This document summarizes key topics in developing and validating predictive classifiers based on gene expression profiling. It discusses the importance of clear study objectives, feature selection methods, model types, and proper evaluation of classifiers using cross-validation to estimate prediction accuracy, rather than overfitting to the training data. Complex feature selection and model fitting are unlikely to help for high-dimensional genomic data. Simple classification methods like linear discriminant analysis often perform best.

1. Topics in the Development and
Validation of Gene Expression
Profiling Based Predictive Classifiers
Richard Simon, D.Sc.
Chief, Biometric Research Branch
National Cancer Institute
Linus.nci.nih.gov/brb

2. BRB Website
http://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

3. 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

4. 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

5. 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

6. Class Comparison and Class
Prediction
• Not clustering problems
• Supervised methods

7. 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

8. 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

9. Feature Selection
• Genes that are differentially expressed among the
classes at a significance level α (e.g. 0.01)
– The α 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

11. 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

12. 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

13. 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 i
i F
i
l x w x
x
F
w
l x
ε
=
=
=
=
∑

14. 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

15. 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

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

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

18. Other Simple Methods
• Nearest neighbor classification
• Nearest k-neighbors
• Nearest centroid classification
• Shrunken centroid classification

19. 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

22. 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

23. 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

24. 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

25. 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

26. • 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

27. 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.

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

30. 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

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

33. Comparison of Internal Validation Methods
Molinaro, 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

35. Simulated Data
40 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

36. Simulated Data
40 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

37. 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

39. • 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

43. 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

46. 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

47. 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 zα β γ= + − +

48. 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

49. Myth
• Huge sample sizes are needed to develop
effective predictive classifiers

50. 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)

51. Sample Size Planning for Classifier
Development
• The expected value (over training sets) of
the probability of correct classification
PCC(n) should be within γ of the maximum
achievable PCC(∞)

52. 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 2δ
• Common variance for differentially expressed genes and
for noise genes

53. Classifier
• Feature selection based on univariate t-
tests for differential expression at
significance level α
• 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 2δ is 1-β(n)

54. • For 2 classes of equal prevalence, let λ1 denote
the largest eigenvalue of the covariance matrix
of informative genes. Then
1
( )
m
PCC
δ
σ λ
  
∞ ≤ Φ  
  

55. ( )
( ) ( )1
1
( ) 1
1
mm
PCC n
m p m
βδ
β
σ λ β α
 − 
≥ Φ − 
− + −  

56. 1.0 1.2 1.4 1.6 1.8 2.0
406080100
2 delta/sigma
Samplesize
gamma=0.05
gamma=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.

58. 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
0.750.800.85
Proportion in under-represented class
PCC(60)

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

62. 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

63. 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

64. 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

65. 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

66. 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

67. 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

68. BRB-ArrayTools
December 2006
• 6635 Registered users
• 1938 Distinct institutions
• 68 Countries
• 311 Citations