Combining gene expression and molecular marker information for mapping complex trait genes: a simulation study

A method for mapping complex trait genes using cDNA microarray and molecular marker data jointly is presented and illustrated via simulation. We introduce a novel approach for simulating phenotypes and genotypes conditionally on real, publicly available, microarray data. The model assumes an underlying continuous latent variable (liability) related to some measured cDNA expression levels. Partial least-squares logistic regression is used to estimate the liability under several scenarios where the level of gene interaction, the gene effect, and the number of cDNA levels affecting liability are varied. The results suggest that: (1) the usefulness of microarray data for gene mapping increases when both the number of cDNA levels in the underlying liability and the QTL effect decrease and when genes are coexpressed; (2) the correlation between estimated and true liability is large, at least under our simulation settings; (3) it is unlikely that cDNA clones identified as significant with partial least squares (or with some other technique) are the true responsible cDNAs, especially as the number of clones in the liability increases; (4) the number of putatively significant cDNA levels increases critically if cDNAs are coexpressed in a cluster (however, the proportion of true causal cDNAs within the significant ones is similar to that in a no-coexpression scenario); and (5) data reduction is needed to smooth out the variability encountered in expression levels when these are analyzed individually.     

Joint Multipoint Linkage Analysis of Multivariate Qualitative and Quantitative Traits. I. Likelihood Formulation and Simulation Results

We describe a variance-components method for multipoint linkage analysis that allows joint consideration of a discrete trait and a correlated continuous biological marker (e.g., a disease precursor or associated risk factor) in pedigrees of arbitrary size and complexity. The continuous trait is assumed to be multivariate normally distributed within pedigrees, and the discrete trait is modeled by a threshold process acting on an underlying multivariate normal liability distribution. The liability is allowed to be correlated with the quantitative trait, and the liability and quantitative phenotype may each include covariate effects. Bivariate discrete-continuous observations will be common, but the method easily accommodates qualitative and quantitative phenotypes that are themselves multivariate. Formal likelihood-based tests are described for coincident linkage (i.e., linkage of the traits to distinct quantitative-trait loci [QTLs] that happen to be linked) and pleiotropy (i.e., the same QTL influences both discrete-trait status and the correlated continuous phenotype). The properties of the method are demonstrated by use of simulated data from Genetic Analysis Workshop 10. In a companion paper, the method is applied to data from the Collaborative Study on the Genetics of Alcoholism, in a bivariate linkage analysis of alcoholism diagnoses and P300 amplitude of event-related brain potentials.     

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.     

An argument for the use of total side frequencies of bilateral nonmetric skeletal traits in population distance analysis: The regression of symmetry on incidence

In population studies based on frequencies of bilateral nonmetric skeletal traits, the choice between sampling by individuals or by sides should depend less on the exigencies imposed by fragmentary remains than on fundamental assumptions about the biological meaning of symmetry/asymmetry. Though the latter has been interpreted in various ways, little attention has focused on the possibility that bilateral correlation is meaningful in quantifying genetic liability for a trait. Analysis of two independent mandibular features, mylohyoid bridge and suppressed third molar, in Indian and Eskimo population samples (total N ? 1,200) reveals a statistically significant pattern of increasing bilateral occurrence with increasing population incidence. This pattern is consistent with the theory that liability for a “quasi-continuous” variant is normally distributed with constant increment between thresholds on an underlying scale. According to theory, phenotypes with more pronounced expression (bilateral occurrence) have greater genetic potential than those with less pronounced expression (unilateral occurrence). Therefore, scoring traits in total left and right sides, by giving greater weight to bilaterally affected individuals, may provide a better estimate of the liability for the trait in the population. Viewed in a theoretical context broader than that of the sampling debate, this pattern of positive regression of symmetry on incidence means that prevalence of unilateral occurrence probably cannot be used to assess the relative strength of genetic versus nongenetic control of threshold variants.     

How meaningful are heritability estimates of liability?

It is commonly acknowledged that estimates of heritability from classical twin studies have many potential shortcomings. Despite this, in the post-GWAS era, these heritability estimates have come to be a continual source of interest and controversy. While the heritability estimates of a quantitative trait are subject to a number of biases, in this article we will argue that the standard statistical approach to estimating the heritability of a binary trait relies on some additional untestable assumptions which, if violated, can lead to badly biased estimates. The ACE liability threshold model assumes at its heart that each individual has an underlying liability or propensity to acquire the binary trait (e.g., disease), and that this unobservable liability is multivariate normally distributed. We investigated a number of different scenarios violating this assumption such as the existence of a single causal diallelic gene and the existence of a dichotomous exposure. For each scenario, we found that substantial asymptotic biases can occur, which no increase in sample size can remove. Asymptotic biases as much as four times larger than the true value were observed, and numerous cases also showed large negative biases. Additionally, regions of low bias occurred for specific parameter combinations. Using simulations, we also investigated the situation where all of the assumptions of the ACE liability model are met. We found that commonly used sample sizes can lead to biased heritability estimates. Thus, even if we are willing to accept the meaningfulness of the liability construct, heritability estimates under the ACE liability threshold model may not accurately reflect the heritability of this construct. The points made in this paper should be kept in mind when considering the meaningfulness of a reported heritability estimate for any specific disease.     

QTL mapping of complex binary traits in an advanced intercross line

An advanced intercross line (AIL) is an easier and more cost-effective approach compared to recombinant inbred lines for fine mapping of quantitative trait loci (QTL) identified by F(2) designs. In an AIL, a complex binary trait can be mapped through analysis of either continuously distributed proxy traits for the liability of the binary trait or the liability itself, the latter presenting the greater statistical challenge. In another work, we successfully applied both approaches in an AIL to fine map previously identified QTL underlying anatomical parameters of the cardiac inter-atrial septum including patent foramen ovale. Here, we describe the statistical methods that we used to analyse complex binary traits in our AIL design. This is achieved using a likelihood-based method, with the expectation-maximisation algorithm allowing use of standard logistic regression methods for model fitting.     

Mapping quantitative trait loci for binary trait in the F2:3 design

In the analysis of inheritance of quantitative traits with low heritability, an F_2:3 design that genotypes plants in F₂ and phenotypes plants in F_2:3 progeny is often used in plant genetics. Although statistical approaches for mapping quantitative trait loci (QTL) in the F_2:3 design have been well developed, those for binary traits of biological interest and economic importance are seldom addressed. In this study, an attempt was made to map binary trait loci (BTL) in the F_2:3 design. The fundamental idea was: the F₂ plants were genotyped, all phenotypic values of each F_2:3 progeny were measured for binary trait, and these binary trait values and the marker genotype informations were used to detect BTL under the penetrance and liability models. The proposed method was verified by a series of Monte-Carlo simulation experiments. These results showed that maximum likelihood approaches under the penetrance and liability models provide accurate estimates for the effects and the locations of BTL with high statistical power, even under of low heritability. Moreover, the penetrance model is as efficient as the liability model, and the F_2:3 design is more efficient than classical F₂ design, even though only a single progeny is collected from each F_2:3 family. With the maximum likelihood approaches under the penetrance and the liability models developed in this study, we can map binary traits as we can do for quantitative trait in the F_2:3 design.     

THE EVOLUTION OF THRESHOLD TRAITS: A QUANTITATIVE GENETIC ANALYSIS OF THE PHYSIOLOGICAL AND LIFE-HISTORY CORRELATES OF WING DIMORPHISM IN THE SAND CRICKET

Many traits are phenotypically discrete but polygenically determined. Such traits can be understood using the threshold model of quantitative genetics that posits a continuously distributed underlying trait, called the liability, and a threshold of response, individuals above the threshold displaying one morph and individuals below the threshold displaying the alternate morph. For many threshold traits the liability probably consists of a hormone or a suite of hormones. Previous experiments have implicated juvenile hormone esterase (JHE), a degratory enzyme of juvenile hormone, as a physiological determinant of wing dimorphism in the crickets Gryllus rubens and G. firmus. The present study uses a half-sib experiment to measure the heritability of JHE in the last nymphal stadium of G. firmus and its genetic correlation with fecundity, a trait that is itself genetically correlated with wing morph. The phenotypic and genetic parameters are consistent with the hypothesis that JHE is a significant component of the liability. Comparison of sire and dam estimates suggest that nonadditive effects may be important. Two models have been proposed to account for the fitness differences between morphs: the dichotomy model, which assumes that each morph can be characterized by a particular suite of traits, and the continuous model, which assumes that the associated fitness traits are correlated with the liability rather than the morphs themselves. The latter model predicts that the fitness differences will not be constant but change with the morph frequencies. Variation in fecundity and flight muscle histolysis are shown to be more consistent with the continuous model. Data from the present experiment on JHE are inconclusive, but results from a previous selection experiment also suggest that variation in JHE is consistent only with the continuous model.     

Genetic analysis of idiopathic hemochromatosis using both qualitative (disease status) and quantitative (serum iron) information.

An ongoing family study of idiopathic hemochromatosis in Brittany, France, allowed us to investigate the segregation of this trait and its linkage and association to the HLA-A locus in 147 pedigrees, comprising 1,408 individuals with over 900 characterized for relevant biological parameters and typed for HLA. The joint consideration of affection status and serum iron concentration reveals no dominance effect on the latter trait and documents the increased information afforded by the consideration of a biological correlate of liability to affection for disease exhibiting incomplete penetrance. Our overall results are in general agreement with published results on a Utah family study.     

Genetic analysis of the cause of exencephaly in the SELH/Bc mouse stock

A new mouse stock, SELH/Bc, having a high liability to exencephaly has been developed. About 17% of SELH fetuses are exencephalic. The genetic cause of this exencephaly was investigated in a cross to a normal related ICR/Bc strain and in subsequent classical genetic crosses (F2, first and second backcrosses). The data were compared with a number of genetic models, including that of a single recessive mutation with 17% penetrance. The data did not fit single-locus inheritance. The expectations from the multifactorial threshold model based on an underlying quantitative liability trait with additive inheritance were found to fit the data very well. The number of loci involved was estimated to be about two or three. About 70% of exencephalic SELH fetuses are female, and there is no overall deficiency of males. The relatively higher risk in females was constant across the genetic backgrounds in the experiment. In summary, the liability to exencephaly in SELH mice appears to be a multifactorial threshold trait, and it therefore resembles human neural tube defects in type of genetic etiology. SELH therefore may be a valuable animal model in the study of neural tube defects.     

Genetic linkage analysis of a dichotomous trait incorporating a tightly linked quantitative trait in affected sib pairs

Many complex diseases are usually considered as dichotomous traits but are also associated with quantitative biological markers or quantitative risk factors. For such dichotomous traits, although their associated quantitative traits may not directly underly the diagnosis of the disease status, if the associated quantitative trait is also linked to the chromosomal regions linked to the dichotomous trait, then joint analysis of dichotomous and quantitative traits should be more efficient than consideration of them separately. Previous studies have focused on the situation when a dichotomous trait can be modeled by a threshold process acting on a single underlying normal liability distribution. However, for many complex disorders, including most psychiatric disorders, diagnosis is generally based on a set of binary or discrete criteria. These traits cannot be modeled on the basis of a threshold process acting on an underlying continuous trait. We propose a likelihood-based method that efficiently combines such a discrete trait and an associated quantitative trait in the analysis, using affected-sib-pair data. Our simulation studies suggest that joint analysis increases the power to detect linkage of dichotomous traits. We also apply the proposed new method to an asthma genome-scan data set and incorporate the total serum immunoglobulin E level in the analysis.     

Bayesian mapping of quantitative trait loci for complex binary traits

A complex binary trait is a character that has a dichotomous expression but with a polygenic genetic background. Mapping quantitative trait loci (QTL) for such traits is difficult because of the discrete nature and the reduced variation in the phenotypic distribution. Bayesian statistics are proved to be a powerful tool for solving complicated genetic problems, such as multiple QTL with nonadditive effects, and have been successfully applied to QTL mapping for continuous traits. In this study, we show that Bayesian statistics are particularly useful for mapping QTL for complex binary traits. We model the binary trait under the classical threshold model of quantitative genetics. The Bayesian mapping statistics are developed on the basis of the idea of data augmentation. This treatment allows an easy way to generate the value of a hypothetical underlying variable (called the liability) and a threshold, which in turn allow the use of existing Bayesian statistics. The reversible jump Markov chain Monte Carlo algorithm is used to simulate the posterior samples of all unknowns, including the number of QTL, the locations and effects of identified QTL, genotypes of each individual at both the QTL and markers, and eventually the liability of each individual. The Bayesian mapping ends with an estimation of the joint posterior distribution of the number of QTL and the locations and effects of the identified QTL. Utilities of the method are demonstrated using a simulated outbred full-sib family. A computer program written in FORTRAN language is freely available on request.     

Genetically determined transient edema found in the WB/ReJ mouse strain in a teratogenic survey with acetazolamide

A continuing survey of the genetic variability of different mouse strains to acetazolamide teratogenesis demonstrated the WB/ReJ strain expresses a high frequency of induced subcutaneous edema in day 15 fetuses. In treated WB/ReJ fetuses, the probability of expression of edema is independent of the expression of forelimb ectrodactyly and, with the dosage regime, there is no significant increase in acetazolamide-induced resorption. It was surprising to find a high frequency of spontaneous edema on day 15 in untreated WB/ReJ fetuses. The spontaneous edema is a transient trait with maximum expression (56%) on day 14, and it is resolved by day 17 without apparent consequence to the survival of previously affected fetuses. There is no sex dimorphism in the liability to express the transient edema. Preliminary genetic crosses to investigate the spontaneous edema were made between WB/ReJ and the C57BL/6J strain, which historically had not be observed to express spontaneous edema. A low frequency of spontaneous edema was observed on day 14 in both C57BL/6J and the reciprocal F1 fetuses. The trait is not additive because there is dominance deviation of the BC1 fetuses in the direction of the F1 fetuses. The data were fitted to a threshold model suggesting that the developmental liability to express the difference in transient edema is determined by more than one gene, but the data can be interpreted by a minimum of two loci with duplicate epistasis. The observed differences in frequencies of edema suggest the genetic model can be tested with relatively few test crosses.     

Alternative developmental pathways and the propensity to migrate: a case study in the Atlantic salmon

Migratory behaviour with its associated phenotypic changes is generally viewed as an adaptive strategy because it incurs survival or reproductive advantages to migrants. The development of a migrant phenotype is believed to be controlled by threshold mechanisms, where individuals emigrate only after surpassing a particular body size but delay migration if below. For such a strategy to respond to natural selection, part of the phenotypic variance in the propensity to migrate must be explained by variation in additive genetic effects. Here, we use data gathered in the field and from a common rearing experiment to test for a genetic basis associated with seaward migration in Atlantic salmon (Salmo salar L.). We document a high heritability of the liability trait underlying the propensity to emigrate in juvenile salmon, and significant differences between offspring grouped according to their sires in body-size threshold values above which emigration takes place. The presence of additive genetic variance in both the liability and thresholds makes the onset of migration a process sensitive to selection and may therefore constitute an important explanatory mechanism for the interpopulation differences in the size at seaward migration observed in this species.     

A recoding scheme for X-linked and pseudoautosomal loci to be used with computer programs for autosomal LOD-score analysis

We present a recoding scheme that allows for a parametric multipoint X-chromosomal linkage analysis of dichotomous traits in the context of a computer program for autosomes that can use trait models with imprinting. Furthermore, with this scheme, it is possible to perform a joint multipoint analysis of X-linked and pseudoautosomal loci. It is required that (1) the marker genotypes of all female nonfounders are available and that (2) there are no male nonfounders who have daughters in the pedigree. The second requirement does not apply if the trait locus is pseudoautosomal. The X-linked marker loci are recorded by adding a dummy allele to the males' hemizygous genotypes. For modelling an X-linked trait locus, five different liability classes are defined, in conjunction with a paternal imprinting model for male nonfounders. The formulation aims at the mapping of a diallelic trait locus relative to an arbitrary number of codominant markers with known genetic distances, in cases where a program for a genuine X-chromosomal analysis is not available.     

The evolutionary ecology of alternative migratory tactics in salmonid fishes

Extensive individual variation in spatial behaviour is a common feature among species that exhibit migratory life cycles. Nowhere is this more evident than in salmonid fishes; individual fish may complete their entire life cycle in freshwater streams, others may migrate variable distances at sea and yet others limit their migrations to larger rivers or lakes before returning to freshwater streams to spawn. This review presents evidence that individual variation in migratory behaviour and physiology in salmonid fishes is controlled by developmental thresholds and that part of the variation in proximal traits activating the development of alternative migratory tactics is genetically based. We summarize evidence that alternative migratory tactics co‐exist within populations and that all individuals may potentially adopt any of the alternative phenotypes. Even though intra‐specific genetic divergence of migratory tactics is uncommon, it may occur if female competition for oviposition sites results in spawning segregation of alternative phenotypes. Because of their polygenic nature, alternative migratory tactics are considered as threshold traits. Threshold traits have two characteristics: an underlying 'liability' trait that varies in a continuous fashion, and a threshold value which is responsible for the discreetness observed in phenotypic distribution. We review evidence demonstrating that body size is an adequate proxy for the liability trait controlling the decision to migrate, but that the same phenotypic outcome (anadromy or residency) may be reached by different developmental pathways. The evidence suggesting a significant heritable component in the development of alternative migratory tactics is subsequently reviewed, leading us to conclude that alternative migratory tactics have considerable potential to respond to selection and evolve. We review what is known about the proximal physiological mechanisms mediating the translation of the continuous value of the liability trait into a discontinuous migratory tactic. We conclude by identifying several avenues for future research, including testing the frequency‐dependent selection hypothesis, establishing the relative importance of adaptive phenotypic plasticity in explaining some geographic gradients in migratory behaviour and identifying the physiological and genetic basis of the switching mechanisms responsible for alternative migratory tactics.     

Why are rare traits unilaterally expressed?: Trait frequency and unilateral expression for cranial nonmetric traits in humans

Based on an analysis of nonmetric trait databases from several large skeletal series in Northern Europe and South America, representing 27 bilateral traits, we report a predictable relationship between the frequency of nonmetric traits and the probability that they are expressed bilaterally. In a wider sampling of traits and populations, this study thus confirms the findings of an earlier study by Ossenberg ([1981] Am. J. Phys. Anthropol. 54:471-479), which reported the same relationship for two mandibular traits. This trend was previously explained by extending the multifactorial threshold model for discontinuous traits to incorporate either separate thresholds for unilateral or bilateral expression, or by a fuzzy threshold in which the probability of bilateral expression increases away from the median threshold value. We show that the trend is produced under the standard multifactorial threshold model for discontinuous traits simply if the within-individual or developmental instability variance remains relatively constant across the range of liability. Under this assumption, the number of individuals in which one side but not the other is pushed over the threshold for trait formation will be a larger proportion of the number of individuals expressing the trait when the trait frequency is low. As trait frequency increases, the significance of within-individual variance as a determinant of trait formation decreases relative to the genetic and among-individual environmental variance. These results have implications for interpreting nonmetric trait data as well as for understanding the prevalence of unilateral vs. bilateral expression of a wide variety of discontinuous traits, including dysmorphologies in humans.     

Prediction of Selection Response for Threshold Dichotomous Traits

This paper presents a formula to predict expected response to one generation of truncation selection for a dichotomous trait under polygenic additive inheritance. The derivation relies on the threshold liability concept and on the normality assumption of the joint distribution of additive genetic values and their predictors used as selection criteria. This formula accounts for asymmetry of response when both the prevalence of the trait and the selection rate differ from 1/2 via a bivariate normal integral term. The relationship with the classical formula R = iota rho sigma G is explained with a Taylor expansion about a zero value of the correlation factor. Properties are illustrated with an example of sire selection based on progeny test performance which shows a departure from usual predictions up to 15-20% at low (0.05) or high (0.95) selection rates. Univariate approximations and extensions to several paths of genetic change are also discussed.     

Estimating missing heritability for disease from genome-wide association studies

Genome-wide association studies are designed to discover SNPs that are associated with a complex trait. Employing strict significance thresholds when testing individual SNPs avoids false positives at the expense of increasing false negatives. Recently, we developed a method for quantitative traits that estimates the variation accounted for when fitting all SNPs simultaneously. Here we develop this method further for case-control studies. We use a linear mixed model for analysis of binary traits and transform the estimates to a liability scale by adjusting both for scale and for ascertainment of the case samples. We show by theory and simulation that the method is unbiased. We apply the method to data from the Wellcome Trust Case Control Consortium and show that a substantial proportion of variation in liability for Crohn disease, bipolar disorder, and type I diabetes is tagged by common SNPs.