Powerful allele sharing statistics for nonparametric linkage analysis

Nonparametric linkage analysis is widely used to map susceptibility genes for complex diseases. This paper introduces six nonparametric statistics for measuring marker allele sharing among the affected members of a pedigree. We compare the power of these new statistics and three previous statistics to detect linkage with Mendelian diseases having recessive, additive, and dominant modes of inheritance. The nine statistics represent all possible combinations of three different IBD scoring functions and three different schemes for sampling genes among affecteds. Our results strongly suggest that the statistic T(rec)(blocks) is best for recessive traits, while the two statistics T(kin)(pairs) and T(all)(kin) vie for best for an additive trait. The best statistic for a dominant trait is less clear. The statistics T(kin)(pairs) and T(all)(kin) are equally promising for small sibships, but in extended pedigrees the statistics T(dom)(blocks) and T(dom)(pairs) appear best. For a complex trait, we advocate computing several of these statistics.     

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.     

A FORTRAN program for the calculation and analysis of two-locus linkage disequilibrium coefficients

Summary A FORTRAN program was written that calculates composite linkage disequilibrium coefficients from genotypic data. Chi-square tests determine whether coefficients calculated for allele and locus pairs are significantly greater than zero. A subroutine is provided that partitions the variance in linkage disequilibrium into within- and between-subpopulation components. Output obtained from analysis of allozyme data collected from natural subpopulations of the house fly (Musca domestica L.) are included to illustrate features of the program.Journal Paper No. J-11345 of the Iowa Agriculture and Home Economics Experiment Station, Ames, Iowa. Project No. 2411     

Parametric and nonparametric linkage analysis: a unified multipoint approach.

In complex disease studies, it is crucial to perform multipoint linkage analysis with many markers and to use robust nonparametric methods that take account of all pedigree information. Currently available methods fall short in both regards. In this paper, we describe how to extract complete multipoint inheritance information from general pedigrees of moderate size. This information is captured in the multipoint inheritance distribution, which provides a framework for a unified approach to both parametric and nonparametric methods of linkage analysis. Specifically, the approach includes the following: (1) Rapid exact computation of multipoint LOD scores involving dozens of highly polymorphic markers, even in the presence of loops and missing data. (2) Non-parametric linkage (NPL) analysis, a powerful new approach to pedigree analysis. We show that NPL is robust to uncertainty about mode of inheritance, is much more powerful than commonly used nonparametric methods, and loses little power relative to parametric linkage analysis. NPL thus appears to be the method of choice for pedigree studies of complex traits. (3) Information-content mapping, which measures the fraction of the total inheritance information extracted by the available marker data and points out the regions in which typing additional markers is most useful. (4) Maximum-likelihood reconstruction of many-marker haplotypes, even in pedigrees with missing data. We have implemented NPL analysis, LOD-score computation, information-content mapping, and haplotype reconstruction in a new computer package, GENEHUNTER. The package allows efficient multipoint analysis of pedigree data to be performed rapidly in a single user-friendly environment.     

Inherent bias toward the null hypothesis in conventional multipoint nonparametric linkage analysis

Traditional nonparametric "multipoint" statistical procedures have been developed for assigning allele-sharing values at a locus of interest to pairs of relatives for linkage studies. These procedures attempt to accommodate a lack of informativity, nongenotyped loci, missing data, and related issues concerning the genetic markers used in a linkage study. However, such procedures often cannot overcome these phenomena in compelling ways and, as a result, assign relevant relative pairs allele-sharing values that are "expected" for those pairs. The practice of assigning expected allele-sharing values to relative pairs in the face of a lack of explicit allele-transmission information can bias traditional nonparametric linkage test statistics toward the null hypothesis of no locus effect. This bias is due to the use of expected values, rather than to a lack of information about actual allele sharing at relevant marker loci. The bias will vary from study to study on the basis of the DNA markers, sample size, relative-pair types, and pedigree structures used, but it can be extremely pronounced and could contribute to a lack of consistent success in the application of traditional nonparametric linkage analyses to complex human traits and diseases. There are several potential ways to overcome this problem, but their foundations deserve greater research. We expose many of the issues concerning allele sharing with data from a large affected-sibling-pair study investigating the genetic basis of autism.     

Rates of decay of linkage disequilibrium under two-locus models of selection

The effect of selection and linkage on the decay of linkage disequilibrium, D, is investigated for a hierarchy of two-locus models. The method of analysis rests upon a qualitative classification of the dynamic of D under selection relative to the neutral dynamic. To eliminate the confounding effects of gene frequency change, the behavior of D is first studied with gene frequencies fixed at their invariant values. Second, the results are extended to certain special situations where gene frequencies are changing simultaneously.A wide variety of selection regimes can cause an acceleration of the rate of decay of D relative to the neutral rate. Specifically, the asymptotic rate of decay is always faster than the neutral rate in the neighborhood of a stable equilibrium point, when viabilities are additive or only one locus is selected. This is not necessarily the case for models in which there is nonzero additive epistasis. With multiplicative viabilities, decay is always accelerated near a stable boundary equilibrium, but decay is only faster near the stable central equilibrium (with = 0) if linkage is sufficiently loose. In the symmetric viability model, decay may even be retarded near a stable boundary equilibrium. Decay is only accelerated near a stable corner equilibrium when the double homozygote is more fit than the double heterozygotes. Decay near a stable edge equilibrium may be retarded if there is loose linkage. With symmetric viabilities there is usually an acceleration of the decay process for gene frequencies near 1/2 when the central equilibrium (with = 0) is stable. This is always the case when the sign of the epistasis is negative or zero.Conversely, the decay ofD is retarded in the neighborhood of a stable equilibrium in the multiplicative and symmetric viability models if any of the conditions above are violated. Near an unstable equilibrium of any of the models considered,D may either increase or decay at a rate slower than, equal to, or faster than the neutral rate. These analytic results are supplemented by numerical studies of the symmetric viability model.     

Strategies and sample-size considerations for mapping a two-locus autosomal recessive disorder.

The problem of detecting linkage, by using the LOD-score method, of polymorphic marker loci to a disorder that is determined by recessive alleles at two independent autosomal diallelic loci has been considered. The expected LOD score and the distribution of the LOD score have been worked out for various scenarios. It is found that the expected numbers of families to be sampled for detection of linkage are within feasible limits if the recombination fractions between the marker loci and the disorder loci are less than or equal to .1. The strategy of studying affected offspring only is shown to be more efficient than the strategy of studying both affected and normal offspring. The efficiency of the "affecteds-only" strategy (1) increases with increase in sibship size, (2) decreases with increase in population prevalence of the disorder, and (3) increases with increase in recombination distances between the marker and the disorder loci. From various considerations, it is found that sampling families of sibship size three with at least one affected, and adopting the affecteds-only strategy for analysis, may be an optimal strategy.     

Expected behavior of conditional linkage disequilibrium.

The ubiquitousness of RFLPs in the human genome has greatly helped the mapping of human disease genes, and it has been suggested that population measures of association between disease and marker loci could help with this mapping. For rare diseases, random samples are taken from within disease genotypes in order to obtain reasonable sample sizes, but this sampling strategy requires a modification of the usual measures of association. We present theoretical predictions for the mean and variance of such a modified measure, under the assumption that the disease gene is maintained at a constant low frequency in the population. The coefficient of variation of this modified measure is large enough that caution is needed in using the measure to locate disease genes, and, furthermore, the coefficient of variation cannot be made arbitrarily small by increasing sample size. The modified association measure is calculated for recently published data on cystic fibrosis.     

A simple nonparametric test for linkage with sib-pair censored event time observations

Consider the case that individual phenotype and genotype observations were collected from a large or moderate number of pedigrees. Some of the pedigrees have multi-generation nuclear families. For each nuclear family, the phenotype trait value of each sibling is the time to onset for a specific event (e.g., disease). Often, this event time may be right censored, that is, an individual is event-free at the study examination time point. In this article, we propose a purely nonparametric test for testing if the distribution of a Haseman-Elston distance measure between two siblings' event times is independent of their mean genetic sharing identical by descent at a genetic marker based on such incomplete observations from all the nuclear families. The new test can be implemented easily and is illustrated with a data set from the Genetic Analysis Workshop 12. The validity of the new test is examined via a simulation study.     

Prediction of empirical p values from asymptotic p values for conditional logistic affected relative pair linkage analysis

OBJECTIVE: p Values are inaccurate for model-free linkage analysis using the conditional logistic model if we assume that the LOD score is asymptotically distributed as a simple mixture of chi-square distributions. When analyzing affected relative pairs alone, permuting the allele sharing of relative pairs does not lead to a useful permutation distribution. As an alternative, we have developed regression prediction models that provide more accurate p values. METHODS: Let E(alpha) be the empirical p value, which is the proportion of statistical tests whose LOD score under the null hypothesis exceeds a threshold determined by alpha, the nominal single test significance value. We used simulated data to obtain values of E(alpha) and compared them with alpha. We also developed a regression model, based on sample size, number of covariates in the model, alpha and marker density, to derive predicted p values for both single-point and multipoint analyses. To evaluate our predictions we used another set of simulated data, comparing the Ealpha for these data with those obtained by using the prediction model, referred to as predicted p values (P(alpha)). RESULTS: Under almost all circumstances the values of P(alpha) were closer to the E(alpha) than were the values of alpha. CONCLUSION: The regression models suggested by our analysis provide more accurate alternative p values for model-free linkage analysis when using the conditional logistic model.     

Parametric and nonparametric multipoint linkage analysis with imprinting and two-locus-trait models: application to mite sensitization

We present two extensions to linkage analysis for genetically complex traits. The first extension allows investigators to perform parametric (LOD-score) analysis of traits caused by imprinted genes-that is, of traits showing a parent-of-origin effect. By specification of two heterozygote penetrance parameters, paternal and maternal origin of the mutation can be treated differently in terms of probability of expression of the trait. Therefore, a single-disease-locus-imprinting model includes four penetrances instead of only three. In the second extension, parametric and nonparametric linkage analysis with two trait loci is formulated for a multimarker setting, optionally taking imprinting into account. We have implemented both methods into the program GENEHUNTER. The new tools, GENEHUNTER-IMPRINTING and GENEHUNTER-TWOLOCUS, were applied to human family data for sensitization to mite allergens. The data set comprises pedigrees from England, Germany, Italy, and Portugal. With single-disease-locus-imprinting MOD-score analysis, we find several regions that show at least suggestive evidence for linkage. Most prominently, a maximum LOD score of 4.76 is obtained near D8S511, for the English population, when a model that implies complete maternal imprinting is used. Parametric two-trait-locus analysis yields a maximum LOD score of 6.09 for the German population, occurring exactly at D4S430 and D18S452. The heterogeneity model specified for analysis alludes to complete maternal imprinting at both disease loci. Altogether, our results suggest that the two novel formulations of linkage analysis provide valuable tools for genetic mapping of multifactorial traits.     

The effects of linkage and gene flow on local adaptation: a two-locus continent-island model

Population subdivision and migration are generally considered to be important causes of linkage disequilibrium (LD). We explore the combined effects of recombination and gene flow on the amount of LD, the maintenance of polymorphism, and the degree of local adaptation in a subdivided population by analyzing a diploid, deterministic continent–island model with genic selection on two linked loci (i.e., no dominance or epistasis). For this simple model, we characterize explicitly all possible equilibrium configurations. Simple and intuitive approximations for many quantities of interest are obtained in limiting cases, such as weak migration, weak selection, weak or strong recombination. For instance, we derive explicit expressions for the measures and r² (the squared correlation in allelic state) of LD. They depend in qualitatively different ways on the migration rate. Remarkably high values of r² are maintained between weakly linked loci, especially if gene flow is low. We determine how the maximum amount of gene flow that admits preservation of the locally adapted haplotype, hence of polymorphism at both loci, depends on recombination rate and selection coefficients. We also investigate the evolution of differentiation by examining the invasion of beneficial mutants of small effect that are linked to an already present, locally adapted allele. Mutants of much smaller effect can invade successfully than predicted by naive single-locus theory provided they are at least weakly linked. Finally, the influence of linkage on the degree of local adaptation, the migration load, and the effective migration rate at a neutral locus is explored. We discuss possible consequences for the evolution of genetic architecture, in particular, for the emergence of clusters of tightly linked, slightly beneficial mutations and the evolution of recombination and chromosome inversions.     

Some epistatic two-locus models of disease. II. The confounding of linkage and association

This paper continues to examine the model discussed in the preceding paper. Specifically, it will be shown how a linkage analysis performed in the presence of a disease-marker association can give rise to erroneous and misleading results.     

Power comparison of parametric and nonparametric linkage tests in small pedigrees

When the mode of inheritance of a disease is unknown, the LOD-score method of linkage analysis must take into account uncertainties in model parameters. We have previously proposed a parametric linkage test called "MFLOD," which does not require specification of disease model parameters. In the present study, we introduce two new model-free parametric linkage tests, known as "MLOD" and "MALOD." These tests are defined, respectively, as the LOD score and the admixture LOD score, maximized (subject to the same constraints as MFLOD) over disease-model parameters. We compared the power of these three parametric linkage tests and that of two nonparametric linkage tests, NPLall and NPLpairs, which are implemented in GENEHUNTER. With the use of small pedigrees and a fully informative marker, we found the powers of MLOD, NPLall, and NPLpairs to be almost equivalent to each other and not far below that of a LOD-score analysis performed under the assumption the correct genetic parameters. Thus, linkage analysis is not much hindered by uncertain mode of inheritance. The results also suggest that both parametric and nonparametric methods are suitable for linkage analysis of complex disorders in small pedigrees. However, whether these results apply to large pedigrees remains to be answered.     

Conclusions of segregation analysis for family data generated under two-locus models.

Susceptibility to a disease may involve the interactive effect of two genes. What conclusions will be drawn by segregation analysis in such a case? To answer this question, we considered a set of two-locus models and the corresponding exact distribution for 300 families. We investigated the conclusions and parameter estimations obtained for this sample, by comparing the likelihood expectations of the unified model and of more restricted models. In many cases, segregation analysis leads to the conclusion of a major gene effect, with or without a polygenic component--usually without a polygenic component in multiplicative models (i.e., where two genes have a multiplicative effect) and with such a component in nonmultiplicative models. For all the models considered, existence of a major gene effect is supported by transmission probability tests; there is evidence for transmission and agreement with the hypothesis of Mendelian transmission. Accordingly, there is no means of detecting that the effect of a major gene, with or without a polygenic component, does not correspond to the correct model. In addition, the parameter estimates for the major gene do not correspond to the characteristics of either of the two genes of the true model. This may substantially affect further linkage analysis.