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Introduction to some basic concepts
in quantitative genetics
Course “Study Design and Data Analysis for Genetic Studies”, Universidad ded Zulia, Maracaibo, Venezuela, 6 April 2005
Harald H.H. Göring
100% genetic contribution 0%
0% environmental contribution 100%
“Nature vs. nurture”
trait
genes environment
“mendelian” traitsinfections,
accidental injuries“complex” traits
“Marker” loci
There are many different types of polymorphisms, e.g.:
• single nucleotide polymorphism (SNP):
AAACATAGACCGGTT
AAACATAGCCCGGTT
• microsatellite/variable number of tandem repeat (VNTR):
AAACATAGCACACA----CCGGTT
AAACATAGCACACACACCGGTT
• insertion/deletion (indel):
AAACATAGACCACCGGTT
AAACATAG--------CCGGTT
• restriction fragment length polymorphism (RFLP)
…
Genetic variation in numbers
There are ~6 x 109 humans on earth, and thus ~12 x 109 copies of each autosomal chromosome. Assuming a mutation rate of ~1 x 108, every single nucleotide will be mutated (~12 x 109) / (~1 x 108) = ~120 in each new generation of earthlings. Thus, every nucleotide will be polymorphic in Homo sapiens, except for those where variation is incompatible with life.
Any 2 chromosomes differ from each other every ~1,000 bp. The 2 chromosomal sets inherited from the mother and the father (each with a length of 3 x 109 bp) therefore differ from each other at ~3 x 109 / ~1,000 = ~ 3 x 106, or ~3 million, locations.
locus: a position in the DNA sequence, defined relative to others; in different contexts, this might mean a specific polymorphism or a very large region of DNA sequence in which a gene might be located
gene: the sum total of the DNA sequence in a given region related to transcription of a given RNA, including introns, exons, and regulatory regions
polymorphism: the existence of 2 or more variants of some locus
allele the variant forms of either a gene or a polymorphism
neutral allele: any allele which has no effect on reproductive fitness; a neutral allele could affect a phenotype, as long as the phenotype itself has no effect on fitness
silent allele: any allele which has no effect on the phenotype under study; a silent allele can affect other phenotype(s) and reproductive fitness
disease-predisposing allele: any allele which increases susceptibility to a given disease; this should not be called a mutation
mutation:the process by which the DNA sequence is altered, resulting in a different allele
Definitions of some important terms
Genetics vs. epidemiology:aggregate effects
• The sharing of environmental factors among related (as well as unrelated) individuals is hard to quantify as an aggregate.
• In contrast, the sharing of genetic factors among related (as well as unrelated) individuals is easy to quantify, because inheritance of genetic material follows very simple rules.
• Aggregate sharing of genetic material can therefore be predicted fairly accurately w/o measurements: e.g.
– a parent and his/her child share exactly 50% of their genetic material (autosomal DNA)
– siblings share on average 50% of their genetic material– a grandparent and his/her grandchild (or half-sibs or avuncular individuals)
share on average 25% of their genetic material
• genome as aggregate “exposure”: While it is not clear whether an individual has been “exposed” to good or bad factors, “co-exposure” among relatives is predictable.
Use of genetic similarity of relatives
• The genetic similarity of relatives, a result of inheritance of copies of the same DNA from a common ancestor, is the basis for– heritability analysis– segregation analysis– linkage analysis– linkage disequilibrium analysis– relationship inference
• between close relatives (e.g., identification of human remains, paternity disputes)
• between distant groups of individuals from the same species (e.g., analysis of migration pattern)
• between different species (e.g., analysis of phylogenetic trees)
– identification of conserved DNA sequences through sequence alignment
– …
Relatives are not i.i.d.
• Unlike many random variables in many areas of statistics, the phenotypes and genotypes of related individuals are not independent and identically distributed (i.i.d.).
• Many standard statistical tests can and/or should therefore not be applied in the analysis of relatives.
• Most analyses on related individuals use likelihood-based statistical approaches, due to the modeling flexibility of this very general statistical framework.
“Mendelian” vs. “complex” traits“simple mendelian” disease
•genotypes of a single locus cause disease
•often little genetic (locus) heterogeneity (sometimes even little allelic heterogeneity); little interaction between genotypes at different genes
•often hardly any environmental effects
•often low prevalence
•often early onset
•often clear mode of inheritance
•“good” pedigrees for gene mapping can often be found
•often straightforward to map
“complex multifactorial” disease
•genotypes of a single locus merely increase risk of disease
•genotypes of many different genes (and various environmental factors) jointly and often interactively determine the disease status
•important environmental factors
•often high prevalence
•often late onset
•no clear mode of inheritance
•not easy to find “good” pedigrees for gene mapping
•difficult to map
Genetic heterogeneity
timelocus homogeneity, allelic homogeneity
locus homogeneity, allelic heterogeneity
locus heterogeneity, allelic homogeneity (at each locus)
time
locus heterogeneity, allelic heterogeneity (at each locus)
Study design
different traits
different study designs
different analytical methods
How to simplify the etiological architecture?
• choose tractable trait– Are there sub-phenotypes within trait?
• age of onset• severity• combination of symptoms (syndrome)
– “endophenotype” or “biomarker ” vs. disease• quantitative vs. qualitative (discrete)• Dichotomizing quantitative phenotypes leads to loss of information.• simple/cheap measurement vs. uncertain/expensive diagnosis• not as clinically relevant, but with simpler etiology
• given trait, choose appropriate study design/ascertainment protocol– study population
• genetic heterogeneity• environmental heterogeneity
– “random” ascertainment vs. ascertainment based on phenotype of interest• single or multiple probands• concordant or discordant probands• pedigrees with apparent “mendelian” inheritance?• inbred pedigrees?
– data structures• singletons, small pedigrees, large pedigrees
– account for/stratify by known genetic and environmental risk factors
Qualitative and quantitative traits
• qualitative or discrete traits:– disease (often dichotomous; assessed by
diagnosis): Huntington’s disease, obesity, hypertension, …
– serological status (seropositive or seronegative)– Drosophila melanogaster bristle number
• quantitative or continuous traits:– height, weight, body mass index, blood pressure,
…– assessed by measurement
discrete trait(e.g. hypertension)
continuous trait(e.g. blood pressure)
0 1
Why use a quantitative trait?Why not?
Pros and cons of disease vs. quantitative trait
disease
• for rare disease, limited variation in random sample; need for non-random ascertainment
• for late-onset diseases, it is difficult/impossible to find multigenerational pedigrees
• diagnosis: often difficult, subjective, arbitrary
• treatment may cure disease or weaken symptoms, but original disease status is generally still known
• of great clinical interest
• often more complex etiologically
continuous trait
• sufficient variation in random sample; non-random ascertainment may not be necessary or advisable
• as no special ascertainment is necessary, any pedigree is suitable
• measurement: often straight-forward, reliable
• medications and other covariates may influence phenotype
• often only of limited/indirect clinical interest
• often simpler etiologically
Dichotomizing quantitative phenotypes generally leads to loss
of information
phenotype
probability density
unaffected affected
Characterization of a quantitative trait
mean : μ =E X[ ] =Xi∑
n
variance: σ 2 =E X−E X[ ]( )2⎡
⎣⎤⎦=
Xi −μ( )∑ 2
n
skewness :E X−E X[ ]( )3⎡
⎣⎤⎦=
Xi −μ( )∑ 3
n
kurtosis :E X−E X[ ]( )4⎡
⎣⎤⎦=
Xi −μ( )∑ 4
n
center of distribution
spread around center
symmetry
thickness of tails
How can a continuous trait result from discrete genetic variation?
Suppose 4 genes influence the trait, each with 2 equally frequent alleles. Assume that at each locus allele 1 decreases the phenotype of an individual by 1, and that allele 2 increases the phenotype by 1.
Now, let us obtain a random sample from the population - by coin tossing. Take 2 coins and toss them. 2 tails mean genotype 11, and phenotype of -2. 2 heads mean genotype 22, and phenotype contribution of +2. 1 head and 1 head is a heterozygote (genotype 12), with phenotype of 0. Repeat this experiment 4 times (once for each locus). Sum up the results to obtain the overall phenotype.
Variance decomposition
σ p2 = σ g
2 + σ e2
phenotypic variance due to all causes
phenotypic variance due to genetic
variation
phenotypic variance due to
environmental variation
σ g2 = σ a
2 + σ d2
phenotypic variance due to genetic
variation
phenotypic variance due to additive
effects of genetic variation
phenotypic variance due to dominant effects of genetic
variation
Decomposition of phenotypic variance attributable to genetic variation
-a 0
phenotypic means of genotypes
+a
AA AB BB
d
σ a2 = 2pq a + d(q − p)( )
2
σ d2 = 2pqd( )
2
-a d=0
phenotypic means of genotypes
+a
AA AB BB
If the phenotypic mean of the heterozygote is half way between the two homozygotes, there is “dose-response” effect, i.e. each dose of allele B increases the phenotype by the same amount. In this case, d = 0, and there is no dominance (interaction between alleles at the same polymorphism).
σ d2 = 2pqd( )
2= 2pq × 0( )
2= 0
σ e2 = σ c
2 + σ u2
phenotypic variance due to
environmental variation
phenotypic variance due to
environmental variation common among individuals
(e.g., culture, household)
phenotypic variance due to
environmental variation unique to
an individual
Decomposition of phenotypic variance attributable to environmental variation
The proportion of the phenotypic variance in a trait that is attributable to the effects of genetic variation.
Definition of heritability
h2 =σ g
2
σ p2
The absolute values of variance attributable to a specific factor are not important, as they depend on the scale of the phenotype. It is the relative values of variance matter.
The proportion of the phenotypic variance in a trait that is attributable to:
Broad sense andnarrow-sense heritability
- effects of genetic variation (broad sense)
- additive effects of genetic variation (narrow sense)
h2 =σ g
2
σ p2
h2 =σa
2
σ p2
100% genetic contribution 0%
0% environmental contribution 100%
“Nature vs. nurture”
trait
genes environment
Different degrees of relationship havedifferent phenotypic covariance/correlation
relative pairphenotypic covariance
phenotypic correlation
parent child
full sibs
half sibs
first cousins
1
2σa
2
1
2σa
2 +14σd
2
1
4σa
2
1
8σ a
2 1
8h2
1
4h2
>1
2h2
1
2h2
(assuming absence of effect of shared environment)
MZ and DZ twins havedifferent phenotypic covariance/correlation
relative pairphenotypic covariance
phenotypic correlation
identical twins
fraternal twins
2x difference
σ a2 + σ d
2 ( + σ c2 )
1
2σa
2 +14σd
2 ( +σ c2 ) >
1
2h2
> h2
σ a2 +
3
2σ d
2 > h2
(assuming equal effect of
shared environment)
Normal distribution
f x( ) =12πσ
e−12(x−μ)2
σ 2
x
f(x)
Variance components approach:multivariate normal distribution (MVN)
In variance components analysis, the phenotype is generally assumed to follow a multivariate normal distribution:
f x( ) =1
2π( )n Ω( )12
exp12
x−μ( )'Ω−1 x−μ( )⎛
⎝⎜⎞
⎠⎟
ln f x( ) =−n2
ln 2π( )−12Ω −
12
x−μ( )'Ω−1 x−μ( )
no. of individuals (in a pedigree)
nn covariance matrix
phenotype vector
mean phenotype
vector
Ω= Ωii∑ σ i
2
Variance-covariance matrix
The variance-covariance matrix describes the phenotypic covariance among pedigree members.
nn structuring
matrix
scalar variance component
(random effect)
Ω=Iσu2
“Sporadic” model:no phenotypic resemblance
between relativesIn the simplest model, the phenotypic covariance among pedigree members is only influenced by environmental exposure unique to each individual. Shared factors among relatives, such as genetic and environmental factors, do not influence the trait.
identity matrix: I=
12...n
1 0 0 00 1 0 00 0 1 00 0 1
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
1 2 ... n
Identity matrix
f m
321
f m 1 2 3
f 1 0 0 0 0
m 0 1 0 0 0
1 0 0 1 0 0
2 0 0 0 1 0
3 0 0 0 0 1
Ω=Iσu2 + 2Φσa
2
Modeling phenotypic resemblance between
relatives:“polygenic” model
kinship matrix
ΦKinship and relationship
matrixkinship matrix:
Each element in the kinship matrix contains probability that the allele at a locus randomly drawn from the 2 chromosomal sets in a person is a copy of the same allele at the same locus randomly drawn from the 2 chromosomal sets in another person. For one individual, = 0.5, assuming absence of inbreeding.
relationship matrix:
This provides the probability that a given locus is shared identical-by-descent among 2 individuals. This is equivalent to the expected proportion of the genome that 2 individuals share in common due to common ancestry. For one individual, 2 = 1, assuming absence of inbreeding.
2Φ
2Φ
Relationship matrix and 7 matrix
relationship
self 1 1
MZ twin pair 1 1
DZ twin pair 0.5 0.25
full sibs 0.5 0.25
half sibs 0.25 0
grandparent - grandchild
0.25 0
avuncular 0.25 0
first cousin 1/8 0
second cousin 1/32 0
7
Relationship matrix:nuclear family
f m
321
f m 1 2 3
f 1 0 0.5 0.5 0.5
m 0 1 0.5 0.5 0.5
1 0.5 0.5 1 0.5 0.5
2 0.5 0.5 0.5 1 0.5
3 0.5 0.5 0.5 0.5 1
Relationship matrix:half-sibs
f1 m
21
f1 m f2 1 2
f1 1 0 0 0.5 0
m 0 1 0 0.5 0.5
f2 0 0 1 0 0.5
1 0.5 0.5 0 1 0.25
2 0 0.5 0.5 0.25 1
f2
• The likelihood of a hypothesis (e.g. specific parameter value(s)) on a given dataset, L(hypothesis|data), is defined to be proportional to the probability of the data given the hypothesis, P(data|hypothesis):
L(hypothesis|data) = constant * P(data|hypothesis)
• Because of the proportionality constant, a likelihood by itself has no interpretation.
• The likelihood ratio (LR) of 2 hypotheses is meaningful if the 2 hypotheses are nested (i.e., one hypothesis is contained within the other):
• Under certain conditions, maximum likelihood estimates are asymptotically unbiased and asymptotically efficient. Likelihood theory describes how to interpret a likelihood ratio.
€
LR =L H1 | data( )L H0 | data( )
=cP data |H1( )cP data |H0( )
=P data |H1( )P data |H0( )
Likelihood
Inference in heritability analysis
H0: (Additive) genetic variation does not contribute to phenotypic variation
H1: (Additive) genetic variation does contribute to phenotypic variation
Λ =−2 lnL H0( )
L H1( )
= −2 lnL σ a
2 = 0, %σ u2( )
L σ a2 ,σ u
2( ) h2 =
σa2
σa2 + σu
2
heritability:
Ω=Iσu2 + 2Φσa
2 + 7σd2
Modeling phenotypic resemblance between
relatives:“polygenic” model allowing
for dominance
matrix of probabilities that 2 individuals inherited the
same alleles on both chromosomes from 2 common ancestors
2Φrelationship
self 1 1
MZ twin pair 1 1
DZ twin pair 0.5 0.25
full sibs 0.5 0.25
half sibs 0.25 0
grandparent - grandchild
0.25 0
avuncular 0.25 0
first cousin 1/8 0
second cousin 1/32 0
7
Relationship matrix and 7 matrix
7 matrix:nuclear family
f m
321
f m 1 2 3
f 1 0 0 0 0
m 0 1 0 0 0
1 0 0 1 0.25 0.25
2 0 0 0.25 1 0.25
3 0 0 0.25 0.25 1
Inference in heritability analysis
H0: (Additive) genetic variation does not contribute to phenotypic variation
H1: (Additive) genetic variation does contribute to phenotypic variation
Λ =−2 lnL H0( )
L H1( )
= −2 lnL σ a
2 = 0,σ d2 = 0, %σ e
2( )
L σ a2 ,σ d
2 ,σ e2( )
2 degrees of freedom
Is it reasonable to assume that the only source for phenotypic resemblance among
relatives is genetic?
No. To overcome this problem, one can try to model shared environment, either in aggregate or broken into specific environmental factors.
Ω=Iσu2 +Hσ c
2 + 2Φσa2
household matrix: accounts for aggregate of environmental factors shared among individuals living in the same household
Household matrix
f m
321
f m 1 2 3
f 1 1 1 0 0
m 1 1 1 0 0
1 1 1 1 0 0
2 0 0 0 1 0
3 0 0 0 0 1
“Household” effect
c2 =σ c
2
σ p2
Nested models for heritability analysis
model
“sporadic” + - -
“household” + + -
“additive polygenic” + - +
“general” + + +
σ u2 σ c
2 σ a2
non-nested hypotheses
Inclusion of covariates
Measured covariates can easily be incorporated as “fixed effects” in the multivariate normal model of the phenotype, by making the expected phenotype different for different individuals as a function of the measured covariates.
ln f x( ) =−n2
ln 2π( )−12Ω −
12
x−μ( )'Ω−1 x−μ( )
μ =μoverall +Yβ
μ i = μ overall + β jYijj
∑
Inclusion of covariates
If covariates are not of interest in and of themselves, one can “regress them out” before pedigree analysis.
Xi =β0 + β jYijj∑
X =Yβ
Xi −Xi =ei
X −X =eThen use residuals as phenotype of interest in pedigree analysis.
Inference regarding covariates
in heritability analysisH0: measured covariate Y does not influence phenotype.
H1: measured covariate Y does influence phenotype.
Λ =−2 lnL H 0( )
L H1( )
= −2 lnL σ a
2 ,σ u2 ,β = 0( )
L σ a2 ,σ u
2 , β( )
Inference regarding covariates
in heritability analysisH0: measured covariate Y does not influence phenotype.
H1: measured covariate Y does influence phenotype.
Λ =−2 lnL H0( )
L H1( )= −2 ln
L σ u2 ,β = 0( )
L σ u2 , β( )
CAUTION:
Related individuals in pedigrees are treated as unrelated. This can easily lead to false positive findings regarding the effect of the covariate!
Choice of covariates
Covariates ought to be included in the likelihood model if they are known to influence the phenotype of interest and if their own genetic regulation does not overlap the genetic regulation of the target phenotype.
Typical examples include sex and age.
In the analysis of height, information on nutrition during childhood should probably be included during analysis. However, known growth hormone levels probably should not be.
Choice of covariates
σ p2 σ p
2
h2
without cov =σa
2
σ p2 ≈0.5 > 0.25 ≈
σa2 − σa
2 I σ cov2( )
σ p2 −σ cov
2 =h2withcov
σ a2
σ cov2
σ a2
Choice of covariates
σ p2 σ p
2
σ cov2
σ a2 σ a
2
h2
without cov =σa
2
σ p2 ≈0.2 < 0.3 ≈
σa2 − σa
2 I σ cov2( )
σ p2 −σ cov
2 =h2withcov
Choice of covariates:special case of treatment/medication
Before treatment/medicationof affected individuals
phenotype
probability density
unaffected affected
After (partially effective) treatment / medication of affected individuals
phenotype
probability density
unaffected affected
apparent effect of covariate
Choice of covariates:special case of treatment/medication
• If medication is ineffective/partially effective, including treatment as a covariate is worse than ignoring it in the analysis.
• If medication is very effective, such that the phenotypic mean of individuals after treatment is equal to the phenotypic mean of the population as a whole, then including medication as a covariate has no effect.
• If medication is extremely effective, such that the phenotypic mean of individuals after treatment is “better” than the phenotypic mean of the population as a whole, then including medication as a covariate is better than ignoring it, but still far from satisfying.
• Either censor individuals or, better, infer or integrate over their phenotypes before treatment, based on information on efficacy etc.
Be careful ininterpretation of heritability estimates
While one can attempt to account for shared environmental factors individually or in aggregate, it is notoriously difficult to do so. In contrast to genetics where “co-exposure” among relatives is predictable due to inheritance rules, this is not the case with environmental factors of interest in epidemiology. If environmental co-exposure is not adequately modeled, shared environmental effects tend to inflate the heritability estimate, because shared exposure is generally greater among relatives, such as mimicking the effects of genetic similarity among relatives. Heritability estimates thus are often overestimates.
Be careful ininterpretation of heritability estimates
Keep in mind that heritability estimates are applicable only to a specific population at a specific point in time.
Heritability of adult height(additive heritability, adjusted for sex and age)
study sample sizeheritability estimate
TOPS 2199 0.78
FLS 705 0.83
GAIT 324 0.88
SAFHS 903 0.76
SAFDS 737 0.92
SHFS
AZ 643 0.80
DK 675 0.81
OK 647 0.79
Jiri 616 0.63
total 7449
Be careful ininterpretation of heritability estimates
Heritability is a population level parameter, summarizing the strength of genetic influences on variation in a trait among members of the population. It does not provide any information regarding the phenotype in a given individual, such as risk of disease.
Relative risk
The risk of disease (or another phenotype) in a relative of an affected individual as compared to the risk of disease in a randomly chosen person from the population.
λrelationship =prelationship
p
Relative risk as a function of heritability
λsib = 1 +1− p
p0.5ha
2 + 0.25hd2( )
p → 0 (rare disease) : λ sib → ∞
p → 1 (common disease) : λ sib → 1
Heritability of adult height(additive heritability, adjusted for sex and age)
phenotype p λsib
autism 0.0004 75
IDDM 0.004 15
schizophrenia 0.01 9
NIDDM 0.2 3
obesity 0.4 <2
Be careful ininterpretation of heritability estimates
A heritability estimate is applicable only to a specific trait. If you alter the trait in any way, such as inclusion of additional/different covariates, this may alter the estimate and/or alter the interpretation of the finding.
Example:
•left ventricular mass not adjusted for blood pressure
•left ventricular mass adjusted for blood pressure
