Bias correction for segregation ratio estimation in human genetics

Segregation ratio estimation has long been important in human genetics. A simple truncated binomial model is considered that assumes complete ascertainment and a deterministic genotype-phenotype relationship. A simple but intuitively appealing estimator of the segregation ratio, previously proposed, is shown to have a negative bias. It is also shown that the bias of this estimator can be largely reduced via a randomization device, resulting in a new estimator that has the same large-sample behavior but with a negligible bias (decaying at a geometric rate). Numerical results are given to show the small-sample performance of this new estimator. An extension to incomplete ascertainment is also considered.     

Segregation analysis in Shwachman-Diamond syndrome: evidence for recessive inheritance

Shwachman-Diamond syndrome is a rare disorder of unknown cause. Reports have indicated the occurrence of affected siblings, but formal segregation analysis has not been performed. In families collected for genetic studies, the mean paternal age and mean difference in parental ages were found to be consistent with the general population. We determined estimates of segregation proportion in a cohort of 84 patients with complete sibship data under the assumption of complete ascertainment, using the Li and Mantel estimator, and of single ascertainment with the Davie modification. A third estimate was also computed with the expectation-maximization (EM) algorithm. All three estimates supported an autosomal recessive mode of inheritance, but complete ascertainment was found to be unlikely. Although there are no overt signs of disease in adult carriers (parents), the use of serum trypsinogen levels to indicate exocrine pancreatic dysfunction was evaluated as a potential measure for heterozygote expression. No consistent differences were found in levels between parents and a normal control population. Although genetic heterogeneity cannot be excluded, our results indicate that simulation and genetic analyses of Shwachman-Diamond syndrome should consider a recessive model of inheritance.     

Reconstituting the frequency spectrum of ascertained single-nucleotide polymorphism data

Most of the available SNP data have eluded valid population genetic analysis because most population genetical methods do not correctly accommodate the special discovery process used to identify SNPs. Most of the available SNP data have allele frequency distributions that are biased by the ascertainment protocol. We here show how this problem can be corrected by obtaining maximum-likelihood estimates of the true allele frequency distribution. In simple cases, the ML estimate of the true allele frequency distribution can be obtained analytically, but in other cases computational methods based on numerical optimization or the EM algorithm must be used. We illustrate the new correction method by analyzing some previously published SNP data from the SNP Consortium. Appropriate treatment of SNP ascertainment is vital to our ability to make correct inferences from the data of the International HapMap Project.     

Multipoint mapping of viability and segregation distorting loci using molecular markers

In line-crossing experiments, deviations from Mendelian segregation ratios are usually observed for some markers. We hypothesize that these deviations are caused by one or more segregation-distorting loci (SDL) linked to the markers. We develop both a maximum-likelihood (ML) method and a Bayesian method to map SDL using molecular markers. The ML mapping is implemented via an EM algorithm and the Bayesian method is performed via the Markov chain Monte Carlo (MCMC). The Bayesian mapping is computationally more intensive than the ML mapping but can handle more complicated models such as multiple SDL and variable number of SDL. Both methods are applied to a set of simulated data and real data from a cross of two Scots pine trees.     

The effect of proband designation on segregation analysis

In many family studies, it is often difficult to know exactly how the families were ascertained. Even if known, the circumstances under which the families came to the attention of the study may violate the assumptions of classical ascertainment bias correction. The purpose of this work was to investigate the effect on segregation analysis of violations of the assumptions of the classical ascertainment model. We simulated family data generated under a simple recessive model of inheritance. We then ascertained families under different "scenarios." These scenarios were designed to simulate actual conditions under which families come to the attention of-and then interact with-a clinic or genetic study. We show that how one designates probands, which one must do under the classical ascertainment model, can influence parameter estimation and hypothesis testing. We demonstrate that, in some cases, there may be no "correct" way to designate probands. Further, we show that interactions within the family, the conditions under which the genetic study must function, and even social influences can have a profound effect on segregation analysis. We also propose a method for dealing with the ascertainment problem that is applicable to almost any study situation.     

Multilist population estimation with incomplete and partial stratification

Multilist capture-recapture methods are commonly used to estimate the size of elusive populations. In many situations, lists are stratified by distinguishing features, such as age or sex. Stratification has often been used to reduce biases caused by heterogeneity in the probability of list membership among members of the population; however, it is increasingly common to find lists that are structurally not active in all strata. We develop a general method to deal with cases when not all lists are active in all strata using an expectation maximization (EM) algorithm. We use a flexible log-linear modeling framework that allows for list dependencies and differential probabilities of ascertainment in each list. Finally, we apply our method of estimating population size to two examples.     

The Estimation of the Segregation Parameter when Ascertainment is not Complete

Procedures to estimate the genetic segregation parameter when ascertainment of families is incomplete, have previously relied on iterative computer algorithms since estimators with closed form are lacking. We now present the Minimum Variance Unbiased Estimator for the segregation parameter under any ascertainment probability. This estimator assumes a simple form when ascertainment is complete. We also present a simple estimator, akin to Li and Mantel's (1968) estimator, but without the restriction that ascertainment be complete. The performance of these estimators is compared with respect to asymptotic efficiency. We also provide tables that define the required number of families of a given size that need to be sampled to achieve a specific power for testing simple hypothesis on the segregation parameter.     

Comparing observed average numbers of affected sibs with those expected under geometric ascertainment bias in an extended range in both simplex and multiple sibships.

Detection bias in recessive ascertainment is generally considered to be confined in a narrow range between unbiased truncate ascertainment and single ascertainment, where methods of segregation analysis are established. While there are arguments for an extended range of analysis, a deflated detection progression below the unbiased level is still being considered as theoretical ground or ignored as sporadics. I show here a method of gauging the ascertainment levels of surveyed data in a geometric continuum. The method is valid for recessive segregation at any ascertainment level and in simplex or multiplex sibships of whatever degree of truncation. Four previously published surveys are used to show conformation with real data and the existence of detection trends spanning the range from the unsuspected very depressed bias level to the inflated level above single ascertainment.     

Segregation analysis with uncertain ascertainment: application to Fanconi anemia.

A Bayesian solution for making inferences about segregation parameters with no information about the ascertainment is presented. Inferences about the segregation probability and the probability of being sporadic are made through the posterior marginal distribution of these parameters after integrating out the ascertainment probability, the nuisance parameter. The method was tested with real and simulated data and performed well. Original Fanconi anemia data, for which no information about the ascertainment was available, were then analyzed, with results that confirmed a monogenic autosomal recessive mode of inheritance.     

Simulation studies of segregation analysis: application to two-locus models

We tested the power of a segregation analysis method (first proposed by Elandt-Johnson) to distinguish between single-locus and two-locus models, with and without environmentally caused reduced penetrance. We also looked at the effect of ascertainment probability on the analysis and at the proband-conditioned ascertainment correction proposed by Cannings and Thompson. We found that: (1) the segregation analysis has sufficient power to distinguish between the fully-penetrant double-recessive (RR) model and the fully-penetrant single-locus dominant and recessive models; (2) the method can also distinguish fairly well between the dominant-recessive (DR) and RR models, even when one does not take into account the population prevalence; (3) the method has much less power to distinguish between the fully-penetrant RR model and the single-locus models with reduced penetrance; (4) when environmental penetrance is taken account of in the analysis, the power of the method to distinguish between the one- and two-locus models improved substantially; (5) the estimates of ascertainment probability, pi, were robust, regardless of the model under which the data were generated; and (6) the Cannings-Thompson approach to ascertainment correction worked well only when the pi used to generate the data was less than .1.     

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.     

The effects of conditioning on probands to correct for multiple ascertainment.

Cannings and Thompson suggested conditioning on the phenotypes of the probands to correct for ascertainment in the analysis of pedigree data. The method assumes single ascertainment and can be expected to yield asymptotically biased parameter estimates except in this specific case. However, because the method is easy to apply, we investigated the degree of bias in the more typical situation of multiple ascertainment, in the hope that the bias might be small and that the method could be applied more generally. To explore the utility of conditioning on probands to correct for multiple ascertainment, we calculated the asymptotic value of the segregation ratio for two versions of the simple Mendelian segregation model on sibship data. For both versions, we found that this asymptotic value decreased approximately linearly as the ascertainment probability increased. When ascertainment was complete, the segregation-ratio estimates were zero, not just asymptotically but for finite sample size as well. In some cases, conditioning on probands actually resulted in greater parameter bias than no ascertainment correction at all. These results hold for a variety of sibship-size distributions, several modes of inheritance, and a wide range of population prevalences of affected individuals.     

Medico-genetic study of the population of Kostroma Province. VII. Approaches to the study of the hereditary pathology load in the population of Kostroma

Complex medical-genetic study was performed in the center of Kostroma Province with the population of about 250 thousands. A method for ascertainment of patients was developed and the information value of different sources of a registration system available is given. The complete algorithm of calculation of both the segregation frequency and the fraction of sporadic cases for autosomal recessive diseases is presented.     

Ascertainment Bias and the Pattern of Nucleotide Diversity at the Human ALDH2 Locus in a Japanese Population

Many East Asian human populations harbor a high-frequency deficiency allele for the aldehyde dehydrogenase 2 (ALDH2) enzyme, a critical protein involved in the metabolism of ethanol. Here we use resequencing and long-range SNP haplotype data from a Japanese sample to test whether patterns of nucleotide diversity and linkage disequilibrium at this locus are compatible with a standard neutral model of evolution. Examination of the pattern of polymorphism at a locus such as this, where the frequency of a common allele is known a priori, introduces an ascertainment bias that must be corrected for in analyses of the frequency spectrum of polymorphisms. We apply a flexible and generally applicable simulation approach to correct for this bias in our ALDH2 data and, also, to explore the effect of bias on the commonly used summary statistics Tajima’s D, Fu and Li’s D, and Fay and Wu’s H. Our study finds no evidence that the pattern of genetic variation at ALDH2 differs from that expected under a standard neutral model. However, our general examination of ascertainment bias indicates that a priori knowledge of segregating alleles greatly affects the expected distributions of summary statistics. Under many parameter combinations we find that ascertainment bias introduces an elevated rate of false positives when summary statistics are used to test for deviations from a standard neutral model. However, we also show that over a wide range of conditions the power of all summary statistics can be greatly increased by incorporating prior knowledge of segregating alleles. [Reviewing Editor: Dr. Martin Kreitman]     

Mapping viability loci using molecular markers

In genetic mapping experiments, some molecular markers often show distorted segregation ratios. We hypothesize that these markers are linked to some viability loci that cause the observed segregation ratios to deviate from Mendelian expectations. Although statistical methods for mapping viability loci have been developed for line-crossing experiments, methods for viability mapping in outbred populations have not been developed yet. In this study, we develop a method for mapping viability loci in outbred populations using a full-sib family as an example. We develop a maximum likelihood (ML) method that uses the observed marker genotypes as data and the proportions of the genotypes of the viability locus as parameters. The ML solutions are obtained via the expectation-maximization algorithm. Application and efficiencies of the method are demonstrated and tested using a set of simulated data. We conclude that mapping viability loci can be accomplished using similar statistical techniques used in quantitative trait locus mapping for quantitative traits.     

Testing for differences in survival with delayed ascertainment

During the interim stages of most large-scale clinical trials, knowledge that a patient is alive or dead is usually not up-to-date. This is due to the pattern of patient visits to hospitals as well as the administrative set-up used by the study to obtain information on vital status. On a two-armed study, if the process of ascertaining vital status is not the same in both treatment groups, then the standard method of testing based on the logrank statistic may not be applicable. Instead, an ad hoc modification to the logrank test, which artificially truncates follow-up prior to the time of analysis, is often used. These approaches have not been formally addressed in the literature. In the early stages of a clinical trial, severe bias or loss of power may result. For this situation, we propose a class of test statistics that extends the usual class of U statistics. Asymptotic normality is derived by reformulating the statistics in terms of counting processes and employing the theory of U statistics along with martingale techniques. For early interim analyses, a numerical study indicates that the new tests can be more powerful than the current practice when differential ascertainment is present. To illustrate the potential loss of information when lagging follow-up to control for ascertainment delays, we reanalyze an AIDS clinical trial with the truncated logrank and the new statistics.     

Microsatellite Evolution: Testing the Ascertainment Bias Hypothesis

Previous studies suggest the median allele length of microsatellites is longest in the species from which the markers were derived, suggesting that an ascertainment bias was operating. We have examined whether the size distribution of microsatellite alleles between sheep and cattle is source dependent using a set of 472 microsatellites that can be amplified in both species. For those markers that were polymorphic in both species we report a significantly greater number of markers (P < 0.001) with longer median allele sizes in sheep, regardless of microsatellite origin. This finding suggests that any ascertainment bias operating during microsatellite selection is only a minor contributor to the variation observed. Received: 6 January 1997 / Accepted: 19 May 1997     

Multinomial Distribution and Ascertainment Models

In this paper general ascertainment models are studied relaxing the strong assumption of complete dominance. Probabilitis of ascertaiment for both the complete and incomplete models depending on family size and register size for two types of affected individuals are derived.     

Ascertainment correction for a population tree via a pruning algorithm for likelihood computation

We present a method for correcting ascertainment-bias in a coalescent-based likelihood for population trees. Our method is computationally simple and fast. To correct for the bias we compute the probability of allele-counts conditioned on the locus being included. This conditional probability is simply the uncorrected likelihood divided by the inclusion probability. A modification of a pruning algorithm is introduced so that the inclusion probability can be computed with a single run of the algorithm. Our computation is exact and avoids Monte-Carlo based methods.     

An EM algorithm for mapping binary disease loci: application to fibrosarcoma in a four-way cross mouse family

Many diseases show dichotomous phenotypic variation but do not follow a simple Mendelian pattern of inheritance. Variances of these binary diseases are presumably controlled by multiple loci and environmental variants. A least-squares method has been developed for mapping such complex disease loci by treating the binary phenotypes (0 and 1) as if they were continuous. However, the least-squares method is not recommended because of its ad hoc nature. Maximum Likelihood (ML) and Bayesian methods have also been developed for binary disease mapping by incorporating the discrete nature of the phenotypic distribution. In the ML analysis, the likelihood function is usually maximized using some complicated maximization algorithms (e.g. the Newton-Raphson or the simplex algorithm). Under the threshold model of binary disease, we develop an Expectation Maximization (EM) algorithm to solve for the maximum likelihood estimates (MLEs). The new EM algorithm is developed by treating both the unobserved genotype and the disease liability as missing values. As a result, the EM iteration equations have the same form as the normal equation system in linear regression. The EM algorithm is further modified to take into account sexual dimorphism in the linkage maps. Applying the EM-implemented ML method to a four-way-cross mouse family, we detected two regions on the fourth chromosome that have evidence of QTLs controlling the segregation of fibrosarcoma, a form of connective tissue cancer. The two QTLs explain 50-60% of the variance in the disease liability. We also applied a Bayesian method previously developed (modified to take into account sex-specific maps) to this data set and detected one additional QTL on chromosome 13 that explains another 26% of the variance of the disease liability. All the QTLs detected primarily show dominance effects.