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  • Statistical Analysis of Microarray Datapedagogix-tagc.univ-mrs.fr/.../pdf_files/11.discriminant_analysis_6sp… · Statistical Analysis of Microarray Data Jacques van Helden [email protected]
11
Supervised classification Statistical Analysis of Microarray Data Jacques van Helden [email protected] Application: ALL versus AML (data from Golub et al., 1999) ! Data source: Golub et al (1999). First historical publication searching for molecular signatures of cancer type. ! Training set " 38 samples from 2 types of leukemia • 27 Acute lymphoblastic leukemia (note: 2 subtypes: ALL-T and ALL-B) • 11 Acute myeloid leukemia " Original data set contains ~7000 genes " Filtering out poorly expressed genes retains 3051 genes ! We re-analyze the data using different methods. ! Selection of differentially expressed genes (DEG) " Welch t-test with robust estimators (median, IQR) retains 367differentially expressed genes with E- value <= 1. " Top plot: circle radius indicates T-test significance. " Bottom plot (volcano plot): • sig = -log10(E-value) >= 0 ! Golub, T. R., Slonim, D. K., Tamayo, P., Huard, C., Gaasenbeek, M., Mesirov, J. P., Coller, H., Loh, M. L., Downing, J. R., Caligiuri, M. A., Bloomfield, C. D. and Lander, E. S. (1999). Molecular classification of cancer: class discovery and class prediction by gene expression monitoring. Science 286, 531-7. ! ! ! ! ! ! !! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! !! !! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! !! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 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! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! −2 −1 0 1 2 0.1 0.2 0.3 0.4 0.5 0.6 Difference between the means Standard error on the difference ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! Golub 1999 - Profiles of selected genes ! The 367 gene selected by the T- test have apparently different profiles. " Some genes seem greener for the ALL patients (27 leftmost samples) " Some genes seem greener for the AML patients (11 rightmost samples) AML ALL Golub – hierarchical vlustering of DEG genes / profiles ! Hierarchical clustering perfectly separates the two cancer types (AML versus ALL). ! This perfect separation is observed for various metrics (Euclidian, correlation, dot product) and agglomeration rules (complete, average, Ward). ! Sample clustering further reveals subgroups of ALL. ! Gene clustering reveals 4 groups of profiles: " AML red, ALL green " AML green, ALL red " Overall green, stronger in AML " Overall red, stronger in ALL golub ; eu distance; complete linkage AML ALL Principal component analysis (PCA) ! Principal component analysis (PCA) relies on a transformation of a multivariate table into a multi- dimensional table of “components”. ! With Golub dataset, " Most variance is captured by the first component. " The first component (Y axis) clearly separates ALL from AML patients. " The second component splits the AML set into two well- separated groups, which correspond almost perfectly to T-cells and B-cells, resp. −0.2 −0.1 0.0 0.1 0.2 0.3 0.4 −0.2 −0.1 0.0 0.1 0.2 0.3 0.4 PC1 PC2 X1_ALL_B X2_ALL_T X3_ALL_T X4_ALL_B X5_ALL_B X6_ALL_T X7_ALL_B X8_ALL_B X9_ALL_T X10_ALL_T X11_ALL_T X12_ALL_B X13_ALL_B X14_ALL_T X15_ALL_B X16_ALL_B X17_ALL_B X18_ALL_B X19_ALL_B 20_ALL_B X21_ALL_B X22_ALL_B X23_ALL_T X24_ALL_B X25_ALL_B X26_ALL_B X27_ALL_B X28_AML X29_AML X30_AML X31_AML X32_AML X33_AML X34_AML X35_AML X36_AML X37_AML X38_AML −5 0 5 10 −5 0 5 10 11 23 55 66 96 108 117 126 135 158 168 171 172 174 178 182 187 192 202 205 237 239 248 253 259 283 297 304 307 320 323 329 330 337 344 345 364 376 377 378 389 394 400 401 419 422 436 453 462 475 479 489 494 515 519 522 523 542 546 557 560 561 566 617 621 622 624 648 666 686 695 701 703 704 713 717 725 735 738 746 763 766 773 785 789 792 801 803 807 808 829 830 839 841 849 858 860 862 866 877 896 904 906 922 932 937 940 963 968 971 984 988 995 1005 1006 1009 1010 1016 1019 1025 1030 1037 1038 1042 1045 1048 1057 1060 1062 1066 1069 1070 1078 1081 1086 1094 1101 1110 1112 1122 1124 11261145 1162 1171 1181 1193 1202 1220 1221 1225 1229 1238 1242 1253 1271 1282 1293 1298 1327 1329 1334 1337 1348 1368 1378 1381 1383 1390 1391 1396 1406 1411 1413 1417 1425 1430 1439 1441 1445 1448 1453 1455 1456 1459 1468 1470 1489 1493 1497 1513 1523 1524 1542 1553 1556 1559 1564 1585 1594 1598 1601 1604 1611 1616 1629 1638 1640 1647 1648 1652 1653 1665 1668 1671 1676 1702 1712 1719 1723 1732 1754 1766 1768 1774 1778 1784 1787 1797 1807 1811 1817 1821 1829 1834 1849 1869 1881 1882 1883 1884 1901 1903 1907 1909 1910 1911 1916 1920 1926 1938 1939 1948 1955 1959 1963 1975 1977 1978 1979 1985 1993 1995 1998 2002 2020 2052 2056 2065 2072 2079 2080 2087 2124 2132 2155 2174 2179 2180 2182 2197 2198 2208 2216 2235 2236 2265 2266 2276 2289 2306 2307 2322 2329 2347 2348 2355 2356 2364 2365 2386 2402 2410 2418 2430 2438 2444 2456 2459 2466 2489 2500 2506 2512 2553 2570 2589 2593 2600 2602 2616 2627 2631 2636 2645 2647 2656 2661 2663 2664 2670 2673 2681 2702 2734 2736 2743 2749 2750 2753 2761 2786 2791 2794 2800 2801 2802 2813 2829 2851 2860 2870 2874 2889 2902 2920 2921 2922 2939 2950 2952 2958 2977 2989 3011 3031 3046 golub.pca Variances 0 10 20 30 40 50 60

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Page 1: Statistical Analysis of Microarray Datapedagogix-tagc.univ-mrs.fr/.../pdf_files/11.discriminant_analysis_6sp… · Statistical Analysis of Microarray Data Jacques van Helden Jacques.van.Helden@ulb.ac.be

Supervised classification

Statistical Analysis of Microarray Data

Jacques van Helden [email protected]

Application: ALL versus AML (data from Golub et al., 1999)

!  Data source: Golub et al (1999). First historical publication searching for molecular signatures of cancer type.

!  Training set "  38 samples from 2 types of leukemia

•  27 Acute lymphoblastic leukemia (note: 2 subtypes: ALL-T and ALL-B)

•  11 Acute myeloid leukemia "  Original data set contains ~7000 genes "  Filtering out poorly expressed genes retains 3051

genes !  We re-analyze the data using different methods. !  Selection of differentially expressed genes (DEG)

"  Welch t-test with robust estimators (median, IQR) retains 367differentially expressed genes with E-value <= 1.

"  Top plot: circle radius indicates T-test significance. "  Bottom plot (volcano plot):

•  sig = -log10(E-value) >= 0

!  Golub, T. R., Slonim, D. K., Tamayo, P., Huard, C., Gaasenbeek, M., Mesirov, J. P., Coller, H., Loh, M. L., Downing, J. R., Caligiuri, M. A., Bloomfield, C. D. and Lander, E. S. (1999). Molecular classification of cancer: class discovery and class prediction by gene expression monitoring. Science 286, 531-7.

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−2 −1 0 1 2

0.1

0.2

0.3

0.4

0.5

0.6

Difference between the means

Stan

dard

erro

r on

the

diffe

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e

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Golub 1999 - Profiles of selected genes

!  The 367 gene selected by the T-test have apparently different profiles.

"  Some genes seem greener for the ALL patients (27 leftmost samples)

"  Some genes seem greener for the AML patients (11 rightmost samples)

AML!ALL!

Golub – hierarchical vlustering of DEG genes / profiles

!  Hierarchical clustering perfectly separates the two cancer types (AML versus ALL).

!  This perfect separation is observed for various metrics (Euclidian, correlation, dot product) and agglomeration rules (complete, average, Ward).

!  Sample clustering further reveals subgroups of ALL.

!  Gene clustering reveals 4 groups of profiles:

"  AML red, ALL green "  AML green, ALL red "  Overall green, stronger in

AML "  Overall red, stronger in ALL

X33_AM

LX37_AM

LX34_AM

LX35_AM

LX30_AM

LX31_AM

LX29_AM

LX28_AM

LX36_AM

LX32_AM

LX38_AM

LX1_ALL

X16_ALL

X19_ALL

X26_ALL

X13_ALL

X15_ALL

X5_ALL

X4_ALL

X24_ALL

X20_ALL

X21_ALL

X18_ALL

X12_ALL

X25_ALL

X7_ALL

X22_ALL

X8_ALL

X27_ALL

X14_ALL

X9_ALL

X11_ALL

X10_ALL

X3_ALL

X6_ALL

X23_ALL

X2_ALL

X17_ALL

240286625318072753114529027462438158182156024667356246179687385152266166847990639418348491202101611717231489785178718691025236519552386248910421883118113291368171542207266621127119933201101239422329192389839105714411782079155964823071811156416531425241826453231070255327342920292129221829937101014174001112175462289614481066112426815669321162188110302306377206524591766922241018280714702179152415427043641221197819592851198529502132280176313482592702129322761963713133720256111931920801223524816381702278670112829712829462159823642794106023482347111023557283126145916482512139030452212982971081108630726272889145519791817191698414453762216877100625934941045235670316401351094160415941712226572511262602220810192087344138134515852020273623552051253940250616291327261696318821513173233010786861948171920801456280254622361909523717337267324442372052133419394898621037199528609669521801741616293930468582155551671123819381721926187453145321821225146819032430200222892663266482917682656297710692589100527491556179726311682056184924564191907789904161143613781009267014131778295827503788601391250010622952111396808197726471220232221742329199825701439166513831652279114971493274384111227926661411197517842761152310898816011676773475155310382124260019117668031048190126611774280028131430117116471406303140183051912421884287028741910263612292197219830119952989

golub ; eu distance; complete linkage

AML! ALL!

Principal component analysis (PCA)

!  Principal component analysis (PCA) relies on a transformation of a multivariate table into a multi-dimensional table of “components”.

!  With Golub dataset, "  Most variance is captured by

the first component. "  The first component (Y axis)

clearly separates ALL from AML patients.

"  The second component splits the AML set into two well-separated groups, which correspond almost perfectly to T-cells and B-cells, resp.

−0.2 −0.1 0.0 0.1 0.2 0.3 0.4

−0.2

−0.1

0.0

0.1

0.2

0.3

0.4

PC1

PC2

X1_ALL_B

X2_ALL_T

X3_ALL_T

X4_ALL_B

X5_ALL_B

X6_ALL_T

X7_ALL_B

X8_ALL_B

X9_ALL_T

X10_ALL_T

X11_ALL_T

X12_ALL_BX13_ALL_B

X14_ALL_T

X15_ALL_B

X16_ALL_B

X17_ALL_B

X18_ALL_B

X19_ALL_B

X20_ALL_B

X21_ALL_B

X22_ALL_B

X23_ALL_T

X24_ALL_B X25_ALL_BX26_ALL_B

X27_ALL_B

X28_AML

X29_AML

X30_AML

X31_AMLX32_AMLX33_AMLX34_AML

X35_AML

X36_AMLX37_AML

X38_AML

−5 0 5 10

−50

510

1123556696 108

117126

135

158 168171

172174178182

187

192

202

205237

239248253259

283297304307

320

323329

330

337344345 364

376

377

378389

394

400

401

419422436

453

462

475

479

489

494

515

519

522

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542

546

557560561

566

617

621 622624

648

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686695701

703704713

717725

735

738

746

763

766 773

785 789

792

801

803807808

829

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839841

849858

860

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877 896

904

906922

932937

940963 968

971

984988

9951005

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1009

10101016

1019 1025

1030

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1042 10451048

1057

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10621066 1069

1070

1078

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109411011110

1112

1122

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13271329

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1391

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149314971513 1523

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15561559156415851594

15981601

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16471648

16521653

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17021712

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1754

1766

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1784

17871797

18071811

1817

1821

1829

1834

1849

1869

1881

1882

1883

1884

19011903

1907

1909

1910

1911

19161920

19261938

1939

1948

1955

19591963

1975

1977

19781979

19851993

1995

1998

2002

20202052

2056

2065

2072

20792080

2087

2124

21322155

217421792180 2182 2197

2198220822162235

2236

2265

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22762289

2306

2307 23222329

234723482355

2356

2364

2365

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24102418

24302438

2444

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2459

24662489

2500

25062512

2553

257025892593

2600

2602

26162627

2631

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2645

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2656

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2663

2664

2670

2673

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2702

2734

2736 27432749 27502753

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28132829

28512860 2870

287428892902

292029212922

2939

2950

29522958

2977

29893011

30313046

golub.pca

Variances

010

2030

4050

60

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Motivation

!  The article by Golub et al. (1999) was motivated by the need to develop efficient diagnostics to predict the cancer type from blood samples of patients.

!  They proposed a “molecular signature” of cancer type, allowing to discriminate ALM from ALL.

!  This first “historical” study relied on somewhat arbitrary criteria to select the genes composing this signature, and the way to apply them to classify new patients.

!  We will present here the classical methods used in statistics to classify “objects” (patients, genes) in pre-defined classes.

Supervised classification - Introduction

!  Clustering consists in grouping objects without any a priori definition of the groups. The group definition emerge from the clustering itself. Clustering is thus unsupervised.

!  In some cases, one would like to focus on some pre-defined classes : "  classifying tissues as cancer or non-cancer "  classifying tissues between different cancer types "  classifying genes according to pre-defined functional classes (e.g. metabolic

pathway, different phases of the cell cycle, ...) !  The classifier can be built with a training set, and used later for

classifying new objects. This is called supervised classification.

Supervised classification methods

!  There are many alternative methods for supervised classification "  Discriminant analysis (linear or quadratic) "  Bayesian classifiers "  K-nearest neighbours (KNN) "  Support Vector Machines (SVM) "  Neural networks "  ...

!  Some methods rely on strong assumptions. "  Discriminant analysis is based on an assumption of multivariate normality. "  In addition, linear discriminant analysis assumes that all the classes have the

same variance. !  Some methods require a large training set, to avoid over-fitting. !  The choice of the method thus depends on the structure and on the size

of the data sets.

Global versus local classifiers

!  As we saw for regression, classifiers can be global or local. "  Global classifiers use the same classification rule in the whole data space. The rule is

built on the whole training set. •  Example: discriminant analysis

"  For local classifiers, a rule is made in the different sub-spaces on the basis of the neighbouring training points. •  Example: KNN

Discriminant analysis

Statistical Analysis of Microarray Data

Jacques van Helden [email protected]

Multivariate data with a nominal criterion variable

!  One disposes of a set of objects (the sample) which have been previously assigned to predefined classes.

!  Each object is characterized by a series of quantitative variables (the predictors), and its class is indicated in a separated column (the criterion variable).

Criterion variablevariable 1 variable 2 ... variable p class

object 1 x1,1 x2,1 ... xp,1 A

object 2 x1,2 x2,2 ... xp,2 A

object 3 x1,3 x2,3 ... xp,3 A... ... ... ... ... ...object i x1,i x2,i ... xp,i B

object i+1 x1,i+1 x2,i+1 ... xp,i+1 B

object i+2 x1,i+2 x2,i+2 ... xp,i+2 B... ... ...object n-1 x1,n-1 x2,n-1 ... xp,n-1 K

object n x1,n x2,n ... xp,n K

Predictor variables

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Discriminant analysis - training and prediction

!  Training phase (training + evaluation) "  The sample is used to build a discriminant function "  The quality of the discriminant function is evaluated

!  Prediction phase "  The discriminant function is used to predict the value of the criterion variable

for new objects

Criterion variablevariable 1 variable 2 ... variable p class

object 1 x11 x21 ... xp1 A

object 2 x12 x22 ... xp2 A

object 3 x13 x23 ... xp3 B... ... ... ... ... ...object ntrain x1n x2n ... xpn K

Criterion variablevariable 1 variable 2 ... variable p class

object 1 x11 x21 ... xp1 ?

object 2 x12 x22 ... xp2 ?

object 3 x13 x23 ... xp3 ?... ... ... ... ... ...object npred x1n x2n ... xpn ?

Predictor variables

Predictor variables

Objects of known class

Discriminant analysis

Training set

Calibration

Discriminant function

Testing set

Prediction Predicted class Confusion table (for evaluation) Comparison

Objects of unknown class

Prediction Predicted class

Conceptual illustration with a single predictor variable

!  Given two predefined classes (A and B), try intuitively to assign a class to each new object (X positions denoted by vertical black bars).

!  How confident do you feel for each of your predictions ?

!  What is the effect of the respective means ?

!  What is the effect of the respective standard deviations ?

!  What is the effect of the population sizes ?

Conceptual illustration with a single variable

!  In this conceptual example, the two populations have the same mean and variance.

!  To which group (A or B) would you assign the points at coordinate x, y, z, t, respectively ?

Conceptual illustration with a single variable

!  Same exercise. !  This example shows that the

assignation is affected by the position of the group centres.

Conceptual illustration with a single variable

!  Same exercise. !  When the centres become too

close, some uncertainty is attached to some points (y, but also partly z).

!  There is thus an effect of group distance.

Page 4: Statistical Analysis of Microarray Datapedagogix-tagc.univ-mrs.fr/.../pdf_files/11.discriminant_analysis_6sp… · Statistical Analysis of Microarray Data Jacques van Helden Jacques.van.Helden@ulb.ac.be

Conceptual illustration with a single variable

!  Same exercise. !  The centres are in the same

position as in the first example, but the variance is larger.

!  This affects the level of separation of the groups, and raises some uncertainty about the group membership of z.

!  The group variance thus affects the assignation.

Conceptual illustration with a single variable

!  Same exercise. !  This illustrates the effect of the

sample size: if a sample has a much larger size than another one, it will increase the likelihood that some observations were issued from this group.

Conceptual illustration with a single variable

!  Same exercise. !  This is the symmetric

situation of the preceding figure.

!  Although the group centres and variances are identical, the change of sample sizes completely modifies the group assignations.

!  This is an effect of prior probability.

Conceptual illustration with a single variable

!  Same exercise. !  If the two groups have

different dispersions, it will affect their likelihood to be the originators of some observations.

!  The relative dispersion of the groups affects the assignation.

Conceptual illustration with a single variable

!  Same exercise. !  Symmetrical situation of

the preceding one: same centres, same sample sizes, but the relative variances vary in the opposite way.

!  The relative dispersion of the groups affects the assignation.

Conceptual illustration with a single variable

!  Same exercise. !  When the dispersion of one

group becomes too high, a simple boundary is not sufficient anymore to separate the two groups.

!  In this example, we would classify the leftmost (x) and rightmost (t, and maybe z) objects as B, and the central ones (y) as A. `

!  We need thus two boundaries to separate these groups.

!  The relative dispersion of the groups affects the assignation.

Page 5: Statistical Analysis of Microarray Datapedagogix-tagc.univ-mrs.fr/.../pdf_files/11.discriminant_analysis_6sp… · Statistical Analysis of Microarray Data Jacques van Helden Jacques.van.Helden@ulb.ac.be

Conceptual illustration with a single variable

!  Same exercise. !  Symmetrical situation of the

preceding figure. !  The relative dispersion of

the groups affects the assignation.

Conceptual illustration with two predictor variables !  Given two predefined classes (A and B), try intuitively to assign a class to each new object

(black dots). !  How confident do you feel for each of your predictions ?

!  What is the effect of the respective means ?

!  What is the effect of the respective standard deviations ?

!  What is the effect of the correlations between the two variables ?

!  Note that the two population can have distinct correlations (orientations of the clouds)

Conceptual illustration with two variable

!  The same concepts can be illustrated in a two-dimensional feature space.

!  Some additional concepts will appear.

!  Try to assign intuitively the points 1 to 6 to either group A or group B.

Conceptual illustration with two variable

!  Effect of the group centre location.

Conceptual illustration with two variable

!  Effect of the group variance.

Conceptual illustration with two variable

!  Effect of the group variance.

Page 6: Statistical Analysis of Microarray Datapedagogix-tagc.univ-mrs.fr/.../pdf_files/11.discriminant_analysis_6sp… · Statistical Analysis of Microarray Data Jacques van Helden Jacques.van.Helden@ulb.ac.be

Conceptual illustration with two variable

!  Effect of the relative variances.

Conceptual illustration with two variable

!  Effect of the relative variances.

Conceptual illustration with two variable

!  Effect of the covariance. between columns of the group.

Conceptual illustration with two variable

!  Effect of the covariance. between columns of the group.

Conceptual illustration with two variable

!  Effect of the relative covariance. between columns of the group.

!  The two groups have now different covariance matrices: the clouds are elongated in different directions.

!  This affects group assignments (example point 2).

Training set

!  There is a subset of objects (in the case below, genes) which can be assigned to predefined classes (e.g. “phosphate”, “methionine” or “control”), on the basis of external information (e.g. biological knowledge).

!  These classes will be used as criterion variable. !  Note : the sample class labels might contain some errors (misclassified objects).

# ORF Gene name Family # ORF Gene name Family # ORF Gene name Family1 YBR093C PHO5 PHO 14 YBR213W MET8 MET 33 YAL038W CDC19 CTL2 YDR481C PHO8 PHO 15 YDR253C MET32 MET 34 YBL005W PDR3 CTL3 YAR071W PHO11 PHO 16 YDR502C SAM2 MET 35 YBL005W-A YBL005W-A CTL4 YHR215W PHO12 PHO 17 YER091C MET6 MET 36 YBL005W-B YBL005W-B CTL5 YOL001W PHO80 PHO 18 YFR030W MET10 MET 37 YBL030C PET9 CTL6 YGR233C PHO81 PHO 19 YHL036W MUP3 MET 38 YBR006W UGA5 CTL7 YML123C PHO84 PHO 20 YIL046W MET30 MET 39 YBR018C GAL7 CTL8 YPL031C PHO85 PHO 21 YIR017C MET28 MET 40 YBR020W GAL1 CTL9 YJL117W PHO86 PHO 22 YJR010W MET3 MET 41 YBR115C LYS2 CTL

10 YCR037C PHO87 PHO 23 YJR137C ECM17 MET 42 YBR184W YBR184W CTL11 YBR106W PHO88 PHO 24 YKL001C MET14 MET 43 YCL018W LEU2 CTL12 YBR296C PHO89 PHO 25 YKR069W MET1 MET 44 YDL131W LYS21 CTL13 YHR136C SPL2 PHO 26 YLR180W SAM1 MET 45 YDL182W LYS20 CTL

27 YLR303W MET17 MET 46 YDL205C HEM3 CTL28 YLR396C VPS33 MET 47 YDL210W UGA4 CTL29 YNL241C ZWF1 MET 48 YDR011W SNQ2 CTL30 YNL277W MET2 MET 49 YDR044W HEM13 CTL31 YOL064C MET22 MET 50 YDR234W LYS4 CTL32 YPL038W MET31 MET 51 YDR285W ZIP1 CTL

... ... ... ...112 YPR065W ROX1 CTL113 YPR138C MEP3 CTL114 YPR145W ASN1 CTL

Phosphate-responding genes Methionine-responding genes Control genes

Page 7: Statistical Analysis of Microarray Datapedagogix-tagc.univ-mrs.fr/.../pdf_files/11.discriminant_analysis_6sp… · Statistical Analysis of Microarray Data Jacques van Helden Jacques.van.Helden@ulb.ac.be

Sample profiles - gene expression data

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phosphate expression data - PHO genesvariable

valu

e

YBR093CYDR481CYAR071WYHR215WYOL001WYGR233CYML123CYPL031CYJL117WYCR037CYBR106WYBR296CYHR136C246810

-4-2 0246

phosphate expression data - MET genesvariable

valu

e

YBR213WYDR253CYDR502CYER091CYFR030WYHL036WYIL046WYIR017CYJR010WYJR137CYKL001CYKR069WYLR180WYLR303WYLR396CYNL241CYNL277WYOL064CYPL038W246810

-4-2 0246

phosphate expression data - CTL genesvariable

valu

e

YAL038WYBL005WYBL005W-AYBL005W-BYBL030CYBR006WYBR018CYBR020WYBR115CYBR184WYCL018WYDL131WYDL182WYDL205CYDL210WYDR011WYDR044WYDR234WYDR285WYDR406WYEL021WYEL039CYER026CYER086WYER179W

2-dimensional visualization of the sample

!  If there are many variables, PCA can be used to visualize the sample on the planed formed by the two principal components.

!  Example: gene expression data "  MET genes seem

undistinguishable from CTL genes (they are indeed not expected tor espond to phosphate)

"  Most PHO genes are clearly distant from the main cloud of points.

"  Some PHO genes are mixed with the CTL genes.

Classification rules

!  New units can be classified on the basis of rules based on the calibration sample !  Several alternative rules can be used

"  Maximum likelihood rule: assign unit u to group g if

"  Inverse probability rule: assign unit u to group g if

"  Posterior probability rule: assign unit u to group g if

Where X is the unit vector g,g’ are two groups f(X|g) is the density function of the value X for group g P(X|g) is the probability to emit the value X given the group g P(g|X) is the probability to belong to group g, given the value X

!

P(X | g) > P(X | g') for g'" g

!

P(g | X) > P(g' | X) for g'" g

!

f (X | g) > f (X | g') for g'" g

Posterior probability rule

!  The posterior probability can be obtained by application of Bayes' theorem

!

P(g | X) =P(X | g)P(g)

P(X)

!

P(g | X) =P(X | g)" g

P(X | g')" g'g '=1

k

#

Where "  X is the unit vector "  g is a group "  k is the number of groups "  !g is the prior probability of group g

Maximum likelihood rule - multivariate normal case

!  If the predictor variable is univariate normal

!

f (X | g) =1

2" # g2e$12X $µg

# g

%

& ' '

(

) * *

2

!

f (X | g) =1

2"( )p #g

e$12X $µg( )'#g$1 X $µg( )%

& ' (

) *

!  If the predictor variable is multivariate normal

Where "  X is the unit vector "  p is the number of variables "  µg is the mean vector for group g "  "g is the covariance matrix for group g

Bayesian classification in case of normality

!  Each object is assigned to the group which minimizes the function

( )

( ) ( )!"#

$%& '('' '

(=

ggg XX

gp

egPfµµ

)

1'21

2

1)(

Page 8: Statistical Analysis of Microarray Datapedagogix-tagc.univ-mrs.fr/.../pdf_files/11.discriminant_analysis_6sp… · Statistical Analysis of Microarray Data Jacques van Helden Jacques.van.Helden@ulb.ac.be

Linear versus quadratic classification rule

!  There is one covariance matrix per group g. "  This matrix indicates the covariance between each column (variable) of the

data set, for the considered group. "  The diagonals of this matrix represent the variance (=covariance between a

variable and itself) !  When all covariance matrix are assumed to be identical

"  The classification rule can be simplified to obtain a linear function. This is referred to as Linear Discriminant Analysis (LDA)

"  In this case,the boundary between groups will be a plane (2 variables) or a hyper-plane (more than 2 variables).

!  If the variances and covariances are expected to differ between groups "  A specific covariance matrix has to be used for each group. "  The boundary between two groups is a curve (with two variables) or a hyper-

surface (more than 2 variables). "  This is referred to as Quadratic Discriminant Analysis (QDA)

Evaluation of the discriminant function - confusion table

!  One way to evaluate the accuracy of the discriminant function is to apply it to the sample itself. This approach is called internal analysis.

!  The known and predicted class are then compared for each sample unit. !  Warning : internal analysis is too optimistic. This approach is not recommended.

variable 1 variable 2 ... variable p known predictedobject 1 x1,1 x2,1 ... xp,1 A A

object 2 x1,2 x2,2 ... xp,2 A B

object 3 x1,3 x2,3 ... xp,3 A A... ... ... ... ... ... ...object i x1,i x2,i ... xp,i B K

object i+1 x1,i+1 x2,i+1 ... xp,i+1 B B

object i+2 x1,i+2 x2,i+2 ... xp,i+2 B B... ... ...object n-1 x1,n-1 x2,n-1 ... xp,n-1 K K

object n x1,n x2,n ... xp,n K K

Predictor variables Criterion variable

Evaluation of the discriminant function - confusion table

!  The results of the evaluation are summarized in a confusion table, which contains the count of the predicted/known combinations.

!  The confusion table can be used to calculate the accuracy of the predictions.

PHO MET CTL SUMPHO 8 0 0 8MET 0 1 1 2CTL 5 18 81 104SUM 13 19 82 114Errors 24 21.05%Correct 90 78.95%

Pre

dict

ed

Confusion table

Known

Evaluation of the discriminant function - plot

!  The two first discriminant functions can be used as X and Y axes for plotting the result.

!  In the same way as for PCA, X and Y axes represent linear combinations of variables

!  However, these combinations are not the same as the first factors obtained by PCA.

"  When comparing with PCA figure, the PHO genes are now all located nearby the X axis.

PPPPCCPCPCCPPCCCCCMCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCMCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC-6-4-20

-2-1 0123LDA Calibration - phosphate expression dataLinear discriminant function 1

Line

ar d

iscr

imin

ant f

unct

ion

2

Letters indicate the predicted class, colors the known class

External analysis

!  Using the sample itself for evaluation is problematic, because the evaluation is biased (too optimistic). To obtain an independent evaluation, one needs two separate sets : one for calibration, and one for evaluation. This approach is called external analysis.

!  The simplest setting is to split randomly the sample into two sets (holdout approach) : "  the training set is used to build a discriminant function "  the testing set is used for evaluation

Criterion variablevariable 1 variable 2 ... variable p class

object 1 x11 x21 ... xp1 A

object 2 x12 x22 ... xp2 A

object 3 x13 x23 ... xp3 B... ... ... ... ... ...object ntrain x1n x2n ... xpn K

variable 1 variable 2 ... variable p known predictedobject 1 x11 x21 ... xp1 A A

object 2 x12 x22 ... xp2 B A

object 3 x13 x23 ... xp3 B B... ... ... ... ... ... ...object ntest x1n x2n ... xpn K K

Criterion variable

Predictor variables

Tra

inin

g s

et

Predictor variables

Tes

tin

g s

et

Leave-one-out (LOO) validation

!  When the sample is too small, it is problematic to loose half of it for testing.

!  In such a case, the leave-one-out (LOO) approach is recommended : 1.  Discard a single object from the sample. 2.  With the remaining objects, build a discriminant function. 3.  Use this discriminant function to predict the class of the discarded object. 4.  Compare known and predicted class for the discarded object. 5.  Iterate the above steps with each object of the sample.

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Profiles after prediction

!  Example: "  Gene expression data "  Linear discriminant analysis "  Leave-one-out cross-validation.

!  Genes predicted as "PHO" have generally high levels of response (but this is not true for all of them)

!  A very few genes are predicted as MET.

!  Most genes predicted as control have a low levels of regulation.

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Predicted as PHOYBR093CYDR481CYAR071WYHR215WYML123CYJL117WYBR296CYHR136C246810

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Predicted as METYDR234WYGR088WYLR263W246810

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Predicted as CTLYOL001WYGR233CYPL031CYCR037CYBR106WYBR213WYDR253CYDR502CYER091CYFR030WYHL036WYIL046WYIR017CYJR010WYJR137CYKL001CYKR069WYLR180WYLR303WYLR396CYNL241CYNL277W

Analysis of the misclassified units

!  The sample might itself contain classification errors. The apparent misclassifications can actually represent corrections of these labelling errors.

!  Example : gene expression data - linear discriminant analysis All the genes "mis"classified as control have actually a flat expression profile.

"  Most of them are MET genes (indeed, these are not expected to respond to phosphate) "  the 4 PHO genes (blue) have a flat profile

246810

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valu

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YOL001WYGR233CYPL031CYCR037CYBR106WYBR213WYDR253CYDR502CYER091CYFR030WYIL046WYIR017CYJR010WYJR137CYKL001CYKR069WYLR180WYLR303WYLR396CYNL241CYNL277WYOL064CYPL038W

Evaluation with leave-one-out

!  Leave-one-out is more severe for evaluating the accuracy of predictions.

Choice of the prior probabilities

!  The classes may have different proportions between the sample and the population

!  For example, we could decide, on the basis of our biological knowledge, that it is likely to have 1% rather than 11% of yeast gene responding to phosphate.

Class Sample Priors from sample

Arbitrary priors

PHO 13 659 5811% 11% 1%

MET 19 964 5817% 17% 1%

CTL 82 4160 566772% 72% 98%

TOTAL 114 5783 5783

Population

Prediction phase Over-fitting

!  A typical application of supervised classification is to classify experiments (e.g. patient types) on the basis of the expression profiles.

!  In this case, the objects are the experiments, and the variables the genes.

!  This raises a problem of over-fitting: the number of variables is much larger than the number of objects in the training set.

!  In such situations, the classifier will tend to build a classification rule which perfectly fits the training set, but fails to generalize to other observations.

!!!!!!!!!!!

!!

!!

!!!!!!!

!

!!

!!

!!

0 100 200 300 400 500

0.0

0.2

0.4

0.6

0.8

1.0

Number of variables and over−fittingrandom tests

Number of variables (p)

Erro

r rat

e

!!!!!!!!!!

!

!

!

!

!

!!!!!!!!

!

!!

!

!

!

LOOinternalexpected (balanced classes)expected (majority class)

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Feature selection

!  One approach to circumvent this problem is to select a subset of variables only. !  This subset of variables can be selected according to different rules.

"  Variable ordering: variables are ordered according to some criterion, and the topmost variables are retained. •  Inter-group distances calculated in each variable separately. This inter-group

distance can be calculated with the t-test. •  P-value of the t-test (the P-value is not always linear with the t statistics, since the

number of observations can vary from row to row if there are missing values). "  Selection of a subset of variables and estimation of the capability of each subset to

classify correctly. The number of possible combinations of variables increases exponentially with the number of variables. Some heuristics allow to select a subset in a quadratic time •  Forward selection •  Backward selection •  Forward-backward selection

Variable ordering with the t-test

Forward stepwise feature selection Summary - discriminant analysis

!  Discriminant analysis is based on a set of quantitative predictor variables, and a single nominal criterion variable.

!  A sample is used to build a set of discriminant functions (calibration), which is then used to assign additional units to classes (prediction).

!  The discriminant function can be either linear or quadratic. Linear discriminant analysis relies on the assumption that the different classes have similar covariance matrices.

!  The accuracy of the discriminant function can be evaluated in different ways. "  On the whole sample (internal approach) "  Splitting of the sample into training and testing set (holdout approach) "  Successively discard each sample unit, build a discriminant function and predict the

discarded unit (leave-one-out) !  The efficiency decreases with the p/N ratio. When this ratio is too low, there is a

problem of over-fitting. !  Stepwise approaches consist in selecting the subset of variables which raises

the highest efficiency.

KNN classifiers

Statistical Analysis of Microarray Data

Jacques van Helden [email protected]

K nearest neighbours

!  Discriminant analysis is a global approach to classification: the discriminant rule is established in the same way for the whole data space, on the basis of group centres and covariance matrices. Discriminant analysis is thus a global classifier.

!  K nearest neighbour (KNN) classifiers takes a very different approach: at each position of the feature space

"  The K closest neighbour points from the training set are identified; "  A vote is established as a function of the relative proportions of the respective training

groups in this set of neighbours. !  KNN is thus a local classifier. !  The choice of K drastically affects group assignments.

Page 11: Statistical Analysis of Microarray Datapedagogix-tagc.univ-mrs.fr/.../pdf_files/11.discriminant_analysis_6sp… · Statistical Analysis of Microarray Data Jacques van Helden Jacques.van.Helden@ulb.ac.be

Support Vector Machines

Statistical Analysis of Microarray Data

Jacques van Helden [email protected]

Web resources

!  Gist "  Download http://microarray.cpmc.columbia.edu/gist/ "  Web interface http://svm.sdsc.edu/cgi-bin/nph-SVMsubmit.cgi


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