Confounding from Cryptic Relatedness in Association Studies

1
Confounding from Cryptic Relatedness in
Association Studies

• Benjamin F. Voight
• (work jointly with JK Pritchard)

2
Importance

• Case/control association tests are becoming
  increasingly popular to identify genes
  contributing to human disease.
• These tests can be susceptible to false positives
  if the underlying statistical assumptions are
  violated, i.e. independence among all sampled
  alleles used in the test for association.
• It is well appreciated that population structure
  results in false positives (Knowler et al., 1988
  Lander and Schork, 1994).
• Methods exist which correct for this effect
  (Devlin and Roeder, 1999 Pritchard and
  Rosenberg, 1999 Pritchard et al. 2000).

• Case/control association tests are becoming
  increasingly popular to identify genes
  contributing to human disease.
• These tests can be susceptible to false positives
  if the underlying statistical assumptions are
  violated, i.e. independence among all sampled
  alleles used in the test for association.
• It is well appreciated that population structure
  results in false positives (Knowler et al., 1988
  Lander and Schork, 1994).
• Methods exist which correct for this effect
  (Devlin and Roeder, 1999 Pritchard and
  Rosenberg, 1999 Pritchard et al. 2000).

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Your (favorite) Population
Obtain a sample of affected cases from
the population.
Obtain a sample of affected cases from
the population.
Cases are not independent draws from the
population allele frequencies.
Cases are not independent draws from the
population allele frequencies.
Problem the relatedness is cryptic, so the
investigator does not know about
the relationships in advance.
Problem the relatedness is cryptic, so the
investigator does not know about
the relationships in advance.
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Importance

• Devlin and Roeder (1999) have argued that if one
  is doing a genetic association study, then surely
  one must believe that the trait of interest has a
  genetic basis that is at least (partially) shared
  among affected individuals.
• Given that cases share a set of risk factors by
  descent, then presumably they are more related to
  one another than to random controls.
• These authors presented numerical examples which
  suggested that this effect may be an important
  factor, in practice.
• However, these examples were artificially
  constructed, and not modeled on any
  population-based process.
• Few empirical data to suggest if cryptic
  relatedness negatively impacts association
  studies. In a founder population,
  non-independence resulting from relatedness does
  matter. (Newman et al., 2001).

• Devlin and Roeder (1999) have argued that if one
  is doing a genetic association study, then surely
  one must believe that the trait of interest has a
  genetic basis that is at least (partially) shared
  among affected individuals.
• Given that cases share a set of risk factors by
  descent, then presumably they are more related to
  one another than to random controls.
• These authors presented numerical examples which
  suggested that this effect may be an important
  factor, in practice.
• However, these examples were artificially
  constructed, and not modeled on any
  population-based process.
• Few empirical data to suggest if cryptic
  relatedness negatively impacts association
  studies. In a founder population,
  non-independence resulting from relatedness does
  matter. (Newman et al., 2001).

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Goals

• Determine whether, or when, cryptic relatedness
  is likely to be a problem for general
  applications.
• Develop a formal model for cryptic relatedness in
  a population genetics framework.
• In a founder population, estimate the inflation
  factor due to (cryptic) relatedness, and compare
  to analytical results.
• Avoid staring at x in front of a chalkboard.

• Determine whether, or when, cryptic relatedness
  is likely to be a problem for general
  applications.
• Develop a formal model for cryptic relatedness in
  a population genetics framework.
• In a founder population, estimate the inflation
  factor due to (cryptic) relatedness, and compare
  to analytical results.
• Avoid staring at x in front of a chalkboard.

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Modeling Definitions

• m affected individuals and m random controls,
  sampled in the current generation.
• Pairs of chromosomes coalesce in a previous
  generation t 1, 2, t with the usual
  probabilities.
• All samples are typed at a single bi-allelic
  locus, unlinked to disease, with alleles B and b,
  at frequencies p and (1-p) in the population.

• m affected individuals and m random controls,
  sampled in the current generation.
• Pairs of chromosomes coalesce in a previous
  generation t 1, 2, t with the usual
  probabilities.
• All samples are typed at a single bi-allelic
  locus, unlinked to disease, with alleles B and b,
  at frequencies p and (1-p) in the population.

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Definitions

• Define
• Kp population prevalence of disease.
• Kt probability that an relative of type t (or t
  ) of an affected proband is also affected.
• lt recurrence risk ratio, Kt/Kp (Risch, 1990).
• Gi(a) indicator (0 or 1) for the B allele on
  homologous chromosome a for the i-th case. (with
  a Î 0, 1 for diploid individuals)
• Hj(a) as above, but for a j-th random control.

• Define
• Kp population prevalence of disease.
• Kt probability that an relative of type t (or t
  ) of an affected proband is also affected.
• lt recurrence risk ratio, Kt/Kp (Risch, 1990).
• Gi(a) indicator (0 or 1) for the B allele on
  homologous chromosome a for the i-th case. (with
  a Î 0, 1 for diploid individuals)
• Hj(a) as above, but for a j-th random control.

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• Define a test statistic which measure the
  difference in allele counts between cases and
  controls (slightly modified from Devlin and
  Roeder, 1999)

• Define a test statistic which measure the
  difference in allele counts between cases and
  controls (slightly modified from Devlin and
  Roeder, 1999)

• Under the null hypothesis of no association
  between the marker and phenotype, an allele has a
  genotype B with probability p, independently for
  all alleles in the sample. If so,

• Under the null hypothesis of no association
  between the marker and phenotype, an allele has a
  genotype B with probability p, independently for
  all alleles in the sample. If so,

• If cryptic relatedness exists in the sample, then
  the variance of the test call this VarT
  may exceed the variance under the null. We
  measure the deviation from the null variance
  using the inflation factor d

• If cryptic relatedness exists in the sample, then
  the variance of the test call this VarT
  may exceed the variance under the null. We
  measure the deviation from the null variance
  using the inflation factor d

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• Recall that we want the variance to our test, T,
  under a model of cryptic relatedness

• Recall that we want the variance to our test, T,
  under a model of cryptic relatedness

• Use the following non-dodgy assumptions
• 1. Draws of alleles from the population are
  simple Bernoulli trials. (Variance terms)
• 2. Controls are a random sample from the
  population. (Covariance terms with Hjs are 0)
• 3. Allow the possibility that cases and controls
  depart from Hardy-Weinberg proportions by some
  factor, call this F. (Covariance terms for
  alleles in the same individual)
• 4. For the mutational model,
• a. Suppose the mutation process is the same for
  cases and random controls.
• b. Conditional on a case and random chromosome
  having a very recent coalescent time (on the
  order of 1-10 generations), assume that the
  chance that the alleles are in different states
  is 0.

• Use the following non-dodgy assumptions
• 1. Draws of alleles from the population are
  simple Bernoulli trials. (Variance terms)
• 2. Controls are a random sample from the
  population. (Covariance terms with Hjs are 0)
• 3. Allow the possibility that cases and controls
  depart from Hardy-Weinberg proportions by some
  factor, call this F. (Covariance terms for
  alleles in the same individual)
• 4. For the mutational model,
• a. Suppose the mutation process is the same for
  cases and random controls.
• b. Conditional on a case and random chromosome
  having a very recent coalescent time (on the
  order of 1-10 generations), assume that the
  chance that the alleles are in different states
  is 0.

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Then after
JKP attempts desperately to keep me honest.
Smoke from my brain
Me, after many hours of intensive
thought processing
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• VarT can be simplified to

where i?i.

• And now, we evaluate the covariance term under a
  model of cryptic relatedness. This covariance
  term is fairly complicated, but it is related to
  the following probability

13

• Apply some Bayesian Trickery

• and after some plug and play we finally get

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Under an additive model

• Handy relationship between any lrs and the
  sibling recurrence risk ratio, a single parameter
  under an additive model (Risch, 1990)

where fr is the kinship coefficient for type-r
relatives, which is ¼ for r 1, and decays by ½
for each increment to r. Using this relationship
we can simplify
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Simulations

• Use Wright-Fisher forward simulation to assess
  analytical results
• Simulate 1,000 bi-allelic unlinked loci forward
  in time 4N generations, with mutation parameter q
  4Nm 1. ()
• Choose a single locus with the desired disease
  allele frequency, and assign phenotypes to all
  members of the population under an additive
  genetic model.
• Select m cases and m random controls, use all
  non-disease loci to infer the inflation factor
  based on the mean of all tests.

() because WF simulations are notoriously slow
to simulate, we use a speed-up by simulating a
smaller population with a proportionally higher
mutation rate, and then rescale the population
size and mutation rate to the desired levels.
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Simulation Results
95 central interval about the mean was at least
.001 in each case.
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Tautological Hutterite Analysis

• Quick-note on the Hutterites
• 13,000 member pedigree where the genealogy is
  known, with 800 members phenotyped/genotyped at
  many markers across the genome.
• Target (for each phenotype)
• a. Estimate coalescent probabilities for cases
  and random controls based on the genealogy
  allele-walking simulations
• b. Calculate the inflation factor (d) for each
  phenotype, and compare to the analytic
  prediction.

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Note increased probabilities in cases over random
controls for recent coalescent times
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Hutterite Analysis

• Quick-note on the Hutterites
• 13,000 member pedigree where the genealogy is
  known, with 800 members phenotyped/genotyped at
  many markers across the genome.
• Target (for each phenotype)
• a. Estimate coalescent probabilities for cases
  and random controls based on the genealogy
  allele-walking simulations
• b. Calculate the inflation factor (d) for each
  phenotype, and compare to the analytic
  prediction.

21
Empirical ds in a Founder Population
The inbreeding coefficient (F) was estimated at
.048 and was included in the calculation.
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Summary

• We modeled cryptic relatedness using
  population-based processes. Surprisingly, these
  expressions are functions of directly observable
  parameters (population size, sample size, and the
  genetic model parameterized by lr).
• Our analytical results indicate that increased
  false positives due to cryptic relatedness will
  usually be negligible for outbred populations.
• We applied out technique to a founder population
  as an example. For six different phenotypes we
  found evidence for inflation, which matched
  analytic predictions.

• We modeled cryptic relatedness using
  population-based processes. Surprisingly, these
  expressions are functions of directly observable
  parameters (population size, sample size, and the
  genetic model parameterized by lr).
• Our analytical results indicate that increased
  false positives due to cryptic relatedness will
  usually be negligible for outbred populations.
• We applied out technique to a founder population
  as an example. For six different phenotypes we
  found evidence for inflation, which matched
  analytic predictions.

23
Acknowledgements

• JK Pritchard and NJ Cox (thesis advisors)
• Carole Ober (access to the empirical data)
• /
• NIH, NIH/NIGMS Genetics Training Grant

Fine, name that tune from memory, recite of the
first 1677 words of Kingmans 1982 paper and Ill
get the next round.
In the bar at the conference during the week