Inherent intractability of the ascertainment problem for pedigree data: a general likelihood framework.

The problem of ascertainment in segregation analysis arises when families are selected for study through ascertainment of affected individuals. In this case, ascertainment must be corrected for in data analysis. However, methods for ascertainment correction are not available for many common sampling schemes, e.g., sequential sampling of extended pedigrees (except in the case of "single" selection). Concerns about whether ascertainment correction is even required for large pedigrees, about whether and how multiple probands in the same pedigree can be taken into account properly, and about how to apply sequential sampling strategies have occupied many investigators in recent years. We address these concerns by reconsidering a central issue, namely, how to handle pedigree structure (including size). We introduce a new distinction, between sampling in such a way that observed pedigree structure does not depend on which pedigree members are probands (proband-independent [PI] sampling) and sampling in such a way that observed pedigree structure does depend on who are the probands (proband-dependent [PD] sampling). This distinction corresponds roughly (but not exactly) to the distinction between fixed-structure and sequential sampling. We show that conditioning on observed pedigree structure in ascertained data sets obtained under PD sampling is not in general correct (with the exception of "single" selection), while PI sampling of pedigree structures larger than simple sibships is generally not possible. Yet, in practice one has little choice but to condition on observed pedigree structure. We conclude that the problem of genetic modeling in ascertained data sets is, in most situations, literally intractable. We recommend that future efforts focus on the development of robust approximate approaches to the problem.     

A resolution of the ascertainment sampling problem. II. Generalizations and numerical results

The ascertainment problem arises when families are sampled by a nonrandom process and some assumption about this sampling process must be made in order to estimate genetic parameters. Under classical ascertainment assumptions, estimation of genetic parameters cannot be separated from estimation of the parameters of the ascertainment process, so that any misspecification of the ascertainment process causes biases in estimation of the genetic parameters. Ewens and Shute proposed a resolution to this problem, involving conditioning the likelihood of the sample on the part of the data which is "relevant to ascertainment." The usefulness of this approach can only be assessed by examining the properties (in particular, bias and standard error) of the estimates which arise by using it for a wide range of parameter values and family size distributions and then comparing these biases and standard errors with those arising under classical ascertainment procedures. These comparisons are carried out in the present paper, and we also compare the proposed method with procedures which condition on, or ignore, parts of the data.     

Sensitivity of lod scores to changes in diagnostic status.

This paper investigates effects on lod scores when one individual in a data set changes diagnostic or recombinant status. First we examine the situation in which a single offspring in a nuclear family changes status. The nuclear-family situation, in addition to being of interest in its own right, also has general theoretical importance, since nuclear families are "transparent"; that is, one can track genetic events more precisely in nuclear families than in complex pedigrees. We demonstrate that in nuclear families log10 [(1-theta)/theta] gives an upper limit on the impact that a single offspring's change in status can have on the lod score at that recombination fraction (theta). These limits hold for a fully penetrant dominant condition and fully informative marker, in either phase-known or phase-unknown matings. Moreover, log10 [(1-theta)/theta] (where theta denotes the value of theta at which Zmax occurs) gives an upper limit on the impact of a single offspring's status change on the maximum lod score (Zmax). In extended pedigrees, in contrast to nuclear families, no comparable limit can be set on the impact of a single individual on the lod score. Complex pedigrees are subject to both stabilizing and destabilizing influences, and these are described. Finally, we describe a "sensitivity analysis," in which, after all linkage analysis is completed, every informative individual in the data set is changed, one at a time, to see the effect which each separate change has on the lod scores. The procedure includes identifying "critical individuals," i.e., those who would have the greatest impact on the lod scores, should their diagnostic status in fact change. To illustrate use of the sensitivity analysis, we apply it to the large bipolar pedigree reported by Egeland et al. and Kelsoe et al. We show that the changes in lod scores observed there, on the order of 1.1-1.2 per person, are not unusual. We recommend that investigators include a sensitivity analysis as a standard part of reporting the results of a linkage analysis.     

A cooperative binomial ascertainment model.

It has been shown that the classical binomial form of ascertainment, assuming a constant probability pi that any affected individual may become a proband for his pedigree, cannot describe a rather wide range of ascertainment procedures that might arise in practice. Some more general heuristic ascertainment formulas might then be preferred, and in this paper we consider the probabilistic basis for these formulas. We retain the binomial assumption of the classical scheme but allow the ascertainment probability to depend on the number of potential probands per pedigree. This probability can be expressed by an increasing or a decreasing function of that number. Various illustrations are given and situations where the "cooperative" binomial scheme should be valuable are discussed.     

Between-generation differences in ascertainment and penetrance: relevance to genetic hypotheses in fragile X

Between-generation differences in ascertainment were examined in 54 extended fragile X pedigrees, where all available members were clinically, psychometrically, and cytogenetically investigated. In 24 families a diagnosis was verified by molecular characterization using the pfxa3 fragile X-specific probe. We found considerable differences between generations in relative proportions of affected fragile X subjects versus non-penetrant carriers. We also found deviation in the segregation ratio in unbiased samples of relatives in pedigrees. We claim that these irregularites are influenced by different rates of ascertainment, depending on the clinical expression of the condition (penetrance) and the fertility of fragile X individuals in a pedigree, as well as by the thoroughness of clinical investigation in individual families. Penetrance and fertilty were estimated in fragile X females assessed by psychometric tests, and they were compared with earlier estimates based on a subjective judgement of their intellectual status. We suggest that the standard correction for ascertainment bias, such as has been applied in segregation analysis of this condition, is not sufficient to adjust for all types of bias.     

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.     

A resolution of the ascertainment sampling problem I. Theory

We consider the "ascertainment problem" arising when families are sampled by a nonrandom sampling process and, for the purpose of estimating genetic parameters, some assumption must be made about the process by which families enter the sample. A resolution of this problem, involving conditioning the likelihood of the sample on that part of the data relevant to ascertainment, is put forward. Numerical examples illustrating the properties of the procedure are provided.     

Detection of parent-of-origin effects for quantitative traits using general pedigree data

Genomic imprinting is a genetic phenomenon in which certain alleles are differentially expressed in a parent-of-origin-specific manner, and plays an important role in the study of complex traits. For a diallelic marker locus in human, the parental-asymmetry tests Q-PAT(c) with any constant c were developed to detect parent-of-origin effects for quantitative traits. However, these methods can only be applied to deal with nuclear families and thus are not suitable for extended pedigrees. In this study, by making no assumption about the distribution of the quantitative trait, we first propose the pedigree parental-asymmetry tests Q-PPAT(c) with any constant c for quantitative traits to test for parent-of-origin effects based on nuclear families with complete information from general pedigree data, in the presence of association between marker alleles under study and quantitative traits. When there are any genotypes missing in pedigrees, we utilize Monte Carlo (MC) sampling and estimation and develop the Q-MCPPAT(c) statistics to test for parent-of-origin effects. Various simulation studies are conducted to assess the performance of the proposed methods, for different sample sizes, genotype missing rates, degrees of imprinting effects and population models. Simulation results show that the proposed methods control the size well under the null hypothesis of no parent-of-origin effects and Q-PPAT(c) are robust to population stratification. In addition, the power comparison demonstrates that Q-PPAT(c) and Q-MCPPAT(c) for pedigree data are much more powerful than Q-PAT(c) only using two-generation nuclear families selected from extended pedigrees.     

Monte Carlo pedigree disequilibrium test for markers on the X chromosome

Because of the need for fine mapping of disease loci and the availability of dense single-nucleotide-polymorphism markers, many forms of association tests have been developed. Most of them are applicable only to triads, whereas some are amenable to nuclear families (sibships). Although there are a number of methods that can deal with extended families (e.g., the pedigree disequilibrium test [PDT]), most of them cannot accommodate incomplete data. Furthermore, despite a large body of literature on association mapping, only a very limited number of publications are applicable to X-chromosomal markers. In this report, we first extend the PDT to markers on the X chromosome for testing linkage disequilibrium in the presence of linkage. This method is applicable to any pedigree structure and is termed "X-chromosomal pedigree disequilibrium test" (XPDT). We then further extend the XPDT to accommodate pedigrees with missing genotypes in some of the individuals, especially founders. Monte Carlo (MC) samples of the missing genotypes are generated and used to calculate the XMCPDT (X-chromosomal MC PDT) statistic, which is defined as the conditional expectation of the XPDT statistic given the incomplete (observed) data. This MC version of the XPDT remains a valid test for association under linkage with the assumption that the pedigrees and their associated affection patterns are drawn randomly from a population of pedigrees with at least one affected offspring. This set of methods was compared with existing approaches through simulation, and substantial power gains were observed in all settings considered, with type I error rates closely tracking their nominal values.     

Analysis of genealogical structure of populations. I. A method of collecting genealogical data in numerical form

A method for collecting genealogical data with respect to an individual, a family, and members of the whole population is suggested. The essence of vertical pedigree construction consists of the same type of steps for filling in data (in the fixed order which excludes skips in the enumeration of lines of descent) about the father and the mother of the next ancestor. Each number in the received ordered list of ancestors uniquely determines a path (line of descent) to the given pedigree member. The path is explicitly described by a sequence of digits 0 and 1 (that corresponds to the sequence of fathers and mothers in the line of descent) at binary notation of this number. As a result, a pedigree is presented as a set of numbered rows that contain information, which uniquely identifies direct ancestors as individual persons. Results of joining separate pedigrees are recorded as a family list that contains lists of children for each parental pair. A pair of parents (more exactly, pointers of their families in the previous generation and numbers of pair members in their families) plays the role of the family "heading." Such a family list permits one to trace lines of descent and relationships for any population members presented in the list. It contains all genealogical information within the bounds of the study in a compact form. Here the process of collection requires considerably less time than traditional graphic representation of pedigrees. In addition, due to repeated checks of data during accumulation of material, error is minimized. Using pedigrees that have been collected, it is possible to calculate the coefficient of inbreeding manually. In connection with the wide prevalence of personal computers at present, it is also important that the data received are in fact ready to direct input to a computer for further automated data processing.     

Aspects of parameter estimation in ascertainment sampling schemes.

It has recently been suggested that ascertainment sampling estimation procedures commonly used are not fully efficient in that the number of unobserved families is an unknown parameter that should be estimated (contrary to common practice) along with the genetic parameters for fully efficient estimation. It has also been suggested that the frequency distribution of family size contains unknown parameters that should similarly be estimated with the genetic parameters. These two suggestions are considered in this paper. It is shown by means of an equivalence theorem that in both cases the estimates and their variances obtained by adopting the suggested procedure are identical with those found by ignoring the unobserved families and by ignoring the family-size distribution. This demonstration leads to a formal justification of further procedures, in particular: (1) use of "method-of-moments" estimators, (2) ignoring the ascertainment scheme in some cases when estimating parameters, and (3) forming estimates of parameters when various parts of the data are obtained by different ascertainment schemes.     

Genetic investigation of quantitative traits related to autism: use of multivariate polygenic models with ascertainment adjustment

Autism is a severe developmental disorder of unknown etiology but with evidence for genetic influences. Here, we provide evidence for a genetic basis of several quantitative traits that are related to autism. These traits, from the Broader Phenotype Autism Symptom Scale (BPASS), were measured in nuclear families, each ascertained through two probands affected by autism spectrum disorder. The BPASS traits capture the continuum of severity of impairments and may be more informative for genetic studies than are the discrete diagnoses of autism that have been used by others. Using a sample of 201 nuclear families consisting of a total of 694 individuals, we implemented multivariate polygenic models with ascertainment adjustment to estimate heritabilities and genetic and environmental correlations between these traits. Our ascertainment adjustment uses conditioning on the phenotypes of probands, requires no modeling of the ascertainment process, and is applicable to multiplex ascertainment and multivariate traits. This appears to be the first such implementation for multivariate quantitative traits. The marked difference between heritability estimates of the trait for language onset with and without an ascertainment adjustment (0.08 and 0.22, respectively) shows that conclusions are sensitive to whether or not an ascertainment adjustment is used. Among the five BPASS traits that were analyzed, the traits for social motivation and range of interest/flexibility show the highest heritability (0.19 and 0.16, respectively) and also have the highest genetic correlation (0.92). This finding suggests a shared genetic basis of these two traits and that they may be most promising for future gene mapping and for extending pedigrees by phenotyping additional relatives.     

Genetic interrelationship between insulin-dependent diabetes mellitus, the autoimmune thyroid diseases, and rheumatoid arthritis.

To investigate the possible coinheritance of autoimmune diseases that are associated with the same HLA antigen, we studied 70 families in which at least two siblings had either type I diabetes mellitus (IDDM), autoimmune thyroid disease (ATD), rheumatoid arthritis (RA), or a combination of these diseases. HLA-A, B, and C typing was performed on all affected sibs in one generation or more. First, we estimated by sib-pair analysis the disease allele frequency (pD) and the mode of inheritance for each disease. According to the method of ascertainment entered into the analysis, the pD for ATD ranged from .120 to .180, for an additive (dominant) mode of inheritance. For RA, the pD ranged from .254 to .341, also for additive inheritance, although recessive inheritance could not be excluded. For IDDM, the pD ranged from .336 to .337 for recessive inheritance; additive inheritance was rejected. Second, we examined the distribution of shared parental haplotypes in pairs of siblings that were discordant for their autoimmune diseases. The results suggested that the same haplotype may predispose to both IDDM and ATD, or IDDM and RA, but not to both RA and ATD. Analysis of pedigrees supported this hypothesis. In 16 families typed for HLA-DR also, the haplotype predisposing to both IDDM and ATD was assigned from pedigree information to DR3 (44%), DR4 (39%), or DR5, DR6, or DR7 (5.5% each). In some families, these haplotypes segregated over several generations with ATD only (either clinical or subclinical), suggesting that in such families, ATD was a marker for a susceptibility to IDDM. In several families, an IDDM haplotype segregated with RA but not with ATD. This suggests that ATD- and RA-associated susceptibilities to IDDM may be biologically different and thus independently increase the risk of IDDM.     

The problem of ascertainment for linkage analysis.

It is generally believed that ascertainment corrections are unnecessary in linkage analysis, provided individuals are selected for study solely on the basis of trait phenotype and not on the basis of marker genotype. The theoretical rationale for this is that standard linkage analytic methods involve conditioning likelihoods on all the trait data, which may be viewed as an application of the ascertainment assumption-free (AAF) method of Ewens and Shute. In this paper, we show that when the observed pedigree structure depends on which relatives within a pedigree happen to have been the probands (proband-dependent, or PD, sampling) conditioning on all the trait data is not a valid application of the AAF method and will result in asymptotically biased estimates of genetic parameters (except under single ascertainment). Furthermore, this result holds even if the recombination fraction R is the only parameter of interest. Since the lod score is proportional to the likelihood of the marker data conditional on all the trait data, this means that when data are obtained under PD sampling the lod score will yield asymptotically biased estimates of R, and that so-called mod scores (i.e., lod scores maximized over both R and parameters theta of the trait distribution) will yield asymptotically biased estimates of R and theta. Furthermore, the problem appears to be intractable, in the sense that it is not possible to formulate the correct likelihood conditional on observed pedigree structure. In this paper we do not investigate the numerical magnitude of the bias, which may be small in many situations. On the other hand, virtually all linkage data sets are collected under PD sampling. Thus, the existence of this bias will be the rule rather than the exception in the usual applications.     

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.     

Estimating the power of a proposed linkage study: a practical computer simulation approach.

I describe a simulation method to estimate the power to detect linkage given a set of pedigrees of known structure and for which family history data may be available. This method can be applied to autosomal and X-linked dominant diseases; depending on the pedigrees under consideration, it will often be applicable for autosomal and X-linked recessive diseases. This power calculation can most usefully be undertaken after family history data are gathered, but prior to examination and testing of pedigree members to obtain marker information. Of key importance, the power calculation is straightforward to carry out and not too time-consuming; it is practical even on a microcomputer. The result of the power calculation is an objective answer to the question: Will my families be sufficient to demonstrate linkage?     

Pedigree reconstruction using identity by descent

Can we find the family trees, or pedigrees, that relate the haplotypes of a group of individuals? Collecting the genealogical information for how individuals are related is a very time-consuming and expensive process. Methods for automating the construction of pedigrees could stream-line this process. While constructing single-generation families is relatively easy given whole genome data, reconstructing multi-generational, possibly inbred, pedigrees is much more challenging. This article addresses the important question of reconstructing monogamous, regular pedigrees, where pedigrees are regular when individuals mate only with other individuals at the same generation. This article introduces two multi-generational pedigree reconstruction methods: one for inbreeding relationships and one for outbreeding relationships. In contrast to previous methods that focused on the independent estimation of relationship distances between every pair of typed individuals, here we present methods that aim at the reconstruction of the entire pedigree. We show that both our methods out-perform the state-of-the-art and that the outbreeding method is capable of reconstructing pedigrees at least six generations back in time with high accuracy. The two programs are available at http://cop.icsi.berkeley.edu/cop/.     

FamSeq: A Variant Calling Program for Family-Based Sequencing Data Using Graphics Processing Units

Various algorithms have been developed for variant calling using next-generation sequencing data, and various methods have been applied to reduce the associated false positive and false negative rates. Few variant calling programs, however, utilize the pedigree information when the family-based sequencing data are available. Here, we present a program, FamSeq, which reduces both false positive and false negative rates by incorporating the pedigree information from the Mendelian genetic model into variant calling. To accommodate variations in data complexity, FamSeq consists of four distinct implementations of the Mendelian genetic model: the Bayesian network algorithm, a graphics processing unit version of the Bayesian network algorithm, the Elston-Stewart algorithm and the Markov chain Monte Carlo algorithm. To make the software efficient and applicable to large families, we parallelized the Bayesian network algorithm that copes with pedigrees with inbreeding loops without losing calculation precision on an NVIDIA graphics processing unit. In order to compare the difference in the four methods, we applied FamSeq to pedigree sequencing data with family sizes that varied from 7 to 12. When there is no inbreeding loop in the pedigree, the Elston-Stewart algorithm gives analytical results in a short time. If there are inbreeding loops in the pedigree, we recommend the Bayesian network method, which provides exact answers. To improve the computing speed of the Bayesian network method, we parallelized the computation on a graphics processing unit. This allowed the Bayesian network method to process the whole genome sequencing data of a family of 12 individuals within two days, which was a 10-fold time reduction compared to the time required for this computation on a central processing unit.

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