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Statistics for Microarrays
Discrimination
Class web site: http://statwww.epfl.ch/davison/teaching/Microarrays/
cDNA gene expression data
Data on G genes for n samples
mRNA samples
sample1 sample2 sample3 sample4 sample5 …
Genes
1
2
3
4
5
0.46
-0.10
0.15
-0.45
-0.06
0.30
0.49
0.74
-1.03
1.06
0.80
0.24
0.04
-0.79
1.35
1.51
0.06
0.10
-0.56
1.09
0.90
0.46
0.20
-0.32
-1.09
...
...
...
...
...
Gene expression level of gene i in mRNA sample j
= (normalized) Log( Red intensity / Green intensity)
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
Discrimination
• Objects (e.g. arrays) are to be classified as
belonging to one of a number of predefined
classes {1, 2, …, K}
• 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.
Example: Tumor Classification
• Reliable and precise classification essential for
successful cancer treatment
• Current methods for classifying human malignancies
rely on a variety of morphological, clinical and
molecular variables
• Uncertainties in diagnosis remain; likely that
existing classes are heterogeneous
• Characterize molecular variations among tumors by
monitoring gene expression (microarray)
• Hope: that microarrays will lead to more reliable
tumor classification (and therefore more
appropriate treatments and better outcomes)
Tumor Classification Using Gene
Expression Data
Three main types of statistical problems
associated with tumor classification:
• Identification of new/unknown tumor classes
using gene expression profiles (unsupervised
learning – clustering)
• Classification of malignancies into known
classes (supervised learning – discrimination)
• Identification of “marker” genes that
characterize the different tumor classes
(feature or variable selection).
Classifiers
• A predictor or classifier partitions the space of
gene expression profiles into K disjoint subsets,
A1, ..., AK, such that for a sample with expression
profile X=(X1, ...,XG)  Ak the predicted class is k
• Classifiers are built from a learning set (LS)
L = (X1, Y1), ..., (Xn,Yn)
• Classifier C built from a learning set L:
C( . ,L): X  {1,2, ... ,K}
• Predicted class for observation X:
C(X,L) = k if X is in Ak
Decision Theory (I)
• Can view classification as statistical
decision theory: must decide which of the
classes an object belongs to
• Use the observed feature vector X to aid
in decision making
• Denote population proportion of objects
of class k as pk = p(Y = k)
• Assume objects in class k have feature
vectors with density pk(X) = p(X|Y = k)
Decision Theory (II)
• One criterion for assessing classifier
quality is the misclassification rate,
p(C(X)Y)
• A loss function L(i,j) quantifies the loss
incurred by erroneously classifying a
member of class i as class j
• The risk function R(C) for a classifier is
the expected (average) loss:
R(C) = E[L(Y,C(X))]
Decision Theory (III)
• Typically L(i,i) = 0
• In many cases can assume symmetric loss
with L(i,j) = 1 for i  j (so that different
types of errors are equivalent)
• In this case, the risk is simply the
misclassification probability
• There are some important examples, such
as in diagnosis, where the loss function is
not symmetric
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)
Fisher Linear Discriminant Analysis
First applied in 1935 by M. Barnard at the
suggestion of R. A. Fisher (1936), Fisher
linear discriminant analysis (FLDA):
1. finds linear combinations of the gene
expression profiles X=X1,...,XG with large ratios
of between-groups to within-groups sums of
squares - discriminant variables;
2. predicts the class of an observation X by the
class whose mean vector is closest to X in
terms of the discriminant variables
Gaussian ML Discriminant Rules
• For multivariate Gaussian (normal) class
densities X|Y= k ~ N(k, k), the ML
classifier is
C(X) = argmink {(X - k) k-1 (X - k)’ + log| k |}
• In general, this is a quadratic rule
(Quadratic discriminant analysis, or QDA)
• In practice, population mean vectors k
and covariance matrices k are estimated
by corresponding sample quantities
Gaussian ML Discriminant Rules
• When all class densities have the same
covariance matrix, k = the discriminant
rule is linear (Linear discriminant analysis, or
LDA; FLDA for k = 2):
C(X) = argmink (X - k) -1 (X - k)’
• When all class densities have the same
diagonal covariance matrix =diag(12… G2),
the discriminant rule is again linear (Diagonal
linear discriminant analysis, or DLDA)
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)
Classification Trees
• 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
Three Aspects of Tree
Construction
• Split Selection Rule
• Split-stopping Rule
• Class assignment Rule
Different approaches to these three
issues (e.g. CART: Classification And
Regression Trees, Breiman et al. (1984);
C4.5 and C5.0, Quinlan (1993)).
Three Rules (CART)
• Splitting: At each node, choose split maximizing
decrease in impurity (e.g. Gini index, entropy,
misclassification error)
• Split-stopping: Grow large tree, prune to obtain
a sequence of subtrees, then use crossvalidation to identify the subtree with lowest
misclassification rate
• Class assignment: For each terminal node,
choose the class minimizing the resubstitution
estimate of misclassification probability, given
that a case falls into this node
Other Classifiers Include…
• Support vector machines (SVMs)
• Neural networks
• Bayesian regression methods
(BREAK)
Features
• Feature selection
– Automatic with trees
– For DA, NN need preliminary selection
– Need to account for selection when
assessing performance
• Missing data
– Automatic imputation with trees
– Otherwise, impute (or ignore)
Performance assessment (I)
• Resubstitution estimation: error rate
on the learning set
– Problem: downward bias
• Test set estimation: divide cases in
learning set into two sets, L1 and L2;
classifier built using L1, error rate
computed for L2. L1 and L2 must be
iid.
– Problem: reduced effective sample size
Performance assessment (II)
• V-fold cross-validation (CV) estimation:
Cases in learning set randomly divided
into V subsets of (nearly) equal size.
Build classifiers leaving one set out; test
set error rates computed on left out set
and averaged.
– Bias-variance tradeoff: smaller V can give
larger bias but smaller variance
• Out-of-bag estimation: covered below
Performance assessment (III)
• Common to do feature selection using all
of the data, 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 should be selected only from
the learning set used to build the model
(and not the entire learning set)
Aggregating classifiers
• Breiman (1996, 1998) found that gains in accuracy
could be obtained by aggregating predictors built
from perturbed versions of the learning set; the
multiple versions of the predictor are aggregated
by voting.
• Let C(., Lb) denote the classifier built from the
bth perturbed learning set Lb, and let wb denote
the weight given to predictions made by this
classifier. The predicted class for an observation
x is given by
argmaxk ∑b wbI(C(x,Lb) = k)
Bagging
• Bagging = Bootstrap aggregating
• Nonparametric Bootstrap (standard
bagging): perturbed learning sets drawn at
random with replacement from the learning
sets; predictors built for each perturbed
dataset and aggregated by plurality voting
(wb = 1)
• Parametric Bootstrap: perturbed learning
sets are multivariate Gaussian
• Convex pseudo-data (Breiman 1996)
Aggregation By-products: Outof-bag estimation of error rate
• Out-of-bag error rate estimate: unbiased
• Use the left out cases from each bootstrap
sample as a test set
• Classify these test set cases, and compare
to the class labels of the learning set to get
the out-of-bag estimate of the error rate
Aggregation By-products:
Case-wise information
• Class probability estimates (votes) (0,1):
the proportion of votes for the “winning”
class; gives a measure of prediction
confidence
• Vote margins (–1,1) : the proportion of
votes for the true class minus the
maximum of the proportion of votes for
each of the other classes; can be used to
detect mislabeled (learning set) cases
Aggregation By-products:
Variable Importance Statistics
• Measure of predictive power
• For each tree, randomly permute the
values of the jth variable for the out-ofbag cases, use to get new classifications
• Several possible importance measures
Aggregation By-products:
Intrinsic Case Proximities
• Proportion of trees for which cases i and
j are in the same terminal node
• “Clustering”
• Outlier detection:
1/sum(squared proximities of cases in same class)
Boosting
• Freund and Schapire (1997), Breiman
(1998)
• Data resampled adaptively so that the
weights in the resampling are increased
for those cases most often misclassified
• Predictor aggregation done by weighted
voting
Comparison of classifiers
• Dudoit, Fridlyand, Speed (JASA, 2002)
• FLDA
• DLDA
• DQDA
• NN
• CART
• Bagging and boosting
Comparison study datasets
• Leukemia – Golub et al. (1999)
n = 72 samples, G = 3,571 genes
3 classes (B-cell ALL, T-cell ALL, AML)
• Lymphoma – Alizadeh et al. (2000)
n = 81 samples, G = 4,682 genes
3 classes (B-CLL, FL, DLBCL)
• NCI 60 – Ross et al. (2000)
N = 64 samples, p = 5,244 genes
8 classes
Leukemia data, 2 classes: Test set error rates;150 LS/TS runs
Leukemia data, 3 classes: Test set error rates;150 LS/TS runs
Lymphoma data, 3 classes: Test set error rates; N=150 LS/TS runs
NCI 60 data :Test set error rates;150 LS/TS runs
Results
• In the main comparison, NN and DLDA had the
smallest error rates, FLDA had the highest
• Aggregation improved the performance of CART
classifiers, the largest gains being with boosting
and bagging with convex pseudo-data
• For the lymphoma and leukemia datasets,
increasing the number of genes to G=200 didn't
greatly affect the performance of the various
classifiers; there was an improvement for the
NCI 60 dataset.
• More careful selection of a small number of genes
(10) improved the performance of FLDA
dramatically
Comparison study – Discussion (I)
• “Diagonal” LDA: ignoring correlation between genes
helped here
• Unlike classification trees and nearest neighbors,
LDA is unable to take into account gene
interactions
• Although nearest neighbors are simple and intuitive
classifiers, their main limitation is that they give
very little insight into mechanisms underlying the
class distinctions
Comparison study – Discussion (II)
• Classification trees are capable of handling and
revealing interactions between variables
• Useful by-product of aggregated classifiers:
prediction votes, variable importance statistics
• Variable selection: A crude criterion such as
BSS/WSS may not identify the genes that
discriminate between all the classes and may not
reveal interactions between genes
• With larger training sets, expect improvement in
performance of aggregated classifiers
Acknowledgements
• Sandrine Dudoit
• Jane Fridlyand
• Yee Hwa (Jean) Yang
• Terry Speed