Thoughts%20about%20the%20TDT

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Thoughts about the TDT
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Contribution of TDT FindingGenes for 3 Complex
Diseases

• PPAR-gamma in Type 2 diabetes
• Altshuler et al. Nat Genet 2676-80, 2000
• NOD2 in Crohns Disease
• Hugot et al., Nature 411 599-603, 2001
• ADAM33 in asthma
• Van Eerdewegh et al., Nature 418 426-430, 2002

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The common PPAR-gamma Pro12Ala polymorphism is
associated with decreased risk of type 2 diabetes

Altshuler et al. Nat Genet 2676-80, 2000
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NOD2 Variants and Susceptibility to Crohns
Disease
Chrom 16q
SNP13 p6x10-6
Hugot et al., Nature 411 599-603, 2001
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ADAM33 Gene Asthma and Bronchial
Hyperresponsiveness
Chrom 20p
P 3x10-6 to 0.04

• Van Eerdewegh et al., Nature 418 426-430, 2002

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Population distributions of (a) disease given
genotype, and (b) genotype given disease.
Affected Affected Affected Affected Affected
Genotype Yes No Genotype Yes No No
M1M1 a 1 a M1M1 d g
M1M2 b 1 b M1M2 e h
M2M2 c 1 c M2M2 f i
(a) (a) (a) (b) (b) (b) (b)
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Clayton
Odds Ratio
Ott
He calls this the relative risk. Confusing!
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D1 M1 D2 M2
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• Null hypothesis ? ½
• (Disease and marker loci unlinked)
• Alternative hypothesis ? lt ½
• (Disease and marker loci linked)

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freq (D1 M1) ? freq (D1) freq (M1) d freq
(D1 M1) freq (D1) freq (M1)
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• We assume that we observe the marker locus
  genotypes, either M1M1, M1M2, or M2M2, of both
  parents and the affected sibs in all families in
  the data.

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Probabilities for transmitted and non-transmitted
marker alleles M1 and M2 from any parent of an
affected child. Non-transmitted
allele Transmitted Allele M1 M2 Total
M1 P(11) P(12) P(1.) M2 P(21) P(22)
P(2.) Total P(.1) P(.2) 1
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P(11) q2 q d / p P(12) q (1 q) (1 ?
q) d / p P(21) q (1 q) (? q) d / p P(22)
(1 q)2 (1 q) d / p
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Numbers of transmitted and non-transmitted
marker alleles M1 and M2 among the parents of
the affected sibs Non-transmitted allele
Transmitted Allele M1 M2
M1 n11 n12 M2 n21 n22
Put n12 n21 n
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Only P(12) and P(21) depend on ? . Also, when ?
½, P(12)P(21) So the natural (TDT) test
statistic is
This (McNemar statistic) has an asymptotic 1 df
?2 distribution when the null hypothesis is true.
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• Note that this statistic depends only on n12 and
  n21 only, and ignores n11 and n22.
• This makes sense the statistic uses data only
  from M1M2 parents, and only these are informative
  for linkage.
• We call these informative parents.
• So at the end of the day we consider only
  transmissions from informative parents.

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• We will focus entirely on the denominator, n, of
  the TDT statistic.
• It is remarkable how many questions one can ask
  about this.
• But before we ask these, we first ask, where does
  this denominator come from?

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• Assuming the null hypothesis is true, n12 has a
  binomial (n, ½) distribution.
• Note this is true even if the data contain
  several affected children from the same family.
• Thus the variance of n12 - n21 ( 2n12 n) is
  4n/4 n.

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• We will examine three situations, all focusing on
  the question Is n the correct (variance)
  denominator for the situation at hand?.

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• Situation 1. Testing for association.
• Here the null hypothesis is no association, or

The problem here is that transmissions to
different affected sibs in the same family are
not independent under this null hypothesis. Thus
when there are several families in the data with
more than one affected sib, n12 does not have a
binomial distribution.
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If H0, d 0, is true, the cell probabilities for
the simple random-mating case are P(11) q2
, P(12) q(1 q) , P(21) q(1 q) , P(22)
(1 q)2 (Thus should we not be testing this H0
by using both n11 n22 n12 n21 and n12 n21
and a 2 degrees of freedom test?) Lets ignore
this point for now.
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P(11) (Si ai (pi2 qi2 di pi qi )) / (Si ai
pi2) P(12) (Si ai (pi2 qi (1 qi) di pi (1
? qi))) / (Si ai pi2) P(21) (Si ai (pi2 qi
(1 qi) di pi (? qi))) / (Si ai pi2) P(22)
(Si ai (pi2 (1 qi)2 di pi (1 qi))) / (Si ai
pi2) ai relative size of subpopulation i di
linkage disequilibrium in subpopulation i pi
frequency of D1 in subpopulation i qi frequency
of M1 in subpopulation i
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Suppose that in family j, M1 is transmitted n12j
times, M2 is transmitted n21j times, from M1M2
parents. Define Dj as n12j n21j The test
statistic is
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Suppose that there is only one affected child in
each family. Then Dj 1 (for all j)
T2 TDT ?2
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• Situation 2. Suppose we have families in the data
  where both parents are dead, (so we do not know
  their marker locus genotypes), but where there
  are two affected sibs, one being M1M1, the other
  M2M2.
• We therefore can infer that both parents were
  informative.
• Should we use the data from these families in the
  analysis, using the standard TDT statistic?

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• The answer is no. Why is this so?
• Because the very fact that we can infer the
  parental genotypes unambiguously means that one
  sib MUST be M1M1 and the other MUST be M1M1.
• In such families there is zero variance, rather
  than some binomial variance, for the number of M1
  genes in the two sibs.

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• Philosophical question is there any difference
  between the actions you take in directly
  observing an event and having unambiguous
  evidence that the event occurred?
• In this case, yes there is.

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Situation 3. Suppose that we have two affected
sibs, one informative (i.e. M1M2) parent, in each
family in the data.
Numbers of transmission from the informative parents Numbers of transmission from the informative parents Numbers of transmission from the informative parents Numbers of transmission from the informative parents
2M1 1M1 , 1M2 2M2 Total
families i j k n
H0 means n/4 n/2 n/4 n
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Suppose that a sharing ?2 has been carried out,
correctly, as a one-sided test. Given i k s,
what is the distribution of ?2TDT ?
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,
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One affected sib, two informative (M1M2) parents
Genotype of affected child Genotype of affected child Genotype of affected child Genotype of affected child
M1 M1 M1 M2 M2M2 Total
families p q r n
Expected when H0 (?½) n/4 n/2 n/4 n
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Subpopulation 1 2 i k Relative
Size a1 a2 ai ak Coefficient of
gametic Disequilibrium d1 d2 di dk
Generation 0 Generation
1 Generation 2 Generation 3
Parents of generation 1 mate only within their
subpopulation
Gametic Disequilibrium ?1
Parents of generation 2 mate at random throughout
population
Gametic Disequilibrium ?2 ?1 (1?)
Parents of generation 3 mate at random throughout
population
Gametic Disequilibrium ?3 ?2 (1?)
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Generation 0
Generation 1 Gametic Disequilibrium ?1
Generation 2 Gametic Disequilibrium ?2
Generation 3 Gametic Disequilibrium ?3
Generation 4, etc
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The value of the TDT statistic in two
models 1. Immediate admixture Generation
1 1.48 Generation 2 2.07 Generation
3 15.34 Generation 4 12.43 2. Gradual
admixture Generation 1 1.48 Generation
2 2.07 Generation 3 8.53 Generation
4 6.99