An autologistic model for the genetic analysis of familial binary data.

Regressive logistic models specify the probability distribution of familial binary traits by conditioning each individual's phenotype on those of preceding relatives; therefore, the expression of the joint probability of the familial data necessitates ordering the observations. In the present paper, we propose an autologistic model of this familial dependence structure, which does not require specification of a particular ordering of the phenotypic observations. Genetic effects are introduced into the model in order to perform segregation analysis that is aimed at detecting the role of a major locus in the expression of familial phenotypes. In this model, the conditional probabilities have a logistic form, and large patterns of dependence between relatives can be considered with a simple interpretation of the parameters measuring the relationship between two phenotypes. The model is compared with the regressive logistic approach in terms of odds ratios and by using a simulation study.     

Genetic Segregation Analysis of Early-Onset Recurrent Unipolar Depression

Major depression is a relatively common psychiatric disorder that can be quite debilitating. Family, twin, and adoption studies indicate that unipolar depression has both genetic and environmental components. Early age at onset and recurrent episodes in the proband each increase the familiarity of the illness. To investigate the potential genetic underpinnings of the disease, we have performed a complex segregation analysis on 832 individuals from 50 multigenerational families ascertained through a proband with early-onset recurrent unipolar major depression. The analysis was conducted by use of regressive models, to test a variety of hypotheses to explain the familial aggregation of recurrent unipolar depression. Analyses were conducted under two alternative definitions of affection status for the relatives of probands: (1) "narrow," in which relatives were assumed to be affected only if they were diagnosed with recurrent unipolar depression; and (2) "broad," in which relatives were assumed to be affected if diagnosed with any major affective illness. Under the narrow-definition assumption, the model that best explains these family data is a transmitted (although non-Mendelian) recessive major effect with significant residual parental effects on affection status. Under the broad-definition assumption, the best-fitting model is a Mendelian codominant major locus with significant residual parental and spousal effects.     

A comparative method for both discrete and continuous characters using the threshold model

The threshold model developed by Sewall Wright in 1934 can be used to model the evolution of two-state discrete characters along a phylogeny. The model assumes that there is a quantitative character, called liability, that is unobserved and that determines the discrete character according to whether the liability exceeds a threshold value. A Markov chain Monte Carlo algorithm is used to infer the evolutionary covariances of the liabilities for discrete characters, sampling liability values consistent with the phylogeny and with the observed data. The same approach can also be used for continuous characters by assuming that the tip species have values that have been observed. In this way, one can make a comparative-methods analysis that combines both discrete and continuous characters. Simulations are presented showing that the covariances of the liabilities are successfully estimated, although precision can be achieved only by using a large number of species, and we must always worry whether the covariances and the model apply throughout the group. An advantage of the threshold model is that the model can be straightforwardly extended to accommodate within-species phenotypic variation and allows an interface with quantitative-genetics models.     

Regressive logistic models for familial disease and other binary traits

The simple Markovian structures of dependence, defined previously for continuous traits, are extended here to familial disease and other binary traits through the use of the logistic function. The regressive models so formulated can incorporate explanatory variables and major gene effects for segregation and linkage analyses. Thus, the goals of epidemiology and genetics in the analysis of familial disease can be accomplished in the same computational scheme.     

Mapping Quantitative Trait Loci for Complex Binary Diseases Using Line Crosses

A composite interval gene mapping procedure for complex binary disease traits is proposed in this paper. The binary trait of interest is assumed to be controlled by an underlying liability that is normally distributed. The liability is treated as a typical quantitative character and thus described by the usual quantitative genetics model. Translation from the liability into a binary (disease) phenotype is through the physiological threshold model. Logistic regression analysis is employed to estimate the effects and locations of putative quantitative trait loci (our terminology for a single quantitative trait locus is QTL while multiple loci are referred to as QTLs). Simulation studies show that properties of this mapping procedure mimic those of the composite interval mapping for normally distributed data. Potential utilization of the QTL mapping procedure for resolving alternative genetic models (e.g., single- or two-trait-locus model) is discussed.     

Segregation analysis comparing liability and quantitative trait models for hypertension using the Genetic Analysis Workshop 13 simulated data

Discrete (qualitative) data segregation analysis may be performed assuming the liability model, which involves an underlying normally distributed quantitative phenotype. The appropriateness of the liability model for complex traits is unclear. The Genetic Analysis Workshop 13 simulated data provides measures on systolic blood pressure, a highly complex trait, which may be dichotomized into a discrete trait (hypertension). We perform segregation analysis under the liability model of hypertensive status as a qualitative trait and compare this with results using systolic blood pressure as a quantitative trait (without prior knowledge at that stage of the true underlying simulation model) using 1050 pedigrees ascertained from four replicates on the basis of at least one affected member. Both analyses identify models with major genes and polygenic components to explain the family aggregation of systolic blood pressure. Neither of the methods estimates the true parameters well (as the true model is considerably more complicated than those considered for the analysis), but both identified the most complicated model evaluated as the preferred model. Segregation analysis of complex diseases using relatively simple models is unlikely to provide accurate parameter estimates but is able to indicate major gene and/or polygenic components in familial aggregation of complex diseases.     

A parametric copula model for analysis of familial binary data

Modeling the joint distribution of a binary trait (disease) within families is a tedious challenge, owing to the lack of a general statistical model with desirable properties such as the multivariate Gaussian model for a quantitative trait. Models have been proposed that either assume the existence of an underlying liability variable, the reality of which cannot be checked, or provide estimates of aggregation parameters that are dependent on the ordering of family members and on family size. We describe how a class of copula models for the analysis of exchangeable categorical data can be incorporated into a familial framework. In this class of models, the joint distribution of binary outcomes is characterized by a function of the given marginals. This function, referred to as a "copula," depends on an aggregation parameter that is weakly dependent on the marginal distributions. We propose to decompose a nuclear family into two sets of equicorrelated data (parents and offspring), each of which is characterized by an aggregation parameter (alphaFM and alphaSS, respectively). The marginal probabilities are modeled through a logistic representation. The advantage of this model is that it provides estimates of the aggregation parameters that are independent of family size and does not require any arbitrary ordering of sibs. It can be incorporated easily into segregation or combined segregation-linkage analysis and does not require extensive computer time. As an illustration, we applied this model to a combined segregation-linkage analysis of levels of plasma angiotensin I-converting enzyme (ACE) dichotomized into two classes according to the median. The conclusions of this analysis were very similar to those we had reported in an earlier familial analysis of quantitative ACE levels.     

The multifactorial/threshold concept -- uses and misuses.

The common congenital malformations have familial distributions that cannot be accounted for by simple Mendelian models, but can be explained in terms of a continuous variable, "liability," with a threshold value beyond which individuals will be affected. Both genetic and environmental factors determine liability, making the system multifactorial. Cleft palate is a useful experimental model, illustrating a number of factors that contribute to palate closure, the nature of a developmental threshold, and how genes and teratogens can alter the components of liability to increase the probability of cleft palate. The nature of the genetic component to liability in human malformations in not clear, and various possibilities, ranging from polygenic in the strict sense to a major gene with reduced penetrance are compatible with the data -- but the important feature is the threshold. Much of the confusion over the concept results from inconsistent use of terminology. The term "multifactorial" should be used for "determined by a combination of genetic and environmental factors," without reference to the nature of the genetic factor(s). "Polygenic" should be reserved for "a large number of genes, each with a small effect, acting additively." When several genes, with more major effects are involved, "multilocal" can be used. When it is not clear which of these is applicable the term "plurilocal" is suggested, in the sense of "genetic variation more complex than a simple Mendelian difference." Since teratological data often represent threshold characters the concept also has important implications for the interpretation of data on dose-response curves, synergisms, and strain differences in response to teratogens.     

Complex segregation analysis of nonsyndromic cleft lip and palate.

This study was undertaken to examine the inheritance pattern of nonsyndromic cleft lip with or without cleft palate (CL/P). Complex segregation analysis using the unified model as in POINTER and the regressive model as in REGD programs were applied to analyze a midwestern U.S. Caucasian population of 79 families ascertained through a proband with CL/F. In REGD, the dominant or codominant Mendelian major locus models of inheritance were the most parsimonious fit. In POINTER, besides the Mendelian major locus model, the multifactorial threshold (MF/T) model and the mixed model were also consistent with the observed data. However, the high heritability parameter of .93 (SD .063) in the MF/T model suggests that any random exogenous factors are unlikely to be the underlying mechanisms, and the mixed model indicates that this high heritability is accounted for by a major dominant locus component. These findings indicate that the best explanation for the etiology of CL/P in this study population is a putative major locus associated with markedly decreased penetrance. Molecular studies may provide further insight into the genetic mechanism underlying CL/P.     

Models of comorbidity for multifactorial disorders.

We develop several formal models for comorbidity between multifactorial disorders. Based on the work of D. N. Klein and L. P. Riso, the models include (i) alternate forms, where the two disorders have the same underlying continuum of liability; (ii) random multiformity, in which affection status on one disorder abruptly increases risk for the second; (iii) extreme multiformity, where only extreme cases have an abruptly increased risk for the second disorder; (iv) three independent disorders, in which excess comorbid cases are due to a separate, third disorder; (v) correlated liabilities, where the risk factors for the two disorders correlate; and (vi) direct causal models, where the liability for one disorder is a cause of the other disorder. These models are used to make quantitative predictions about the relative proportions of pairs of relatives who are classified according to whether each relative has neither disorder, disorder A but not B, disorder B but not A, or both A and B. For illustration, we analyze data on major depression (MD) and generalized anxiety disorder (GAD) assessed in adult female MZ and DZ twins, which enable estimation of the relative impact of genetic and environmental factors. Several models are rejected--that comorbid cases are due to chance; multiformity of GAD; a third independent disorder; and GAD being a cause of MD. Of the models that fit the data, correlated liabilities, MD causes GAD, and reciprocal causation seem best. MD appears to be a source of liability for GAD. Possible extensions to the models are discussed.     

Logistic regression for dependent binary observations

The likelihood of a set of binary dependent outcomes, with or without explanatory variables, is expressed as a product of conditional probabilities each of which is assumed to be logistic. The models are called regressive logistic models. They provide a simple but relatively unknown parametrization of the multivariate distribution. They have the theoretical and practical advantage that they can be analyzed and fitted as in logistic regression for independent outcomes, and with the same computer programs. The paper is largely expository and is intended to motivate the development and usage of the regressive logistic models. The discussion includes serially dependent outcomes, equally predictive outcomes, more specialized patterns of dependence, multidimensional tables, and three examples.     

Complex segregation analysis of dilated cardiomyopathy (DCM) in Irish wolfhounds

The objective of the present study was to analyse the mode of inheritance for dilated cardiomyopathy (DCM) in Irish wolfhounds using regressive logistic models by testing for mechanisms of genetic transmission. Insights from this spontaneous animal model should aid importantly in understanding basic pathogenic mechanisms with regard to genetics and molecular biology of DCM in humans. Moreover, a procedure for the simultaneous prediction of breeding values and the estimation of genotype probabilities for DCM is expected to markedly improve breeding programmes. Results of cardiovascular examinations of 1018 dogs carried out between 1987 and 2003 by one veterinarian were analysed. Data of 878 dogs from 531 litters in 147 different kennels were used for complex segregation analyses. Pedigree information was available for more than 15 generations. Male dogs were affected significantly more often by DCM than female dogs. The segregation analysis showed that among all other tested models a mixed monogenic-polygenic model including a sex-dependent allele effect best explained the segregation of affected animals in the pedigrees. A pure monogenic inheritance of DCM could be significantly rejected in favour of the major gene and most general model. The gene action of the major gene was significantly different between female and male dogs.     

ANCESTRAL CHARACTER ESTIMATION UNDER THE THRESHOLD MODEL FROM QUANTITATIVE GENETICS

Evolutionary biology is a study of life's history on Earth. In researching this history, biologists are often interested in attempting to reconstruct phenotypes for the long extinct ancestors of living species. Various methods have been developed to do this on a phylogeny from the data for extant taxa. In the present article, I introduce a new approach for ancestral character estimation for discretely valued traits. This approach is based on the threshold model from evolutionary quantitative genetics. Under the threshold model, the value exhibited by an individual or species for a discrete character is determined by an underlying, unobserved continuous trait called “liability.” In this new method for ancestral state reconstruction, I use Bayesian Markov chain Monte Carlo (MCMC) to sample the liabilities of ancestral and tip species, and the relative positions of two or more thresholds, from their joint posterior probability distribution. Using data simulated under the model, I find that the method has very good performance in ancestral character estimation. Use of the threshold model for ancestral state reconstruction relies on a priori specification of the order of the discrete character states along the liability axis. I test the use of a Bayesian MCMC information theoretic criterion based approach to choose among different hypothesized orderings for the discrete character. Finally, I apply the method to the evolution of feeding mode in centrarchid fishes.     

Ionizing radiation and genetic risks. V. Multifactorial diseases: A review of epidemiological and genetic aspects of congenital abnormalities in man and of models on maintenance of quantitative traits in populations

This paper discusses (a) data on the epidemiological and etiological aspects of human congenital abnormalities, (b) the multifactorial threshold model and other models which have been proposed to explain their inheritance patterns and recurrence risks in families and (c) current concepts on mechanisms on the prevalence of heritable variation for quantitative traits in populations.Congenital abnormalities, which afflict an estimated 6% of all live births, are etiologically heterogeneous. The majority of these do not follow Mendelian transmission patterns, but do ‘run’ in families. The multifactorial threshold model is an extension of genetic principles developed for quantitative traits to all-or-none traits; in its simplest formulation, it assumes the existence in the population of an underlying normally distributed ‘liability’ (which is due to numerous genetic and environmental factors acting additively, each contributing a small amount of liability) and of a ‘threshold’ beyond which the individual is affected. For most congenital abnormalities, the nature of these factors remains unknown. Other models assume fewer causal factors although, again, these remain to be identified.The question of how considerable heritable variation for most quantitative / polygenic traits has come to exist is a long-standing one in evolutionary population genetics. Models postulating that its existence is consistent with a balance between recurrent mutation and stabilizing selection or suggesting the possible operation of other mechanisms have been published in the literature.     

Compound regressive models for family data.

The regressive models for the analysis of family data are extended to include cases in which the within-sibship covariation may exceed that implied by the class A regressive model, but for which birth order is not required. In addition to specified major genes, if any, and common parental phenotypes, the excess within-sibship covariation may come from a common cumulative risk from unspecified factors such as a shared environment, and other genes. The within-sibship cumulative risk has a probability distribution in the population. The sib-sib correlation (more generally within-sibship statistical dependence) is equal for all pairs within a given sibship. The compound regressive model is thus a version of the class D regressive model with the property of within-sibship interchangeability. The work is motivated here by comparing and contrasting the Elston-Stewart algorithm and the Morton-MacLean algorithm for the mixed model of inheritance. This points the way to derive practical algorithms for the compound regressive models proposed, with easy extensions to pedigrees of arbitrary structure, and to multilocus problems.     

Refutation of the general single-locus model for the etiology of schizophrenia.

All published studies on the familial incidence of schizophrenia appropriate for testing the applicability of the general single-locus two-allele model are examined under the assumption of a unitary etiology for all schizophrenia. We show that the single major locus model is inadequate to predict the incidence in four classes of relatives of schizophrenic probands (parents, siblings, monozygotic, and dizygotic cotwins). In addition, the observed proportion of affected offspring from dual matings differ significantly from the model''s prediction. The lack of an overall fit between the published familial distributions and the monogenic model suggests that a single major locus is insufficient for the etiology of schizophrenia. Further efforts in examining multifactorial models, mixed models, and other transmission models may be fruitful.     

Complex Segregation Analysis of Pedigrees from the Gilda Radner Familial Ovarian Cancer Registry Reveals Evidence for Mendelian Dominant Inheritance

Background

Familial component is estimated to account for about 10% of ovarian cancer. However, the mode of inheritance of ovarian cancer remains poorly understood. The goal of this study was to investigate the inheritance model that best fits the observed transmission pattern of ovarian cancer among 7669 members of 1919 pedigrees ascertained through probands from the Gilda Radner Familial Ovarian Cancer Registry at Roswell Park Cancer Institute, Buffalo, New York.

Methodology/Principal Findings

Using the Statistical Analysis for Genetic Epidemiology program, we carried out complex segregation analyses of ovarian cancer affection status by fitting different genetic hypothesis-based regressive multivariate logistic models. We evaluated the likelihood of sporadic, major gene, environmental, general, and six types of Mendelian models. Under each hypothesized model, we also estimated the susceptibility allele frequency, transmission probabilities for the susceptibility allele, baseline susceptibility and estimates of familial association. Comparisons between models were carried out using either maximum likelihood ratio test in the case of hierarchical models, or Akaike information criterion for non-nested models. When assessed against sporadic model without familial association, the model with both parent-offspring and sib-sib residual association could not be rejected. Likewise, the Mendelian dominant model that included familial residual association provided the best-fitting for the inheritance of ovarian cancer. The estimated disease allele frequency in the dominant model was 0.21.

Conclusions/Significance

This report provides support for a genetic role in susceptibility to ovarian cancer with a major autosomal dominant component. This model does not preclude the possibility of polygenic inheritance of combined effects of multiple low penetrance susceptibility alleles segregating dominantly.     

Evolutionary genetics of partial migration – the threshold model of migration revis(it)ed

Partial migration is a common and widespread phenomenon in animal populations. Even though the ecological causes for the evolution and maintenance of partial migration have been widely discussed, the consequences of the genetics underlying differences in migration patterns have been little acknowledged. Here, I revise current ideas on the genetics of partial migration and identify open questions, focussing on migration in birds. The threshold model of migration describing the inheritance and phenotypic expression of migratory behaviour is strongly supported by experimental results. As a consequence of migration being a threshold trait, high levels of genetic variation can be preserved, even under strong directional selection. This is partly due to strong environmental canalization. This cryptic genetic variation may explain rapid de novo evolution of migratory behaviour in resident populations and the high prevalence of partial migration in animal populations. To date the threshold model of migration has been tested only under laboratory conditions. For obtaining a more realistic representation of migratory behaviour in the wild, the simple threshold model needs to be extended by considering that the threshold of migration or the liability may be modified by environmental effects. This environmental threshold model is valid for both facultative and obligate migration movements, and identifies genetic accommodation as an important process underlying evolutionary change in migration status. Future research should aim at identifying the major environmental variables modifying migration propensity and at determining reaction norms of the threshold and liability across variation in these variables.     

Multivariate Bayesian analysis of Gaussian,right censored Gaussian,ordered categorical and binary traits using Gibbs sampling

A fully Bayesian analysis using Gibbs sampling and data augmentation in a multivariate model of Gaussian, right censored, and grouped Gaussian traits is described. The grouped Gaussian traits are either ordered categorical traits (with more than two categories) or binary traits, where the grouping is determined via thresholds on the underlying Gaussian scale, the liability scale. Allowances are made for unequal models, unknown covariance matrices and missing data. Having outlined the theory, strategies for implementation are reviewed. These include joint sampling of location parameters; efficient sampling from the fully conditional posterior distribution of augmented data, a multivariate truncated normal distribution; and sampling from the conditional inverse Wishart distribution, the fully conditional posterior distribution of the residual covariance matrix. Finally, a simulated dataset was analysed to illustrate the methodology. This paper concentrates on a model where residuals associated with liabilities of the binary traits are assumed to be independent. A Bayesian analysis using Gibbs sampling is outlined for the model where this assumption is relaxed.