Clustering and Classification In Gene Expression Data

1
Clustering and Classification In Gene Expression
Data

• Carlo Colantuoni
• ccolantu_at_jhsph.edu

Slide Acknowledgements Elizabeth Garrett-Mayer,
Rafael Irizarry, Giovanni Parmigiani, David
Madigan, Kevin Coombs, Richard Simon, Ingo
Ruczinski.Classification based in part on Chapter
10 of Hand, Manilla, Smyth and Chapter 7 of Han
and Kamber
2
Data from Garber et al. PNAS (98), 2001.
3
(No Transcript)
4
Clustering

• Clustering is an exploratory tool to see who's
  running with who Genes and Samples.
• Unsupervized
• NOT for classification of samples.
• NOT for identification of differentially
  expressed genes.

5
Clustering

• Clustering organizes things that are close into
  groups.
• What does it mean for two genes to be close?
• What does it mean for two samples to be close?
• Once we know this, how do we define groups?
• Hierarchical and K-Means Clustering

6
Distance

• We need a mathematical definition of distance
  between two points
• What are points?
• If each gene is a point, what is the mathematical
  definition of a point?

7
Points
1 2 . . . . . . . N
1 2 . . . . . . . . G

• Gene1 (E11, E12, , E1N)
• Gene2 (E21, E22, , E2N)
• Sample1 (E11, E21, , EG1)
• Sample2 (E12, E22, , EG2)
• Egiexpression gene g, sample i

DATA MATRIX
8
Most Famous Distance

• Euclidean distance
• Example distance between gene 1 and 2
• Sqrt of Sum of (E1i -E2i)2, i1,,N
• When N is 2, this is distance as we know it

Baltimore
Distance
DC
When N is 20,000 you have to think abstractly
9
Correlation can also be used to compute distance

• Pearson Correlation
• Spearman Correlation
• Uncentered Correlation
• Absolute Value of Correlation

10

• The difference is that, if you have two vectors X
  and Y with identical
• shape, but which are offset relative to each
  other by a fixed value,
• they will have a standard Pearson correlation
  (centered correlation)
• of 1 but will not have an uncentered correlation
  of 1.

11
The similarity/distance matrices
1 2 ...G
1 2 .N
1 2 . . . . . . . . G
1 2 . . . . . . . . G
GENE SIMILARITY MATRIX
DATA MATRIX
12
The similarity/distance matrices
1 2 ..N
1 2 .N
1 2 . . . . . . . . G
1 2 . . . N
SAMPLE SIMILARITY MATRIX
DATA MATRIX
13
Gene and Sample Selection

• Do you want all genes included?
• What to do about replicates from the same
  individual/tumor?
• Genes that contribute noise will affect your
  results.
• Including all genes dendrogram cant all be
  seen at the same time.
• Perhaps screen the genes?

14
Two commonly seen clustering approaches in gene
expression data analysis

• Hierarchical clustering
• Dendrogram (red-green picture)
• Allows us to cluster both genes and samples in
  one picture and see whole dataset organized
• K-means/K-medoids
• Partitioning method
• Requires user to define K of clusters a
  priori
• No picture to (over)interpret

15
Hierarchical Clustering

• The most overused statistical method in gene
  expression analysis
• Gives us pretty red-green picture with patterns
• But, pretty picture tends to be pretty unstable.
• Many different ways to perform hierarchical
  clustering
• Tend to be sensitive to small changes in the data
• Provided with clusters of every size where to
  cut the dendrogram is user-determined

16
Choose clustering direction

• Agglomerative clustering (bottom-up)
• Starts with as each gene in its own cluster
• Joins the two most similar clusters
• Then, joins next two most similar clusters
• Continues until all genes are in one cluster
• Divisive clustering (top-down)
• Starts with all genes in one cluster
• Choose split so that genes in the two clusters
  are most similar (maximize distance between
  clusters)
• Find next split in same manner
• Continue until all genes are in single gene
  clusters

17
Choose linkage method (if bottom-up)

• Single Linkage join clusters whose distance
  between closest genes is smallest (elliptical)
• Complete Linkage join clusters whose distance
  between furthest genes is smallest (spherical)
• Average Linkage join clusters whose average
  distance is the smallest.

18
Dendrogram Creation Interpretation
19
Dendrogram Creation Interpretation
20
Dendrogram Creation Interpretation
21
Cluster Assignment
22
Simulated Data with 4 clusters 1-10, 11-20,
21-30, 31-40
450 relevant genes 450 noise genes.
450 relevant genes.
23
K-means and K-medoids

• Partitioning Method
• Dont get pretty picture
• MUST choose number of clusters K a priori
• More of a black box because output is most
  commonly looked at purely as assignments
• Each object (gene or sample) gets assigned to a
  cluster
• Begin with initial partition
• Iterate so that objects within clusters are most
  similar

24
K-means (continued)

• Euclidean distance most often used
• Spherical clusters.
• Can be hard to choose or figure out K.
• Not unique solution clustering can depend on
  initial partition
• No pretty figure to (over)interpret

25
K-means Algorithm

• 1. Choose K centroids at random
• 2. Make initial partition of objects into k
  clusters by assigning objects to closest centroid
• Calculate the centroid (mean) of each of the k
  clusters.
• a. For object i, calculate its distance to each
  of
• the centroids.
• b. Allocate object i to cluster with closest
  
• centroid.
• c. If object was reallocated, recalculate
  centroids based
• on new clusters.
• 4. Repeat 3 for object i 1,.N.
• Repeat 3 and 4 until no reallocations occur.
• Assess cluster structure for fit and stability

26
K-means

• We start with some data
• Interpretation
• We are showing expression for two samples for 14
  genes
• We are showing expression for two genes for 14
  samples
• This is with 2 genes.

Iteration 0
27
K-means

• Choose K centroids
• These are starting values that the user picks.
• There are some data driven ways to do it

Iteration 0
28
K-means

• Make first partition by finding the closest
  centroid for each point
• This is where distance is used

Iteration 1
29
K-means

• Now re-compute the centroids by taking the middle
  of each cluster

Iteration 2
30
K-means

• Repeat until the centroids stop moving or until
  you get tired of waiting

Iteration 3
31
K-means Limitations

• Final results depend on starting values
• How do we chose K? There are methods but not much
  theory saying what is best.
• Where are the pretty pictures?

32
Assessing cluster fit and stability

• Most often ignored.
• Cluster structure is treated as reliable and
  precise
• Can be VERY sensitive to noise and to outliers
• Homogeneity and Separation
• Cluster Silhouettes how similar genes within a
  cluster are to genes in other clusters (Rousseeuw
  Journal of Computation and Applied Mathematics,
  1987)

33
Silhouettes

• Silhouette of gene i is defined as
• ai average distance of gene i to other gene in
  same cluster
• bi average distance of gene i to genes in its
  nearest neighbor cluster

34
WADP Weighted Average Discrepancy Pairs

• Add perturbations to original data
• Calculate the number of paired samples that
  cluster together in the original cluster that
  didnt in the perturbed
• Repeat for every cutoff (i.e. for each k)
• Do iteratively
• Estimate for each k the proportion of discrepant
  pairs.

35
(No Transcript)
36
Classification

• Diagnostic tests are good examples of classifiers
• A patient has a given disease or not
• The classifier is a machine that accepts some
  clinical parameters as input, and spits out an
  prediction for the patient
• D
• Not-D
• Classes must be mutually exclusive and exhaustive

37
Components of Class Prediction

• Select features (genes)
• Which genes will be included in the model
• Select type of classifier
• E.g. (D)LDA, SVM, k-Nearest-Neighbor,
• Fit parameters for model (train the classifier)
• Quantify predictive accuracy Cross-Validation

38
Feature Selection

• Goal is to identify a small subset of genes which
  together give accurate predictions.
• Methods will vary depending on nature of
  classification problem
• Choose genes with significant t-statistics to
  distinguish between two simple classes e.g.

39
Classifier Selection

• In microarray classification, the number of
  features is (almost) always much greater than the
  number of samples.
• Overfitting is a distinct risk, and increases
  with more complicated methods.

40
How microarrays differ from the rest of the world

• Complex classification algorithms such as neural
  networks that perform better elsewhere dont do
  as well as simpler methods for expression data.
• Comparative studies have shown that simpler
  methods work as well or better for microarray
  problems because the number of candidate
  predictors exceeds the number of samples by
  orders of magnitude.
• (Dudoit, Fridlyand and Speed JASA 2001)

41
Statistical Methods Appropriate for Class
Comparison may not be Appropriate for Class
Prediction

• Demonstrating statistical significance of
  prognostic factors is not the same as
  demonstrating predictive accuracy.
• Demonstrating goodness of fit of a model to the
  data used to develop it is not a demonstration
  of predictive accuracy.
• Most statistical methods were not developed for
  pgtgtn prediction problems

42
Linear discriminant analysis

• If there are K classes, simply draw lines
  (planes) to divide the space of expression
  profiles into K regions, one for each class.
• If profile X falls in region K, predict class K.

43
Nearest Neighbor Classification

• To classify a new observation X, measure the
  distance d(X,Xi) between X and every sample Xi
  in training set
• Assign to X the class label of its nearest
  neighbor in the training set.

44
Random Forests

• Build several random decision trees and have
  them vote to determine final classification

45
Evaluating a classifier

• Want to estimate the error rate when classifier
  is used to predict class of a new observation
• The ideal approach is to get a set of new
  observations, with known class label and see how
  frequently the classifier makes the correct
  prediction.
• Performance on the training set is a poor
  approach, and will deflate the error estimate.
• Cross validation methods are used to get less
  biased estimates of error using only the training
  data.

46
Split-Sample Evaluation

• Training-set
• Used to select features, select model type,
  determine parameters and cut-off thresholds
• Test-set
• Withheld until a single model is fully specified
  using the training-set.
• Fully specified model is applied to the
  expression profiles in the test-set to predict
  class labels.
• Number of errors is counted

47
V-fold cross validation

• Divide data into V groups.
• Hold one group back, train the classifier on
  other V-1 groups, and use it to predict the last
  one.
• Rotate through all V points, holding each back.
• Error estimate is total error rate on all V test
  groups.

48
Leave-one-out Cross Validation

• Hold one data point back, train the classifier
  on other n-1 data points, and use it to predict
  the last one.
• Rotate through all n points, holding each back.
• Error estimate is total error rate on all n test
  values.

49
Non-cross-validated Prediction
1. Prediction rule is built using full data
set. 2. Rule is applied to each specimen for
class prediction.
Cross-validated Prediction (Leave-one-out method)
1. Full data set is divided into training and
test sets (test set contains 1 specimen). 2.
Prediction rule is built from scratch using the
training set. 3. Rule is applied to the specimen
in the test set for class prediction. 4.
Process is repeated until each specimen has
appeared once in the test set.
50
Which to use depends mostly on sample size

• If the sample is large enough, split into test
  and train groups.
• If sample is barely adequate for either testing
  or training, use leave one out
• In between consider V-fold. This method can give
  more accurate estimates than leave one out, but
  reduces the size of training set.

51
Beware

• Cross-validation of a model cannot occur after
  selecting the genes to be used in the model

52
Incomplete (incorrect) Cross-Validation

• Publications are using all the data to select
  genes and then cross-validating only the
  parameter estimation component of model
  development
• Highly biased
• Many published complex methods which make strong
  claims based on incorrect cross-validation.
• Frequently seen in complex feature set selection
  algorithms
• Some software encourages inappropriate
  cross-validation

53
Gene-Expression Profiles in Hereditary Breast
Cancer

• Breast tumors studied
• 7 BRCA1 tumors
• 8 BRCA2 tumors
• 7 sporadic tumors
• Log-ratios measurements of 3226 genes for each
  tumor after initial data filtering

RESEARCH QUESTION Can we distinguish BRCA1 from
BRCA1 cancers and BRCA2 from BRCA2 cancers
based solely on their gene expression profiles?
54
BRCA1
55
BRCA2