Statistics for Nonparametric Linkage Analysis of X-Linked Traits in General Pedigrees

We have compared the power of several allele-sharing statistics for "nonparametric" linkage analysis of X-linked traits in nuclear families and extended pedigrees. Our rationale was that, although several of these statistics have been implemented in popular software packages, there has been no formal evaluation of their relative power. Here, we evaluate the relative performance of five test statistics, including two new test statistics. We considered sibships of sizes two through four, four different extended pedigrees, 15 different genetic models (12 single-locus models and 3 two-locus models), and varying recombination fractions between the marker and the trait locus. We analytically estimated the sample sizes required for 80% power at a significance level of.001 and also used simulation methods to estimate power for a sample size of 10 families. We tried to identify statistics whose power was robust over a wide variety of models, with the idea that such statistics would be particularly useful for detection of X-linked loci associated with complex traits. We found that a commonly used statistic, S(all), generally performed well under various conditions and had close to the optimal sample sizes in most cases but that there were certain cases in which it performed quite poorly. Our two new statistics did not perform any better than those already in the literature. We also note that, under dominant and additive models, regardless of the statistic used, pedigrees with all-female siblings have very little power to detect X-linked loci.     

Robustness and power of the maximum-likelihood-binomial and maximum-likelihood-score methods,in multipoint linkage analysis of affected-sibship data.

The maximum-likelihood-binomial (MLB) method, based on the binomial distribution of parental marker alleles among affected offspring, recently was shown to provide promising results by two-point linkage analysis of affected-sibship data. In this article, we extend the MLB method to multipoint linkage analysis, using the general framework of hidden Markov models. Furthermore, we perform a large simulation study to investigate the robustness and power of the MLB method, compared with those of the maximum-likelihood-score (MLS) method as implemented in MAPMAKER/SIBS, in the multipoint analysis of different affected-sibship samples. Analyses of multiple-affected sibships by means of the MLS were conducted by consideration of all possible sib pairs, with (weighted MLS [MLSw]) or without (unweighted MLS [MLSu]) application of a classic weighting procedure. In simulations under the null hypothesis, the MLB provided very consistent type I errors regardless of the type of family sample (sib pairs or multiple-affected sibships), as did the MLS for samples with sib pairs only. When samples included multiple-affected sibships, the MLSu led to inflation of low type I errors, whereas the MLSw yielded very conservative tests. Power comparisons showed that the MLB generally was more powerful than the MLS, except in recessive models with allele frequencies <.3. Missing parental marker data did not strongly influence type I error and power results in these multipoint analyses. The MLB approach, which in a natural way accounts for multiple-affected sibships and which provides a simple likelihood-ratio test for linkage, is an interesting alternative for multipoint analysis of sibships.     

Powerful regression-based quantitative-trait linkage analysis of general pedigrees

We present a new method of quantitative-trait linkage analysis that combines the simplicity and robustness of regression-based methods and the generality and greater power of variance-components models. The new method is based on a regression of estimated identity-by-descent (IBD) sharing between relative pairs on the squared sums and squared differences of trait values of the relative pairs. The method is applicable to pedigrees of arbitrary structure and to pedigrees selected on the basis of trait value, provided that population parameters of the trait distribution can be correctly specified. Ambiguous IBD sharing (due to incomplete marker information) can be accommodated in the method by appropriate specification of the variance-covariance matrix of IBD sharing between relative pairs. We have implemented this regression-based method and have performed simulation studies to assess, under a range of conditions, estimation accuracy, type I error rate, and power. For normally distributed traits and in large samples, the method is found to give the correct type I error rate and an unbiased estimate of the proportion of trait variance accounted for by the additive effects of the locus-although, in cases where asymptotic theory is doubtful, significance levels should be checked by simulations. In large sibships, the new method is slightly more powerful than variance-components models. The proposed method provides a practical and powerful tool for the linkage analysis of quantitative traits.     

The possibility of modern segregation analysis to discriminate monogenic and multifactorial types of the inheritance of traits. The reliability of models and the power of the analysis

Using computer simulation of family data sets within segregation analysis limits the comparative research of the autosomal major locus model (MLM) and the multifactorial model (MFM) which described the basic types of inheritance of the alternative traits (affected--nonaffected) has been conducted. Robustness of models (statistical aspect) and the power of the analysis (its possibility to discriminate MLM and MFM) are tested. It is shown that permissible level of relative errors for estimated population's frequency is increased, when the values of parameters of models are increased (with decrease of sporadic cases' portion). The power of the analysis is the greatest in the field of greatest robustness of models: the higher the parameters of models, the portion of sporadic cases being low, the stronger the power of the analysis. MFM satisfactorily describes family distribution given by additive MLM. On the contrary, if the MLM corresponds to the multifactorial traits, this model is additive. It is shown that the possibility of the analysis to reject alternative model is decreased when determination of model grows down.     

Likelihood-based association analysis for nuclear families and unrelated subjects with missing genotype data

Missing data occur in genetic association studies for several reasons including missing family members and uncertain haplotype phase. Maximum likelihood is a commonly used approach to accommodate missing data, but it can be difficult to apply to family-based association studies, because of possible loss of robustness to confounding by population stratification. Here a novel likelihood for nuclear families is proposed, in which distinct sets of association parameters are used to model the parental genotypes and the offspring genotypes. This approach is robust to population structure when the data are complete, and has only minor loss of robustness when there are missing data. It also allows a novel conditioning step that gives valid analysis for multiple offspring in the presence of linkage. Unrelated subjects are included by regarding them as the children of two missing parents. Simulations and theory indicate similar operating characteristics to TRANSMIT, but with no bias with missing data in the presence of linkage. In comparison with FBAT and PCPH, the proposed model is slightly less robust to population structure but has greater power to detect strong effects. In comparison to APL and MITDT, the model is more robust to stratification and can accommodate sibships of any size. The methods are implemented for binary and continuous traits in software, UNPHASED, available from the author.     

Further segregation analysis of the fragile X syndrome with special reference to transmitting males

Summary A new series of 96 pedigrees with the fra(X) syndrome was analysed using complex segregation analysis with pointers, defining affection as any degree of mental impairment. These families were found to exhibit the same segregation pattern as the first series of 110 pedigrees (Sherman et al. 1984). The best estimate for penetrance of mental impairment in males was 79% and in females was 35% for the combined data. Again, there was little evidence for sporadic cases among affected males.Many more intellectually normal transmitting males have been observed since the existence of such males and the concomitant need to investigate the paternal side of pedigrees was recognized. On further investigation of all 206 pedigrees from the old and new data sets, the sibships of nonexpressing males appeared to be different from those of expressing males. Our analysis, using mental impairment as the phenotype, suggested that obligate carrier mothers and daughters of intellectually normal transmitting males are rarely, if ever, mentally impaired and that the sibs of transmitting males are much less likely to be retarded than the sibs of mentally impaired males. Though mothers and daughters of transmitting males are similar in phenotype, the expression of the gene in their offspring appears to be different: the penetrance of mental impairment is higher in offspring of intellectually normal daughters of transmitting males than in offspring of intellectually normal mothers of transmitting males. The implications of these observations for genetic counseling and for genetic models of the fra(X) syndrome are discussed.     

Genomewide rapid association using mixed model and regression: a fast and simple method for genomewide pedigree-based quantitative trait loci association analysis

For pedigree-based quantitative trait loci (QTL) association analysis, a range of methods utilizing within-family variation such as transmission-disequilibrium test (TDT)-based methods have been developed. In scenarios where stratification is not a concern, methods exploiting between-family variation in addition to within-family variation, such as the measured genotype (MG) approach, have greater power. Application of MG methods can be computationally demanding (especially for large pedigrees), making genomewide scans practically infeasible. Here we suggest a novel approach for genomewide pedigree-based quantitative trait loci (QTL) association analysis: genomewide rapid association using mixed model and regression (GRAMMAR). The method first obtains residuals adjusted for family effects and subsequently analyzes the association between these residuals and genetic polymorphisms using rapid least-squares methods. At the final step, the selected polymorphisms may be followed up with the full measured genotype (MG) analysis. In a simulation study, we compared type 1 error, power, and operational characteristics of the proposed method with those of MG and TDT-based approaches. For moderately heritable (30%) traits in human pedigrees the power of the GRAMMAR and the MG approaches is similar and is much higher than that of TDT-based approaches. When using tabulated thresholds, the proposed method is less powerful than MG for very high heritabilities and pedigrees including large sibships like those observed in livestock pedigrees. However, there is little or no difference in empirical power of MG and the proposed method. In any scenario, GRAMMAR is much faster than MG and enables rapid analysis of hundreds of thousands of markers.     

Comparisons of different sampling designs for the determination of genetic transmission mechanisms in quantitative traits

The power provided by several sampling designs to detect segregation at a major locus was investigated in a simulation study using phenotypes constructed from a major-locus genotypic mean, a background polygenic effect, and an individual-specific environmental effect. Questions of which relatives, how many relatives, and how many independent pedigrees to collect were considered, using configurations ranging from nuclear families of size 5 to 4-generation pedigrees of size 45. Each configuration contained a single proband whose phenotype exceeded the 95th percentile in a population where 2.5% carry the disease susceptibility allele. Results suggest that, under the conditions simulated, when total sample size is fixed, samples composed of 3-generation pedigrees of intermediate size provide a greater magnitude of support for the presence of a major locus than do samples composed of nuclear families or 4-generation pedigrees. This study is the first to consider both the discriminatory power and estimation efficiency in comparing alternative sampling strategies for pedigree data.     

The promises and problems of linkage analysis by using the current canine genome map

There have been major advances in the canine genome map over recent years, with an agreed karyotype for all 38 pairs of autosomes and an integrated linkage-radiation-hybrid map now available. An individual dog breed represents a closed, isolated population, and many suffer genetic diseases. Some are homologs of human conditions, and dogs represent naturally occurring, potentially useful large mammal models. Other conditions may be unique to this species, but study of these offers new insights into metabolic pathways. Linkage analysis may identify a locus linked to a particular condition. Dog pedigrees facilitate these studies by being relatively more informative than human families, although planned breeding strategies to maximize the informativeness may be required. A study of a late-onset, acquired heart disease, dilated cardiomyopathy (DCM), in Newfoundland dogs highlights some of the problems. The disease occurs naturally in pedigrees within the UK in dogs kept as pets and breeding stock. Problems encountered include confirmation of the mode of inheritance and determination of phenotype. A genome-wide linkage analysis study with over 200 markers failed to detect linkage. The study indicates that more detailed linkage maps are required, and further synteny information between dog and human gained before study of diseases affecting naturally occurring families of dogs from inbred pedigrees may have maximal chance of success in order to utilize this model.     

Multipoint quantitative-trait linkage analysis in general pedigrees.

Multipoint linkage analysis of quantitative-trait loci (QTLs) has previously been restricted to sibships and small pedigrees. In this article, we show how variance-component linkage methods can be used in pedigrees of arbitrary size and complexity, and we develop a general framework for multipoint identity-by-descent (IBD) probability calculations. We extend the sib-pair multipoint mapping approach of Fulker et al. to general relative pairs. This multipoint IBD method uses the proportion of alleles shared identical by descent at genotyped loci to estimate IBD sharing at arbitrary points along a chromosome for each relative pair. We have derived correlations in IBD sharing as a function of chromosomal distance for relative pairs in general pedigrees and provide a simple framework whereby these correlations can be easily obtained for any relative pair related by a single line of descent or by multiple independent lines of descent. Once calculated, the multipoint relative-pair IBDs can be utilized in variance-component linkage analysis, which considers the likelihood of the entire pedigree jointly. Examples are given that use simulated data, demonstrating both the accuracy of QTL localization and the increase in power provided by multipoint analysis with 5-, 10-, and 20-cM marker maps. The general pedigree variance component and IBD estimation methods have been implemented in the SOLAR (Sequential Oligogenic Linkage Analysis Routines) computer package.     

Parentage and Sibship Inference From Multilocus Genotype Data Under Polygamy

Likelihood methods have been developed to partition individuals in a sample into sibling clusters using genetic marker data without parental information. Most of these methods assume either both sexes are monogamous to infer full sibships only or only one sex is polygamous to infer full sibships and paternal or maternal (but not both) half sibships. We extend our previous method to the more general case of both sexes being polygamous to infer full sibships, paternal half sibships, and maternal half sibships and to the case of a two-generation sample of individuals to infer parentage jointly with sibships. The extension not only expands enormously the scope of application of the method, but also increases its statistical power. The method is implemented for both diploid and haplodiploid species and for codominant and dominant markers, with mutations and genotyping errors accommodated. The performance and robustness of the method are evaluated by analyzing both simulated and empirical data sets. Our method is shown to be much more powerful than pairwise methods in both parentage and sibship assignments because of the more efficient use of marker information. It is little affected by inbreeding in parents and is moderately robust to nonrandom mating and linkage of markers. We also show that individually much less informative markers, such as SNPs or AFLPs, can reach the same power for parentage and sibship inferences as the highly informative marker simple sequence repeats (SSRs), as long as a sufficient number of loci are employed in the analysis.     

Estimating quantitative genetic parameters using sibships reconstructed from marker data

Previous techniques for estimating quantitative genetic parameters, such as heritability in populations where exact relationships are unknown but are instead inferred from marker genotypes, have used data from individuals on a pairwise level only. At this level, families are weighted according to the number of pairs within which each family appears, hence by size rather than information content, and information from multiple relationships is lost. Estimates of parameters are therefore not the most efficient achievable. Here, Markov chain Monte Carlo techniques have been used to partition the population into complete sibships, including, if known, prior knowledge of the distribution of family sizes. These pedigrees have then been used with restricted maximum likelihood under an animal model to estimate quantitative genetic parameters. Simulations to compare the properties of parameter estimates with those of existing techniques indicate that the use of sibship reconstruction is superior to earlier methods, having lower mean square errors and showing nonsignificant downward bias. In addition, sibship reconstruction allows the estimation of population allele frequencies that account for the relationships within the sample, so prior knowledge of allele frequencies need not be assumed. Extensions to these techniques allow reconstruction of half sibships when some or all of the maternal genotypes are known.     

A survey of affected-sibship statistics for nonparametric linkage analysis

We have compared the power of a large number of allele-sharing statistics for "nonparametric" linkage analysis with affected sibships. Our rationale was that there is an extensive literature comparing statistics for sibling pairs but that there has not been much guidance on how to choose statistics for studies that include sibships of various sizes. We concentrated on statistics that can be described as assigning scores to each identity-by-descent-sharing configuration that a pedigree might take on (Whittemore and Halpern 1994). We considered sibships of sizes two through five, 27 different genetic models, and varying recombination fractions between the marker and the trait locus. We tried to identify statistics whose power was robust over a wide variety of models. We found that the statistic that is probably used most often in such studies-S(all)-performs quite well, although it is not necessarily the best. We also found several other statistics (such as the R criterion, S(robdom), and the Sobel-and-Lange statistic C) that perform well in most situations, a few (such as S(-#geno) and the Feingold-and-Siegmund version of S(pairs)) that have high power only in very special situations, and a few (such as S(-#geno), the N criterion, and the Sobel-and-Lange statistic B) that seem to have low power for the majority of the trait models. For the most part, the same statistics performed well for all sibship sizes. We also used our results to give some suggestions regarding how to weight sibships of different sizes, in forming an overall statistic.     

X-linked agammaglobulinemia and the red blood cell determinants Xg and 12E7 are not closely linked

Summary Studies on the segregation of the red blood cell determinant Xg in 12 families with X-linked inheritance of agammaglobulinemia (XLA) in 3–4 generations suggested linkage of Xg with XLA. One extensive pedigree of a Dutch family with XLA in eight generations was investigated for Xg and the quantitative polymorphism 12E7. LIPED analysis indicated linkage disproven up to 25cM distance within this pedigree. Taken together with data obtained from two other informative XLA pedigrees and with published data, the results indicate no close linkage between XLA and Xg:12E7, the distance between XLA and Xg being more than 20cM.     

Linkage analysis of systolic blood pressure: a score statistic and computer implementation

A genome-wide linkage analysis was conducted on systolic blood pressure using a score statistic. The randomly selected Replicate 34 of the simulated data was used. The score statistic was applied to the sibships derived from the general pedigrees. An add-on R program to GENEHUNTER was developed for this analysis and is freely available.     

Robustness and power of the unified model in the analysis of quantitative measurements.

The resolution between skewness in the distribution of a quantitative trait and segregation of a major gene is a difficult issue in family studies. Quantitative data were simulated on six-member nuclear families in order to study the behavior of the unified model under these circumstances. Replicates of 100 nuclear families were generated assuming a multifactorial model with skewness. In the range where a major gene was falsely detected in 80%-100% of the simulations analyzed under the transmission probability or mixed models, use of the unified model reduces the frequency of false inference to between 10% and 40%. This protection against a false conclusion requires estimation of the three transmission probabilities and testing hypotheses of Mendelian transmission and equal transmission probabilities. Alternatively, it was shown that use of a transformation to remove skewness induced by a major gene leads to a decrease of power of approximately 55%. These results suggest that the unified model may obviate the need to compare analyses performed on transformed and untransformed data, particularly when skewness is low (less than 0.2) or high (greater than 0.4). For intermediate skewness (0.2-0.4), estimating segregation parameters under the mixed model simultaneously with a transformation to remove residual skewness can be considered as an alternative method.     

A test for linkage and association in general pedigrees: the pedigree disequilibrium test

Family-based tests of linkage disequilibrium typically are based on nuclear-family data including affected individuals and their parents or their unaffected siblings. A limitation of such tests is that they generally are not valid tests of association when data from related nuclear families from larger pedigrees are used. Standard methods require selection of a single nuclear family from any extended pedigrees when testing for linkage disequilibrium. Often data are available for larger pedigrees, and it would be desirable to have a valid test of linkage disequilibrium that can use all potentially informative data. In this study, we present the pedigree disequilibrium test (PDT) for analysis of linkage disequilibrium in general pedigrees. The PDT can use data from related nuclear families from extended pedigrees and is valid even when there is population substructure. Using computer simulations, we demonstrated validity of the test when the asymptotic distribution is used to assess the significance, and examined statistical power. Power simulations demonstrate that, when extended pedigree data are available, substantial gains in power can be attained by use of the PDT rather than existing methods that use only a subset of the data. Furthermore, the PDT remains more powerful even when there is misclassification of unaffected individuals. Our simulations suggest that there may be advantages to using the PDT even if the data consist of independent families without extended family information. Thus, the PDT provides a general test of linkage disequilibrium that can be widely applied to different data structures.     

The quantitative LOD score: test statistic and sample size for exclusion and linkage of quantitative traits in human sibships.

We present a test statistic, the quantitative LOD (QLOD) score, for the testing of both linkage and exclusion of quantitative-trait loci in randomly selected human sibships. As with the traditional LOD score, the boundary values of 3, for linkage, and -2, for exclusion, can be used for the QLOD score. We investigated the sample sizes required for inferring exclusion and linkage, for various combinations of linked genetic variance, total heritability, recombination distance, and sibship size, using fixed-size sampling. The sample sizes required for both linkage and exclusion were not qualitatively different and depended on the percentage of variance being linked or excluded and on the total genetic variance. Information regarding linkage and exclusion in sibships larger than size 2 increased as approximately all possible pairs n(n-1)/2 up to sibships of size 6. Increasing the recombination (theta) distance between the marker and the trait loci reduced empirically the power for both linkage and exclusion, as a function of approximately (1-2theta)4.     

The efficiency of designs for fine-mapping of quantitative trait loci using combined linkage disequilibrium and linkage

In a simulation study, different designs were compared for efficiency of fine-mapping of QTL. The variance component method for fine-mapping of QTL was used to estimate QTL position and variance components. The design of many families with small size gave a higher mapping resolution than a design with few families of large size. However, the difference is small in half sib designs. The proportion of replicates with the QTL positioned within 3 cM of the true position is 0.71 in the best design, and 0.68 in the worst design applied to 128 animals with a phenotypic record and a QTL explaining 25% of the phenotypic variance. The design of two half sib families each of size 64 was further investigated for a hypothetical population with effective size of 1000 simulated for 6000 generations with a marker density of 0.25 cM and with marker mutation rate 4 × 10^-4 per generation. In mapping using bi-allelic markers, 42~55% of replicated simulations could position QTL within 0.75 cM of the true position whereas this was higher for multi allelic markers (48~76%). The accuracy was lowest (48%) when mutation age was 100 generations and increased to 68% and 76% for mutation ages of 200 and 500 generations, respectively, after which it was about 70% for mutation ages of 1000 generations and older. When effective size was linearly decreasing in the last 50 generations, the accuracy was decreased (56 to 70%). We show that half sib designs that have often been used for linkage mapping can have sufficient information for fine-mapping of QTL. It is suggested that the same design with the same animals for linkage mapping should be used for fine-mapping so gene mapping can be cost effective in livestock populations.     

Power of the joint segregation analysis method for testing mixed major-gene and polygene inheritance models of quantitative traits

Understanding the genetic architecture of quantitative traits can greatly assist the design of strategies for their manipulation in plant-breeding programs. For a number of traits, genetic variation can be the result of segregation of a few major genes and many polygenes (minor genes). The joint segregation analysis (JSA) is a maximum-likelihood approach for fitting segregation models through the simultaneous use of phenotypic information from multiple generations. Our objective in this paper was to use computer simulation to quantify the power of the JSA method for testing the mixed-inheritance model for quantitative traits when it was applied to the six basic generations: both parents (P₁ and P₂), F₁, F₂, and both backcross generations (B₁ and B₂) derived from crossing the F₁ to each parent. A total of 1968 genetic model-experiment scenarios were considered in the simulation study to quantify the power of the method. Factors that interacted to influence the power of the JSA method to correctly detect genetic models were: (1) whether there were one or two major genes in combination with polygenes, (2) the heritability of the major genes and polygenes, (3) the level of dispersion of the major genes and polygenes between the two parents, and (4) the number of individuals examined in each generation (population size). The greatest levels of power were observed for the genetic models defined with simple inheritance; e.g., the power was greater than 90% for the one major gene model, regardless of the population size and major-gene heritability. Lower levels of power were observed for the genetic models with complex inheritance (major genes and polygenes), low heritability, small population sizes and a large dispersion of favourable genes among the two parents; e.g., the power was less than 5% for the two major-gene model with a heritability value of 0.3 and population sizes of 100 individuals. The JSA methodology was then applied to a previously studied sorghum data-set to investigate the genetic control of the putative drought resistance-trait osmotic adjustment in three crosses. The previous study concluded that there were two major genes segregating for osmotic adjustment in the three crosses. Application of the JSA method resulted in a change in the proposed genetic model. The presence of the two major genes was confirmed with the addition of an unspecified number of polygenes. Received: 18 August 2000 / Accepted: 9 March 2001