Statistical Methods in GeneticEpidemiology

Abstract

Genetic epidemiology combines the scientific disciplines of human genetics, epidemiology, and biostatistics and has close relationships with the fields of medicine, molecular genetics, and molecular epidemiology. The latter traditionally has been concerned more with the study of molecular markers of exposure, susceptibility, and disease (see chapter Molecular Epidemiology of this handbook). The field is also a specialized subdiscipline of biometry and mathematical population genetics with major biometrical contributions to human genetics and the development of statistical methods including segregation, linkage, and association analysis; simulation methods; and computer algorithms. Rather than focusing on cells or molecules (as in molecular genetics) or on individual patients (as in clinical genetics), genetic epidemiology research is conducted using populations or large series of systematically collected families (Khoury et al. 1993).

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1 Introduction

Genetic epidemiology combines the scientific disciplines of human genetics, epidemiology, and biostatistics and has close relationships with the fields of medicine, molecular genetics, and molecular epidemiology. The latter traditionally has been concerned more with the study of molecular markers of exposure, susceptibility, and disease (see chapter Molecular Epidemiology of this handbook). The field is also a specialized subdiscipline of biometry and mathematical population genetics with major biometrical contributions to human genetics and the development of statistical methods including segregation, linkage, and association analysis; simulation methods; and computer algorithms. Rather than focusing on cells or molecules (as in molecular genetics) or on individual patients (as in clinical genetics), genetic epidemiology research is conducted using populations or large series of systematically collected families (Khoury et al. 1993).

Genetic epidemiology aims to detect the genetic origin of phenotypic variability in humans (Vogel 2000) and unravel genetic components that contribute to the development or the course of a disease (or more generally a phenotype, the observed trait), along with environmental or other risk factors that may modify the effects of genes. Thus, the International Society of Genetic Epidemiology (IGES 2012) describes the field as a marriage between the disciplines of genetics and epidemiology, emphasizing the need to join the fields. Whereas genetics tends to focus on the genotype-phenotype correlation neglecting the environment and epidemiology tends to focus on environmental and demographic factors, a full understanding of the etiology of complex traits can only be achieved by considering both explaining how genes are expressed in the presence of different environmental contexts and how genetic and environmental factors act together in shaping a phenotype.

In contrast to classical risk factor epidemiology, the three main complications in genetic epidemiology are dependencies, large data sets, and the use of indirect evidence. The structure of chromosomes and families or populations leads to major dependencies within the data, thus requiring customized models and tests. Modern technologies can yield millions of genotypes per subject for many thousands of subjects at an affordable cost, and even higher density sequencing platforms are now becoming available, along with a plethora of other data types (e.g., expression, proteomics) and repositories of biological knowledge (ontologies). In many studies, the disease-related functionally relevant DNA variant(s) in a gene are not directly observed, and hence the evidence on them is only indirectly given through correlated variant(s).

This chapter is solely devoted to methods dissecting the genotype-phenotype correlation with a binary phenotype (affected/unaffected). It does not specifically cover quantitative phenotypes, although many of the techniques discussed below can be applied to such problems. Section 38.2 presents an overview of major study designs and types of analysis. Section 38.3 introduces the most important genetic models. Sections 38.4–38.6 cover the three major types of analysis, segregation, linkage, and association analysis. Section 38.7 describes recent developments and looks to the future. We offer some general conclusions in Sect. 38.8. Detailed information on diseases used as examples in this chapter may be found in the standard reference of McKusick (1998) or its online version, Online Mendelian Inheritance in Man (OMIM 2012).

2 Study Types

Genetic epidemiology investigations are usually triggered by epidemiological studies that demonstrate a positive family history as a risk factor for disease, suggesting the existence of genetic or shared environmental factors. Often the goal of initial studies is to estimate the relative risk for relatives of affected individuals compared to the general population, such as λ _S in the case of siblings of affected individuals, in order to support a genetic hypothesis.

To further investigate familial aggregation, a segregation analysis may be carried out in pedigrees. This aims to determine whether a major gene is influencing a given phenotype in these families and if so to estimate the parameters of the underlying genetic model. All methods for segregation analysis are based on probability calculations for the observed phenotypes conditional on hypothetical genetic model parameters and on family structure, that is genealogies. Parameter estimation is often based on likelihood-ratio tests in order to select the most plausible model nested within a hypothetical general model. Sometimes family studies are also used solely (twin studies) or jointly with a major gene to estimate the heritability h ² of a trait, that is, the proportion of the variance explained by (additional) genetic components. Hence, λ _S or h ² are often used to indicate the genetic basis of a phenotype in a population (or enriched families) before marker studies are performed.

The primary cause of a monogenic disease such as cystic fibrosis is a mutation within a single gene that segregates according to Mendelian laws (see below). The predisposing variants (the alleles carrying the risk) of this major gene are usually rare in the population. For complex or multifactorial diseases, there may still be Mendelian subforms such as breast cancer caused by the major gene BRCA1. For rare monogenic diseases and rare Mendelian subforms of complex diseases, segregation analysis and subsequent further analyses perform well. However, complex diseases in general require more sophisticated methods of analysis. For example, in Alzheimer’s disease, there are at least three major genes and several susceptibility genes conferring moderate risk (oligogenes). Oligogenes can have relatively common alleles carrying the risk. Polygenic effects at many loci across the whole genome, each with a minor effect, may contribute to disease.

If there is evidence for the existence of genetic factors contributing to a disease, the next step is to identify susceptibility genes in order to quantify the genetic influence and to understand the underlying genetic model and pathway to the phenotype. To this end, measures of correlation between a genetic marker and the (unknown) disease locus are used. A genetic marker is a DNA segment for which the chromosomal localization is known and multiple alleles can be determined. In general, methods assume Mendelian segregation of the marker (see Sect. 38.3.2). Frequently used markers are multiallelic microsatellites and biallelic single nucleotide polymorphisms (SNPs). A marker is termed a polymorphism if the frequency of the most common variant is less than 99%.

Two types of correlation between a genetic marker and the susceptibility locus are used:

• Linkage (cosegregation at the family level): Linkage is present if the transmissions of DNA at marker and disease susceptibility loci from a parent to a child are not independent. Relatives with a similar disease status (e.g., both affected) are then more similar at a marker close to the disease susceptibility locus than expected under independence.

• Linkage disequilibrium (LD, association at the population level): LD is present if in a gamete the joint probability for a specific marker allele and a specific disease allele differs from the product of individual probabilities. In affected individuals, certain marker alleles will then be more or less frequent than in randomly selected individuals from the population.

Linkage analysis in families is based on linkage; association analysis in populations or families uses linkage disequilibrium. Some designs and corresponding statistical methods are capable of integrating both types of information into the analysis.

For the analysis of complex diseases with genetic markers, we can distinguish two major approaches: A candidate gene investigation focuses on genes (or genomic regions) whose function in the pathway to the phenotype is thought to be known. A prominent example of a candidate gene system is the HLA (human leukocyte antigen) complex on chromosome 6. HLA is involved in immune resistance and is thus a natural candidate gene region for all autoimmune diseases. The genotypes of the relevant functional component of the candidate genes are not always observed. In this case, we use the information on genetic markers that lie in or in close proximity to the candidate gene in question. In contrast, a genome scan – a systematic coarse grid search of the whole genome with genetic markers – aims to localize one or more regions harboring susceptibility genes. A typical scan might investigate approximately 350–700 microsatellites with an average distance of 5–10 cM (centiMorgan, see Sect. 38.3.3) or 500,000 or more SNPs along the genome, depending on the type of genetic information and design used (linkage or association, respectively).

3 Genetic Models

3.1 Terminology

The genome is the complete collection of an individual’s genetic material present in every cell, consisting of chromosomes (long DNA strands). A gene is a piece of a chromosome coding for a function that can be seen as the heritable unit. The locus is the position of a piece along the chromosome. The locus might denote the position of, for example, a gene, a gene complex, or a marker. The different variants of a gene are called alleles.

The human genome is diploid, that is, chromosomes are paired (homologous chromosomes) with the exception of the sex chromosomes in males. Each human somatic cell contains 22 autosomal pairs and 1 pair of sex chromosomes. In a pair, the autosomal chromosomes contain the same gene with possibly different alleles at the same location. During meiosis, a diploid chromosome set is reduced to a haploid chromosome set of a germ cell, the gamete.

A pair of an individual’s alleles at a locus is called a g enotype. If the alleles are identical, the individual is called homozygous at the locus, otherwise heterozygous. Two copies of a gene are called identical by descent (IBD) if both copies are identical and are copies of the same gene in a common ancestor. An individual is homozygous by descent (HBD) when its gene pair is IBD. When considering several loci simultaneously, the multilocus alleles inherited from the same parent constitute a haplotype.

3.2 Mendelian Single-Locus Model

Mendelian segregation (Mendel 1865) is the simplest and most commonly used model for the mode of inheritance for a single locus. An individual randomly and independently inherits one allele from father and mother, respectively. All segregation events from parents to offspring are independent. This implies that copies of some alleles are frequently present in offspring and other alleles are lost in subsequent generations, hence leading to random changes in population allele frequencies over time (genetic drift).

Consider the phenotype affected/unaffected by a certain disease. Let S denote a susceptibility gene with n alleles S ₁, S ₂, …, S _n . The distribution of allele frequencies in the population is denoted by p _r = P(S _r ) (r = 1, …, n). Under Hardy-Weinberg equilibrium (HWE) (Hardy 1908; Weinberg 1980), the (unordered) genotype frequencies are given by

$$\begin{array}{*{20}l} &&P\left (S_{r}S_{s}\right ) = p_{r}p_{s} = p_{r}^{2} & \ \ \mathrm{for}\;r = s \\ &&P\left (S_{r}S_{s}\right ) = 2p_{r}p_{s} &\ \ \mathrm{for}\;r\neq s\\ \end{array}.$$

These frequencies follow from independence of the corresponding allele frequencies, combining two ordered genotypes for heterozygotes. Its maintenance in a population can be derived by applying Mendelian segregation to each possible parental mating type, assuming random mating (Khoury et al. 1993).

Penetrance describes the relation between genotype and phenotype. It is the conditional probability that an individual will be affected given its genotype: f _rs = P(affected | S _r S _s ).

Classical monogenic diseases are those caused by a single major gene, for which the penetrances take only the values 0 or 1. Often a locus S is assumed to be biallelic, that is, to have only two different alleles. Let S ₁ denote the “susceptibility” allele (mutation) and S ₂ the “normal” allele (wild type). For a classical dominant disease, all carriers of the susceptibility allele are affected (f ₁₁ = f ₁₂ = f ₂₁ = 1, f ₂₂ = 0); for a classical recessive disease, only homozygous carriers are affected (f ₁₁ = 1, f ₁₂ = f ₂₁ = f ₂₂ = 0).

Many classical hereditary diseases follow a Mendelian mode of inheritance. Often, the prevalence is below 1 in 1,000 live births. Examples are cystic fibrosis (autosomal recessive gene, cystic fibrosis transmembrane regulator (CFTR)) and Huntington chorea (autosomal dominant gene, huntingtin (HTT)). Both diseases can be caused by any of many different mutations. However, the assumption of a gene with a normal and a susceptibility allele (group) worked well in identifying these genes as disease causes, even though the true inheritance is much more complicated. The aim in statistical genetics is not to specify the correct and complete model but to address the scientific question adequately with a parsimonious mathematical model. If this model is too simple, then more complex or new biologically motivated models need to be considered.

The genotype-phenotype relation is complicated for many diseases. Individuals with a susceptible genotype can stay unaffected (incomplete penetrance) and those with a non-susceptible genotype can become affected (phenocopies). The penetrances at a specific gene locus may be different for different alleles and may depend on age, sex, environmental exposures, or other factors. For a general single locus with susceptibility allele S ₁, we generally assume that 1 ≥ f ₁₁ ≥ f ₁₂ = f ₂₁ ≥ f ₂₂ ≥ 0 and specifically for a recessive mode of inheritance that f ₁₂ = f ₂₁ = f ₂₂ or for a dominant one that f ₁₁ = f ₁₂ = f ₂₁. It is usually assumed that the parental origin of an allele has no influence on a disease, that is, f ₁₂ = f ₂₁, although there is a growing literature on imprinting and parent-of-origin effects that violate this assumption (Zhou et al. 2010; Ainsworth et al. 2011).

3.3 Linkage

For the joint inheritance at two loci, independent Mendelian segregation does not generally hold, owing to crossover events and recombinations. Gametes (comprising one allele from each pair of chromosomes) are formed during meiosis, when homologous chromosomes are arranged next to each other and partly overlap. A chromosome breakage and a crossover – an exchange between homologous chromosomal segments – can occur. A recombination between loci A and B occurs when a gamete has a haplotype comprising a combination of alleles different from that on the same grandparental chromosome due to crossovers between the loci.

Consider the formation of gametes during meiosis displayed in Fig. 38.1. Between distant loci A and B (see Fig. 38.1a), a crossover is likely to result in a recombination of the haplotypes A₁B₁ and A₂B₂ to give the new haplotypes A₁B₂ and A₂B₁. If the two loci A and B are very close (see Fig. 38.1b), this is very unlikely. Map distance is defined as the expected number of crossovers between two loci (Haldane 1919). In expectation, the number of crossovers is roughly proportional to physical distance between two loci, so this distance measure is additive, that is, for three (ordered) loci A, B, and C, the map distance between A and C is the sum of the distances between A and B and between B and C. The map unit is called a Morgan (M), named after T.H. Morgan, a Nobel prize winning geneticist (1866–1945), who discovered the importance of chromosomes for the inheritance process. The human genome contains approximately 3.3 billion base pairs with a total length of approximately 33 M, so as a rough guide, 1 centimorgan (cM = 0. 01 M) corresponds to one million base pairs in the physical map.

Fig. 38.1

Formation of gametes during meiosis from one parental pair of chromosomes with a single crossover. Left: parental chromosome pair, middle: crossover event (crossover point denoted by the circle), right: gametes for offspring formation. At the two loci A and B, the parent is double heterozygous A₁A₂ and B₁B₂. (a) The crossover occurred between locus A and B. The two middle gametes show recombination. (b) The crossover occurred above locus A and B so that the gametes do not show recombination

By genotyping, it is possible to observe recombinations between two loci, but crossovers are not directly observable. Figure 38.1a shows recombination due to a single crossover. For a double crossover (two chromosomal exchanges between loci), no recombination would be observed. This holds true for any even number of crossovers. Thus, a recombination is defined as an uneven number of crossovers between a pair of loci. The recombination rate θ – the ratio of the number of recombinant gametes to the total number of gametes formed – is used as a measure of genetic distance between two loci. If loci are on different chromosomes or far away on the same chromosome, they segregate independently. By definition, there is linkage between the loci if 0 ≤ θ < 0. 5, and no linkage if θ = 0. 5. The closer loci are to each other, the less likely there will be crossovers and hence a recombination. Complete linkage (complete cosegregation) implies no recombination, and thus θ = 0.

In Fig. 38.2, a double heterozygous parent with haplotypes A₁B₁ and A₂B₂ and a double homozygous parent with haplotype A₃B₃ are considered. For the double heterozygous parent, a meiosis can create the non-recombinant haplotypes A₁B₁ and A₂B₂ or the recombinant haplotypes A₁B₂ and A₂B₁. In order to determine recombination, a parent homozygous even at one locus is not informative. Given that recombination is present, each of the two recombinant haplotypes occurs with probability 0.5. Given that no recombination is present, each of the two non-recombinant haplotypes occurs with probability 0.5. For θ = 0. 5, there is independent segregation so that all four possible haplotypes are equally likely.

Fig. 38.2

Formation of recombinant and non-recombinant haplotypes by meiosis

If three ordered close loci A, B, and C are considered, θ _AC ≈ θ _AB + θ _BC. In contrast to the map distance in Morgans, recombination distances are not additive. Mapping functions provide a translation of recombination distances into map distance in Morgans. In the majority of chromosomal regions, recombination rates for women are higher than for men.

The potential informativeness of a single marker chosen from an existing marker map (without consideration of the disease locus) is determined by its genetic variability, that is, allele distribution. A measure of marker informativitiy is the heterozygosityH, defined as follows (Weiss 1993; Ott 1999):

$$H =\displaystyle\sum \limits _{ r\neq s}^{n}p_{ r}p_{s}.$$

3.4 Linkage Disequilibrium

Linkage and linkage disequilibrium (LD) are different concepts. As linkage describes the coinheritance at two loci, it can only be observed in families, and it is independent of the specific alleles. LD describes the relation between alleles at two loci in a population.

Let S denote a locus with n alleles S ₁, S ₂, …, S _n and allele frequencies p _r = P(S _r ) and M a locus with m alleles M ₁, M ₂, …, M _m and allele frequencies q _i = P(M _i ). A common measure of LD is the haplotype probability minus its expectation under no association. For two biallelic loci, it is denoted by D or δ. For multiallelic markers, the parameter δ _ir is often used to define the linkage disequilibrium between M _i and S _r as

$$\delta _{\mathit{ir}} = P(M_{i},S_{r}) - P(S_{r})P(M_{i}),\qquad \qquad \mathit{i} = 1,\ldots,\mathit{m};\,\mathit{r} = 1,\ldots,n.$$

LD is present if δ _ir ≠ 0 for any pair of alleles M _i and S _r . Under LD, the allele distribution at locus M is dependent of the S allele present. Often used measures of LD are D’ (Devlin et al. 1996), defined as δ divided by its theoretical maximum for the observed allele frequencies, that is, a rescaling of δ to range between 0 and 1, and R ², the square of the correlation coefficient.

LD can arise in several different ways (Suarez and Hampe 1994). At linked loci, complete LD can be caused by a recent mutation at one locus. However, disequilibrium is also possible without linkage between the loci (the term gametic disequilibrium is preferable in this case). One important mechanism for the development of disequilibrium at unlinked loci, even on different chromosomes, is population stratification. For example, through immigration or non-random mating (e.g., by religion or social status), populations may admix with different allele distributions in the populations.

Under random mating, LD decays over generations g according to δ _g = (1 − θ)^g δ ₀, where δ ₀ is the initial LD at generation 0 (Maynard Smith 1989). Thus, whatever the origin of LD, in the presence of tight linkage, it can stay strong during many generations. Without tight linkage, LD will degrade rapidly. Thus, LD provides indirect evidence for linkage.

4 Segregation Analysis

The aim of segregation analysis is to test for the existence of a major gene influencing a phenotype and to estimate its mode of inheritance. The pattern of inheritance may be investigated in a few large families or in many small families.

Consider a Mendelian single-locus model for a major gene with susceptibility allele S ₁ and normal allele S ₂. In the classical Mendelian disease model, the penetrances P(affected | genotype) are only 0 or 1, so the genotype directly translates to a phenotype, and the families segregating the S ₁ display characteristic disease patterns.

The simplest segregation tests (see, e.g., Sham (1998)) are based on segregation ratios, the proportion of affecteds among offspring of particular parental mating types. For illustration, consider a rare autosomal dominant disease and matings between an affected and an unaffected individual. These will usually be S ₁ S ₂ ×S ₂ S ₂ matings. Let r be the observed number of affecteds among n offspring and q the probability for a child to be affected. Then the segregation ratio is the unknown parameter q of a binomial distribution with sample size n. If the null hypothesis q = 0. 5 is not rejected, it may be concluded that the data are compatible with an autosomal dominant disease pattern.

For each of six possible mating types ($S_{1}S_{1} \times S_{1}S_{1},S_{1}S_{1} \times S_{1}S_{2},S_{1}S_{1} \times S_{2}S_{2},S_{1}S_{2} \times S_{1}S_{2},S_{1}S_{2} \times S_{2}S_{2},S_{2}S_{2} \times S_{2}S_{2}$), the distribution of genotypes and phenotypes in the offspring is determined by various genetic models. However, families are often recruited non-randomly according to particular ascertainment criteria, such as enrichment for disease, yielding an oversampling for particular parental genotypes, so for a valid test, the probability distribution needs to be corrected for this ascertainment bias. For example, if all families with at least one affected offspring are recruited (truncate ascertainment), the distribution of the number of affected offspring can be corrected for ascertainment by considering a truncated binomial distribution conditioning on r ≥ 1 per family. If instead each case has an equal probability of being ascertained (single ascertainment), then multiple case families are represented proportional to the number affected and simply excluding the proband from the analysis may suffice. In practice, ascertainment schemes may be complex or unsystematic (Elston 1995), and misspecification of ascertainment might cause serious bias in the estimation of genetic parameters (see, e.g., Shute and Ewens (1988)).

For an extended pedigree with N individuals, a numerical procedure is needed. Let L denote the likelihood for the observed vector of phenotypes Y = (Y ₁, …, Y _N ), given a genetic model and the pedigree structure. L can be calculated by summing over all possible genotype vectors G = (G ₁, …, G _N ) _i , i = 1, …, N, in a given family, a particular one denoted by g = (g ₁, …, g _N). We assume that the phenotype Y _i only depends on genotype G _i of that individual i. Thus, we get

$$L\left (Y \right ) =\displaystyle\sum \limits _{g_{1}}\displaystyle\sum \limits _{g_{2}}\cdots \displaystyle\sum \limits _{g_{N}}P(Y \vert G = g)P(G = g).$$

The Elston-Stuart algorithm (Elston and Stewart 1971) provides an efficient recursive formula

$$L(Y ) =\displaystyle\sum \limits _{g_{1}}\displaystyle\sum \limits _{g_{2}}\cdots \displaystyle\sum \limits _{g_{N}}\displaystyle\prod \limits _{j=1}^{N}P(Y _{ j}\vert g_{j})\displaystyle\prod \limits _{k\,=\,1}^{N_{1} }P(g_{k})\displaystyle\prod \limits _{m\,=\,1}^{N_{2} }\tau \left (g_{m}\vert g_{m1}g_{m2}\right )$$

where N ₁ denotes the number of founders (individuals without specified parents in the pedigree, i.e., members of the oldest generation and married-in spouses) and N ₂ the number of non-founders. The parameters of the genetic model are (1) the genotype distribution P(g _k ), k = 1, …, N ₁, for the founders in the population for which HWE is usually assumed, (2) the transmission probabilities from parents m ₁ and m ₂ to offspring

$$m,\uptau(g_{m}\vert g_{m_{1}},g_{m_{2}})$$

, according to Mendelian segregation (τ(S ₁ | S ₁ S ₁) = 1; τ(S ₁ | S ₁ S ₂) = 0. 5; τ(S ₁ | S ₂ S ₂) = 0), and (3) the penetrances P(Y _j | g _j ) relating genotypes to disease. This recursive formula works well on a simple pedigree of arbitrary size. As several unrelated pedigrees are independent from each other, their likelihoods can be multiplied to yield the total likelihood of a sample of pedigrees. Computations on complex pedigrees with marriage chains and inbreeding loops (such as consanguineous marriages) are often only possible with approximation methods.

Segregation analysis works well for monogenic diseases. Due to the unclear genotype-phenotype relationship, they are much more difficult in complex diseases. Several genetic and non-genetic factors, such as age, sex, and exposure factors, may have an influence on disease. Genetic heterogeneity may be caused by different alleles of the same gene or different genes or modifying factors that lead to different phenotypes segregating within a family (Evans and Harris 1992). In addition, there are further types of genetic heterogeneity such as genomic imprinting. In the presence of heterogeneity, considering homogeneous subgroups (defined by, say, severity, age of onset, family history, ethnicity) can lead to a clearer genotype-phenotype relation and thus to identify a possibly Mendelian subform of the disease.

An example of a highly successful segregation analysis for a complex disease is breast cancer (Newman et al. 1988). The families were ascertained through a population-based cancer registry. The ascertainment criteria for index cases were women with breast cancer, Caucasian, diagnosed before the age of 55 during a specified period with a histologically confirmed primary tumor. There was no selection on family history. Thousand five hundred and seventy-nine nuclear families were recruited, along with one large extended pedigree. Complex segregation analysis was applied using the program POINTER (Lalouel et al. 1983). It models an underlying unobserved quantitative trait called liability as a mixture of three normal distributions with different means for each of the genotypes, allowing for polygenes and an environmental component, with disease corresponding to the liability exceeding a certain threshold (Morton and MacLean 1974). For a predisposing genotype, the mean liability is shifted compared to the mean for non-disposing genotypes such that more individuals will exceed the threshold. Evaluation by direct modeling of the transmission probabilities allows the identification of the major factor as a major Mendelian gene. Population-based liability classes were taken from cumulative incidence rates estimated from a large epidemiological study in the region.

Segregation analysis is based on likelihood-ratio tests comparing different models. First, one investigates the consistency of the data with a major gene model; next, one considers which mode of inheritance fits the data best. To avoid false-positive results, the likelihood for Mendelian transmission probabilities can be compared against more general models corresponding to environmental or cultural transmission. For breast cancer, an autosomal dominant major gene provided the best fit, although the general single-locus model with three penetrances resulted in a comparable fit, and the non-Mendelian transmission models were strongly rejected.

Segregation analysis for a disease without a single major gene but only a few oligogenes may not be particularly rewarding. Many genetic marker studies are nowadays carried out without specification of the genetic model, but it is still worth establishing that a disease has some genetic basis by estimating the heritability h ² before embarking extensive marker studies.

5 Linkage Analysis

In linkage analysis, the cosegregation between marker and disease within families is investigated to find evidence for linkage and often to estimate the recombination rate θ. The classical lod score method (Morton 1955) is a test for linkage between a susceptibility gene and a marker (null hypothesis H ₀: θ = 0. 5 vs. alternative H ₁: θ < 0. 5) under a parametric model for the genetic effect, allowing estimation of θ. For a detailed description, see Ott (1999). Let L(θ) denote the likelihood for the observed phenotypes at a particular value for θ conditional on the (assumed) model, the marker allele distribution, and the given pedigrees. In the usual notation, the underlying conditioning is sometimes left out. The lod score function (“log odds”) is the log likelihood ratio

$$Z(\theta ) = \mathrm{LOD}\left (\theta \right ) =\log _{10} \frac{L\left (\theta \right )} {L\left (0.5\right )}$$

as a function of θ. Z(θ) compares the likelihood under linkage with recombination rate θ with the likelihood under no linkage, that is, θ = 0. 5. Z(θ) will be maximized over all possible values for θ, that is, 0 ≤ θ ≤ 0. 5, to yield Z _max. Values of Z _max > 3 are taken as evidence for linkage. The recombination rate is estimated by θ _max, the θ-value corresponding to Z _max. If Z _max < − 2, linkage can be excluded. The limits 3 and − 2 are based on a sequential Wald test, such that the posterior probability for linkage when rejecting H ₀ is 95% for a single alternative θ. As logarithms of base 10 are used, the limits correspond to stopping limits of 1,000 and 0.01 in the sequential testing procedure yielded by setting the probabilities of types I and II errors at α = 0. 001 and β = 0. 01 (Morton 1955).

The likelihood L(θ) for linkage between two loci A and B for a sibship can be derived easily when the genotypes at both loci are directly observable. Let the mothers’ genotypes be A₁A₂, B₁B₂ and the fathers’ A₁A₁, B₁B₁; here, only the double heterozygous mother is informative. Without knowing the maternal grandparent’s genotypes, one cannot determine the phase, that is, whether the mother’s haplotypes are A₁B₁ and A₂B₂ (phase I) or A₁B₂ and A₂B₁ (phase II). If the phase were known, L(θ) would be given by a binomial distribution with parameters n and θ, where n is the number of informative meioses. For an unknown phase, consider first phase I: Let n _x and n _y denote the number of meioses from the mother to the n children, of which n _x are non-recombinants (A₁B₁ or A₂B₂) and n _y are recombinants (A₁B₂ or A₂B₁). Under phase II, let n _x and n _y denote instead the number of recombinants and non-recombinants. Assuming LD phases I and II are both equally likely, then

$$L\left (\theta \right ) = \left (\begin{array}{*{20}c} n_{x} + n_{y} \\ \quad n_{x}\\ \end{array} \right )\left [\frac{1} {2}{\left (1-\theta \right ){}^{n_{x} }\theta }^{n_{y} } + \frac{1} {2}\theta {\left (1-\theta \right ){}^{n_{x} }\theta }^{n_{y} }\right ].$$

For the sibship in Fig. 38.3, let us now determine the likelihood L(θ), the lod score function Z(θ), Z _max, and θ _max. Assume an autosomal dominant gene S with a rare susceptibility allele S ₁ and a normal allele S ₂. Thus, the affected father and all affected siblings have genotype S ₁ S ₂. The marker M has alleles M ₁, M ₂, and M ₃. The mother is homozygous and uninformative for linkage. She will not be considered further.

Fig. 38.3

Pedigree with a sibship of size six with marker information and with genotype information concerning the susceptibility locus, owing to the clear-cut rare autosomal dominant mode of inheritance (individuals: square = male, circle = female, black = affected, white = unaffected)

As a result of the genotyped grandparents, the father’s haplotypes are known: S ₁ M ₁ and S ₂ M ₂. Thus, the phase is known and the likelihood is

$$L\left (\theta \right ) = \left (\begin{array}{*{20}c} 6\\ 0\\ \end{array} \right ){\left (1-\theta \right ){}^{6}\theta }^{0} ={ \left (1-\theta \right )}^{6}.$$

The lod score function is

$$\begin{array}{ll} Z(\theta ) = \mathrm{LOD}\left (\theta \right ) =\log _{10} \frac{L\left (\theta \right )} {L\left (0.5\right )} & =\log _{10}\frac{{\left (1-\theta \right )}^{6}} {{\left (0.5\right )}^{6}} \\ & = 6\log _{10}\left (1-\theta \right ) + 6\log _{10}2 \\ & = 6\log _{10}\left (1-\theta \right ) + C \end{array}$$

where C denotes a constant independent of θ. The maximum of the lod score function is Z _max = 1. 8 for θ _max = 0. This corresponds to complete linkage as supported by no observed recombinations.

Missing information on grandparental genotypes in Fig. 38.3 results in an unknown phase. Then the lod score function would be

$$\begin{array}{ll} Z(\theta )& = \mathrm{LOD}\left (\theta \right ) =\log _{10} \frac{L\left (\theta \right )} {L\left (0.5\right )} =\log _{10}\frac{0.{5\theta }^{6}+0.5{\left (1-\theta \right )}^{6}} {{\left (0.5\right )}^{6}} \\ & =\log _{10}\left(\theta^{6} +{ \left (1-\theta \right )}^{6}\right ) + 5\log _{10}\end{array}$$

(2.)

In this case, the maximum of the lod score function is Z _max = 1. 5 for θ _max = 0. Due to the uncertain phase, the maximum lod score is reduced. However, the estimate for the recombination rate stays at θ = 0.

In Fig. 38.3, assume now that the second affected child has the genotype M ₂ M ₃ (and the genotype S ₁ S ₂). With the phase as indicated in the figure, one recombination needs to be taken into account now. Thus,

$$\begin{array}{ll} Z(\theta )& = \mathrm{LOD}\left (\theta \right ) =\log _{10} \frac{L\left (\theta \right )} {L\left (0.5\right )} =\log _{10}\frac{6\theta {\left (1-\theta \right )}^{5}} {6{\left (0.5\right )}^{6}} \\ & =\log _{10}\theta + 5\log _{10}\left (1-\theta \right ) + 6\log _{10}\end{array}$$

(2.)

With one recombination, the maximum of the lod score function is Z _max = 0. 63 for θ _max = 1 ∕ 6 = 0. 17. Now linkage is estimated as incomplete, and Z _max is markedly reduced.

If in Fig. 38.3 the genotypes of the father and his parents are unknown, the father’s genotype can be inferred as either M ₁ M ₁ or M ₁ M ₂. If HWE can be assumed, the likelihood of the recombination rate L(θ) can be calculated as a function of the marker allele frequencies in offspring. A detailed calculation will show that in this case, a rare marker allele M ₁ will result in a high lod score and a more common marker allele M ₁ will result in a lower lod score.

For a larger pedigree, L(θ) can be computed by the Elston-Stuart algorithm (Elston and Stewart 1971) described earlier, where now the g _j are the joint genotypes formed by the two loci S and M and θ is part of the transmission probabilities for the formation of gametes as recombinants or non-recombinants. Across families, the segregation process is independent, so L(θ) is simply the product of the individual family-specific likelihoods.

The lod score method has been very successful in localizing major genes, such as BRCA1 for breast cancer (Hall et al. 1990), which was facilitated by focusing on early-onset families for which the genetic relative risk (RR) is strong and by having available a good segregation model. However, in complex diseases, the mode of inheritance is usually unclear, leading to false-positive and false-negative results as well as biased estimation of θ (Risch 1991; Lander and Schork 1994; Terwilliger and Ott 1994; Ott 1999). Joint segregation and linkage analysis often leads to biased parameter estimation, particularly if families are not systematically ascertained or the segregation model is misspecified. In MOD-score analysis (Risch 1984; Clerget-Darpoux et al. 1986), the LOD score is maximized over θ and the parameters of a biallelic single-locus model, that is, allele frequency and three penetrances. Non-parametric methods or model-free methods have been developed to avoid assumptions about the underlying genetic model. Their aim is to provide evidence for linkage without specifying the parameters of the underlying mode of inheritance and without estimating the recombination rate (Lander and Schork 1994; Elston 1998). They are often based on the identity-by-descent (IBD) status (Penrose 1953). For example, the IBD status of a patient and one of his/her siblings can take on the values 0, 1, or 2, according to the number of marker alleles that have been transmitted to both siblings from exactly the same grandparental copy of a parent’s gene and are thus identical.

Allele sharing methods test whether relatives with a similar disease status (e.g., both affected) are more frequently similar in IBD at the marker than expected in the absence of linkage. In the affected-sib-pair (ASP) method (Day and Simons 1976), the observed counts of n ASPs with 0, 1, or 2 marker alleles IBD are compared with the expected ones assuming no linkage (0.25n, 0.5n, or 0.25n) using a χ²-goodness-of-fit-test.

The literature on IBD methods is extensive and more powerful methods have been developed (e.g., Holmans 1993; Whittemore and Tu 1998). To determine IBD unambiguously, the marker must be sufficiently polymorphic, and the parents must be genotyped, or neighboring loci must be genotyped to yield sufficient information on the grandpaternal inheritance of the alleles. Often, IBD needs to be estimated. Sometimes, the identity-by-state (IBS) status (the number of identical marker alleles without considering ancestry) is used instead. In the Lander-Green algorithm for multipoint linkage analysis (Lander and Green 1987), the inheritance vector and thus the IBD status at a particular locus can be determined much more precisely even when some parents are not genotyped. The algorithm calculates the inheritance vector using a hidden Markov model walking from marker to marker, where pedigree size that a software can handle is limited by the length of this vector determined by the number of founders and non-founders in the pedigree (Lander and Green 1987; Kruglyak et al. 1996).

6 Association Analysis

The aim of association studies is to provide evidence for association or linkage disequilibrium in a population. LD results in an association between marker alleles and alleles of a susceptibility gene, such that certain marker alleles will be present more often in affected individuals than in a random sample of individuals from the population.

In classic case-control studies, marker allele frequencies or genotype frequencies in a group of unrelated affected individuals are compared to those in a group of unrelated unaffected individuals. Numerous associations have been identified with case-control studies, for example, associations of autoimmune diseases (e.g., diabetes, multiple sclerosis) with the HLA system or of apolipoprotein E (APOE) allele ε4 with Alzheimer’s disease (Corder et al. 1993). The APOE ε4 allele frequency is approximately 35% in Alzheimer’s patients, but only 15% in the older population not suffering from dementia. If a positively associated marker allele is frequent in a population, such as APOE ε4, then it is by itself not a good predictor for disease status, and the proportion of homozygotes for the allele is high. Linkage analysis methods are in general not very powerful in this situation.

Besides the usual limitations of classical case-control studies in epidemiology (cf. chapter Case-Control Studies of this handbook), case-control studies to investigate linkage disequilibrium in genetic epidemiology must take a particular form of confounding known as population stratification into account. Population stratification denotes the presence of different ancestry populations, that is, discrete subpopulations or admixture of populations. If individuals are descended from populations with different allele frequencies and this is not taken into account, then spurious associations can be induced. To avoid this confounding, cases and controls must originate from the same homogeneous (including ethnically homogeneous) source population, or an appropriate design and analysis strategy needs to be employed.

If an association is found that is not considered spurious, this may have two causes (Lander and Schork 1994):

• The disease-associated allele is the susceptibility allele itself. If so, this association is expected to occur in all populations harboring this allele.

• The associated allele is in linkage disequilibrium with the susceptibility allele at the disease locus. If this is the case, then different associations can occur in different populations due to different haplotype frequencies at the two loci.

In the first case, marker and disease loci are identical, so θ = 0, and LD is complete. In the second case, marker and disease locus are in general very close to each other. For this reason, association studies are highly valuable for the investigation of candidate genes.

As mentioned above, uncontrolled population stratification can result in spurious associations. For case-control studies, there are methods for taking the existence of subpopulations into account during statistical testing. All methods require many markers along the genome to be genotyped. In the genomic control method (Devlin and Roeder 1999), a variance inflation factor is used to adjust the test statistic, taking into account correlations between individuals in subpopulations. The structured association method (Pritchard et al. 2000) estimates the population structure and either assigns individuals to the most likely subgroup or better, estimates the proportion of each individuals’ genome derived from each subgroup. Association is subsequently tested within subgroups or adjusted for ancestral source proportions. Both methods typically use panels of hundreds of “ancestrally informative” markers. Principal component analysis (Price et al. 2006) generally uses all the markers from a genome-wide scan and adjusts the association of any particular marker for the first few dozen principal components.

Family-based association studies avoid bias due to inadequate controls and population stratification by design. The concept of internal controls was first proposed by Falk and Rubinstein (1987). For the original design, nuclear families with at least one affected child are recruited, and the two parental alleles not transmitted to the affected child are used as internal controls (Fig. 38.4). With this design, information on both linkage and association between a marker and the susceptibility gene is used.

Fig. 38.4

Nuclear family with one affected child. Alleles transmitted from the parents to the affected child are denoted in white. Alleles not transmitted from the parents to the affected child are denoted in black (individuals: square = male, circle = female, black = affected, white = unaffected, inside: alleles)

For a biallelic marker, the data resulting from this study design can be presented in various ways as 2 ×2 contingency tables and analyzed with standard statistical tests to investigate whether certain alleles are transmitted from the parents to an affected child more often than not (Terwilliger and Ott 1992; Schaid and Sommer 1994). Although in principle, all these procedures test for association (H ₀: δ = 0 vs. H ₁: δ≠0) and most for linkage as well, the most appropriate test respecting the matched nature of the transmitted and non-transmitted allele data is the McNemar test, which in this context is known as the Transmission/Disequilibrium Test (TDT) (Spielman et al. 1993). The TDT is a haplotype-based analysis of the matched sample (Table 38.1). The test statistic is

$$\mathrm{TDT} = {(b - c)}^{2}/(b + c).$$

The TDT compares whether the M ₁ allele is more often transmitted to an affected child (b) than the M ₂ allele (c) from heterozygous parents or vice versa. The test only considers M ₁ M ₂ parents, since homozygous parents are not informative for preferential transmission of either allele.

Table 38.1 2 ×2 contingency table for family-based association studies based on a sample of N families with one affected child and both parents showing the matching of the two alleles of a parent. Consider a biallelic marker with alleles M ₁, M ₂. Small letters (a, b, c, d) denote allele counts. 2N denotes the total number of parental genotypes (i.e., of pairs of transmitted and non-transmitted alleles) to the affected child from the 2N parents

The literature on family-based association analysis is vast (see, e.g., Whittaker and Morris (2001)). Important extensions of the above methods allow the application to multi-allelic markers, to tightly linked loci, and to quantitative traits. In addition, the design also allows for other types of nuclear families, such as sibships with affected and unaffected individuals (Spielman and Ewens 1998; Laird and Lange 2006). If a particular mode of inheritance is suspected, specialized versions of the TDT or related likelihood methods may yield higher power (Schaid 1999). If a candidate gene is to be investigated in detail, then a haplotype analysis can be carried out considering several biallelic polymorphisms (SNPs) in the same gene. The first step in a haplotype analysis is the estimation of the haplotype frequency in a population or the estimation of the most probable haplotype pair in an individual. For cohort or case-control studies, see Excoffier and Slatkin (1995), Stephens and Donnelly (2003), and Browning and Browning (2009); for family samples, see Rohde and Fuerst (2001) and Qian and Beckmann (2002). In the second step, linkage disequilibrium is investigated on the basis of the estimated haplotypes or haplotype frequencies. Some of these LD measures have already been described above (Devlin and Risch 1995).

Besides the analysis of main effects, gene-gene and gene-environment interactions can also be investigated in association analysis using standard tools. For gene-environment interaction, the case-only design enables the analysis of multiplicative interactions of factors, required to be independent in the population, on the basis of a sample of diseased individuals (Albert et al. 2001). If the independence assumption is valid, it is very efficient; if it is violated, such as for smoking-addiction genes and smoking, results can be severely distorted.

7 Current Methods and Outstanding Challenges

Genetics – and all its subdisciplines – has been an amazingly fast-moving field, with outstanding developments in both technology and biological insights. Methodological developments in statistical genetics and bioinformatics have had to scramble to keep pace; not the least of these challenges has been the daunting computational challenges posed by the massive data sets these advances have provided. In the remainder of this chapter, we review the current state of the art in the design of modern genetic epidemiology studies and the analysis of ultra-high-dimensional data and attempt to anticipate some of the novel developments that will be required in the foreseeable future.

7.1 Genome-Wide Association Studies

Fifteen years ago, Risch and Merikangas (1996) published a farsighted article in Science on the failure of traditional linkage analysis to uncover the genetics of complex diseases and made the then-radical suggestion that it would soon be possible to explore the entire genome by direct association methods. Their prediction came true, enabled by two developments in particular. The first was the advent of chip-based genotyping platforms that made it possible to assay hundreds of thousands to millions of SNP genotypes at a cost of under $1,000 per sample with high reliability. The second was a concerted effort by the public and private sectors to map the entire human genome (the Human Genome Project) and then to assemble a catalogue of known variants in a sample of Caucasian, African, and Asian populations (the International HapMap Project). In combination, these two advances provide a feasible way to directly genotype a large proportion of common variants and to predict many variants located next to the genotyped markers that are not typed directly.

It was nearly a decade before the first success of this approach was published in the form of a trio of papers on age-related macular degeneration in Science, one using this approach (Klein et al. 2005), which implicated a gene CFH in the complement pathway, a finding that has subsequently been confirmed numerous times. Since then, associations of about 210 diseases or quantitative traits and 1,300 genetic loci at genome-wide levels of significance (p < 5 ×10^{− 8}) have been published that have been independently replicated; the most recent version of the catalogue of published GWAS is available at the website of National Humane Genome Research Institute (NHGRI), National Institutes of Health (Hindorff et al. 2012; see also Hindorff et al. 2009). Study design and statistical analysis methods for such genome-wide association studies (GWAS) have been discussed in greater depth than is possible here (Hirschhorn and Daly 2005; Wang et al. 2005; Kraft and Cox 2008; McCarthy et al. 2008; Thomas 2010a,b; Witte 2010). Instead, we briefly review some of the recurring themes.

Multistage study designs. Early on, it was recognized that the multiple comparisons burden might be alleviated by some kind of staged design, in which only a portion of the sample would be used for screening the entire genome using one of the expensive high-density commercial platforms offering a fixed array of SNPs, followed by genotyping the remainder of the sample using a custom panel of only the most promising SNPs (Satagopan and Elston 2003). The final analysis then combined the information from both stages rather than treating it as a discovery and independent replication design (Skol et al. 2006); furthermore, it was possible to optimize the allocation of subjects to the two (or more) stages and the selection of the threshold for selecting SNPs to be genotyped in the later stages (Wang et al. 2006; Skol et al. 2007). With rapidly declining costs and increasing density of coverage of commercial panels, the interest in multistage designs has declined as investigators have recognized the advantage of having genome-wide data on the entire available sample as a resource for testing a broad range of hypotheses. However, the basic concept remains in spirit in the requirement for independent replication, as discussed in Sect. 38.7.3, and with the need for selecting a manageable number of individuals for next-generation sequencing technologies (Thomas et al. 2009).

Multiple testing and replication. With hundreds of thousands, if not millions, of associations being tested in a single study, there is an obvious need to avoid false-positive claims by adopting a stringent level of significance. A simple Bonferroni correction for one million SNPs would suggest a threshold of 0. 05 ∕ 10⁶ = 5 ×10^{− 8}, which has become the conventional criterion for claiming genome-wide significance, nominally ensuring a 5% probability of making at least one false-positive (family-wise error rate). This calculation fails to take into account the correlation among these tests due to linkage disequilibrium but is not a bad approximation for even the more recent platforms that allow testing of 2.5–5 million SNPs, as 1 million turns out to be roughly the effective number of independent tests in populations of European descent (or roughly double that in African-descent populations) (Pe’er et al. 2008). Fast asymptotic approximations have been developed (Conneely and Boehnke 2007) that allow for LD within a region, or permutation tests can be used as a gold standard for more complex dependency structures.

Despite the stringency of the genome-wide significance threshold, there are many factors than can lead to increased false-positives, some of which are discussed further below. But the key point is that a single study, however significant, is not usually considered convincing evidence of a genuine association without independent replication (Ioannidis 2007). Such replication is not simply to guard against chance variation (which can always be avoided simply by adopting an even more stringent significance threshold) but due to various sources of uncontrolled bias. Hence, real scientific replication should involve some elements of validation in different populations, by different investigators, using different methods. This is already demanded by the standard Bradford-Hill criteria in epidemiology used to establish validity of the association and strengthen belief in causality. This is not always possible in practice, however, such as for rare diseases where the discovery comes from a consortium of virtually all the available data or for studies conducted in unique settings (suggesting a need for some flexibility in the replication demands of granting agencies and journal editors!). Here in particular, the support by additional biological evidence is highly warranted.

Population substructure and study designs. One of the most pervasive sources of bias in GWAS is population substructure, as discussed in Sect. 38.6. Most GWAS therefore adjust instead for either an estimate of global ancestry from a finite set of founding populations using ancestry informative markers, typically using the STRUCTURE program (Pritchard et al. 2000) or using the top principal components from all or a subset of the markers, typically using the EIGENSTRAT program (Patterson et al. 2006). Either of these approaches tends to be quite effective at controlling the overall false-positive rate at least in homogeneous populations such as those of European descent. As a diagnostic for whether residual overdispersion due to uncontrolled population substructure remains after such adjustment, Quantile-Quantile plots of observed versus expected p-values of the single marker test statistics are generally used, and the genomic control overdispersion factor (Devlin and Roeder 1999) is checked to see if it is close to one. Control of population substructure can be more difficult in admixed populations like African-Americans and Hispanics, but these offer the advantage of being able to use within-individual comparisons for admixture mapping (Patterson et al. 2004; Freedman et al. 2006). For conventional GWAS scans in admixed populations, further adjustment for local ancestry (i.e., the ancestral origins of individual’s chromosomes in the specific region of interest) may be necessary; the LAMP (Sankararaman et al. 2008) and HAPMIX (Price et al. 2009) programs can be used for this purpose.

Imputation. From the beginning, it has been understood that most associations discovered in a GWAS were unlikely to be directly causal, because only a small fraction of the genetic variation in the genome was being tested, but would hopefully reflect indirect associations with nearby causal variants through linkage disequilibrium between the causal and the marker loci. With the availability of the much more extensive catalogue of variation from the HapMap project, it has now become possible to infer the genotypes for most of the common variants in the genome by using imputation techniques from one of these standard reference panels (Li et al. 2010; Marchini and Howie 2010). Although programs such as MACH provide an assessment of the most likely genotype at each untyped locus, their use in association analysis would fail to account for the uncertainty in these imputations, essentially a form of measurement error leading to biased tests. A more appropriate procedure is therefore to use the estimated genotype probabilities or (under an additive model) the expected “gene dosage” as a continuous variable in a logistic regression analysis (Hu and Lin 2010).

Reporting. Given the potential range of problems and the approaches different investigators might take to addressing them, there is a need for some systematic guidance for how GWAS should be reported, if only to avoid subsequent problems in synthesizing the literature. Several authoritative statements have been issued by various groups to address this need, without imposing a straightjacket that would stifle investigators’ creativity (Ehm et al. 2005; Chanock et al. 2007; Ioannidis et al. 2008; Hudson and Cooper 2009; Khoury et al. 2009; Little et al. 2009).

GWAS Summary. Over roughly the last 5 years, a consensus has emerged that most of the discovered novel associations of common, complex disorders with common SNPs have been relatively weak, with odds ratios typically in the range of about 1.2–1.6. Furthermore, even in the aggregate, these associations account for only a small portion of the total heritability estimated from classical twin or family studies (McCarthy and Hirschhorn 2008). While it is possible that these heritability estimates may be somewhat biased, it is certain that there remains a large portion of undiscovered genetic variation (“dark matter”) to be accounted for (Hindorff et al. 2009; Manolio et al. 2009). Even for such a strongly related genetic and well-measured quantitative trait as height, the 180 loci that have so far been discovered based on meta-analysis of studies totaling over 183,000 individuals account for only about 10% of the total variability. Furthermore, it has been estimated that even with astronomical sample sizes, the number of loci of comparable effect sizes might rise to 600 but would still account for only about 20% of the heritability, which has been estimated at greater than 80% of the total phenotypic variation (Lango Allen et al. 2010). A variety of hypotheses have been advanced to account for this unexplained heritability, including rare variants, copy number variants, gene-environment and gene-gene interactions, and epigenetic effects, which will feature prominently as we move into the “post-GWAS” era.

7.2 Post-GWAS

Meta-analysis. Given the small effect sizes being sought and the enormous multiple comparisons penalty, most successful GWASs have required thousands of subjects. Nevertheless, as the experience with height demonstrates, no one study is likely to uncover more than a small fraction of the loci involved in a complex trait, and even larger sample sizes will be needed. Hence, the field has moved into a “Big Science” era, in which many investigators studying a given trait have formed consortia to pool all the available data for analyses of tens or hundreds of thousands of subjects (de Bakker et al. 2008; Zeggini and Ioannidis 2009). Some of these consortia have functioned simply to meet the replication requirement for each other’s discoveries, but the more important purpose is to try to identify additional and weaker associations through a much larger sample size. This could in principle be accomplished by either a reanalysis of their combined raw data (“mega-analysis”) or by meta-analysis of their summary statistics (Lin and Zeng 2010). In practice, the latter is usually much easier to accomplish, particularly if the different studies have used different genotyping platforms but can effectively impute genotypes for a larger, common set of HapMap SNPs (Zaitlen and Eskin 2010).

Fine mapping. Having identified one or more genome-wide significant and replicated regions, one might proceed to try to localize the region where a causal variant or variants might lie before attempting deep sequencing or functional studies. Here, the obvious strategy is simply to retest the available samples (or better, additional samples) with a higher density of markers in the region(s) of interest, but there are obvious trade-offs between sample size, number of regions that can be fine-mapped, region sizes, and density of markers (or criteria for selecting specific markers). These issues are amenable to methodological research, but there does not seem to be any generally agreed guidelines as of this writing. Given the speed the field is moving and the rapidly dropping costs of sequencing, many groups have decided to proceed directly to sequencing, bypassing the intermediate step of fine mapping.

Interactions and pathway analyses. Following an initial scan for main effects of SNPs (in either a single study or meta-analysis of several), much more remains to be explored. One possibility is that there could be larger gene-environment or gene-gene interaction effects that do not produce significant marginal effects. The obvious problem is that the number of possible interactions can be very much larger than the number of main effects: For a GWAS of one million SNPs, for example, there are half a trillion possible pairwise interactions. A simple Bonferroni correction for multiple comparisons would thus require a significance level of 1 ×10^{− 13}, and of course, interaction tests would require much larger sample sizes than main effects even at the same significance level (Marchini et al. 2005). Power can be enhanced by “case-only” analyses based on an assumption of independence of the interacting factors in the source population. For example, a reanalysis of cleft palate data obtained substantially narrower confidence limits on the interaction between smoking and the TGFα gene (odds ratio (OR) = 5.14, 95% confidence interval (CI) = (1.68,15.7) for the case-only analysis compared with OR = 6. 57, 95% CI = (1.72,20.0) for the case-control analysis), equivalent to a 30% reduction in sample size required for the same precision (Umbach and Weinberg 1997). Case-only analyses are, however, biased if this assumption is violated, as might arise due to LD among nearby pairs of loci, population stratification, or behavioral factors that induce an association between genes and environmental factors. To overcome these difficulty, various staged or hybrid approaches have been introduced (Evans et al. 2006; Kooperberg and Leblanc 2008; Mukherjee and Chatterjee 2008; Li and Conti 2009; Murcray et al. 2011). Although power will still be much lower than for main effects, these various methods generally yield much better power than a simple exhaustive search (Cornelis et al. 2012; Mukherjee et al. 2012). To make systematic study of gene-environment interactions of adequate sample sizes possible in the future, it is essential that investigators planning new GWAS design their studies to have appropriate environmental measurements and appropriate population-based sampling schemes. For example, the recent US National Institutes of Health (NIH) “post-GWAS” initiative aimed at synthesizing all the available data on five cancer sites, replicating findings, and characterizing genetic risks and their modification by environmental exposures has been limited by the fact that many of the available studies have not collected any environmental exposure data, and if collected, the choice of measurements was highly variable and ranged from very crude to very detailed assessment.

Another possibility is that there could be many SNPs that individually fail to be genome-wide significant but that jointly contribute to a common pathway. A variety of methods have been developed for identifying subsets of genes in known pathways that collectively are overrepresented among the top GWAS associations. Of these, the technique of Gene Set Enrichment Analysis (Wang et al. 2010), originally developed for gene expression data, has been most widely used. Hierarchical Bayes methods provide a more flexible regression-based approach to incorporate external knowledge about genomic, pathway, or functional annotation into the analysis of GWAS data (Lewinger et al. 2007). Cantor et al. (2010) provide an extensive review of these and other approaches to prioritizing GWAS associations for further investigation. Key to all these methods is the extent and quality of external databases such as the Kyoto Encyclopedia of Genes and Genomes and the Gene Ontology, which can be used for annotation in a systematic manner (Thomas et al. 2007). It is an interesting phenomenon that, after the failure of most candidate gene studies to yield replicable findings, the enthusiasm for the “agnostic” GWAS approach is now drifting back toward a synthesis of pathway-based and agnostic reasoning!

Functional studies. A somewhat unexpected finding from many GWAS is how few of the discovered associations lie in coding regions of genes (Hindorff et al. 2009). While some of the SNP associations could reflect LD with nearby coding variants that have not yet been discovered, it seems more likely that the majority will reflect variants in promoter regions of genes or long-range enhancers. Molecular techniques for functional characterization of causal associations will depend on the nature of the postulated effect, a topic which is beyond the scope of this article; for a recent set of recommendations, see Freedman et al. (2011). Nevertheless, there is a growing interest in “integrative genomics” approaches that can combine information across different types of data, such as SNPs, mutations in tumor tissues, transcriptomics, methylomics, metabolomics, and proteomics (Schadt et al. 2005; Hawkins et al. 2010).

7.3 Targeted, Whole-Exome, and Whole-Genome Sequencing

GWASs are based on an underlying “common disease, common variant” hypothesis (Reich and Lander 2001), which postulates that complex diseases are caused, at least in part, by common variants that can be effectively tagged by other SNPs in the region. The SNP panels used in most GWAS, based on one million or fewer SNPs, indeed are generally effective at tagging most “common” variants (conventionally defined as those with minor allele frequencies (MAF) of at least 5%). Newer generations of 2.5–5 million SNP panels will enhance the coverage of “uncommon” variants (those in the range of 1–5% MAF), but even these are not expected to provide good coverage of “rare” variants (less than 1% MAF). To discover these, direct sequencing will be necessary. The advent of several different massively parallel and fast “next-generation sequencing” (NGS) technologies (Davey et al. 2011) has now made this cost-effective, at least for targeted regions such as those around selected GWAS hits or the whole-exome. At the time of this writing, costs are typically around $1,000 per sample for whole exome sequencing and $5,000 for the whole genome, but the “$1,000 genome” (Mardis 2006) is anticipated in the near future. However, the data management and data analysis challenges are formidable. In addition to the storage problems of terabytes of raw data produced for even a single subject (petabytes for a typical study), NGS does not directly yield genotype calls but rather a random set of short sequence reads that must be aligned to cover the region of interest. The number of reads at any specific location is thus randomly distributed (with a mean given by the average depth of coverage), so the genotype can only be inferred probabilistically, taking into account possible errors in the reads themselves. Thus, in addition to the trade-offs mentioned earlier about region size and sample size, a further dimension to the design challenge is the necessary depth of sequencing, subject to a constraint on total cost (Sampson et al. 2011). The optimal design will depend upon the purpose – whether for discovery of rare variants or for testing association. One promising option is to perform the sequencing on pooled DNA samples (Futschik and Schlotterer 2010), which can considerably reduce costs, but this adds complexity to the optimization of the numbers of subjects and numbers of pools.

Multiple Rare Variant Analyses. One of the first analytical issues that arises concerns testing of association with rare variants. For feasible sample sizes, it is virtually impossible to test association with any single rare variant, both because of their rarity and the massive multiple comparisons penalty – for a typical whole-genome sequencing study, for example, one might expect to discover on the order of 20 million variants, most appearing only a few times. Interest has therefore tended to shift to tests of the “multiple rare variants” hypothesis (Price et al. 2010). The most commonly used technique is some form of “burden” test, which simply compares the total number of rare variants at a particular locus carried by cases and controls, possibly weighted in some fashion by their frequency or other characteristics (see Basu and Pan (2011) for a comparison of the available methods), but hierarchical Bayes methods that take account of model uncertainty offer a flexible and attractive alternative (Quintana et al. 2011).

Study designs for assessing causality. Most GWASs have used a case-control design with unrelated subjects for greatest statistical power. However, family-based designs offer two important advantages for studying rare variants. First, the sample can be enriched for rare variants by targeting cases with a strong family history. But more important is that looking at the pattern of cosegregation of variants with disease within families offers great potential for distinguishing causal variants from private polymorphisms that are simply circulating in the family but have nothing to do with the disease (Zhu et al. 2010; Shi and Rao 2011). This is essentially a form of linkage analysis, which has long been recognized as the design of choice for mapping rare major genes.

7.4 Risk Models and Translational Significance

Finally, as the number of genetic associations grows, it is natural to ask whether they can be used for genetic risk prediction. For genetic risk prediction, the purpose for the test must be distinguished, for example, a screening test for the general population or a high-risk population or a confirmation or exclusion test for a particular mutation for a particular family member of a disease-enriched family. For major genes, segregation models as described above, using available genetic and non-genetic information and possibly indirect LD measures, can be used for prediction. For screening in populations, clearly a single SNP with relative risk less than 1.5 has very little value, but in the aggregate, one might consider a risk index based on all the known variants. For prostate cancer, for example, there are now more than 50 known GWAS associations, and a substantial portion of the population will carry several of these variants. Unfortunately, attempts to do this so far have not been particularly encouraging (Jostins and Barrett 2011; MacInnis et al. 2011; So et al. 2011; Newcombe et al. 2012). For prostate cancer in African-Americans, Haiman et al. (2011) reported a twofold gradient in predicted risk across quartiles of risk using 40 GWAS SNPs and 3.5-fold using only the 27 that were significantly associated among African-Americans, but the latter comparison may be subject to some overfitting. A clinically useful screening test should have both high sensitivity and high specificity (Kraft and Hunter 2009; Kraft et al. 2009; Janssens et al. 2011). Since both depend on where one draws the line between “normal” and “elevated” risk, a widely used measure of the overall performance of a screening test is the Area Under the Receiver Operating Curve (AUC) (Sanderson et al. 2005; Zou et al. 2007), obtained by plotting sensitivity against one minus specificity across the range of possible cut-points of the index (here, the predicted genetic risk score). For breast and prostate cancer, Machiela et al. (2011) found AUCs of 0.53 and 0.57, respectively, compared with 0.50 for an index that was no better than chance. Clinically useful risk indices would require an AUC of the order of 0.80 or better. For Crohn’s disease, this appears attainable (Pharoah et al. 2002), but not at this point for most cancers (Chatterjee et al. 2011). Perhaps a more important question is what is the additional predictive value of genetic test results on top of established risk factors, including family history (Pencina et al. 2008; So et al. 2011). With the proliferation of direct-to-consumer genetic testing kits, some based on rather flimsy scientific evidence or with somewhat misleading advice about possible lifestyle changes to improve their risks, this question becomes of immediate translational significance (Hudson et al. 2007; Kaye 2008). See Levy et al. (2007) for a discussion of the scientific and ethical significance of the publication of the first complete human genome sequence. (The United States Genetic Information Non-discrimination Act (May 5, 2008) and a similar law in Germany (Gendiagnostikgesetz vom 31. Juli 2009 (BGBl. I S. 2529, 3672)) were enacted to protect individuals from discrimination based on genetic test results.) Although genetic risk prediction for the general population may still be some ways off, there is potentially greater utility in predicting genetic variation in response to treatments because these effects are likely to be much stronger due to the lack of time for evolution to eliminate deleterious variants, considering the recentness of most drug exposures (Altshuler et al. 2008).

8 Conclusions

The field of genetic epidemiology is in the midst of a fundamental paradigm shift. Originally based on methods for describing familial aggregation, testing for the existence of a genetic basis (segregation analysis), and localizing genetic causes (linkage analysis), the mainstay has become testing association with directly measured genotypes. A decade ago, this was feasible only for a modest number of variants in candidate genes, an approach that is now widely viewed as not having been particularly rewarding because of our lack of success in picking good candidates. With the advent of high-density genotyping platforms, agnostic scans for common variants across the entire genome have become popular and have led to many unexpected discoveries, albeit generally with rather small effect sizes that even in the aggregate account for only a modest proportion of the total estimated heritability of most complex diseases. These data are now being mined with sophisticated algorithms in the hope of identifying novel pathways across many of the suggestive, if not genome-wide significant, associations. Future directions are aimed at trying to identify the cause of the remaining unexplained heritability by targeted, whole-exome, or (soon) whole-genome sequencing – technologies that will pose formidable statistical and computational challenges – and by understanding the biological basis of the observed associations through regulatory, epigenetic, or other mechanisms.