Epistasis and a Flexible Framework for Detecting Epistasis

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Epistasis and a Flexible Framework for Detecting
Epistasis

• Yixuan Chen
• 11/9/2007

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• Jason H Moore A global view of epistasis. Nature
  Genetics 37, 13-14 (2005).
• Jason H Moore et al. A flexible computational
  framework for detecting, characterizing, and
  interpreting statistical patterns of epistasis in
  genetic studies of human disease susceptibility.
  Journal of Theoretical Biology 241, 252-261
  (2006).

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Outline

• Epistasis
• The flexible framework
• Experiment Design
• Results
• Discussions
• In the Future

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Epistasis

• Epistasis is a phenomenon whereby the effects of
  a given gene on a biological trait are masked or
  enhanced by one or more other genes.
• For complex traits such as diabetes, asthma,
  hypertension, etc., the presence of epistasis is
  a particular cause for concern.

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Biological and Statistical Epistasis

• The physical interactions among proteins and
  other biomolecules and their impact on phenotype
  constitute biological epistasis.
• Deviations from additivity in a linear
  statistical model define statistical epistasis.
• The relationship between biological and
  statistical epistasis is important and needs
  further research efforts.

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Why is There Epistasis?

• From an evolutionary biology perspective, for a
  phenotype to be buffered against the effects of
  mutations, it must have an underlying genetic
  architecture that is comprised of networks of
  genes that are redundant and robust.
• This creates dependencies among the genes in the
  network and is realized as epistasis.

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A Simple Example
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Disease Model

• Penetrance
• Pr (affected genotype)
• One-locus Dominant Model

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Two-locus Model
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Enumeration of Two-locus Models

• Although there are 29512 possible models,
  because of symmetries in the data, only 50 of
  these are unique.
• Enumeration allows 0 and 1 only for penetrance
  values (fully penetrant).

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A Flexible Framework

• The framework contains four steps to detect,
  characterize, and interpret epistasis
• Select interesting combinations of SNPs
• Construct new attributes from those selected
• Develop and evaluate a classification model using
  the newly constructed attribute(s)
• Interpret the final epistasis model using visual
  methods

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Step 1 Attribute selection

• Use entropy-based measures of IG and interaction
• Evaluate the gain in information about a class
  variable (e.g. case-control status) from merging
  two attributes together
• This measure of IG allows us to gauge the benefit
  of considering two (or more) attributes as one
  unit

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Information

• A discrete random variable X, that can take on
  possible values x1...xn
• The information of observing xi is
• I(xi) log2Pr(xi)
• Example
• An English character has information
  log2(1/26)4.7
• A common Chinese character has information
  log2(1/2500)11.3

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Shannon Entropy

• The information entropy

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Information Gain

• Consider two attributes, A and B, and a class
  label C.

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Information Gain (c1)

• If IG(ABC) gt 0
• Evidence for an attribute interaction that cannot
  be linearly decomposed
• If IG(ABC) lt 0
• The information between A and B is redundant
• If IG(ABC) 0
• Evidence of conditional independence or a mixture
  of synergy and redundancy

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Attribute Selection based on Entropy

• Entropy-based IG is estimated for each individual
  attribute (i.e. main effects) and each pairwise
  combination of attributes (i.e. interaction
  effects).
• Pairs of attributes are sorted and those with the
  highest IG, or percentage of entropy in the class
  removed, are selected for further consideration.

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Step 2 Constructive induction

• Use Multifactor Dimensionality Reduction (MDR)
• A multilocus genotype combination is considered
  high-risk if the ratio of cases to controls
  exceeds given threshold T, else it is considered
  low-risk
• Genotype combinations considered to be high-risk
  are labeled G1 while those considered low-risk
  are labeled G0.
• This process constructs a new one-dimensional
  attribute with levels G0 and G1

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MDR Step 1 2

• Step 1 partition the data into some number of
  equal parts for cross-validation
• Step 2 a set of N genetic and/or discrete
  environmental factors is selected from the list
  of all factors

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MDR Step 3

• The N factors and their multifactor classes or
  cells are represented in N-dimensional space
• The ratio of the number of cases to the number of
  controls is evaluated within each multifactor cell

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MDR Step 4

• Each multifactor cell in N-dimensional space is
  labeled as high-risk if the ratio meets or
  exceeds some threshold T (e.g. T 1.0) and
  low-risk if otherwise
• Those cells labeled high-risk are in one group
  and those low-risk are in another group, which
  reduces the N-dimensional model to one dimension

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MDR Step 5 6

• Step 5 all possible combinations of N factors
  are evaluated sequentially for their ability to
  classify affected and unaffected individuals in
  the training data, and the best N-factor model is
  selected.
• Step 6 the independent test data from the
  cross-validation is used to estimate the
  prediction error of the best model selected.

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MDR Final

• Steps 1 through 6 are repeated for each possible
  cross validation interval
• The final step determine which multifactor
  levels (e.g. genotypes) are high risk and which
  are low risk using the entire dataset.
• Final ratio threshold is determined by the ratio
  of the number of affected individuals in the
  dataset divided by the number of the unaffected

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Step 3 Classification and machine learning

• A naïve Bayes classifier is adopted

• Where vj is one of a set of V classes and ai is
  one of n attributes describing an event or data
  element.
• The class associated with a specific attribute
  list is the one, which maximizes the probability
  of the class and the probability of each
  attribute value given the specified class.

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Step 4 Interpretation Interaction Graphs

• Comprised of a node for each attribute with
  pairwise connections between them.
• Each node is labeled the percentage of entropy
  removed (i.e. IG) by each attribute.
• Each connection is labeled the percentage of
  entropy removed for each pairwise Cartesian
  product of attributes.

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Step 4 Interpretation Dendrograms

• Hierarchical clustering is used to build a
  dendrogram that places strongly interacting
  attributes close together at the leaves of the
  tree.

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Simulation Dataset

• One-locus
• M63

• Two-locus
• M27
• M170

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Simulation Dataset (c1)

• Three-locus Model
• Combination of M170 and M63
• In the form PcaP1bP2
• Generate 200 cases and 200 controls
• For each dataset, created two new datasets
• The functional SNPs plus the class variable
• Just a single attribute constructed using MDR in
  addition to the class variable.

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Atrial Fibrillation Dataset

• 250 patient with AF and 250 matched controls
• The ACE gene insertion/deletion (I/D)
  polymorphism, the T174M, M235T, G-6A, A-20C,
  G-152A, and G-217A polymorphisms of the
  angiotensinogen gene, and the A1166C polymorphism
  of the angiotensin II type I receptor gene were
  studied.

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Experiment Design

• 10-fold cross validation is used to compare
• Accuracy (TPTN)/(TPTNFPFN)
• Sensitivity TP/(TPFN)
• Specificity TN/(TNFP)
• Precision TP/(TPFP)
• (T True F False P Positive N Negative)
• Means were compared using a corrected resampled
  t-test and were considered statistically
  different when the p-value lt 0.05

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Results
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Summary

• Selects interesting combinations of polymorphisms
  based on their interaction information estimated
  using the entropy-based measures
• Interesting subsets of attributes can be reduced
  to single attribute using constructive induction
  methods such as MDR
• Then the single attribute is modeled using
  machine learning and classification
• The interaction graphs and interaction
  dendrograms constructed using these measures are
  useful for model interpretation.

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Discussions

• The flexibility of this framework is the ability
  to plug and play
• Different attribute selection methodsother than
  the entropy-based
• Different constructive induction algorithmsother
  than the MDR
• Different machine learning strategiesother than
  a naïve Bayes classifier

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Challenges

• The challenge is making inferences about
  biological epistasis from statistical epistasis

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In the Future

• Whole-genome association studies with thousands
  of measured genetic variations
• Discover high-order epistasis among genes
• Merge knowledge from genetic studies in human
  populations with detailed descriptions of
  transcriptional networks, biochemical pathways
  and physiological systems in individuals.

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THANKS!