Cluster analysis for microarray data

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Cluster analysis for microarray data

• Anja von Heydebreck

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Aim of clustering Group objects according to
their similarity
Cluster a set of objects that are similar to
each other and separated from the
other objects. Example green/ red data
points were generated from two different normal
distributions
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Clustering microarray data
gene expression data matrix

• Genes and experiments/samples are given as the
  row and column vectors of a gene expression data
  matrix.
• Clustering may be applied either to genes or
  experiments (regarded as vectors in Rp or Rn).

n experiments
p genes
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Why cluster genes?

• Identify groups of possibly co-regulated genes
  (e.g. in conjunction with sequence data).
• Identify typical temporal or spatial gene
  expression patterns (e.g. cell cycle data).
• Arrange a set of genes in a linear order that is
  at least not totally meaningless.

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Why cluster experiments/samples?

• Quality control Detect experimental
  artifacts/bad hybridizations
• Check whether samples are grouped according to
  known categories (though this might be better
  addressed using a supervised approach
  statistical tests, classification)
• Identify new classes of biological samples (e.g.
  tumor subtypes)

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Alizadeh et al., Nature 403503-11, 2000
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Cluster analysis

• Generally, cluster analysis is based on two
  ingredients
• Distance measure Quantification of
  (dis-)similarity of objects.
• Cluster algorithm A procedure to group objects.
  Aim small within-cluster distances, large
  between-cluster distances.

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Some distance measures

• Given vectors x (x1, , xn), y (y1, , yn)
• Euclidean distance
• Manhattan distance
• Correlation
• distance

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Which distance measure to use?

• The choice of distance measure should be based
  on the application area. What sort of
  similarities would you like to detect?
• Correlation distance dc measures trends/relative
  differences
• dc(x, y) dc(axb, y) if a gt 0.

x (1, 1, 1.5, 1.5) y (2.5, 2.5, 3.5, 3.5)
2x 0.5 z (1.5, 1.5, 1, 1)
dc(x, y) 0, dc(x, z) 2. dE(x, z) 1, dE(x,
y) 3.54.
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Which distance measure to use?

• Euclidean and Manhattan distance both measure
  absolute differences between vectors. Manhattan
  distance is more robust against outliers.
• May apply standardization to the observations
• Subtract mean and divide by standard
  deviation
• After standardization, Euclidean and correlation
  distance are equivalent

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K-means clustering

• Input N objects given as data points in Rp
• Specify the number k of clusters.
• Initialize k cluster centers. Iterate until
  convergence
• - Assign each object to the cluster with the
  closest center (wrt Euclidean distance).
• - The centroids/mean vectors of the obtained
  clusters are taken as new cluster centers.
• K-means can be seen as an optimization problem
• Minimize the sum of squared within-cluster
  distances,
• Results depend on the initialization. Use several
  starting points and choose the best solution
  (with minimal W(C)).

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Partitioning around medoids (PAM)

• K-means clustering is based on Euclidean
  distance.
• Partitioning around medoids (PAM) generalizes the
  idea and can be used with any distance measure d
  (objects xi need not be vectors).
• The cluster centers/prototypes are required to be
  observations
• Try to minimize the sum of distances of the
  objects to their cluster centers,
• using an iterative procedure analogous to the
  one in K-means clustering.

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K-means/PAM How to choose K (the number of
clusters)?

• There is no easy answer.
• Many heuristic approaches try to compare the
  quality of clustering results for different
  values of K (for an overview see Dudoit/Fridlyand
  2002).
• The problem can be better addressed in
  model-based clustering, where each cluster
  represents a probability distribution, and a
  likelihood-based framework can be used.

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Self-organizing maps

• K rs clusters are arranged as nodes of a
  two-dimensional grid. Nodes represent cluster
  centers/prototype vectors.
• This allows to represent similarity between
  clusters.
• Algorithm
• Initialize nodes at random positions.
  Iterate
• - Randomly pick one data point (gene) x.
• - Move nodes towards x, the closest node
  most, remote nodes (in terms of the grid) less.
  Decrease amount of movements with no. of
  iterations.

from Tamayo et al. 1999
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Self-organizing maps
from Tamayo et al. 1999 (yeast cell cycle data)
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Hierarchical clustering

• Similarity of objects is represented in a tree
  structure (dendrogram).
• Advantage no need to specify the number of
  clusters in advance. Nested clusters can be
  represented.

Golub data different types of leukemia.
Clustering based on the 150 genes with highest
variance across all samples.
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Agglomerative hierarchical clustering

• Bottom-up algorithm (top-down (divisive) methods
  are less common).
• Start with the objects as clusters.
• In each iteration, merge the two clusters with
  the minimal distance from each other - until you
  are left with a single cluster comprising all
  objects.
• But what is the distance between two clusters?

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Distances between clusters used for hierarchical
clustering

• Calculation of the distance between two
  clusters is based on the pairwise distances
  between members of the clusters.
• Complete linkage largest distance
• Average linkage average distance
• Single linkage smallest distance

Complete linkage gives preference to
compact/spherical clusters. Single linkage can
produce long stretched clusters.
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Hierarchical clustering

• The height of a node in the dendrogram represents
  the distance of the two children clusters.
• Loss of information n objects have n(n-1)/2
  pairwise distances, tree has n-1 inner nodes.
• The ordering of the leaves is not uniquely
  defined by the dendrogram 2n-2 possible choices.

Golub data different types of leukemia.
Clustering based on the 150 genes with highest
variance across all samples.
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Alternative direct visualization of
similarity/distance matrices
Useful if one wants to investigate a specific
factor (advantage no loss of information). Sort
experiments according to that factor.
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The role of feature selection

• Sometimes, people first select genes that appear
  to be differentially expressed between groups of
  samples. Then they cluster the samples based on
  the expression levels of these genes. Is it
  remarkable if the samples then cluster into the
  two groups?
• No, this doesnt prove anything, because the
  genes were selected with respect to the two
  groups! Such effects can even be obtained with a
  matrix of i.i.d. random numbers.

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Clustering of time course data

• Suppose we have expression data from different
  time points t1, , tn, and want to identify
  typical temporal expression profiles by
  clustering the genes.
• Usual clustering methods/distance measures dont
  take the ordering of the time points into account
  the result would be the same if the time points
  were permuted.
• Simple modification Consider the difference
  xi(j1) xij between consecutive timepoints as
  an additional observation yij. Then apply a
  clustering algorithm such as K-means to the
  augmented data matrix.

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Biclustering

• Usual clustering algorithms are based on global
  similarities of rows or columns of an expression
  data matrix.
• But the similarity of the expression profiles of
  a group of genes may be restricted to certain
  experimental conditions.
• Goal of biclustering identify homogeneous
  submatrices.
• Difficulties computational complexity, assessing
  the statistical significance of results
• Example Tanay et al. 2002.

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Conclusions

• Clustering has been very popular in the analysis
  of microarray data. However, many typical
  questions in microarray studies (e.g. identifying
  expression signatures of tumor types) are better
  addressed by other methods (classification,
  statistical tests).
• Clustering algorithms are easy to apply (you
  cannot really do it wrong), and they are useful
  for exploratory analysis. But it is difficult to
  assess the validity/significance of the results.
  Even random data with no structure can yield
  clusters or exhibit interesting looking patterns.

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Clustering in R

• library mva
• - Hierarchical clustering hclust, heatmap
• - k-means kmeans
• library class
• - Self-organizing maps SOM
• library cluster
• - pam and other functions

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References

• T. Hastie, R. Tibshirani, J. Friedman The
  elements of statistical learning (Chapter 14).
  Springer 2001.
• M. Eisen et al. Cluster analysis and display of
  genome-wide expression patterns.
  Proc.Natl.Acad.Sci.USA 95, 14863-8, 1998.
• P. Tamayo et al. Interpreting patterns of gene
  expression with self-organizing maps Methods and
  application to hematopoietic differentiation.
  Proc.Natl.Acad.Sci.USA 96, 2907-12, 1999.
• S. Dudoit, J. Fridlyand A prediction-based
  resampling method for estimating the number of
  clusters in a dataset. Genome Biology 3(7), 2002.
  
• A. Tanay, R. Sharan, R. Shamir Discovering
  statistically significant biclusters in gene
  expression data. Bioinformatics 18, Suppl.1,
  136-44, 2002.