OVERVIEW

1
OVERVIEW

• Elston-Stewart Algorithm
• Quantitative Trait Loci
• Non-parametric analysis

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Model specification in linkage analysis

• In linkage analysis, the parameter of primary
  interest is q, the recomb. fraction.
• This is the only parameter that appears in the
  log-likelihood function
• However, in order to deal with more complex
  pedigrees and loci, it is helpful to introduce
  other parameters in the framework of a general
  model.

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Other parameters

• Penetrance parameters specify the relationship
  between genotype and phenotype.
• Let g and x represent the vectors of observed
  genotypes and phenotypes, respectively, over all
  individuals.
• Then the penetrance function can be written
  P(xg)?P(xigi)
• For example, consider disease allele D and normal
  allele d, in which case four penetrance
  parameters are required. These give the
  conditional probabilities of disease given the
  genotypes AA, Aa, aA, aa. For autosomal
  dominance, the parameters are (1,1,1,0).

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Other parameters

• Transmission parameters give the probability of
  inheriting a genotype given the genotype of the
  parents.
• Transmission is captured by the term p(gi gi,f
  , gi,m) which is a function of the recombination
  fraction q
• Transmission parameters only apply to individuals
  whose parents are included in the pedigree.
• For individuals whose parents are not included,
  these are called founding members. The
  parameters which define the distribution of
  genotypes in the founding members of the pedigree
  are known as population parameters.

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The likelihood function
Gi represents all possible genotypes of
individual i of n. Individuals 1 .. p are
founders (parents). Individuals p1 .. n are
non-founders (children). This function is
prohibitive to evaluate due to the huge number of
products and sums. However, the Elston-Stewart
algorithm gives a method for aggregating terms in
the calculation to reduce the number of products
and sums required.
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Complexity analysis
It is not hard to see that the number of terms in
this expression is enormous! For two loci, with
m1 and m2 alleles each, this corresponds to m1m2
ordered haplotypes and (m1m2)2 possible genotypes
for an individual. There are (m1m2)2n genotype
combinations over n individuals. Therefore, the
likelihood function is a sum over (m1m2)2n terms,
each term being a product of 2n probabilities.
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Elston-Stewart Algorithm
Elston and Stewart suggested a simple recursion
for grouping terms of the likelihood function
which greatly reduces the number of additions and
multiplications. A simple example of how this
works is as follows. Examine this equation (how
many operations?)
Compared to the following re-write, which is
possible because some terms in the sum are
independent of others (how many operations?)
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Analysis of quantitative traits (QTLs)

• Traits that are determined by a single locus are
  necessarily discrete.
• Single-locus traits have been our focus so far,
  and include ABO blood type, HLA antigens, and
  rare dominant and recessive diseases.
• The situation with continuous traits is less
  clear (e.g., height, weight).
• While these clearly exhibit genetic inheritance,
  they cannot solely be determined by the action of
  genes at a single locus, because a single locus
  in discrete, not continuous, in nature.
• The bell-shaped distribution of most of these
  quantitative traits suggests that several or many
  factors, both genetic and environmental, are at
  play.
• Underlying genetic events at a single locus are
  masked (or at best convoluted) by the operation
  of the other factors.
• Unfortunately, most traits are continuous/quantita
  tive.

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Moving to non-parametric approaches

• Complex diseases such as heart disease, diabetes,
  and depression are caused by multiple genetic and
  environmental factors.
• A complete likelihood model would include all
  these factors, their joint probability
  distribution in the population, and their joint
  effect on the penetrance.
• How to proceed in the midst of all of this
  complexity is an open problem.
• However, the main approach has been to abandon
  the so-called parametric method of conventional
  linkage (in which q is the main parameter) and to
  instead measure the association between the
  sharing of marker alleles among siblings and the
  sharing of their disease status.

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Measures of allele sharing by relatives

• The concept of allele sharing is central to
  non-parametric methods of linkage analysis.
• There are two different forms, identical-by-state
  (IBS) and identical-by-descent (IBD).
• Two alleles of the same physical form are IBS.
• If, in addition to being IBS, the two alleles
  descended from the same ancestral allele, they
  are also IBD.

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Example of IBS versus IBD

• Consider the following pedigree with loci A and B
  which are in extremely tight linkage

A2A2B1B2
A1A2B1B1
A1A2B1B2
A1A2B1B1

• At locus A, how many alleles do the siblings have
  that are IBS?
• At locus A, how many alleles do the siblings have
  that are IBD?
• Note the close relationship between IBD and
  recombination events. This is why IBD is more
  relevant than IBS for linkage analysis

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Measuring association with IBD

• Define two indicator variables
• Df 1 if the two siblings have the same paternal
  allele 0 otherwise
• Dm 1 if the two siblings have the same maternal
  allele 0 otherwise
• Let D Df Dm be the total IBD value of the
  sib-pair.
• D is a binomial random variable with possible
  values 0, 1, 2 with probabilities ¼, ½, ¼.
• For two loci A and B, their corresponding IBD
  values DA and DB will be independent for unlinked
  loci, but positively correlated for linked loci.
• The IBD status at locus A will be the same as the
  IBD status at locus B if and only if
• NEITHER haplotype is recombinant between the
  loci OR
• BOTH haplotypes are recombinant between the loci

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Measuring association with IBD

• In terms of the recombination fraction, Y q2
  (1-q)2where Y is the probability that the IBD
  status of A and B is the same.
• The same is true for haplotypes transmitted from
  the other parent.
• It can be shown that corr(DA,DB) 2 Y-1.

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Measuring association between IBD and a
quantitative trait

• For an entire population of individuals, we can
  aggregate the individual D.
• Let p the proportion of alleles at a locus that
  are IBD between pairs of relatives. Note p 0,
  0.5, 1 corresponding to D 0, 1, 2 which
  counts alleles for a single sib-pair.
• Let X1 and X2 be the (continuous) quantitative
  trait values of siblings 1 and 2.
• Haseman/Elston method Regress (X1-X2)2 onto p
• A regression coefficient significantly less than
  zero is evidence for linkage.

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x
x
x
x
x
(X1-X2)2
x
x
x
x
x
p
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Components of the genetic model

• Components of the genetic model include
  inheritance pattern (dominant vs. recessive,
  sex-linked vs. autosomal), trait allele frequency
  (a common or rare disease?), and the frequency of
  new mutation at the trait locus.
• Another important component of the genetic model
  is the penetrance of the trait allele. Knowing
  the penetrance of the disease allele is crucial
  because it specifies the probability that an
  unaffected individual is unaffected because he's
  a non-gene carrier or because he's a
  non-penetrant gene carrier. The frequency of
  phenocopies is an important component, too.
• Rough estimates of the disease allele frequency
  and penetrance can often be obtained from the
  literature or from computer databases, such as
  Online Mendelian Inheritance in Man
  (http//www3.ncbi.nlm.nih.gov/Omim/). Estimates
  of the rate of phenocopies and new mutation are
  frequently guesses, included as a nuisance
  parameter in some cases to allow for the fact
  that these can exist.
• Linkage analysis is relatively robust to modest
  misspecification of the disease allele frequency
  and penetrance, but misspecification of whether
  the disease is dominant or recessive can lead to
  incorrect conclusions of linkage or non-linkage.

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Steps to linkage analysis

• In pedigrees in which the genetic model is known,
  linkage analysis can be broken down into five
  steps
• State the components of the genetic model.
• Assign underlying disease genotypes given
  information in the genetic model.
• Determine putative linkage phase.
• Score the meiotic events as recombinant or
  non-recombinant.
• Calculate and interpret LOD scores.
• Let's take a look at each of these steps in
  detail.

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State the components of the model

• In this example, the disease allele will be
  assumed to be rare and to function in an
  autosomal dominant fashion with complete
  penetrance, and the disease locus will be assumed
  to have two alleles
• N (for normal or wild-type)
• A (for affected or disease)

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Assign underlying disease genotypes

• The assumption of complete penetrance of the
  disease allele allows all unaffected individuals
  in the pedigree to be assigned a disease genotype
  of NN. Since the disease allele is assumed rare,
  the disease genotype for affected individuals can
  be assigned as AN.

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Determine putative phase

• Individual II-1 has inherited the disease trait
  together with marker allele 2 from his affected
  father. Thus, the A allele at the disease locus
  and the 2 allele at the marker locus were
  inherited in the gamete transmitted to II-1.
  Once the putative linkage phase (the disease
  allele "segregates" with marker allele 2) has
  been established, this phase can be tested in
  subsequent generations.

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Score the meiotic events asrecombinant (R) or
non-recombinant (NR)

• There are four possible gametes from the
  affected parent II-1 N1, N2, A1, and A2. Based
  on the putative linkage phase assigned in step 3,
  gametes A2 and N1 are non-recombinant.

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Calculate LOD scores

• In this example, the highest LOD score is -0.09
  at q 0.40. At no value of q is the lod score
  positive, let alone gt3.0, so this pedigree has no
  evidence in favor of linkage between the disease
  and marker loci.