BMI 731 Winter 2005 Chapter1: SNP Analysis

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BMI 731- Winter 2005Chapter1 SNP Analysis

• Catalin Barbacioru
• Department of Biomedical Informatics
• Ohio State University

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Biological Background

• Cells are fundamental working units of every
  living systems
• The nucleus contains a large DNA
  (Deoxyribonucleic acid) molecule, which carries
  the genetic instructions
• A DNA molecule consists of two strands that wrap
  around each other to resemble a twisted ladder.
• Each strand is composed of one sugar molecule,
  one phosphate molecule, and a base.
• Four different bases are present in DNA - adenine
  (A), thymine (T), cytosine (C), and guanine (G).
• The particular order of the bases arranged along
  the sugar - phosphate backbone is called the DNA
  sequence

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Biological Background
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Biological Background

• Each strand of the DNA molecule is held together
  at its base by weak hydrogen bonds.
• The four bases pair in a set manner Adenine (A)
  pairs with thymine (T), while cytosine (C) pairs
  with guanine (G). These pairs of bases are known
  as Base Pairs (bp). 
• The DNA is organized into separate long segments
  called chromosomes, where the number of
  chromosomes differ across organisms (46 for
  humans or 23 pairs, each parent contributes 23
  chromosomes)

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Glossary

• Allele Alternative form of a gene. One of the
  different forms of a gene that can exist at a
  single locus.
• Genotype The specific allelic composition of a
  cell, either of the entire cell or more commonly
  for a certain gene or a set of genes.
• Haplotype A set of closely linked genetic
  markers present on one chromosome which tend to
  be inherited together (not easily separable by
  recombination).

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Glossary

• Locus A point in the genome, identified by a
  marker, which can be mapped by some means.
• Marker Also known as a genetic marker, a segment
  of DNA with an identifiable physical location on
  a chromosome whose inheritance can be followed. A
  marker can be a gene, or it can be some section
  of DNA with no known function.
• Mutation A permanent structural alteration in
  DNA.

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Glossary

• Hardy-Weinberg equilibrium The stable frequency
  distribution of genotypes, AA, Aa, and aa, in the
  proportions p2, 2pq, and q2 respectively (where
  p and q are the frequencies of the alleles, A and
  a) that is a consequence of random mating in the
  absence of mutation, migration, natural
  selection, or random drift.
• Linkage disequilibrium When the observed
  frequencies of haplotypes in a population does
  not agree with haplotype frequencies predicted by
  multiplying together the frequency of individual
  genetic markers in each haplotype.

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A Little Population Genetics

• Population genetics (and evolutionary genetics)
  deal with groups of organisms and families,
  usually natural populations.
• We can discern two strands of thought in the
  area. One is the study of very large ("ideal")
  idealized groups or populations, where models can
  be deterministic.
• The other is dealing with smaller populations,
  where the role of chance can play a larger role
  (so called genetic drift).

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Genotype and allele frequencies

• One question of crucial interest is this how
  common are the different alleles at a given locus
  in a given population.

The percentages are our best estimate of the
probability that an individual will carry that
genotype in the population of London, Oxford and
Cambridge. The observed heterozygosity is 49.6.
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There is another population described in this
table. It is the population of gametes that gave
rise to individuals tested
The percentages here are our best estimate of the
probability that a sperm or egg taken from that
population will carry that particular allele. If
the frequency of the commonest allele at a
particular locus is less than 99, we call this a
polymorphic locus or polymorphism.
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Hardy-Weinberg equilibrium

• Hardy-Weinberg equilibrium describes the
  relationship between the gametic or allele
  frequencies, and the resulting genotypic
  frequencies. It holds if the following properties
  are true for the given locus,
• 1.Random mating or panmixia the choice of a
  mate is not influenced by his/her genotype at the
  locus.
• 2.The locus does not affect the chance of mating
  at all, either by altering fertility or
  decreasing survival to reproductive age.

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• If these properties hold, then the probability
  that two gametes will meet and give rise to a new
  genotype is simply the product of the allele
  frequencies (a la binomial)
• P(AA) P(A) x P(A) pA2
• P(aa) P(a) x P(a) pa2
• P(Aa) 1 - P(AA) - P(aa) 2 x P(A) x P(a)
• 2pApa.

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Tests for HWE

• For a two-allele case, disequilibrium coefficient
  is
• D PAA pA2
• where PAA P(AA) the probability of AA genotype
  and
• pA P(A) is the probability of allele A.
• If nAA, nAa, naa are the numbers of individuals
  with genotypes AA, Aa and aa respectively, from a
  total of n individuals, then estimators of the
  above probabilities are
• PAA nAA/n, PAa nAa/n, Paa naa/n, where n
  nAAnAanaa
• pA (2nAAnAa)/2n, pa (2naanAa)/2n and pa
  pA 1

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Chi-square testfor HWE

• Then under HWE

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Chi-square testfor HWE

• The goodness-of-fit chi-squared statistic is
• XA2 Sgenotypes (Obs-Exp)2/Exp
• (nD)2/npA2 (-2nD)2/2npApa (nD)2/npa2
• nD2/pA2(1-pA)2
• and the test rejects (H0) the assumption of HWE
  if
• XA2 gt 3.84
• The usual problems associated with this test that
  it is sensitive to small expected values. An
  alternative version (Yates), which overcomes
  continuity assumptions is
• XA2 Sgenotypes (Obs-Exp-0.5)2/Exp

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Fisher (exact) test for HWE

• Under HWE hypothesis, the probability of the
  observed set of genotypic counts nAA, nAa and naa
  in a sample of size n is

whereas the allele counts nA and na are
binomially distributed if HWE holds
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Fisher (exact) test for HWE

• Putting together, the probability of the observed
  genotypic frequencies, assuming HWE, conditional
  on the observed allele frequencies is

which can be expressed in terms of the allele A
number and Of the number of heterozygotes nAa. We
reject the HWE hypothesis if the above
conditional probability is less than the
significance level of type I error (a), usually
0.05.
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HWE test - Example
Causes rejection of HWE at 5 significance level
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Power and sample size of tests for HWE

• Statistical tests of hypothesis are subject to
  two kind of errors a true hypothesis may be
  rejected (type I error or a or significance level
  or p-value) or a false hypothesis may not be
  rejected (type II error or ß or 1-power of the
  test).
• For the chi-square test, theory provides that, in
  large samples, X2 is distributed approximately as
  a chi-square with 1 d.f. when the hypothesis is
  true and as a noncentral chi-square when the
  hypothesis is false i.e.
• X2 ?2(1) when H0 is true
• X2 ?2(1, ?) when H0 is false
• where ? is the noncentrality parameter (see
  tables).

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Power and sample size of tests for HWE

• The disequilibrium coefficient, D, required for
  attaining 90 power and a 0.05 significance level
  for the chi-square test is

Alternatively, the number of samples required in
order to attain 90 power and a 0.05
significance level for the chi-square test when
the disequilibrium coefficient is D, is
If the required power is 50 or 80, then 10.5
is replaced by 3.84 or 8.7
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Linkage disequilibriumGametic disequilibrium at
two loci

• Measures the association of two alleles at two
  different loci.
• Given two biallelic loci with alleles A, a and B,
  b respectively, let the disequilibrium
  coefficient be
• DAB pAB pApB.
• The (ML) estimator of DAB is DAB pAB pApB.
• A chi-square statistic for the hypothesis of no
  disequilibrium, H0 DAB0, is the test statistic

and the test rejects H0 if XAB2 gt 3.84 .
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Linkage disequilibriumGametic disequilibrium at
two loci

• An exact test for gametic linkage disequilibrium
  depends on the probabilities of all possible
  samples of gametic numbers for the observed
  allele numbers. Under the assumption of no
  linkage disequilibrium

and the allele probabilities are
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Linkage disequilibriumGametic disequilibrium at
two loci

• Taking the ratio between these quantities gives
  the probability of gametic numbers conditional on
  allele numbers

which depends on n, nAB, nA and nB only. As in
the case of HWE, this probability is compared
with the chosen significance Level (p-value).
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Linkage disequilibrium Genotypic disequilibrium

• When genotypes are scored, it is often not
  possible to distinguish between the two double
  heterozygotes AB/ab and Ab/aB, so that the
  gametic frequencies cannot be inferred. Under the
  assumption of random mating, in which genotypic
  frequencies are assumed to be the products of
  gametic frequencies, it is possible to estimate
  gametic frequencies. A measure of (digenic)
  linkage disequilibrium between alleles A and B
  is

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Linkage disequilibrium Genotypic disequilibrium

• If the 9 genotypic classes are numbered as

then an (ML) estimator for ?AB is
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Linkage disequilibrium Genotypic disequilibrium

• The chi-square test statistics for LD is

Note the explicit way in which departures from HW
are Included in this expresion.
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• ?2 represents the statistical correlation between
  two sites, and takes value 1 if only two
  haplotypes are present. It is arguably the most
  relevant measure for association between
  susceptibility loci and SNPs. For example,
  suppose SNP1 is involved in disease
  susceptibility, but we genotype cases and
  controls at a nearby site SNP2. Then, to achieve
  the same power to detect associations at SNP2 as
  we would have at SNP1, we need to increase our
  sample size by a factor of 1/ ?2.

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• These measures are defined for pairs of sites,
  but for some applications we might instead want
  to measure how strong LD is across an entire
  region that contains many polymorphic sites for
  example, for testing whether the strength of LD
  differs significantly among loci or across
  populations, or whether there is more or less LD
  in a region than predicted under a particular
  model. Measuring LD across a region is not
  straightforward, but one approach is to use the
  measure ?, which measures how much recombination
  would be required under a particular population
  model to generate the LD that is seen in the
  data. The development of methods for estimating
  is now an active research. This type of method
  can potentially also provide a statistically
  rigorous approach to the problem of determining
  whether LD data provide evidence for the presence
  of hotspots.