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.     

Segregation analysis of cryptogenic epilepsy and an empirical test of the validity of the results.

We used POINTER to perform segregation analysis of cryptogenic epilepsy in 1,557 three-generation families (probands and their parents, siblings, and offspring) ascertained from voluntary organizations. Analysis of the full data set indicated that the data were most consistent with an autosomal dominant (AD) model with 61% penetrance of the susceptibility gene. However, subsequent analyses revealed that the patterns of familial aggregation differed markedly between siblings and offspring of the probands. Risks in siblings were consistent with an autosomal recessive (AR) model and inconsistent with an AD model, whereas risks in offspring were inconsistent with an AR model and more consistent with an AD model. As a further test of the validity of the AD model, we used sequential ascertainment to extend the family history information in the subset of families judged likely to carry the putative susceptibility gene because they contained at least three affected individuals. Prevalence of idiopathic/cryptogenic epilepsy was only 3.7% in newly identified relatives expected to have a 50% probability of carrying the susceptibility gene under an AD model. Approximately 30% (i.e., 50% x 61%) were expected to be affected under the AD model resulting from the segregation analysis. These results suggest that the familial distribution of cryptogenic epilepsy is inconsistent with any conventional genetic model. The differences between siblings and offspring in the patterns of familial risk are intriguing and should be investigated further.     

Approaches to familial aggregation: hypothesis testing and estimation when families are selected through parent probands under a variant of single ascertainment

Epidemiologic approaches to testing and estimating familial aggregation of a disease consist of comparing rates of disease in relatives of individuals with the disease (known as case probands) with rates of disease in relatives of individuals without the disease (known as control probands). Gold et al. (J Am Stat Ass 1967;62: 409-420) derived an explicit mathematical model and sampling methods, under which this approach is equivalent to testing the null hypotheses that the disease risk in families is homogenous. A basic assumption of this model is that every family member has the same risk of disease and that disease status is independent among family members, although the disease risk may vary between families. When the disease is suspected of having a genetic component, rather than being purely environmental, this model has been shown to be appropriate for detecting disease aggregation in siblings, when relatives are siblings of probands. This model however is unrealistic for use in nuclear families when the affected status of offspring is not independent of the affected status of parents, and these families are selected through an affected or an unaffected parent, so that a parent is the proband and relatives are offspring of probands. We extend the Gold et al. model to allow for the disease risk in offspring to vary with the affected status of the parent. We assume that families are selected through affected and unaffected parents, under a variation of single ascertainment. Under this study design, we show that the usual test of association between affected status of probands and relatives, performed by comparing sample proportions of affected relatives of affected and unaffected probands, respectively, is no longer equivalent to a test of homogeneity of disease risk in offspring. Instead, it is equivalent to testing that the disease risk in offspring is independent of the number of affected parents. This test reduces to a test of homogeneity if and only if one assumes that the variation in disease risk in offspring, between families, is solely due to the variation in the number of affected parents. As a result, we show that under this study design, the standard chi2 test must be modified in order to obtain a valid test of familial aggregation. In addition the sample proportions of affected relatives of case and control probands, respectively, are shown to provide unbiased estimates of the expected risk of disease in an offspring given an affected/unaffected parent. We apply these results to methods of sample selection and discuss the practical implications of these findings.     

Ascertainment: An Overwiew of the Classical Segregation Analysis Model for Independent Sibships

Several different methodologies for parameter estimation under various ascertainment sampling schemes have been proposed in the past. In this article, some of the methodologies that have been proposed for independent sibships under the classical segregation analysis model are synthesized, and the general likelihoods derived for single, multiple and complete ascertainment. The issue of incorporating the sibship size distribution into the analysis is addressed, and the effect of conditioning the likelihood on the observed sibship sizes is discussed. It is shown that when the number of probands in a sibship is not specified, the corresponding likelihood can be used for a broader class of ascertainment schemes than is subsumed by the classical model.     

Ascertainment adjustment in genetic studies of ordinal traits

Most genetic studies recruit high risk families and the discoveries are based on non-random selected groups. We must consider the consequences of this ascertainment process in order to apply the results of genetic research to the general population. In previous reports, we developed a latent variable model to assess the familial aggregation and inheritability of ordinal-scaled diseases, and found a major gene component of alcoholism after applying the model to the data from the Yale family study of comorbidity of alcoholism and anxiety (YFSCAA). In this report, we examine the ascertainment effects on parameter estimates and correct potential bias in the latent variable model. The simulation studies for various ascertainment schemes suggest that our ascertainment adjustment is necessary and effective. We also find that the estimated effects are relatively unbiased for the particular ascertainment scheme used in the YFSCAA, which assures the validity of our earlier conclusion.     

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.     

Empirical, free of genetic model, estimation of recurrence risks in multifactorial diseases: conditional probability approach]

Conditional probability approach in estimation of recurrence risks in sibships of different parental phenotypic matings with the different set of affected and normal siblings is considered. The formulae are presented for calculation of recurrence risks in cases of equal and different susceptibility of two sexes under different ways of sampling of family data: direct selection of offsprings through the parents; indirect selection of offsprings through affected siblings--the probands, under different ascertainment probability--from pi = 1.0 ("exhaustive selection") up to pi----0 ("single selection"); for the case of different susceptibility of the two sexes a possibility of the differences in the ascertainment probabilities of men (pi m) and of women (pi w) is allowed, unlike "independent ascertainment model", which requires the constancy of pi. The case of multiple incompatible subforms is considered for estimation of the recurrence risks of the specified subforms. The methods of the risks estimation proposed are free of genetic models being universal both for classical mendelian traits (with the constant risks) and for multifactorial ones (with variable risks).     

Genetic epidemiology of vitiligo: multilocus recessivity cross-validated.

Vitiligo is a dermatological disorder characterized by hypopigmentary patches that tend to become progressive over time. There are reports of extensive familial aggregation. A genetic model for this disorder was earlier proposed by us. This model postulates that recessive alleles at multiple unlinked autosomal loci interact epistatically in the pathogenesis of vitiligo. The present family study was primarily undertaken to cross-validate the proposed genetic model. Data on 194 families from the United States were collected. Each family was ascertained through an affected proband. Analyses of these data reveal that approximately 20% of probands have at least one first-degree relative afflicted with vitiligo. All types of first-degree relatives of probands show a significant risk of developing vitiligo. Results of segregation and robustness analyses reveal that the genetic model postulated by us previously is the most parsimonious model for the present family data set.     

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.     

A segregation and linkage study of classical and nonclassical 21-hydroxylase deficiency.

The segregation of classical and nonclassical 21-hydroxylase deficiency (21-OHD) and its linkage to HLA-B was investigated in 220 families. First, the surprisingly high frequency of the nonclassical 21-OHD gene estimated elsewhere was confirmed using a different methodology which avoided particular assumptions concerning the classification of an individual''s genotype. In the present study the gene frequency was found to be .103 +/- .020 in an ethnically pooled sample and was as high as .223 +/- .062 among Ashkenazi Jews. Second, the segregation analysis of families ascertained through a nonclassical 21-OHD proband and those ascertained through a classical 21-OHD proband showed essentially identical results. A partial recessive model with no recombination between 21-OHD and HLA-B fitted the data better than did a complete recessive model with approximately 0.5% recombination between 21-OHD and HLA-B. The support for the partial over the complete recessive model depended on the assumed ascertainment probability, an unknown parameter in these data. Four families provided most of the evidence against the complete recessive model. All these included an unaffected sib who shared both HLA-B specificities in common with the affected proband. Possible explanations for the condition in these families include recombination, gene conversion, mutation in one of the parental gametes, or technical errors.     

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.     

Putative non-Mendelian transmission of retinoblastoma in males: a phenotypic segregation analysis of 150 pedigrees

In a previous genotypic study of eight families, we discribed paternal segregation distortion favoring the transmission of mutant alleles at the retinoblastoma gene locus (RB1). In the current study, we reviewed all published retinoblastoma pedigrees with defined ascertainment (n = 150), to determine whether the phenotypic segregation frequency at the RB1 locus is in general influenced by the sex of the transmitting parent. Segregation analysis under complete ascertainment revealed that 49.1% of the offspring of male transmitters were affected, while 44.3% of the offspring of female transmitters were affected. While this difference is not statistically significant, it is consistent with the previous findings. No significant sex distortion could be detected among the progeny of carrier fathers and mothers. In order to quantify the transmission ratio more precisely further prospective molecular genetic analysis is warranted. We propose a biological mechanism to account for a putative segregation distortion, namely that genetic recombination creates clones of spermatogonia that are homozygous for the mutant RB1 allele leading to a non-Mendelian ratio of sperm. This model can be experimentally tested using amplification of DNA from single sperm cells.     

Inheritance of acute appendicitis: familial aggregation and evidence of polygenic transmission.

We explored familiality as well as the heritability and possible mode(s) of inheritance of acute appendicitis in childhood and early adolescence. Our case-control study showed that a positive family history for reported appendectomy was significantly more frequent in families of 80 consecutive patients eventually proved to have histopathologic acute appendicitis than in families of surgical controls matched for sex, age, and number of siblings. The relative risk was 10.0 (95% confidence limits 4.7-21.4). The pattern of familial aggregation was further supported by the fact that the age-standardized morbidity ratio was four times greater among family members of cases than among controls. We then applied the unified mixed model of segregation analysis, as implemented in the computer program POINTER, to a new set of 100 multigenerational pedigrees of children with histopathologically confirmed acute appendicitis that were broken down into 674 nuclear families. Age-specific morbidity risk and lifetime incidence of acute appendicitis were estimated from relatives of controls matched for age and sex to probands. Complex segregation analysis supported a polygenic or multifactorial model with a total heritability of 56%. There was no evidence to support a major gene, although a rare gene could not be ruled out as the cause of a small proportion of cases. Specific studies to address genetic and environmental factors in this serious disease seem worthwhile; but, for now, a positive family history of appendicitis might join other evidence leading to improved clinical recognition of acute appendicitis.     

Developmental dyslexia--recurrence risk estimates from a german bi-center study using the single proband sib pair design

OBJECTIVE: Several studies have demonstrated a genetic component for dyslexia. However, both segregation and linkage analyses show contradictory results pointing at the necessity of an optimal ascertainment scheme for molecular genetic studies. Previously, we have argued that the single proband sib pair design (SPSP) would be optimal. The aims of this paper therefore are to demonstrate the practicability of the SPSP design and the estimation of recurrence risks for reading and writing. METHODS: We assessed spelling and reading in a family sample ascertained through the SPSP design. 287 families with at least two siblings and their parents were recruited. At least one child was affected with spelling disorder according to a one standard deviation (1SD) discrepancy criterion. RESULTS: Mean values for probands and their siblings were different for both the spelling and the reading phenotype. For the probands, variances of the phenotype spelling were smaller. These effects became stronger with more extreme selection criteria. Both siblings fulfilled the 1SD criterion for spelling and reading in 60.3 and 28.9% of the families, respectively, indicating a low cost efficiency of the double proband sib pair approach. A recurrence risk of 4.52 (CI: 4.07-4.93) was obtained for spelling when the 1SD criterion was applied to both siblings. Recurrence risk estimates were similar for reading. CONCLUSION: The study demonstrates the suitability of the SPSP design for genetic analysis of dyslexia. The recurrence risk estimates may be used for determining sample sizes in gene mapping studies.     

The effects of a known family-size distribution on the estimation of genetic parameters.

We consider the question: In a segregation analysis, can knowledge of the family-size distribution (FSD) in the population from which a sample is drawn improve the estimators of genetic parameters? In other words, should one incorporate the population FSD into a segregation analysis if one knows it? If so, then under what circumstances? And how much improvement may result? We examine the variance and bias of the maximum likelihood estimators both asymptotically and in finite samples. We consider Poisson and geometric FSDs, as well as a simple two-valued FSD in which all families in the population have either one or two children. We limit our study to a simple genetic model with truncate selection. We find that if the FSD is completely specified, then the asymptotic variance of the estimator may be reduced by as much as 5%-10%, especially when the FSD is heavily skewed toward small families. Results in small samples are less clear-cut. For some of the simple two-valued FSDs, the variance of the estimator in small samples of one- and two-child families may actually be increased slightly when the FSD is included in the analysis. If one knows only the statistical form of the FSD, but not its parameter, then the estimator is improved only minutely. Our study also underlines the fact that results derived from asymptotic maximum likelihood theory do not necessarily hold in small samples. We conclude that in most practical applications it is not worth incorporating the FSD into a segregation analysis. However, this practice may be justified under special circumstances where the FSD is completely specified, without error, and the population consists overwhelmingly of small families.     

Evaluating genetic heterogeneity in complex disorders

OBJECTIVES: The Admixture test is routinely used in linkage analysis to take account of genetic heterogeneity, and yields an estimate of the proportion of families (alpha) segregating the linked disease gene. In complex disorders, the assumptions of the Admixture test are violated. We therefore explore how the estimate of alpha relates to the true proportion of linked families with a complex disorder in a population or dataset. METHODS: We simulated a two-locus heterogeneity model and varied genetic parameters, ascertainment scheme and phenocopy frequency. RESULTS: In this model, alpha is almost always overestimated, by as little as 5% to as much as 60%. The bias is largely attributable to (1). intrafamilial heterogeneity arising from ascertainment of families with many affected members or from analysis of dense pedigrees; (2). low informativeness, which occurs in the presence of reduced penetrance; and (3). differences in the evidence for linkage in linked and unlinked families. This bias is also affected by the analysis phenocopy frequency, but only if the linked locus is dominant and the unlinked locus is recessive. CONCLUSIONS: We conclude that, in complex diseases, the Admixture test has greater value in detecting linkage than in estimating the proportion of linked families in a dataset.     

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.     

A major gene for primary hypoalphalipoproteinemia.

Sixteen kindreds were ascertained through probands clinically determined to have primary hypoalphalipoproteinemia, characterized by bottom decile high-density lipoprotein cholesterol (HDL-c), but otherwise normolipidemic. Age- and sex-adjusted, standardized HDL-c levels on 64 individuals in 14 nuclear families in which the proband was a parent were analyzed using the unified mixed model of segregation analysis as implemented in the computer program POINTER. The analysis proceeded by using the likelihood of offspring conditional on the parental phenotypes (conditional likelihood), which appears to overcome the limitation of possible heterogeneity in the selection criteria and provides an appropriate correction for the ascertainment. In these families, the multifactorial contribution to the phenotype appears to be small and significant only in the offspring generation. Although it was not possible to resolve the dominance pattern at the major locus since none of a recessive, additive, or dominant hypothesis could be firmly rejected, these families provided clear evidence for a major gene. Genetic heterogeneity is still a possibility, even within "primary" hypoalphalipoproteinemia.     

Evidence for major gene inheritance of Alzheimer disease in families of patients with and without apolipoprotein E epsilon 4.

Apolipoprotein E (APOE) genotype is the single most important determinant to the common form of Alzheimer disease (AD) yet identified. Several studies show that family history of AD is not entirely accounted for by APOE genotype. Also, there is evidence for an interaction between APOE genotype and gender. We carried out a complex segregation analysis in 636 nuclear families of consecutively ascertained and rigorously diagnosed probands in the Multi-Institutional Research in Alzheimer Genetic Epidemiology study in order to derive models of disease transmission which account for the influences of APOE genotype of the proband and gender. In the total group of families, models postulating sporadic occurrence, no major gene effect, random environmental transmission, and Mendelian inheritance were rejected. Transmission of AD in families of probands with at least one epsilon 4 allele best fit a dominant model. Moreover, single gene inheritance best explained clustering of the disorder in families of probands lacking epsilon 4, but a more complex genetic model or multiple genetic models may ultimately account for risk in this group of families. Our results also suggest that susceptibility to AD differs between men and women regardless of the proband's APOE status. Assuming a dominant model, AD appears to be completely penetrant in women, whereas only 62%-65% of men with predisposing genotypes develop AD. However, parameter estimates from the arbitrary major gene model suggests that AD is expressed dominantly in women and additively in men. These observations, taken together with epidemiologic data, are consistent with the hypothesis of an interaction between genes and other biological factors affecting disease susceptibility.