Supervised Classification

Selection bias in gene extraction on the basis of microarray gene-expression data

Ambroise and McLachlan

Proceedings of the National Academy of SciencesVol. 99, Issue 10, 6562-6566, May 14, 2002

http://www.pnas.org/cgi/content/full/99/10/6562

Supervised Classification of Tissue Samples

AIM: TO CONSTRUCT A CLASSIFIER c(y) FOR PREDICTING THE UNKNOWN CLASS LABEL z OF A TISSUE SAMPLE y.

e.g. g = 2 classes C1 - DISEASE-FREE C2 - METASTASES

We OBSERVE the CLASS LABELS z1, …, zn where zj = i if jth tissue sample comes from the ith class (i=1,…,g).

Sample 1 Sample 2 Sample M

Gene 1Gene 2

Gene N

Expression ProfileE

xpression S

ignature

Sample 1 Sample n. . . . . . .

Gene 1

. . .

. . .

.

Gene p

Class 1(good prognosis)

Class 2(poor prognosis)

Supervised Classification (Two Classes)

Microarray to be used as routine clinical screenby C. M. Schubert

Nature Medicine 9, 9, 2003.

The Netherlands Cancer Institute in Amsterdam is to become the first institution in the world to use microarray techniques for the routine prognostic screening of cancer patients. Aiming for a June 2003 start date, the center will use a panoply of 70 genes to assess the tumor profile of breast cancer patients and to determine which women will receive adjuvant treatment after surgery.

Selection Bias

Bias that occurs when a subset of the variables is selected (dimension reduction) in some “optimal” way, and then the predictive capability of this subset is assessed in the usual way; i.e. using an ordinary measure for a set of variables.

Selection Bias

Discriminant Analysis:McLachlan (1992 & 2004, Wiley, Chapter 12)

Regression:Breiman (1992, JASA)

“This usage (i.e. use of residual of SS’s etc.) has long been a quiet scandal in the statistical community.”

Nature Reviews Cancer, Feb. 2005

LINEAR CLASSIFIER

FORM

yyc Tβ 0)(

for the production of the group label z of a future entity with feature vector y.

pp yβyββ 110

FISHER’S LINEAR DISCRIMINANT FUNCTION

)(sign ycz

)()(2

1

)(

211

210

211

yyyy

yy

T

S

Sβ

and , , 21 yycovariance matrix found from the training data

where

and S are the sample means and pooled sample

Microarrays also to be used in the prediction of breast cancer by Mike West (Duke University) and the Koo Foundation Sun Yat-Sen Cancer Centre, Taipei

Huang et al. (2003, The Lancet, Gene expression predictors of breast cancer).

LINEAR CLASSIFIER

FORM

yy TC β 0)(

for the production of the group label z of a future entity with feature vector y.

pp yβyββ 110

FISHER’S LINEAR DISCRIMINANT FUNCTION

)(sign yCz

)()(2

1

)(

211

210

211

yyyy

yy

S

Sβ

T

and , , 21 yycovariance matrix found from the training data

where

and S are the sample means and pooled sample

SUPPORT VECTOR CLASSIFIERVapnik (1995)

)( yC

n

jj

1 ,

2

2

1

0

min

ββ

subject to

jjj )C(z 1y,0j

,,1

n

),,1( nj

where β0 and β are obtained as follows:

relate to the slack variables

separable case

pp yβyββ 110

jj

n

jj z y

1

ˆˆ β

with non-zero j only for those observations j for which theconstraints are exactly met (the support vectors).

01

01

ˆ ,ˆ

ˆ ˆ)(

n

jjjj

n

j

Tjjj

z

zC

yy

yyy

Support Vector Machine (SVM)

REPLACE )( yy h

01

01

ˆ ),(ˆ

ˆ )(),(ˆ)(

n

jjj

n

jjj

K

hhC

yy

yyy

where the kernel function )(),(),( yyyy hhK jj is the inner product in the transformed feature space.

by

HASTIE et al. (2001, Chapter 12)

The Lagrange (primal function) is

(1) )1()(111

2

2

1

n

jjjjjj

n

jj

n

jjP CzL yβ

which we maximize w.r.t. β, β0, and ξj.

Setting the respective derivatives to zero, we get

).,,1( 0 ,0 ,0

(4) ).,,1(

(3)

(2)

1

1

nj

nj

z

z

jjj

jj

n

jjj

n

jjjj

yβ

with and

(5) 1 11 2

1k

Tjkjk

n

j

n

kj

n

jjD zzL yy

We maximize (5) subject to

n

jjjj z

1

.0 and 0

In addition to (2) to (4), the constraints include

.,,1for

(8) 0)1()(

(7) 0

(6) 0)1()(

j

nj

Cz

Cz

jjj

j

jjjj

x

x

Together these equations (2) to (8) uniquely characterize the solution to the primal and dual problem.

By substituting (2) to (4) into (1), we obtain the Lagrangian dual function

Leo Breiman (2001)

Statistical modeling: the two cultures (with discussion).

Statistical Science 16, 199-231.

Discussants include Brad Efron and David Cox

GUYON, WESTON, BARNHILL & VAPNIK (2002, Machine Learning)

LEUKAEMIA DATA:

Only 2 genes are needed to obtain a zero CVE (cross-validated error rate)

COLON DATA:

Using only 4 genes, CVE is 2%

Since p>>n, consideration given to selection of suitable genes

SVM: FORWARD or BACKWARD (in terms of magnitude of weight βi)

RECURSIVE FEATURE ELIMINATION (RFE)

FISHER: FORWARD ONLY (in terms of CVE)

GUYON et al. (2002)

LEUKAEMIA DATA:

Only 2 genes are needed to obtain a zero CVE (cross-validated error rate)

COLON DATA:

Using only 4 genes, CVE is 2%

GUYON et al. (2002)

“The success of the RFE indicates that RFE has a built in regularization mechanism that we do not understand yet that prevents overfitting the training data in its selection of gene subsets.”

Example: Microarray DataColon Data of Alon et al. (1999)

n=62 (40 tumours; 22 normals)

tissue samples of

p=2,000 genes in a

2,000 62 matrix.

Figure 1: Error rates of the SVM rule with RFE procedure averaged over 50 random splits of colon tissue samples

Figure 2: Error rates of the SVM rule with RFE procedure averaged over 50 random splits of leukemia tissue samples

Figure 3: Error rates of Fisher’s rule with stepwise forward selection procedure using all the colon data

Figure 4: Error rates of Fisher’s rule with stepwise forward selection procedure using all the leukemia data

Figure 5: Error rates of the SVM rule averaged over 20 noninformative samples generated by random permutations of the class labels of the

colon tumor tissues

Aware of selection bias:

SPANG et al. (2001, Silico Biology)

WEST et al. (2001, PNAS)

NGUYEN and ROCKE (2002)

ADDITIONAL REFERENCES

Selection bias ignored:

XIONG et al. (2001, Molecular Genetics and Metabolism)

XIONG et al. (2001, Genome Research)

ZHANG et al. (2001, PNAS)

Error Rate Estimation

(y1, y2, y3,……………, yn)

Suppose there are two groups G1 and G2

c(y) is a classifier formed from the data set

The apparent error is the proportion of the data set misallocated by c(y).

Use c(1)(y1) to allocate y1 to either G1 or G2.

From the original data set, remove y1 to give the reduced set

(y2, y3,……………, yn)Then form the classifier c(1)(y ) from this reduced set.

Cross-Validation

Repeat this process for the second data point, y2.

So that this point is assigned to either G1 or G2 on the basis of the classifier c(2)(y2).

And so on up to yn.

Ten-Fold Cross Validation

1 2 3 4 5 6 7 8 9 10

T r a i n i n gTest

BOOTSTRAP APPROACH

Efron’s (1983, JASA) .632 estimator

B1.632 AE.368 632. B

where B1 is the bootstrap when rule is applied to a point not in the training sample.

A Monte Carlo estimate of B1 is

otherwise 0 esmisallocat * if 1

otherwise 0sample bootstrapth if 1

1

and

with

11

1

jjk

jjk

jkjkjkj

j

x

kx

Rk

K

k

K

k

n

j

Q

I

IQIE

nEB

Rk*

where

Toussaint & Sharpe (1975) proposed the ERROR RATE ESTIMATOR

CV2E )AE 1( A ww-(w)

5.0w

McLachlan (1977) proposed w=wo where wo is chosen to minimize asymptotic bias of A(w) in the case of two homoscedastic normal groups.

Value of w0 was found to range between 0.6 and 0.7, depending on the values of . and , ,

2

1

n

np

where

B1 )AE 1( 632. ww-B

.632+ estimate of Efron & Tibshirani (1997, JASA)

rw

368.1

632.

AE

AE1

B

r

g

i

ii qp1

)1(

where

(relative overfitting rate)

(estimate of no information error rate)

If r = 0, w = .632, and so B.632+ = B.632

r = 1, w = 1, and so B.632+ = B1

Ten-Fold Cross Validation

1 2 3 4 5 6 7 8 9 10

T r a i n i n gTest

MARKER GENES FOR HARVARD DATA

For a SVM based on 64 genes, and using 10-fold CV,we noted the number of times a gene was selected.

No. of genes Times selected 55 1 18 2 11 3 7 4 8 5 6 6 10 7 8 8 12 9 17 10

No. of Timesgenes selected 55 1 18 2 11 3 7 4 8 5 6 6 10 7 8 8 12 9 17 10

tubulin, alpha, ubiquitous

Cluster Incl N90862

cyclin-dependent kinase inhibitor 2C (p18, inhibits CDK4)

DEK oncogene (DNA binding)

Cluster Incl AF035316

transducin-like enhancer of split 2, homolog of Drosophila E(sp1)

ADP-ribosyltransferase (NAD+; poly (ADP-ribose) polymerase)

benzodiazapine receptor (peripheral)

Cluster Incl D21063

galactosidase, beta 1

high-mobility group (nonhistone chromosomal) protein 2

cold inducible RNA-binding protein

Cluster Incl U79287

BAF53

tubulin, beta polypeptide

thromboxane A2 receptor

H1 histone family, member X

Fc fragment of IgG, receptor, transporter, alpha

sine oculis homeobox (Drosophila) homolog 3

transcriptional intermediary factor 1 gamma

transcription elongation factor A (SII)-like 1

like mouse brain protein E46

minichromosome maintenance deficient (mis5, S. pombe) 6

transcription factor 12 (HTF4, helix-loop-helix transcription factors 4)

guanine nucleotide binding protein (G protein), gamma 3, linked

dihydropyrimidinase-like 2

Cluster Incl AI951946

transforming growth factor, beta receptor II (70-80kD)

protein kinase C-like 1

MARKER GENES FOR HARVARD DATA

Breast cancer data set in van’t Veer et al. (van’t Veer et al., 2002, Gene Expression Profiling Predicts Clinical Outcome Of Breast Cancer, Nature 415)

These data were the result of microarray experiments on three patient groups with different classes of breast cancer tumours.

The overall goal was to identify a set of genes that could distinguish between the different tumour groups based upon the gene expression information for these groups.

Breast tumours have a genetic signature. The expressionpattern of a set of 70 genes can predict whether a tumouris going to prove lethal, despite treatment, or not.

“This gene expression profile will outperform all currently used clinical parameters in predicting disease outcome.”

van ’t Veer et al. (2002), van de Vijver et al. (2002)

Number of Genes Error Rate for Top 70 Genes (without correction for Selection Bias as Top 70)

Error Rate for Top 70 Genes (with correction for Selection Bias as Top 70)

Error Rate for 5422 Genes (with correction for Selection Bias)

1 0.50 0.53 0.56

2 0.32 0.41 0.44

4 0.26 0.40 0.41

8 0.27 0.32 0.43

16 0.28 0.31 0.35

32 0.22 0.35 0.34

64 0.20 0.34 0.35

70 0.19 0.33 -

128 - - 0.39

256 - - 0.33

512 - - 0.34

1024 - - 0.33

2048 - - 0.37

4096 - - 0.40

5422 - - 0.44

van de Vijver et al. (2002) considered a further 234 breast cancer tumours but have only made available the data for the top 70 genes based on the previous study of van ‘t Veer et al. (2002)

Number of Genes

From 70 genes From original 24481 genes (set missing values to 0)

From original 24481 genes ( using KNN for missing values, k=10)

1 0.29491525 0.4023327 0.4199797

2 0.17288136 0.3850913 0.3825558

4 0.20000000 0.3747465 0.3756592

8 0.13220339 0.3033469 0.3061866

16 0.10508475 0.2314402 0.2319473

32 0.08474576 0.2038540 0.2240365

64 0.09491525 0.2038540 0.1915822

70 0.09491525

128 0.1634888 0.1600406

256 0.1462475 0.1507099

512 0.1359026 0.1438134

1024 0.1324544 0.1496957

2048 0.1521298 0.1364097

4096 0.1481744 0.1403651

8192 0.1550710 0.1605477

16384 0.1683570 0.1738337

24481 0.1683570 0.1772819

0 5 10 15

0.0

0.1

0.2

0.3

0.4

0.5

log2(Number of Genes)

CV

Err

or

Ra

tes

Unbiased (kNN)biased

Nearest-Shrunken Centroids(Tibshirani et al., 2002)

The usual estimates of the class means are shrunk toward the overall mean of the data, where

n

j

ijiji nyzy1

/

iyy

./1

n

j

j nyyand

The nearest-centroid rule is given by

where yv is the vth element of the feature vector y and .viv yy )(

v

In the previous definition, we replace the sample mean of the vth gene by its shrunken estimate

ivy

where

2

111 )( nnm ii

Comparison of Nearest-Shrunken Centroids with SVM

Apply (i) nearest-shrunken centroids and (ii) the SVM with RFEto colon data set of Alon et al. (1999), withN = 2000 genes and M = 62 tissues (40 tumours, 22 normals)

Nearest-Shrunken Centroids applied to Alon data

(b) Class-specific Error Rates

(a) Overall Error Rates

SVM with RFE applied to Alon data

(a) Overall Error Rates

(b) Class-specific Error Rates