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PubMed1984

On the lod score method in linkage analysis.

Genetic epidemiology deals with the interaction of environmental and genetic determinants in common diseases. Linkage analysis is an important branch of this field. The current practice of claiming linkage between two genetic loci when the maximum lod score z(theta) exceeds 3 has not received theoretical justification, whether considered as a sequential or as a fixed sample size test. Within the framework of significance testing, Wald's (1947) formulae are not applicable to allow this procedure a sequential interpretation. Considered as a fixed sample size test, we find that a chi 2 approximation would instead be very adequate. Since repeated significance testing is performed on linkage data, the nominal significance level should be more stringent for each test than the overall level. Some recent developments in group sequential trials by Pocock (1977) and in repeated significance testing by Woodroofe (1979) seem to indicate that the critical value of the maximum lod score should lie roughly between 0.9 and 3.3, depending on the maximum number of repetitions anticipated, on whether the significance level is desired to be 0.05, 0.01 or 0.001, and on whether the test is derived from a one-sided or a two-sided consideration. In terms of the group sequential approach, if a maximum of twenty repetitions is allowed, if z(theta) greater than log10 A is considered as a one-sided test and assumed to be symmetric when linkage is absent, then the type I error is approximately given by 1/A. We also treat the confidence interval approach for exclusion of unlikely recombination values.

Genetic Linkage↗

PubMed1984

Easy calculations of lod scores and genetic risks on small computers.

A computer program that calculates lod scores and genetic risks for a wide variety of both qualitative and quantitative genetic traits is discussed. An illustration is given of the joint use of a genetic marker, affection status, and quantitative information in counseling situations regarding Duchenne muscular dystrophy.

Computers↗

PubMed2005

Smoothing of the bivariate LOD score for non-normal quantitative traits.

Variance component analysis provides an efficient method for performing linkage analysis for quantitative traits. However, type I error of variance components-based likelihood ratio testing may be affected when phenotypic data are non-normally distributed (especially with high values of kurtosis). This results in inflated LOD scores when the normality assumption does not hold. Even though different solutions have been proposed to deal with this problem with univariate phenotypes, little work has been done in the multivariate case. We present an empirical approach to adjust the inflated LOD scores obtained from a bivariate phenotype that violates the assumption of normality. Using the Collaborative Study on the Genetics of Alcoholism data available for the Genetic Analysis Workshop 14, we show how bivariate linkage analysis with leptokurtotic traits gives an inflated type I error. We perform a novel correction that achieves acceptable levels of type I error.

Alcoholism↗

PubMed2000

Robust LOD scores for variance component-based linkage analysis.

The variance component method is now widely used for linkage analysis of quantitative traits. Although this approach offers many advantages, the importance of the underlying assumption of multivariate normality of the trait distribution within pedigrees has not been studied extensively. Simulation studies have shown that traits with leptokurtic distributions yield linkage test statistics that exhibit excessive Type I error when analyzed naively. We derive analytical formulae relating the deviation from the expected asymptotic distribution of the lod score to the kurtosis and total heritability of the quantitative trait. A simple correction constant yields a robust lod score for any deviation from normality and for any pedigree structure, and effectively eliminates the problem of inflated Type I error due to misspecification of the underlying probability model in variance component-based linkage analysis.

Computer Simulation↗

PubMed2003

Correlation between quantitative traits and correlation between corresponding LOD scores: detection of pleiotropic effects.

BACKGROUND: We address the question of whether statistical correlations among quantitative traits lead to correlation of linkage results of these traits. Five measured quantitative traits (total cholesterol, fasting glucose, HDL cholesterol, blood pressure, and triglycerides), and one derived quantitative trait (total cholesterol divided by the HDL cholesterol) are used for phenotype correlation studies. Four of them are used for linkage analysis. RESULTS: We show that although correlation among phenotypes partially reflects the correlation among linkage analysis results, the LOD-score correlations are on average low. The most significant peaks found by using different traits do not often overlap. CONCLUSION: Studying covariances at specific locations in LOD scores may provide clues for further bivariate linkage analyses.

Adult↗

PubMed2002

Quantification of type I error probabilities for heterogeneity LOD scores.

Locus heterogeneity is a major confounding factor in linkage analysis. When no prior knowledge of linkage exists, and one aims to detect linkage and heterogeneity simultaneously, classical distribution theory of log-likelihood ratios does not hold. Despite some theoretical work on this problem, no generally accepted practical guidelines exist. Nor has anyone rigorously examined the combined effect of testing for linkage and heterogeneity and simultaneously maximizing over two genetic models (dominant, recessive). The effect of linkage phase represents another uninvestigated issue. Using computer simulation, we investigated type I error (P value) of the "admixture" heterogeneity LOD (HLOD) score, i.e., the LOD score maximized over both recombination fraction theta and admixture parameter alpha and we compared this with the P values when one maximizes only with respect to theta (i.e., the standard LOD score). We generated datasets of phase-known and -unknown nuclear families, sizes k = 2, 4, and 6 children, under fully penetrant autosomal dominant inheritance. We analyzed these datasets (1) assuming a single genetic model, and maximizing the HLOD over theta and alpha; and (2) maximizing the HLOD additionally over two dominance models (dominant vs. recessive), then subtracting a 0.3 correction. For both (1) and (2), P values increased with family size k; rose less for phase-unknown families than for phase-known ones, with the former approaching the latter as k increased; and did not exceed the one-sided mixture distribution xi = (1/2) chi1(2) + (1/2) chi2(2). Thus, maximizing the HLOD over theta and alpha appears to add considerably less than an additional degree of freedom to the associated chi1(2) distribution. We conclude with practical guidelines for linkage investigators.

Chi-Square Distribution↗

PubMed1993

Linkage of familial Alzheimer disease to chromosome 14 in two large early-onset pedigrees: effects of marker allele frequencies on lod scores.

Alzheimer disease (AD) is a devastating neurodegenerative disease leading to global dementia. In addition to sporadic forms of AD, familial forms (FAD) have been recognized. Mutations in the amyloid precursor protein (APP) gene on chromosome (CHR) 21 have been shown to cause early-onset AD in a small number of pedigrees. Recently, linkage to markers on CHR 14 has been established in several early-onset FAD pedigrees. We now report lod scores for CHR 14 markers in two large early-onset FAD pedigrees. Pairwise linkage analysis suggested that in these pedigrees the mutation is tightly linked to the loci D14S43 and D14S53. However, assumptions regarding marker allele frequencies had a major and often unpredictable effect on calculated lod scores. Therefore, caution needs to be exercised when single pedigrees are analyzed with marker allele frequencies determined from the literature or from a pool of spouses.

Alleles↗

PubMed1989

Efficient computation of lod scores: genotype elimination, genotype redefinition, and hybrid maximum likelihood algorithms.

Calculation of multilocus lod scores presents challenging problems in numerical analysis, combinatories, programming, and genetics. It is possible to accelerate these computations by exploiting the simple pedigree structure of a CEPH-type pedigree consisting of a nuclear family plus all four grandparents. Lathrop et al. (1986) have done this by introducing likelihood factorization and transformation rules and Lander & Green (1987) by the method of 'hidden Markov chains'. The present paper explores an alternative approach based on genotype redefinition in the grandparents and systematic phase elimination in all pedigree members. All three approaches accelerate the computation of a single likelihood. Equally relevant to multilocus mapping are search strategies for finding the maximum likelihood estimates of recombination fractions. Hybrid algorithms that start with the EM algorithm and switch midway to quasi-Newton algorithms show promise. These issues are investigated in the context of a simulated 10 locus example. This same example allows us to illustrate a simple strategy for determining locus order.

Algorithms↗

PubMed2001

Replication of linkage to quantitative trait loci: variation in location and magnitude of the lod score.

Replication of linkage signals from independent samples is considered an important step toward verifying the significance of linkage signals in studies of complex traits. The purpose of this empirical investigation was to examine the variability in the precision of localizing a quantitative trait locus (QTL) by analyzing multiple replicates of a simulated data set with the use of variance components-based methods. Specifically, we evaluated across replicates the variation in both the magnitude and the location of the peak lod scores. We analyzed QTLs whose effects accounted for 10-37% of the phenotypic variance in the quantitative traits. Our analyses revealed that the precision of QTL localization was directly related to the magnitude of the QTL effect. For a QTL with effect accounting for > 20% of total phenotypic variation, > 90% of the linkage peaks fall within 10 cM from the true gene location. We found no evidence that, for a given magnitude of the lod score, the presence of interaction influenced the precision of QTL localization.

Chromosome Mapping↗

PubMed2004

Using lod scores to detect sex differences in male-female recombination fractions.

Human recombination fraction (RF) can differ between males and females, but investigators do not always know which disease genes are located in genomic areas of large RF sex differences. Knowledge of RF sex differences contributes to our understanding of basic biology and can increase the power of a linkage study, improve gene localization, and provide clues to possible imprinting. One way to detect these differences is to use lod scores. In this study we focused on detecting RF sex differences and answered the following questions, in both phase-known and phase-unknown matings: (1) How large a sample size is needed to detect a RF sex difference? (2) What are "optimal" proportions of paternally vs. maternally informative matings? (3) Does ascertaining nonoptimal proportions of paternally or maternally informative matings lead to ascertainment bias? Our results were as follows: (1) We calculated expected lod scores (ELODs) under two different conditions: "unconstrained," allowing sex-specific RF parameters (theta(female), theta(male)); and "constrained," requiring theta(female) = theta(male). We then examined the DeltaELOD (identical with difference between maximized constrained and unconstrained ELODs) and calculated minimum sample sizes required to achieve statistically significant DeltaELODs. For large RF sex differences, samples as small as 10 to 20 fully informative matings can achieve statistical significance. We give general sample size guidelines for detecting RF differences in informative phase-known and phase-unknown matings. (2) We defined p as the proportion of paternally informative matings in the dataset; and the optimal proportion p(circ) as that value of p that maximizes DeltaELOD. We determined that, surprisingly, p(circ) does not necessarily equal (1/2), although it does fall between approximately 0.4 and 0.6 in most situations. (3) We showed that if p in a sample deviates from its optimal value, no bias is introduced (asymptotically) to the maximum likelihood estimates of theta(female) and theta(male), even though ELOD is reduced (see point 2). This fact is important because often investigators cannot control the proportions of paternally and maternally informative families. In conclusion, it is possible to reliably detect sex differences in recombination fraction.

Female↗

PubMed1984

Interpretation of LOD scores with a set of marker loci.

The problem discussed is that of interpreting LOD scores when multiple markers are tested for linkage with a particular disease locus. Provided the tests are based on independent segregations, both the maximum and the average LOD can be translated into a score that would have the same significance level were it the result of a single test.

Biometry↗

PubMed1997

Hereditary spastic paraplegia: LOD-score considerations for confirmation of linkage in a heterogeneous trait.

Hereditary spastic paraplegia (HSP) is a degenerative disorder of the motor system, defined by progressive weakness and spasticity of the lower limbs. HSP may be inherited as an autosomal dominant (AD), autosomal recessive, or an X-linked trait. AD HSP is genetically heterogeneous, and three loci have been identified so far: SPG3 maps to chromosome 14q, SPG4 to 2p, and SPG4a to 15q. We have undertaken linkage analysis with 21 uncomplicated AD families to the three AD HSP loci. We report significant linkage for three of our families to the SPG4 locus and exclude several families by multipoint linkage. We used linkage information from several different research teams to evaluate the statistical probability of linkage to the SPG4 locus for uncomplicated AD HSP families and established the critical LOD-score value necessary for confirmation of linkage to the SPG4 locus from Bayesian statistics. In addition, we calculated the empirical P-values for the LOD scores obtained with all families with computer simulation methods. Power to detect significant linkage, as well as type I error probabilities, were evaluated. This combined analytical approach permitted conclusive linkage analyses on small to medium-size families, under the restrictions of genetic heterogeneity.

Chromosomes, Human, Pair 14↗

PubMed2003

Mapping loci influencing blood pressure in the Framingham pedigrees using model-free LOD score analysis of a quantitative trait.

This paper presents a method of performing model-free LOD-score based linkage analysis on quantitative traits. It is implemented in the QMFLINK program. The method is used to perform a genome screen on the Framingham Heart Study data. A number of markers that show some support for linkage in our study coincide substantially with those implicated in other linkage studies of hypertension. Although the new method needs further testing on additional real and simulated data sets we can already say that it is straightforward to apply and may offer a useful complementary approach to previously available methods for the linkage analysis of quantitative traits.

Adult Children↗

PubMed1988

Detection of linkage between a quantitative trait and a marker locus by the lod score method: sample size and sampling considerations.

A simulation study is here conducted to measure the power of the lod score method to detect linkage between a quantitative trait and a marker locus in various situations. The number of families necessary to detect such linkage with 80% power is assessed for different sets of parameters at the trait locus and different values of the recombination fraction. The effects of varying the mode of sampling families and the sibship size are also evaluated.

Computer Simulation↗

PubMedDate not supplied

Lod score curves for phase-unknown matings.

For a phase-unknown nuclear family, we show that the likelihood and lod score are unimodal, and we describe conditions under which the maximum occurs at recombination fraction theta = 0, theta = 1/2, and 0 < theta < 1/2. These simply stated necessary and sufficient conditions seem to have escaped the notice of previous statistical geneticists.

Female↗

PubMed2004

Clustering patterns of LOD scores for asthma-related phenotypes revealed by a genome-wide screen in 295 French EGEA families.

A genome-wide scan for asthma phenotypes was conducted in the whole sample of 295 EGEA families selected through at least one asthmatic subject. In addition to asthma, seven phenotypes involved in the main asthma physiopathological pathways were considered: SPT (positive skin prick test response to at least one of 11 allergens), SPTQ score being the number of positive skin test responses to 11 allergens, Phadiatop (positive specific IgE response to a mixture of allergens), total IgE levels, eosinophils, bronchial responsiveness (BR) to methacholine challenge and %predicted FEV(1). Four regions showed evidence for linkage (P</=0.001): 6q14 for %FEV(1), 12p13 for IgE, 17q22-q24 for SPT and 21q21 for both SPTQ and %FEV(1). Nine other regions indicated smaller linkage signals (0.001<P</=0.005). While most of these regions have been reported by previous asthma and lung function screens, 6q14 appears to be a new region potentially linked to %FEV(1). To determine which of these various asthma phenotypes are more likely to share common genetic determinants, a principal component analysis was applied to the genome-wide LOD scores. This analysis revealed clustering of LODs for asthma, SPT and Phadiatop on one axis and clustering of LODs for %FEV(1), BR and SPTQ on the other, while LODs for IgE and eosinophils appeared to be independent from all other LODs. These results provide new insights into the potential sharing of genetic determinants by asthma-related phenotypes.

Adolescent↗

PubMed2002

Impact of parental relationships in maximum lod score affected sib-pair method.

Many studies are done in small isolated populations and populations where marriages between relatives are encouraged. In this paper, we point out some problems with applying the maximum lod score (MLS) method (Risch, [1990] Am. J. Hum. Genet. 46:242-253) in these populations where relationships exist between the two parents of the affected sib-pairs. Characterizing the parental relationships by the kinship coefficient between the parents (f), the maternal inbreeding coefficient (alpha(m), and the paternal inbreeding coefficient (alpha(p)), we explored the relationship between the identity by descent (IBD) vector expected under the null hypothesis of no linkage and these quantities. We find that the expected IBD vector is no longer (0.25, 0.5, 0.25) when f, alpha(m), and alpha(p) differ from zero. In addition, the expected IBD vector does not always follow the triangle constraints recommended by Holmans ([1993] Am. J. Hum. Genet. 52:362-374). So the classically used MLS statistic needs to be adapted to the presence of parental relationships. We modified the software GENEHUNTER (Kruglyak et al. [1996] Am. J. Hum. Genet. 58: 1347-1363) to do so. Indeed, the current version of the software does not compute the likelihood properly under the null hypothesis. We studied the adapted statistic by simulating data on three different family structures: (1) parents are double first cousins (f=0.125, alpha(m)=alpha(p)=0), (2) each parent is the offspring of first cousins (f=0, alpha(m)=alpha(p)=0.0625), and (3) parents are related as in the pedigree from Goddard et al. ([1996] Am. J. Hum. Genet. 58:1286-1302) (f=0.109, alpha(m)=alpha(p)=0.0625). The appropriate threshold needs to be derived for each case in order to get the correct type I error. And using the classical statistic in the presence of both parental kinship and parental inbreeding almost always leads to false conclusions.

Computer Simulation↗

PubMed1997

Allele-sharing models: LOD scores and accurate linkage tests.

Starting with a test statistic for linkage analysis based on allele sharing, we propose an associated one-parameter model. Under general missing-data patterns, this model allows exact calculation of likelihood ratios and LOD scores and has been implemented by a simple modification of existing software. Most important, accurate linkage tests can be performed. Using an example, we show that some previously suggested approaches to handling less than perfectly informative data can be unacceptably conservative. Situations in which this model may not perform well are discussed, and an alternative model that requires additional computations is suggested.

Alleles↗