Detecting and correcting for population structure in casecontrol studies

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Detecting and correcting for population structure
in case-control studies
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Population structure in genetic association
studies

• Population consists of underlying subpopulations.
• Disease prevalence different between
  subpopulations.
• Cases preferentially ascertained from specific
  subpopulations.
• False positive evidence of association will occur
  at genetic markers that differ in genotype
  frequencies between the subpopulations.
• Traditionally, human geneticists have been
  skeptical of case-control studies for this reason.

CASES
CONTROLS
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Example

• Population consists of two equally frequent
  isolated sub-populations.
• In the population overall Pr(Disease MM) lt
  Pr(Disease) Pr(MM)
• If we ascertain individuals without regard to
  subpopulation, cases tend to be selected from
  subpopulation 1, which has a low frequency of the
  MM marker genotype.

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Disease prevalence and marker allele frequencies
vary across populations
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Matching

• One solution to the problem is to allow for
  structure at the design stage, by matching cases
  and controls for ethnic group, for example.
• When a case is selected from a given ethnic
  group, a matched control is selected from the
  same group.
• Matched case-control studies require a matched
  analysis.
• However, there may be fine-scale structure or
  within ethnic groups or population admixture that
  cannot be accounted for by matching.
• Apparent association between SNPs and type 2
  diabetes in Pima Indians.
• Type 2 diabetes occurs with greater prevalence in
  Caucasian individuals.
• Association due to population admixture cases
  tended to have a greater proportion of Caucasian
  ancestry, and allele frequencies vary between the
  ancestral populations.

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Solutions to the problem

• We can eliminate the problem of population
  structure by collecting family data.
• Family-based association designs ascertain
  affected cases and their parents.
• Form internal controls from alleles not
  transmitted from the parents to the child,
  effectively matching for ancestry.
• Less powerful since two parents are required to
  form a single matched control.
• Parental data may not always be available, e.g.
  for late-age onset diseases.
• For unrelated samples of cases and controls, we
  can make use of genotype data across the genome
  to make inferences about and/or adjust for
  population ancestry.
• In the presence of structure, there will be many
  more (false) positive signals of association than
  we would expect by chance.

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Test for mis-matching of case-control samples

• If an apparent association is due to population
  structure, there should be many such associations
  across the genome (because allele frequencies
  vary between populations).
• Pritchard and Rosenberg (1999) suggest a strategy
  of typing additional random markers scattered
  throughout the genome to test for the presence of
  structure.
• Type the sample at L unlinked markers and test
  for disease-marker association at each marker.
  Test statistic for structure
• where Xi2 is the usual Cochran-Armitage trend
  test statistic for the ith marker.
• Under the null hypothesis of no population
  substructure
• We expect no association between disease and
  random markers.
• X2SUB has an approximate chi-squared distribution
  with L degrees of freedom.

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Comments

• This test looks for an average difference in
  ancestry between the case and control samples,
  not for population structure per se.
• Do not assign individuals to specific
  populations, but provides an overall test of
  substructure.
• If you have a large number of candidate loci, you
  may choose to use the candidate data together to
  test for mismatching.
• How many markers are required to detect
  structure?
• Depends on the extent of structure in the
  population.
• Pritchard and Rosenberg (1999) suggest 20
  microsatellite markers or 30 SNPs to detect
  moderate stratification.
• As the price of genotyping decreases it becomes
  more feasible to use a large number of markers to
  test for structure, and hence identify fine-scale
  stratification.

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Genomic control

• Devlin and Roeder (1999) used theoretical
  arguments to propose that with population
  structure, the distribution of Cochran-Armitage
  trend tests, genome-wide, is inflated by a
  constant multiplicative factor ?.
• We can estimate the multiplicative inflation
  factor using the statistic ?
  median(Xi2)/0.465.
• Inflation factor ? gt 1 indicates population
  structure and/or genotyping error.
• We can carry out an adjusted test of association
  that takes account of any mismatching of
  cases/controls at any SNP using the statistic
  Xi2/ ?.

True hits?
Structure?
Inflation factor ? 1.11
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Comments

• Advantages.
• Easy to implement genomic control in whole genome
  association studies.
• Requires relatively small numbers of markers
  (minimum of around 50 SNPs).
• Can be extended to the analysis of quantitative
  traits and to genotype frequencies obtained from
  pooled DNA.
• Disadvantages.
• Limited to relatively simple tests of
  association, and is less robust to haplotype
  tests, for example.
• There will be a loss in power if there are
  different genetic effects acting in the different
  subpopulations.

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Structured association

• A two-phase approach
• Use genotype data from unlinked genetic markers
  to learn about population structure, and to infer
  the ancestry of individuals in the sample
  (Pritchard et al. 2000a).
• Makes use of Bayesian MCMC technology to
  approximate the posterior probability that any
  individual comes from a specific subpopulation,
  or the proportion of ancestry of any individual
  from each subpopulation.
• Implemented in the STRUCTURE software package.
• Test for association at candidate loci taking
  account of the ancestry of cases and controls
  (Pritchard et al. 2000b)
• Implemented in the STRAT software package.

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STRUCTURE

• Consider a sample of unrelated individuals typed
  at many unlinked markers (i.e. not in LD),
  yielding genotypes G.
• Assume that the sampled individuals have been
  ascertained from K discrete subpopulations.
• Goal is to estimate allele frequencies, pK,
  within each subpopulation (under the assumption
  of Hardy-Weinberg equilibrium) and the
  assignments, Z, of each individual to each
  subpopulation.
• Conditional on allele frequencies, pK, we can
  write down an expression for the posterior
  distribution of subpopulation assignments for the
  ith individual using Bayes theorem
• where Pr(ziJ) is the prior probability of
  assignment to subpopulation J, typically taken to
  be 1/K.

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• Similarly, conditional on the subpopulation
  assignments of each individual, Z, we can write
  down an expression for the posterior distribution
  of allele frequencies, Pr(pKZ,G), given a
  pre-specified prior density Pr(pK).
• STRUCTURE uses Bayesian Markov chain Monte Carlo
  (MCMC) techniques to sample, in turn, from the
  densities Pr(pKZ,G) and Pr(ZpK,G). Algorithm
  run for an initial burn-in period to allow for
  convergence. In the subsequent sampling period,
  sampled values of the population allele
  frequencies and subpopulation assignments for
  each individual are recorded to approximate the
  marginal posterior distributions of pK and Z.
• One drawback is the choice of K, the number of
  subpopulations, which will not generally be known
  in advance. Ad-hoc assessment of best choice of
  K by comparison of model fit (likelihoods).
• The basic structure model assumes that each
  individual belongs to just one discrete
  subpopulation. STRUCTURE also allows for admixed
  populations, where the subpopulation assignments,
  Z, are replaced by vectors of ancestry, Q, where
  qik denotes the proportion of the ancestry of the
  ith individual that comes from subpopulation k.

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Example
Study of 1056 individuals from 52 populations
using 377 autosomal microsatellite markers.
Rosenberg et al. (2002).
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Example (continued)
Each individual is a thin vertical line that is
partitioned into K coloured segments according to
its membership coefficients in K clusters.
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Example (continued)
Inferred ancestry within geographic regions. 93
of variability in allele frequencies occurs
within populations.
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STRAT

• In a structured population, the null hypothesis
  of interest is of no association of disease
  phenotype with marker genotypes within each
  subpopulation. The alternative hypothesis is
  that genotype frequencies vary with subpopulation
  and disease phenotype.
• Allows for the possibility of different genetic
  effects in each subpopulation (allelic or genetic
  heterogeneity).
• Conditional on the ancestries (Q) estimated by
  STRUCTURE, we can construct a test statistic by
  computing the likelihood ratio under the two
  hypotheses
• Allele frequencies (p0 and p1) in each
  subpopulation under the two hypotheses estimated
  via maximum likelihood via implementation of the
  expectation-maximisation algorithm.
• Simulation used to assess the significance of ?.

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Comments

• Advantages.
• Versatility can allow for discrete or admixed
  subpopulations, and can utilise SNPs and
  microsatellites.
• Provides a general framework for allowing for
  structure in association tests could be extended
  to multi-locus or haplotype tests.
• Provides detailed information about population
  structure.
• Disadvantages.
• Use and interpretation requires some care (for
  example choice of number of subpopulations).
• Requires more markers than genomic control.
• Too computationally intensive for whole genome
  data requires selection of a subset of unlinked
  markers.

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Multivariate techniques

• Principal components analysis (PCA) has become a
  standard tool in genetics to study geographic
  variation in allele frequencies.
• PCA is used to infer continuous axes of genetic
  variation (eigenvectors) that reduce the data to
  a small number of dimensions, whilst describing
  as much of the variability between individuals as
  possible.
• Patterson et al. (2006) demonstrate that PCA can
  be used to identify population structure in large
  scale data sets with hundreds of thousands of
  genetic markers, and can allow for LD between
  loci.
• Observed genotypes and phenotypes can be
  continuously adjusted by amounts attributable to
  ancestry along each axis (Price et al. 2006),
  effectively matching cases and controls.
• Computing association statistics using the
  ancestry adjusted genotypes and phenotypes will
  take account of population structure.
• Implemented in the EIGENSTRAT software package.

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Example 1. Three African populations.
Eigenvector 1 separates out the San population.
Eigenvector 2 separates the Bantu and Mandenka
populations, although the structure is less
obvious.
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Example 2. Three East Asian populations. The
first two eigenvectors (together) separate the
Japanese from the Chinese and Northern Thai. The
large dispersal of the Thai population along a
line where the Chinese are at an extreme suggests
some gene flow of a Chinese related population
into Thailand
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Simulation study

• Price et al. (2006) generated data at 100,000
  random SNPs for 500 cases and 500 controls from
  two different subpopulations 60 of cases and
  40 of controls came from subpopulation 1.
• Three types of SNPs simulated
• Random SNPs with no association to disease, but
  allele frequency differences between the two
  subpopulations at the level we would expect
  between European populations.
• Differentiated SNPs with no association to
  disease, but allele frequencies of 0.8 in
  subpopulation 1 and 0.2 in subpopulation 2.
• Causal SNPs associated with disease, with allele
  frequencies generated in the same way as random
  SNPs, and a multiplicative relative risk of 1.5
  for the causal allele.
• Cochran-Armitage trend test performed correction
  for structure made using genomic control and
  EIGENSTRAT.

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Simulation study results

• Proportion of SNPs in each category yielding
  significant evidence of association with plt10-4.
• Type I error rate of random SNPs correct for
  genomic control and EIGENSTRAT, but slightly
  inflated without correction due to population
  structure.
• Type I error rate of differentiated SNPs correct
  only for EIGENSTRAT.
• Minimal reduction in power to detect causal SNPs
  for EIGENSTRAT compared to uncorrected test.

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Multi-dimensional scaling

• We can measure the similarity between pairs of
  individuals by means of their identity by state
  (IBS) across the genome. Over M markers, the IBS
  between the ith and jth individuals is given by
• where Gik denotes the number of minor alleles
  (0, 1 or 2) carried by the ith individual at SNP
  k.
• Multi-dimensional scaling aims to detect axes of
  variation between individuals that maximise the
  disimilarities between them.
• In this context, very closely related to
  principal components analysis, and yields similar
  results to EIGENSTRAT.
• Incorporate eigenvectors as covariates in a
  logistic regression model to eliminate the
  effects of population structure.

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References

• Devlin B, Roeder K (1999). Genomic control for
  association studies. Biometrics 55997-1004.
• Patterson N, Price AL, Reich D (2006). Population
  structure and eigenanalysis. PLoS Genetics 2
  2074-2093.
• Price AL, Patterson NJ, Plenge RM, Weinblatt ME,
  Shadick NA, Reich D (2006). Principal components
  analysis corrects for stratification in
  genome-wide association studies. Nature Genetics
  38 904-909.
• Pritchard JK, Rosenberg N (1999). Use of unlinked
  genetic markers to detect population
  stratification in association studies. American
  Journal of Human Genetics 65 220-228.
• Pritchard JK, Stephens M, Donnelly P (2000a).
  Inference of population structure using
  multilocus genotype data. Genetics 155 945-959.
• Pritchard JK, Stephens M, Rosenberg N, Donnelly
  P (2000b). Association mapping in structured
  populations. American Journal of Human Genetics
  67170-181.
• Rosenberg N, Pritchard JK, Weber JL, Cann HM,
  Kidd KK, Zhivotovsky LA, Feldman MW (2002).
  Genetic structure of human populations. Science
  298 2981-2985.

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Comments

• Advantages.
• Multivariate techniques are computationally
  efficient and can be applied in the context of
  whole genome association studies.
• The axes of variation can be interpreted in terms
  of population structure, and with large numbers
  of SNPs can clearly differentiate between even
  relatively similar subpopulations and admixed
  groups.
• Disadvantages.
• Some care is needed in interpretation of the
  eigenvectors (for example may indicate extended
  regions of LD, rather than population structure).

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Summary

• Population structure can lead to spurious
  associations if disease prevalence and allele
  frequencies vary between subpopulations.
• We can use information from markers scattered
  throughout the genome to test for the presence of
  structure, identify groups of individuals with
  similar ancestry, and to correct association
  tests for mismatching of cases and controls.
• STRUCTURE provides a very detailed view of
  population structure, but is limited to a few
  thousand markers because of computational
  constraints.
• Multivariate statistical techniques are less
  computationally intensive, and can be used to
  simply correct association tests for structure in
  a whole genome association study.