on the molecular level. Evidence of association and linkage may indicate which markers in the region are closest to a disease locus. As a test of linkage, transmissions from heterozygous parents to all of their affected children can be included in the TDT; however, the TDT is a valid chi2 test of association only if transmissions to unrelated affected children are used in the analysis. If the sample contains independent nuclear families with multiple affected children, then one procedure that has been used to test for association is to select randomly a single affected child from each sibship and to apply the TDT to those data. As an alternative, we propose two statistics that use data from all of the affected children. The statistics give valid chi2 tests of the null hypothesis of no association or no linkage and generally are more powerful than the TDT with a single, randomly chosen, affected child from each family.