Download Comparison of Discrimination Methods for the

Comparison of Discrimination Methods for
the Classification of Tumors Using Gene
Expression Data
Presented by: Tun-Hsiang Yang
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purpose of this paper
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Compare the performance of different discrimination
methods
Nearest Neighbor classifier
Linear discriminant analysis
Classification tree
Machine learning approaches: bagging, boosting
Investigate the use of prediction votes to assess the
confidence of each prediction
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Statistical problems:
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The identification of new/unknown tumor classes using gene
expression profiles
Clustering analysis/unsupervised learning
The classification of malignancies into known classes
Discriminant analysis/supervised learning
The identification of marker genes that identified different tumor
classes
 Variable (Gene) selection
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Datasets
Gene expression data on p genes for n mRNA samples:
n x p matrix X={x ij},
where x ij denotes the expression level of gene (variable) j
in ith mRNA sample(observation)
Response:
k-dimensional vector Y={yi},
where yi denotes the class of observation i
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Lymphoma dataset (p=4682, n=81,k=3)
Leukemia dataset (p=3571, n=72, k=3 or 2)
NCI 60 dataset (p=5244, n=61, k=8)
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Data preprocessing
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Imputation of missing data (KNN)
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Standardization of data (Euclidean distance)
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preliminary gene selection
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Lymphoma dataset (p=4682  p=50, n=81,k=3)
Leukemia dataset (p=3571p=40, n=72, k=3)
NCI 60 dataset (p=5244p=30, n=61, k=8)
BSS ( j )

WSS ( j )
 I(y
i

i
 k )( x kj  x . j ) 2
k
 I(y
i


i
 k )( x ij  x kj ) 2
k
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Visual presentation of Leukemia dataset
P=3571
p=40
Correlation matrix (72x72) ordered by class
Black: 0 correlation / Red: positive correlation / Green: negative correlation
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Prediction Methods
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Supervised Learning Methods
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Machine learning approaches
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Supervised Learning Methods
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Nearest Neighbor classifier(NN)
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Fisher Linear Discriminant Analysis (LDA)
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Weighted Gene Voting
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Classification trees (CART)
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Nearest Neighbor
The k-NN rule
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Find the k closest observations in the learning set
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Predict the class for each element in the test dataset by majority vote
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K is chosen by minimizing cross-validation error rate
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Linear Discirminantion Analysis
FLDA consists of
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finding linear functions a’x of the gene expression levels
x=(x1, …,xp) with large ratio of between groups to within
groups sum of squares
Predicting the class of an observation by the class whose
mean vector is closest to the discrimination variables
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Maximum likelihood discriminant rules
•Predicts the class of an observation x as C(x)=argmaxkpr(x|y=k)
C ( x)  arg min k {( x  k )k ( x  k )'  log |k |}
1
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Weighted Gene Voting
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An observation x=(x1,…xp) is classified as 1 iff

p

j 1
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
( x1 j  x2 j )
^
^
1 j   2 j

(x j 

( x1 j  x2 j )
2
)0
Prediction strength as the margin of victory(p9)
max( v1, v 2)  min( v1, v 2)
PS 
max( v1, v 2)  min( v1, v 2)
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Classification tree
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Constructed by repeated splits of subsets (nodes)
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Each terminal subset is assigned a class label
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The size of the tree is determined by minimizing the cross
validation error rate
Three aspects to tree construction
 the selection of the splits
 the stopping criteria
 the assignment of each terminal node to a class
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Aggregated Predictors
There are several ways to generate perturbed learning set:
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Bagging
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Boosting
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Convex Pseudo data (CPD)
arg max k  wb I (C ( x,Lb )  k )
b
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Bagging
Predictors are built for each sub-sample and aggregated by
Majority voting with equal wb=1
Non-parametric bootstrap:
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drawing at random with replacement to form a perturbed
learning sets of the same size as the original learning set
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By product: out of bag observations can be used to estimate
misclassification rates of bagged predictors
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A prediction for each observation (xi, yi) is obtained by
aggregating the classifiers in which (xi,yi) is out-of-bag
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Bagging (cont.)
Parametric bootstrap:
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Perturbed learning sets are generated according to a mixture of
MVN distributions
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For each class k, the class sample mean and covariance matrix were
taken as the estimates of distribution parameters
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Make sure at least one observation sampled from each class
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Boosting
The bth step of the boosting algorithm
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Get another learning set Lb of the same size nL
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Build a classifier based on Lb
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Run the learning set L
let di=1 if the ith case is classified incorrectly
di=0 otherwise
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Define
Update by
b=Pidi and Bbdi=(1- b)/ b
pi=piBbdi/  piBbdi
Re-sampling probabilities are reset to equal if b>=1/2 or b=0
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Prediction votes
For aggregated classifiers, prediction votes assessing the strength of
a prediction may be defined for each observation
The prediction vote (PV) for an observation x
PV ( x) 
max k  wb I (C ( x,Lb )  k )
b
w
b
b
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Study Design
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Randomly divide the dataset into a learning and test set
(2:1 scheme)
For each of N=150 runs:
 Select a subset of p genes from the learning set with the
largest BSS/WSS
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Build the different predictors using the learning sets with
p genes
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Apply the predictors to the observations in the test set to
obtain test set error rates
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Results
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Test set error rates: apply classifier build based on learning set
to test set. Summarized by box-plot over runs
Observation-wise error rates: for each observation, record the
proportion of times it was classified incorrectly. Summarized by
means of survival plots
Variable selection: compare the effect of increasing or
decreasing number of genes (variables)
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Leukemia data, two classes
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Leukemia data, three classes
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Lymphoma data
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Conclusions
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In the main comparison, NN and DLDA had the smallest error rates,
while FLDA had the highest error rates
Aggregation improved the performance of CART classifiers, the largest
gains being with boosting and bagging with CPD
For the lymphoma and leukemia datasets, increasing the number of
variables to p=200 did not affect much the performance of the various
classifiers. There was an improvement for the NCI 60 dataset.
A more carefully selection of a small number of genes (p=10)
improved the performance of FLDA dramatically
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