clustering and classification · 2017-10-25 · 6 statistical methods in microarray analysis...

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Statistical Methods in Microarray Analysis Tutorial

Clustering and Classification

Jean Yee Hwa YangUniversity of California, San Francisco

http://www.biostat.ucsf.edu/jean/

Institute for Mathematical Sciences, National University of Singapore

January 2 - 6, 2004


Classification

• Task: assign objects to classes (groups) on the basis of measurements made on the objects

• Unsupervised: classes unknown, want to discover them from the data (cluster analysis)

• Supervised: classes are predefined, want to use a (training or learning) set of labeled objects to form a classifier for classification of future observations


Clustering


Basic principles of clustering

Aim: to group observations that are “similar”based on predefined criteria.

Issues: Which genes / arrays to use?Which similarity or dissimilarity measure?Which clustering algorithm?

It is advisable to reduce the number of genes from the full set to some more manageable number, before clustering. The basis for this reduction is usually quite context specific.

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Expression Data

For each gene, calculate a summary statistics and/or

adjusted p-values

Clustering

Clustering

Set of candidate DE genes.Biological verification

Descriptive interpretation

Similarity metrics

Clustering algorithm


Which similarity or dissimilarity measure?

• A metric is a measure of the similarity or dissimilarity between two data objects

• Used to form data points into clusters

• Two main classes of distance:- Correlation coefficients

- Compares shape of expression curves- Two types of correlation:

- Centered.- Un-centered.

- Distance metrics- City Block (Manhattan) distance- Euclidean distance


• Pearson Correlation CoefficientSx = Standard deviation of x

Sy = Standard deviation of y

• Others include Spearman’s ρ and Kendall’s τ

∑=

−

−

−n

i y

i

x

in S

yyS

xx

11

1

Correlation (a measure between -1 and 1)

Positive correlation Negative correlation

You can use absolute correlation to capture both positive and negative correlation


Potential pitfalls

Correlation = 1

3


Distance metrics

• City Block (Manhattan) distance:- Sum of differences across

dimensions- Less sensitive to outliers - Diamond shaped clusters

• Euclidean distance:- Most commonly used

distance- Sphere shaped cluster- Corresponds to the

geometric distance into the multidimensional space

∑ −=i

ii yxYXd ),( ∑ −=i

ii yxYXd 2)(),(

where gene X = (x1,… ,xn) and gene Y=(y1,… ,yn)

X

Y

Condition 1

Con

ditio

n 2

Condition 1

X

Y

Con

ditio

n 2


Euclidean vs Correlation (I)

• Euclidean distance• Correlation


xx

Complete (minimum)

Distance between centroids

Distance between clustersBetween-cluster dissimilarity measures

Average (Mean) linkage

xx

Single (maximum)


Clustering algorithms

• Clustering algorithm comes in 2 basic flavors

Partitioning Hierarchical

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Partitioning methods

• Partition the data into a pre-specified number k of mutually exclusive and exhaustive groups.

• Iteratively reallocate the observations to clusters until some criterion is met, e.g. minimize within cluster sums of squares.

• Examples:- k-means, self-organizing maps (SOM), PAM, etc.;- Fuzzy: needs stochastic model, e.g. Gaussian

mixtures.


K = 2



K = 4



Hierarchical methods

• Hierarchical clustering methods produce a tree or dendrogram.

• They avoid specifying how many clusters are appropriate by providing a partition for each k obtained from cutting the tree at some level.

• The tree can be built in two distinct ways- bottom-up: agglomerative clustering.- top-down: divisive clustering.

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1 5 2 3 4

1 5 2 3 4

1,2,5

3,41,5

1,2,3,4,5

Agglomerative

Illustration of points In two dimensional space

15

34

2


1 5 2 3 4

1 5 2 3 4

1,2,5

3,41,5

1,2,3,4,5

Agglomerative

Tree re-ordering?

15

34

2

1 52 3 4


Partitioning vs. hierarchical

Partitioning:Advantages• Optimal for certain

criteria.• Genes automatically

assigned to clustersDisadvantages• Need initial k;• Often require long

computation times.• All genes are forced into

a cluster.

HierarchicalAdvantages• Faster computation.• Visual.Disadvantages• Unrelated genes are

eventually joined• Rigid, cannot correct later

for erroneous decisions made earlier.

• Hard to define clusters.


Clustering microarray data

• Clustering leads to readily interpretable figures and can be helpful for identifying patterns in time or space.

Examples:• We can cluster cell samples (cols),

e.g. 1) for identification (profiles). Here, we might want to estimate the number of different neuron cell types in a set of samples, based on gene expression.

2) the identification of new / unknown tumor classes using gene expression profiles.

• We can cluster genes (rows) , e.g. using large numbers of yeast experiments, to identify groups of co-regulated genes.

• We can cluster genes (rows) to reduce redundancy (cf. variable selection) in predictive models.

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Taken from Nature February, 2000Paper by A Alizadeh et alDistinct types of diffuse large B-cell lymphoma identified by Gene expression profiling,

Clustering both cell samplesand genes


Clustering cell samplesDiscovering sub-groups

Taken fromAlizadeh et al(Nature, 2000)


Yeast Cell Cycle(Cho et al, 1998)6 × 5 SOM with 828 genes

Taken from Tamayo et al, (PNAS, 1999)

Clustering genesFinding different patterns in the data


Some other issues on clustering

• Two-way clustering.

• Comparing trees.

• Estimating the number of cluster.- Silhouette width (used in PAM).- Gap statistics

Ref: Tibshirani, Walther and Hastie “Estimating the number of clusters in a dataset via the gap statistic”.

- Clest: Ref: Dudoit & Fridlyand “A prediction-based resampling method for estimating the number of clusters in a dataset.”

• Note: select a subsets of genes using their class association before clustering.

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Summary

Which clustering method should I use?- What is the biological question?- Do I have a preconceived notion of how many clusters there

should be?- Can a gene be in multiple clusters?- Hard or soft boundaries between clusters

Keep in mind:- Clustering cannot NOT work. That is, every clustering methods

will return clusters.- Clustering helps to group / order information and is a

visualization tool for learning about the data. However, clustering results do not provide biological “proof”.


Comparison of clustering and single gene approaches for microarray data analysis

Cluster analysis:1) Usually outside the normal framework of statistical

inference.2) Less appropriate when only a few genes are likely to

change.3) Needs lots of experiments.

Single gene approaches1) May be too noisy in general to show much.2) May not reveal coordinated effects of positively correlated

genes.3) Harder to relate to pathways.


Discrimination

Classification procedureFeature selection

Performance assessmentComparison study


Predefined Class

{1,2,… K}

1 2 K

Objects

Basic principles of discrimination•Each object associated with a class label (or response) Y ∈ {1, 2, … , K} and a feature vector (vector of predictor variables) of G measurements: X = (X1, … , XG)

Aim: predict Y from X.

X = {red, square} Y = ?

Y = Class Label = 2

X = Feature vector{colour, shape}

Classification rule ?

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Classification

Learning SetData with

known classes

ClassificationTechnique

Classificationrule

Data with unknown classes

ClassAssignment

Discrimination

Prediction


?Bad prognosisrecurrence < 5yrs

Good Prognosisrecurrence > 5yrs

ReferenceL van’t Veer et al (2002) Gene expression profiling predicts clinical outcome of breast cancer. Nature, Jan..

ObjectsArray

Feature vectorsGene

expression

Predefine classesClinical

outcome

new array

Learning set

Classificationrule

Good PrognosisMatesis > 5


B-ALL T-ALL AML

ReferenceGolub et al (1999) Molecular classification of cancer: class discovery and class prediction by gene expression monitoring. Science 286(5439): 531-537.

ObjectsArray

Feature vectorsGene

expression

Predefine classes

Tumor type

?

new array

Learning set

ClassificationRule

T-ALL


Classification Rule

-Classification procedure,-Feature selection,

-Parameters [pre-determine, estimable],

Distance measure,Aggregation methods

Performance Assessmente.g. Cross validation

• One can think of the classification rule as a black box, some methods provides more insight into the box.

• Performance assessment needs to be looked at for all classification rule.

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Classification rule Maximum likelihood discriminant rule

• A maximum likelihood estimator (MLE) chooses the parameter value that makes the chance of the observations the highest

• For known class conditional densities pk(X), the maximum likelihood (ML) discriminant rule predicts the class of an observation X by

C(X) = argmaxk pk(X)


Gaussian ML discriminant rules

• For multivariate Gaussian (normal) class densities X|Y= k ~ N(µk, Σk), the ML classifier is

C(X) = argmink {(X - µk) Σk-1 (X - µk)’+ log| Σk |}

• In general, this is a quadratic rule (Quadratic discriminant analysis, or QDA)

• In practice, population mean vectors µk and covariance matrices Σk are estimated by corresponding sample quantities


ML discriminant rules - special cases

[DLDA] Diagonal linear discriminant analysisclass densities have the same diagonal covariance matrix ∇= diag(s1

2, … , sp2)

[DQDA] Diagonal quadratic discriminant analysis)class densities have different diagonal covariance matrix ∇k= diag(s1k

2, … , spk2)

Note. Weighted gene voting of Golub et al. (1999) is a minor variant of DLDA fortwo classes (wrong variance calculation).


Nearest neighbor classification

• Based on a measure of distance between observations (e.g. Euclidean distance or one minus correlation).

• k-nearest neighbor rule (Fix and Hodges (1951)) classifies an observation X as follows:- find the k observations in the learning set closest to X- predict the class of X by majority vote, i.e., choose

the class that is most common among those k observations.

• The number of neighbors k can be chosen by cross-validation (more on this later).

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Nearest neighbor rule


Classification tree

• Partition the feature space into a set of rectangles, then fit a simple model in each one

• Binary tree structured classifiers are constructed by repeated splits of subsets (nodes) of the measurement space X into two descendant subsets (starting with X itself)

• Each terminal subset is assigned a class label; the resulting partition of X corresponds to the classifier


Classification tree

Gene 1Mi1 < -0.67

Gene 2Mi2 > 0.18

0

2

1

yes

yes

no

no 0

1

2

Gene 1

Gene 2

-0.67

0.18


Three aspects of tree construction

• Split selection rule: - Example, at each node, choose split maximizing decrease in

impurity (e.g. Gini index, entropy, misclassification error).

• Split-stopping: - Example, grow large tree, prune to obtain a sequence of

subtrees, then use cross-validation to identify the subtree with lowest misclassification rate.

• Class assignment: - Example, for each terminal node, choose the class minimizing

the resubstitution estimate of misclassification probability, given that a case falls into this node.

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Classification with SVMs


Other classifiers include…

• Neural networks

• Logistic regression

• Projection pursuit

• Bayesian belief networks


Why select features?

Correlation plotData: Leukemia, 3 class

No feature selection

Top 100feature selection

Selection based on variance

-1 +1


Explicit feature selection

• One-gene-at-a-time approaches. Genes are ranked based on the value of a univariate test statistic such as: t- or F-statistic or their non-parametric variants (Wilcoxon/Kruskal-Wallis); p-value.

• Possible meta-parameters include the number of genes G or a p-value cut-off. A formal choice of these parameters may be achieved by cross-validation or bootstrap procedures.

• Multivariate approaches. More refined feature selection procedures consider the joint distribution of the expression measures, in order to detect genes with weak main effects but possibly strong interaction.

• Bo & Jonassen (2002): Subset selection procedures for screening gene pairs to be used in classification.

• Breiman (1999): ranks genes according to importance statistic define in terms of prediction accuracy. Note that tree building itself does not involve explicit feature selection.

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Implicit feature selection

• Feature selection may also be performed implicitly by the classification rule itself.

• In classification trees, features are selected at each step based on reduction in impurity and The number of features used (or size of the tree) is determined by pruning the tree using cross-validation. Thus, feature selection is inherent part of tree-building and pruning deals with over-fitting.

• Shrinkage methods and adaptive distance function. May be used for LDA and kNN.


Performance assessment

• Any classification rule needs to be evaluated for its performance on the future samples. It is almost never the case in microarray studies that a large independent population-based collection of samples is available at the time of initial classifier-building phase.

• One needs to estimate future performance based on what is available: often the same set that is used to build the classifier.

• Assessing performance of the classifier based on- Cross-validation.- Test set- Independent testing on future dataset


Diagram of performance assessment

Training set


TrainingSet

Independenttest set

Classifier

Classifier

Resubstitutionestimation

Test set estimation


Performance assessment (I)

• Resubstitution estimation: error rate on the learning set.- Problem: downward bias

• Test set estimation:

1) divide learning set into two sub-sets, L and T; Build the classifier on L and compute the error rate on T.

2) Build the classifier on the training set (L) and compute the error rate on an independent test set (T). - L and T must be independent and identically distributed (i.i.d).- Problem: reduced effective sample size

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Diagram of performance assessment

Training set


TrainingSet

Independenttest set

(CV) Learningset

(CV) Test set

Classifier

Classifier

Classifier

Resubstitutionestimation

Test set estimation

Cross Validation


Performance assessment (II)

• V-fold cross-validation (CV) estimation: Cases in learning set randomly divided into V subsets of (nearly) equal size. Build classifiers by leaving one set out; compute test set error rates on the left out set and averaged. - Bias-variance tradeoff: smaller V can give larger bias but smaller

variance- Computationally intensive.

• Leave-one-out cross validation (LOOCV).

(Special case for V=n). Works well for stable classifiers (k-NN, LOOCV)


Performance assessment (III)

• Common practice to do feature selection using the learning , then CV only for model building and classification.

• However, usually features are unknown and the intended inference includes feature selection. Then, CV estimates as above tend to be downward biased.

• Features (variables) should be selected only from the learning set used to build the model (and not the entire set)


Another component in classification rule:aggregating classifiers

Training Set

X1, X2, … X100

Classifier 1Resample 1




Examples:BaggingBoosting

Random Forest

Aggregateclassifier

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Aggregating classifiers:Bagging

Training Set (arrays)X1, X2, … X100

Tree 1Resample 1X*1, X*2, … X*100

Lets the treevote




Testsample

Class 1

Class 2

Class 1

Class 1

90% Class 110% Class 2


Comparison study

• Leukemia data – Golub et al. (1999)- n = 72 samples, - G = 3,571 genes,- 3 classes (B-cell ALL, T-cell ALL, AML).

• Reference:S. Dudoit, J. Fridlyand, and T. P. Speed (2002). Comparison of discrimination methods for the classification of tumors using gene expression data. Journal of the American Statistical Association, Vol. 97, No. 457, p. 77-87


Leukemia data, 3 classes: Test set error rates;150 LS/TS runs


Results

• In the main comparison, NN and DLDA had the smallest error rates.

• Aggregation improved the performance of CART classifiers.

• For the leukemia datasets, increasing the number of genes to G=200 didn't greatly affectthe performance of the various classifiers.

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Comparison study – discussion (I)

• “Diagonal”LDA: ignoring correlation between genes helped here. Unlike classification trees and nearest neighbors, DLDA is unable to take into account gene interactions.

• Classification trees are capable of handling and revealing interactions between variables. In addition, they have useful by-product of aggregated classifiers: prediction votes, variable importance statistics.

• Although nearest neighbors are simple and intuitiveclassifiers, their main limitation is that they give very little insight into mechanisms underlying the class distinctions.


Summary (I)

• Bias-variance trade-off. Simple classifiers do well on small datasets. As the number of samples increases, we expect to see that classifiers capable of considering higher-order interactions (and aggregated classifiers) will have an edge.

• Cross-validation . It is of utmost importance to cross-validate for every parameter that has been chosen based on the data, including meta-parameters - what and how many features- how many neighbors- pooled or unpooled variance- classifier itself.

If this is not done, it is possible to wrongly declare having discrimination power when there is none.


Summary (II)

• Generalization error rate estimation. It is necessary to keep sampling scheme in mind.

• Thousands and thousands of independent samples from variety of sources are needed to be able to address the true performance of the classifier.

• We are not at that point yet with microarrays studies. Van Veer et al (2002) study is probably the only study to date with ~300 test samples.


Reference 1Retrospective studyL van’t Veer et al Gene expression profiling predicts clinical outcome of breast cancer. Nature, Jan 2002..

Learning set

Bad Good

ClassificationRule

Reference 2Retrospective studyM Van de Vijver et al. A gene expression signature as a predictor of survival in breast cancer. The New England Jouranl of Medicine, Dec 2002.

Reference 3Prospective trials.Aug 2003Clinical trialshttp://www.agendia.com/

Feature selection.Correlation with class

labels, very similar to t-test.

Using cross validation toselect 70 genes

295 samples selected from Netherland Cancer Institute

tissue bank (1984 – 1995).

Results” Gene expression profile is a morepowerful predictor then standard systems

based on clinical and histologic criteria

Agendia (formed by reseachers from the Netherlands Cancer Institute)Plan to start in Oct, 2003

1) 3000 subjects [Health Council of the Netherlands]2) 5000 subjects New York based Avon Foundation.

Custorm arrays are made by Aglient including70 genes + 1000 controls

Case studies

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Acknowledgements

Clustering

• Agnes Paquet

• David Erle

• Andrea Barczac,

• UCSF SandlerGenomics Core Facility.

Discrimination

• Jane Fridlyand

• Mark Segal

• Terry Speed

• Sandrine Dudoit

clustering and classification · 2017-10-25 · 6 statistical methods in microarray analysis...

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