Topics in the Development and Validation of Gene Expression

Profiling Based Predictive Classifiers

Richard Simon, D.Sc.Chief, Biometric Research Branch

National Cancer InstituteLinus.nci.nih.gov/brb

BRB Websitehttp://linus.nci.nih.gov/brb

• Powerpoint presentations and audio files

• Reprints & Technical Reports

• BRB-ArrayTools software

• BRB-ArrayTools Data Archive

• Sample Size Planning for Targeted Clinical Trials

Simplified Description of Microarray Assay

• Extract mRNA from cells of interest– Each mRNA molecule was transcribed from a single gene and it

has a linear structure complementary to that gene• Convert mRNA to cDNA introducing a fluorescently

labeled dye to each molecule• Distribute the cDNA sample to a solid surface containing

“probes” of DNA representing all “genes”; the probes are in known locations on the surface

• Let the molecules from the sample hybridize with the probes for the corresponding genes

• Remove excess sample and illuminate surface with laser with frequency corresponding to the dye

• Measure intensity of fluorescence over each probe

Resulting Data

• Intensity over a probe is approximately proportional to abundance of mRNA molecules in the sample for the gene corresponding to the probe

• 40,000 variables measured for each case– Excessive hype– Excessive skepticism– Some familiar statistical paradigms don’t work

well

Good Microarray Studies Have Clear Objectives

• Class Comparison (Gene Finding)– Find genes whose expression differs among predetermined

classes, e.g. tissue or experimental condition

• Class Prediction– Prediction of predetermined class (e.g. treatment outcome)

using information from gene expression profile– Survival risk-group prediction

• Class Discovery– Discover clusters of specimens having similar expression

profiles

Class Comparison and Class Prediction

• Not clustering problems

• Supervised methods

Class Prediction ≠ Class Comparison

• A set of genes is not a predictive model• Emphasis in class comparison is often on understanding

biological mechanisms– More difficult than accurate prediction and usually requires a

different experiment

• Demonstrating statistical significance of prognostic factors is not the same as demonstrating predictive accuracy

Components of Class Prediction

• Feature (gene) selection– Which genes will be included in the model

• Select model type – E.g. Diagonal linear discriminant analysis,

Nearest-Neighbor, …

• Fitting parameters (regression coefficients) for model– Selecting value of tuning parameters

Feature Selection

• Genes that are differentially expressed among the classes at a significance level (e.g. 0.01) – The level is a tuning parameter– Number of false discoveries is not of direct relevance for

prediction

• For prediction it is usually more serious to exclude an informative variable than to include some noise variables

Optimal significance level cutoffs for gene selection. 50 differentially expressed genes

out of 22,000 genes on the microarrays 2δ/σ n=10 n=30 n=50

1 0.167 0.003 0.00068

1.25 0.085 0.0011 0.00035

1.5 0.045 0.00063 0.00016

1.75 0.026 0.00036 0.00006

2 0.015 0.0002 0.00002

Complex Gene Selection

• Small subset of genes which together give most accurate predictions– Genetic algorithms

• Little evidence that complex feature selection is useful in microarray problems

Linear Classifiers for Two Classes

( )

vector of log ratios or log signals

features (genes) included in model

weight for i'th feature

decision boundary ( ) > or < d

i ii F

i

l x w x

x

F

w

l x

Linear Classifiers for Two Classes

• Fisher linear discriminant analysis• Diagonal linear discriminant analysis (DLDA)

– Ignores correlations among genes

• Compound covariate predictor• Golub’s weighted voting method• Support vector machines with inner product

kernel• Perceptrons

When p>>n

• It is always possible to find a set of features and a weight vector for which the classification error on the training set is zero.

• There is generally not sufficient information in p>>n training sets to effectively use more complex methods

Myth

• Complex classification algorithms such as neural networks perform better than simpler methods for class prediction.

• Comparative studies have shown that simpler methods work as well or better for microarray problems because they avoid overfitting the data.

Other Simple Methods

• Nearest neighbor classification

• Nearest k-neighbors

• Nearest centroid classification

• Shrunken centroid classification

Evaluating a Classifier

• Most statistical methods were not developed for p>>n prediction problems

• Fit of a model to the same data used to develop it is no evidence of prediction accuracy for independent data

• Demonstrating statistical significance of prognostic factors is not the same as demonstrating predictive accuracy

• Testing whether analysis of independent data results in selection of the same set of genes is not an appropriate test of predictive accuracy of a classifier

Internal Validation of a Classifier

• Re-substitution estimate– Develop classifier on dataset, test predictions

on same data– Very biased for p>>n

• Split-sample validation

• Cross-validation

Split-Sample Evaluation

• Training-set– Used to select features, select model type, determine

parameters and cut-off thresholds

• Test-set– Withheld until a single model is fully specified using

the training-set.– Fully specified model is applied to the expression

profiles in the test-set to predict class labels. – Number of errors is counted

Leave-one-out Cross Validation

• Omit sample 1– Develop multivariate classifier from scratch on

training set with sample 1 omitted– Predict class for sample 1 and record whether

prediction is correct

Leave-one-out Cross Validation

• Repeat analysis for training sets with each single sample omitted one at a time

• e = number of misclassifications determined by cross-validation

• Subdivide e for estimation of sensitivity and specificity

• With proper cross-validation, the model must be developed from scratch for each leave-one-out training set. This means that feature selection must be repeated for each leave-one-out training set.

– Simon R, Radmacher MD, Dobbin K, McShane LM. Pitfalls in the analysis of DNA microarray data. Journal of the National Cancer Institute 95:14-18, 2003.

• The cross-validated estimate of misclassification error is an estimate of the prediction error for model fit using specified algorithm to full dataset

Prediction on Simulated Null Data

Generation of Gene Expression Profiles

• 14 specimens (Pi is the expression profile for specimen i)

• Log-ratio measurements on 6000 genes

• Pi ~ MVN(0, I6000)

• Can we distinguish between the first 7 specimens (Class 1) and the last 7 (Class 2)?

Prediction Method

• Compound covariate prediction

• Compound covariate built from the log-ratios of the 10 most differentially expressed genes.

Number of misclassifications

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Pro

po

rtio

n o

f sim

ula

ted

da

ta s

ets

0.00

0.05

0.10

0.90

0.95

1.00

Cross-validation: none (resubstitution method)Cross-validation: after gene selectionCross-validation: prior to gene selection

Major Flaws Found in 40 Studies Published in 2004

• Inadequate control of multiple comparisons in gene finding– 9/23 studies had unclear or inadequate methods to deal with

false positives• 10,000 genes x .05 significance level = 500 false positives

• Misleading report of prediction accuracy– 12/28 reports based on incomplete cross-validation

• Misleading use of cluster analysis – 13/28 studies invalidly claimed that expression clusters based on

differentially expressed genes could help distinguish clinical outcomes

• 50% of studies contained one or more major flaws

Myth

• Split sample validation is superior to LOOCV or 10-fold CV for estimating prediction error

Comparison of Internal Validation MethodsMolinaro, Pfiffer & Simon

• For small sample sizes, LOOCV is much less biased than split-sample validation

• For small sample sizes, LOOCV is preferable to 10-fold, 5-fold cross-validation or repeated k-fold versions

• For moderate sample sizes, 10-fold is preferable to LOOCV

• Some claims for bootstrap resampling for estimating prediction error are not valid for p>>n problems

Simulated Data40 cases, 10 genes selected from 5000

Method Estimate Std Deviation

True .078

Resubstitution .007 .016

LOOCV .092 .115

10-fold CV .118 .120

5-fold CV .161 .127

Split sample 1-1 .345 .185

Split sample 2-1 .205 .184

.632+ bootstrap .274 .084

Simulated Data40 cases

Method Estimate Std Deviation

True .078

10-fold .118 .120

Repeated 10-fold .116 .109

5-fold .161 .127

Repeated 5-fold .159 .114

Split 1-1 .345 .185

Repeated split 1-1 .371 .065

DLBCL Data

Method Bias Std Deviation MSE

LOOCV -.019 .072 .008

10-fold CV -.007 .063 .006

5-fold CV .004 .07 .007

Split 1-1 .037 .117 .018

Split 2-1 .001 .119 .017

.632+ bootstrap -.006 .049 .004

• Ordinary bootstrap– Training and test sets overlap

• Bootstrap cross-validation (Fu, Carroll,Wang)

– Perform LOOCV on bootstrap samples– Training and test sets overlap

• Leave-one-out bootstrap– Predict for cases not in bootstrap sample– Training sets are too small

• Out-of-bag bootstrap (Breiman)

– Predict for case i based on majority rule of predictions for bootstrap samples not containing case i

• .632+ bootstrap– w*LOOBS+(1-w)RSB

Permutation Distribution of Cross-validated Misclassification Rate of a

Multivariate Classifier• Randomly permute class labels and repeat

the entire cross-validation• Re-do for all (or 1000) random

permutations of class labels• Permutation p value is fraction of random

permutations that gave as few misclassifications as e in the real data

Does an Expression Profile Classifier Predict More Accurately Than Standard

Prognostic Variables?

• Not an issue of which variables are significant after adjusting for which others or which are independent predictors– Predictive accuracy, not significance

• The two classifiers can be compared by ROC analysis as functions of the threshold for classification

• The predictiveness of the expression profile classifier can be evaluated within levels of the classifier based on standard prognostic variables

Does an Expression Profile Classifier Predict More Accurately Than Standard

Prognostic Variables?

• Some publications fit logistic model to standard covariates and the cross-validated predictions of expression profile classifiers

• This is valid only with split-sample analysis because the cross-validated predictions are not independent

log ( ) ( | )i iit p y x i z

Survival Risk Group Prediction

• For analyzing right censored data to develop predictive classifiers it is not necessary to make the data binary

• Can do cross-validation to predict high or low risk group for each case

• Compute Kaplan-Meier curves of predicted risk groups• Permutation significance of log-rank statistic• Implemented in BRB-ArrayTools• BRB-ArrayTools also provides for comparing the risk

group classifier based on expression profiles to one based on standard covariates and one based on a combination of both types of variables

Myth

• Huge sample sizes are needed to develop effective predictive classifiers

Sample Size Planning References

• K Dobbin, R Simon. Sample size determination in microarray experiments for class comparison and prognostic classification. Biostatistics 6:27-38, 2005

• K Dobbin, R Simon. Sample size planning for developing classifiers using high dimensional DNA microarray data. Biostatistics (2007)

Sample Size Planning for Classifier Development

• The expected value (over training sets) of the probability of correct classification PCC(n) should be within of the maximum achievable PCC()

Probability Model

• Two classes• Log expression or log ratio MVN in each class with

common covariance matrix• m differentially expressed genes• p-m noise genes• Expression of differentially expressed genes are

independent of expression for noise genes• All differentially expressed genes have same inter-class

mean difference 2• Common variance for differentially expressed genes and

for noise genes

Classifier

• Feature selection based on univariate t-tests for differential expression at significance level

• Simple linear classifier with equal weights (except for sign) for all selected genes. Power for selecting each of the informative genes that are differentially expressed by mean difference 2 is 1-(n)

• For 2 classes of equal prevalence, let 1 denote the largest eigenvalue of the covariance matrix of informative genes. Then

1

( )m

PCC

1

1( ) 1

1

mmPCC n

m p m

1.0 1.2 1.4 1.6 1.8 2.0

40

60

80

100

2 delta/sigma

Sam

ple

siz

e

gamma=0.05gamma=0.10

Sample size as a function of effect size (log-base 2 fold-change between classes divided by

standard deviation). Two different tolerances shown, . Each class is equally represented in the population. 22000 genes on an array.

b) PCC(60) as a function of the proportion in the under-represented class. Parameter settings same

as a), with 10 differentially expressed genes among 22,000 total genes. If the proportion in the under-represented class is small (e.g., <20%), then the PCC(60) can decline significantly.

0.1 0.2 0.3 0.4 0.5

0.7

50.8

00.8

5

Proportion in under-represented class

PC

C(6

0)

Acknowledgements

• Kevin Dobbin• Alain Dupuy• Wenyu Jiang• Annette Molinaro• Ruth Pfeiffer• Michael Radmacher• Joanna Shih• Yingdong Zhao• BRB-ArrayTools Development Team

BRB-ArrayTools

• Contains analysis tools that I have selected as valid and useful

• Analysis wizard and multiple help screens for biomedical scientists

• Imports data from all platforms and major databases

• Automated import of data from NCBI Gene Express Omnibus

Predictive Classifiers in BRB-ArrayTools

• Classifiers– Diagonal linear discriminant– Compound covariate – Bayesian compound covariate– Support vector machine with

inner product kernel– K-nearest neighbor

– Nearest centroid– Shrunken centroid (PAM)– Random forrest– Tree of binary classifiers for k-

classes

• Survival risk-group– Supervised pc’s

• Feature selection options– Univariate t/F statistic– Hierarchical variance option– Restricted by fold effect– Univariate classification power– Recursive feature elimination– Top-scoring pairs

• Validation methods– Split-sample– LOOCV– Repeated k-fold CV– .632+ bootstrap

Selected Features of BRB-ArrayTools• Multivariate permutation tests for class comparison to control

number and proportion of false discoveries with specified confidence level– Permits blocking by another variable, pairing of data, averaging of

technical replicates• SAM

– Fortran implementation 7X faster than R versions• Extensive annotation for identified genes

– Internal annotation of NetAffx, Source, Gene Ontology, Pathway information

– Links to annotations in genomic databases• Find genes correlated with quantitative factor while controlling

number of proportion of false discoveries• Find genes correlated with censored survival while controlling

number or proportion of false discoveries• Analysis of variance

Selected Features of BRB-ArrayTools

• Gene set enrichment analysis. – Gene Ontology groups, signaling pathways, transcription

factor targets, micro-RNA putative targets– Automatic data download from Broad Institute– KS & LS test statistics for null hypothesis that gene set is not

enriched– Hotelling’s and Goeman’s Global test of null hypothesis that

no genes in set are differentially expressed– Goeman’s Global test for survival data

• Class prediction– Multiple classifiers– Complete LOOCV, k-fold CV, repeated k-fold, .632

bootstrap– permutation significance of cross-validated error rate

Selected Features of BRB-ArrayTools

• Survival risk-group prediction– Supervised principal components with and without clinical

covariates– Cross-validated Kaplan Meier Curves– Permutation test of cross-validated KM curves

• Clustering tools for class discovery with reproducibility statistics on clusters– Internal access to Eisen’s Cluster and Treeview

• Visualization tools including rotating 3D principal components plot exportable to Powerpoint with rotation controls

• Extensible via R plug-in feature• Tutorials and datasets

BRB-ArrayTools

• Extensive built-in gene annotation and linkage to gene annotation websites

• Publicly available for non-commercial use– http://linus.nci.nih.gov/brb

BRB-ArrayToolsDecember 2006

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• 1938 Distinct institutions

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• 311 Citations