Design

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Design Analysis of Microarray Studies for
Diagnostic Prognostic Classification

• Richard Simon, D.Sc.
• Chief, Biometric Research Branch
• National Cancer Institute
• http//linus.nci.nih.gov/brb

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http//linus.nci.nih.gov/brb

• http//linus.nci.nih.gov/brb
• Powerpoint presentations
• Reprints Technical Reports
• BRB-ArrayTools software
• BRB-ArrayTools Data Archive
• Sample Size Planning for Targeted Clinical Trials

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Simon R, Korn E, McShane L, Radmacher M, Wright
G, Zhao Y. Design and analysis of DNA microarray
investigations, Springer-Verlag, 2003.
Radmacher MD, McShane LM, Simon R. A paradigm fo
r class prediction using gene expression
profiles. Journal of Computational Biology
9505-511, 2002. Simon R, Radmacher MD, Dobbin
K, McShane LM. Pitfalls in the analysis of DNA
microarray data. Journal of the National Cancer
Institute 9514-18, 2003. Dobbin K, Simon R. Co
mparison of microarray designs for class
comparison and class discovery, Bioinformatics
181462-69, 2002 19803-810, 2003 212430-37,
2005 212803-4, 2005. Dobbin K and Simon R. Sa
mple size determination in microarray experiments
for class comparison and prognostic
classification. Biostatistics 627-38, 2005.
Dobbin K, Shih J, Simon R. Questions and answers
on design of dual-label microarrays for
identifying differentially expressed genes.
Journal of the National Cancer Institute
951362-69, 2003. Wright G, Simon R. A random v
ariance model for detection of differential gene
expression in small microarray experiments.
Bioinformatics 192448-55, 2003.
Korn EL, Troendle JF, McShane LM, Simon R.Contro
lling the number of false discoveries. Journal of
Statistical Planning and Inference 124379-08,
2004. Molinaro A, Simon R, Pfeiffer R. Predicti
on error estimation A comparison of resampling
methods. Bioinformatics 213301-7,2005.
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Simon R. Using DNA microarrays for diagnostic an
d prognostic prediction. Expert Review of
Molecular Diagnostics, 3(5) 587-595, 2003.
Simon R. Diagnostic and prognostic prediction us
ing gene expression profiles in high dimensional
microarray data. British Journal of Cancer
891599-1604, 2003. Simon R and Maitnourim A. E
valuating the efficiency of targeted designs for
randomized clinical trials. Clinical Cancer
Research 106759-63, 2004. Maitnourim A and S
imon R. On the efficiency of targeted clinical
trials. Statistics in Medicine 24329-339, 2005.
Simon R. When is a genomic classifier ready for
prime time? Nature Clinical Practice Oncology
14-5, 2004. Simon R. An agenda for Clinical Tr
ials clinical trials in the genomic era.
Clinical Trials 1468-470, 2004.
Simon R. Development and Validation of Therapeut
ically Relevant Multi-gene Biomarker Classifiers.
Journal of the National Cancer Institute
97866-867, 2005. Simon R. A roadmap for devel
oping and validating therapeutically relevant
genomic classifiers. Journal of Clinical Oncology
(In Press). Freidlin B and Simon R. Adaptive si
gnature design. Clinical Cancer Research (In
Press). Simon R. Validation of pharmacogenomic
biomarker classifiers for treatment selection.
Disease Markers (In Press). Simon R. Guideline
s for the design of clinical studies for
development and validation of therapeutically
relevant biomarkers and biomarker classification
systems. In Biomarkers in Breast Cancer, Hayes DF
and Gasparini G, Humana Press (In Press).
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Myth

• That microarray investigations should be
  unstructured data-mining adventures without clear
  objectives

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• Good microarray studies have clear objectives,
  but not generally gene specific mechanistic
  hypotheses
  
• Design and analysis methods should be tailored to
  study objectives

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Good Microarray Studies Have Clear Objectives

• Class Comparison
• Find genes whose expression differs among
  predetermined classes
  
• Class Prediction
• Prediction of predetermined class (phenotype)
  using information from gene expression profile
  
• Class Discovery
• Discover clusters of specimens having similar
  expression profiles
  
• Discover clusters of genes having similar
  expression profiles

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Class Comparison and Class Prediction

• Not clustering problems
• Global similarity measures generally used for
  clustering arrays may not distinguish classes
  
• Dont control multiplicity or for distinguishing
  data used for classifier development from data
  used for classifier evaluation
  
• Supervised methods
• Requires multiple biological samples from each
  class

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Levels of Replication

• Technical replicates
• RNA sample divided into multiple aliquots and
  re-arrayed
  
• Biological replicates
• Multiple subjects
• Replication of the tissue culture experiment

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• Biological conclusions generally require
  independent biological replicates. The power of
  statistical methods for microarray data depends
  on the number of biological replicates.
• Technical replicates are useful insurance to
  ensure that at least one good quality array of
  each specimen will be obtained.

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Class Prediction

• Predict which tumors will respond to a particular
  treatment
  
• Predict which patients will relapse after a
  particular treatment

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• Class prediction methods usually have gene
  selection as a component
  
• The criteria for gene selection for class
  prediction and for class comparison are
  different
  
• For class comparison false discovery rate is
  important
  
• For class prediction, predictive accuracy is
  important

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Clarity of Objectives is Important

• Patient selection
• Many microarray studies developing classifiers
  are not therapeutically relevant
  
• Analysis methods
• Many microarray studies use cluster analysis
  inappropriately or misleadingly

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Microarray Platforms for Developing Predictive
Classifiers

• Single label arrays
• Affymetrix GeneChips
• Dual label arrays using common reference design
• Dye swaps are unnecessary

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Common Reference Design
A1
A2
B1
B2
RED
R
R
R
R
GREEN
Array 1
Array 2
Array 3
Array 4
Ai ith specimen from class A
Bi ith specimen from class B
R aliquot from reference pool
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• The reference generally serves to control
  variation in the size of corresponding spots on
  different arrays and variation in sample
  distribution over the slide.
• The reference provides a relative measure of
  expression for a given gene in a given sample
  that is less variable than an absolute measure.
  
• The reference is not the object of comparison.
• The relative measure of expression will be
  compared among biologically independent samples
  from different classes.

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Myth

• For two color microarrays, each sample of
  interest should be labeled once with Cy3 and once
  with Cy5 in dye-swap pairs of arrays.

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Dye Bias

• Average differences among dyes in label
  concentration, labeling efficiency, photon
  emission efficiency and photon detection are
  corrected by normalization procedures
• Gene specific dye bias may not be corrected by
  normalization

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• Dye swap technical replicates of the same two rna
  samples are rarely necessary.
  
• Using a common reference design, dye swap arrays
  are not necessary for valid comparisons of
  classes since specimens labeled with different
  dyes are never compared.
• For two-label direct comparison designs for
  comparing two classes, it is more efficient to
  balance the dye-class assignments for independent
  biological specimens than to do dye swap
  technical replicates

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Balanced Block Design
A1
B2
A3
B4
RED
A2
B1
B3
A4
GREEN
Array 1
Array 2
Array 3
Array 4
Ai ith specimen from class A
Bi ith specimen from class B
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• Detailed comparisons of the effectiveness of
  designs
  
• Dobbin K, Simon R. Comparison of microarray
  designs for class comparison and class discovery.
  Bioinformatics 181462-9, 2002
  
• Dobbin K, Shih J, Simon R. Statistical design of
  reverse dye microarrays. Bioinformatics
  19803-10, 2003
  
• Dobbin K, Simon R. Questions and answers on the
  design of dual-label microarrays for identifying
  differentially expressed genes, JNCI
  951362-1369, 2003

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• Common reference designs are very effective for
  many microarray studies. They are robust, permit
  comparisons among separate experiments, and
  permit many types of comparisons and analyses to
  be performed.
• For simple two class comparison problems,
  balanced block designs require many fewer arrays
  than common reference designs.
  
• Efficiency decreases for more than two classes
• Are more difficult to apply to more complicated
  class comparison problems.
  
• They are not appropriate for class discovery or
  class prediction.
  
• Loop designs are less robust, and dominated by
  either common reference designs or balanced block
  designs, and are not suitable for class
  prediction or class discovery.

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What We Will Not Discuss Today

• Image analysis
• Normalization
• Clustering methods
• Class comparison
• SAM and other methods of gene finding
• FDR (false discovery rate) and methods for
  controlling the number of false positive genes

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Simple Procedures

• If each gene is tested for significance at level
  ? and there are k genes, then the expected number
  of false discoveries is k ? .
  
• To control E(FD) ? u
• Conduct each of k tests at level ? u/k
• Bonferroni control of familywise error (FWE) rate
  at level 0.05
  
• Conduct each of k tests at level 0.05/k
• At least 95 confident that FD 0

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False Discovery Rate (FDR)

• FDR Expected proportion of false discoveries
  among the tests declared significant
  
• Studied by Benjamini and Hochberg (1995)

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Controlling the Expected False Discovery Rate

• Compare classes separately by gene and compute
  significance levels pi
  
• Rank genes in order of significance
• p(1)
• Find largest index i for which
• p(i)k / i ? FDR
• Consider genes with the ith smallest p values
  as statistically significant

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Problems With Simple Procedures

• Bonferroni control of FWE is very conservative
• p values based on normal theory are not accurate
  at extremes quantiles
  
• Difficult to achieve extreme quantiles for
  permutation p values of individual genes
  
• Controlling expected number or proportion of
  false discoveries may not provide adequate
  control because distributions of FD and FDP may
  have large variances
• Multiple comparisons are controlled by adjustment
  of univariate (single gene) p values and so may
  not take advantage of correlation among genes

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Additional Procedures

• SAM - Significance Analysis of Microarrays
• Tusher et al., PNAS, 2001
• Estimate FDR
• Statistical properties unclear
• Empirical Bayes
• Efron et al., JASA, 2001
• Related to FDR
• Ad hoc aspects
• Multivariate permutation tests
• Korn et al., 2001 (http//linus.nci.nih.gov/brb)
• Control number or proportion of false
  discoveries
  
• Can specify confidence level of control

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Multivariate Permutation Procedures(Korn et al.,
2001)

• Allows statements like
• FD Procedure We are 95 confident that the
  (actual) number of false discoveries is no
  greater than 5.
  
• FDP Procedure We are 95 confident that the
  (actual) proportion of false discoveries does not
  exceed .10.

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Class Prediction Model

• Given a sample with an expression profile vector
  x of log-ratios or log signals and unknown class.
  
  
• Predict which class the sample belongs to
• The class prediction model is a function f which
  maps from the set of vectors x to the set of
  class labels 1,2 (if there are two classes).
  
• f generally utilizes only some of the components
  of x (i.e. only some of the genes)
  
• Specifying the model f involves specifying some
  parameters (e.g. regression coefficients) by
  fitting the model to the data (learning the data).

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Problems With Many Diagnostic/Prognostic Marker
Studies

• Are not reproducible
• Retrospective non-focused analysis
• Multiplicity problems
• Inter-laboratory assay variation
• Have no impact
• Not therapeutically relevant questions
• Not therapeutically relevant group of patients
• Black box predictors

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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

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Do Not Confuse Statistical Methods Appropriate
for Class Comparison with Those Appropriate for
Class Prediction

• Demonstrating statistical significance of
  prognostic factors is not the same as
  demonstrating predictive accuracy.
  
• Demonstrating goodness of fit of a model to the
  data used to develop it is not a demonstration
  of predictive accuracy.
  
• Statisticians are used to inference, not
  prediction
  
• Most statistical methods were not developed for
  pn prediction problems
  

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Feature Selection

• Genes that are differentially expressed among the
  classes at a significance level ? (e.g. 0.01)
  
• The ? level is selected only to control the
  number of genes in the model

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t-test Comparisons of Gene Expression

• xjN(?j1 , ?j2) for class 1
• xjN(?j2 , ?j2) for class 2
• H0j ?j1 ?j2

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Estimation of Within-Class Variance

• Estimate separately for each gene
• Limited degrees of freedom
• Gene list dominated by genes with small fold
  changes and small variances
  
• Assume all genes have same variance
• Poor assumption
• Random (hierarchical) variance model
• Wright G.W. and Simon R. Bioinformatics192448-245
  5,2003
  
• Inverse gamma distribution of residual variances
• Results in exact F (or t) distribution of test
  statistics with increased degrees of freedom for
  error variance
  
• For any normal linear model

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Feature Selection

• Small subset of genes which together give most
  accurate predictions
  
• Combinatorial optimization algorithms
• Genetic algorithms
• Little evidence that complex feature selection is
  useful in microarray problems
  
• Failure to compare to simpler methods
• Some published complex methods for selecting
  combinations of features do not appear to have
  been properly evaluated
  

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Linear Classifiers for Two Classes
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Linear Classifiers for Two Classes

• Fisher linear discriminant analysis
• Requires estimating correlations among all genes
  selected for model
  
• y vector of class labels
• Diagonal linear discriminant analysis (DLDA)
  assumes features are uncorrelated
  
• Naïve Bayes classifier
• Compound covariate predictor (Radmacher) and
  Golubs method are similar to DLDA in that they
  can be viewed as weighted voting of univariate
  classifiers

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Linear Classifiers for Two Classes

• Compound covariate predictor
• Instead of for DLDA

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Linear Classifiers for Two Classes

• Support vector machines with inner product kernel
  are linear classifiers with weights determined to
  separate the classes with a hyperplain that
  minimizes the length of the weight vector

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Support Vector Machine
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Perceptrons

• Perceptrons are neural networks with no hidden
  layer and linear transfer functions between input
  output
  
• Number of input nodes equals number of genes
  selected
  
• Number of output nodes equals number of classes
  minus 1
  
• Number of inputs may be major principal
  components of genes or major principal components
  of informative genes
  
• Perceptrons are linear classifiers

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Naïve Bayes Classifier

• Expression profiles for class j assumed normal
  with mean vector mj and diagonal covariance
  matrix D
  
• Likelihood of expression profile vector x is l(x
  mj ,D)
  
• Posterior probability of class j for case with
  expression profile vector x is proportional to pj
  l(x mj ,D)

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Compound Covariate Bayes Classifier

• Compound covariate y ?tixi
• Sum over the genes selected as differentially
  expressed
  
• xi the expression level of the ith selected gene
  for the case whose class is to be predicted
  
• ti the t statistic for testing differential
  expression for the ith gene
  
• Proceed as for the naïve Bayes classifier but
  using the single compound covariate as predictive
  variable
  
• GW Wright et al. PNAS 2005.

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When pn The Linear Model is Too Complex

• 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.
  
• Why consider more complex models?

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Myth

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

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• Artificial intelligence sells to journal
  reviewers and peers who cannot distinguish hype
  from substance when it comes to microarray data
  analysis.
• Comparative studies have shown that simpler
  methods work as well or better for microarray
  problems because they avoid overfitting the data.

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Other Simple Methods

• Nearest neighbor classification
• Nearest k-neighbors
• Nearest centroid classification
• Shrunken centroid classification

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Nearest Neighbor Classifier

• To classify a sample in the validation set as
  being in outcome class 1 or outcome class 2,
  determine which sample in the training set its
  gene expression profile is most similar to.
• Similarity measure used is based on genes
  selected as being univariately differentially
  expressed between the classes
  
• Correlation similarity or Euclidean distance
  generally used
  
• Classify the sample as being in the same class as
  its nearest neighbor in the training set

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Other Methods

• Neural networks
• Top-scoring pairs
• CART
• Random Forrest
• Genetic algorithm based classification

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Apparent Dimension Reduction Based Methods

• Principal component regression
• Supervised principal component regression
• Partial least squares
• Stepwise logistic regression

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When There Are More Than 2 Classes

• Nearest neighbor type methods
• Decision tree of binary classifiers

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Decision Tree of Binary Classifiers

• Partition the set of classes 1,2,,K into two
  disjoint subsets S1 and S2
  
• Develop a binary classifier for distinguishing
  the composite classes S1 and S2
  
• Compute the cross-validated classification error
  for distinguishing S1 and S2
  
• Repeat the above steps for all possible
  partitions in order to find the partition S1and
  S2 for which the cross-validated classification
  error is minimized
• If S1and S2 are not singleton sets, then repeat
  all of the above steps separately for the classes
  in S1and S2 to optimally partition each of them

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Evaluating a Classifier

• Prediction is difficult, especially the
  future.
  
• Neils Bohr
• Fit of a model to the same data used to develop
  it is no evidence of prediction accuracy for
  independent data.
  

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Evaluating a Classifier

• Fit of a model to the same data used to develop
  it is no evidence of prediction accuracy for
  independent data
  
• Goodness of fit vs prediction accuracy
• Demonstrating statistical significance of
  prognostic factors is not the same as
  demonstrating predictive accuracy
  
• Demonstrating stability of identification of gene
  predictors is not necessary for demonstrating
  predictive accuracy
  

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Evaluating a Classifier

• The classification algorithm includes the
  following parts
  
• Determining what type of classifier to use
• Gene selection
• Fitting parameters
• Optimizing with regard to tuning parameters
• If a re-sampling method such as cross-validation
  is to be used to estimate predictive error of a
  classifier, all aspects of the classification
  algorithm must be repeated for each training set
  and the accuracy of the resulting classifier
  scored on the corresponding validation set

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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
• Ideally test set data is from different centers
  than the training data and assayed at a different
  time

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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

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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

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• Cross validation is only valid if the test set is
  not used in any way in the development of the
  model. Using the complete set of samples to
  select genes violates this assumption and
  invalidates cross-validation.
• 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.
• The cross-validated estimate of misclassification
  error is an estimate of the prediction error for
  model fit using specified algorithm to full
  dataset
• If you use cross-validation estimates of
  prediction error for a set of algorithms indexed
  by a tuning parameter and select the algorithm
  with the smallest cv error estimate, you do not
  have a valid estimate of the prediction error for
  the selected model

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

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Invalid Criticisms of Cross-Validation

• You can always find a set of features that will
  provide perfect prediction for the training and
  test sets.
  
• For complex models, there may be many sets of
  features that provide zero training errors.
  
• A modeling strategy that either selects among
  those sets or aggregates among those models, will
  have a generalization error which will be validly
  estimated by cross-validation.

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Simulated Data40 cases, 10 genes selected from
5000
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DLBCL Data
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Simulated Data40 cases
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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
  

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Gene-Expression Profiles in Hereditary Breast
Cancer

• Breast tumors studied
• 7 BRCA1 tumors
• 8 BRCA2 tumors
• 7 sporadic tumors
• Log-ratios measurements of 3226 genes for each
  tumor after initial data filtering

RESEARCH QUESTION Can we distinguish BRCA1 from
BRCA1 cancers and BRCA2 from BRCA2 cancers
based solely on their gene expression profiles?

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BRCA1
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BRCA2
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Classification of BRCA2 Germline Mutations
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Common Problems With Internal Classifier
Validation

• Pre-selection of genes using entire dataset
• Failure to consider optimization of tuning
  parameter part of classification algorithm
  
• Varma Simon, BMC Bioinformatics 2006
• Erroneous use of predicted class in regression
  model

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Incomplete (incorrect) Cross-Validation

• Publications are using all the data to select
  genes and then cross-validating only the
  parameter estimation component of model
  development
• Highly biased
• Many published complex methods which make strong
  claims based on incorrect cross-validation.
  
• Frequently seen in complex feature set selection
  algorithms
  
• Some software encourages inappropriate
  cross-validation

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Incomplete (incorrect) Cross-Validation

• Let M(b,D) denote a classification model
  developed on a set of data D where the model is
  of a particular type that is parameterized by a
  scalar b.
• Use cross-validation to estimate the
  classification error of M(b,D) for a grid of
  values of b Err(b).
  
• Select the value of b that minimizes Err(b).
• Caution Err(b) is a biased estimate of the
  prediction error of M(b,D).
  
• This error is made in some commonly used methods

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Complete (correct) Cross-Validation

• Construct a learning set D as a subset of the
  full set S of cases.
  
• Use cross-validation restricted to D in order to
  estimate the classification error of M(b,D) for a
  grid of values of b Err(b).
  
• Select the value of b that minimizes Err(b).
• Use the mode M(b,D) to predict for the cases in
  S but not in D (S-D) and compute the error rate
  in S-D
  
• Repeat this full procedure for different learning
  sets D1 , D2 and average the error rates of the
  models M(bi,Di) over the corresponding
  validation sets S-Di

82
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 and inference are different
• 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

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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
  

84
External Validation

• Should address clinical utility, not just
  predictive accuracy
  
• Therapeutic relevance
• Should incorporate all sources of variability
  likely to be seen in broad clinical application
  
• Expression profile assay distributed over time
  and space
  
• Real world tissue handling
• Patients selected from different centers than
  those used for developing the classifier

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Survival Risk Group Prediction

• Evaluate individual genes by fitting single
  variable proportional hazards regression models
  to log signal or log ratio for gene
  
• Select genes based on p-value threshold for
  single gene PH regressions
  
• Compute first k principal components of the
  selected genes
  
• Fit PH regression model with the k pcs as
  predictors. Let b1 , , bk denote the estimated
  regression coefficients
  
• To predict for case with expression profile
  vector x, compute the k supervised pcs y1 , ,
  yk and the predictive index ? b1 y1 bk yk

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Survival Risk Group Prediction

• LOOCV loop
• Create training set by omitting ith case
• Develop supervised pc PH model for training set
• Compute cross-validated predictive index for ith
  case using PH model developed for training set
  
• Compute predictive risk percentile of predictive
  index for ith case among predictive indices for
  cases in the training set

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Survival Risk Group Prediction

• Plot Kaplan Meier survival curves for cases with
  cross-validated risk percentiles above 50 and
  for cases with cross-validated risk percentiles
  below 50
• Or for however many risk groups and thresholds is
  desired
  
• Compute log-rank statistic comparing the
  cross-validated Kaplan Meier curves

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Survival Risk Group Prediction

• Repeat the entire procedure for all (or large
  number) of permutations of survival times and
  censoring indicators to generate the null
  distribution of the log-rank statistic
• The usual chi-square null distribution is not
  valid because the cross-validated risk
  percentiles are correlated among cases
  
• Evaluate statistical significance of the
  association of survival and expression profiles
  by referring the log-rank statistic for the
  unpermuted data to the permutation null
  distribution

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Survival Risk Group Prediction

• Other approaches to survival risk group
  prediction have been published
  
• The supervised pc method is 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

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Sample Size Planning References

• K Dobbin, R Simon. Sample size determination in
  microarray experiments for class comparison and
  prognostic classification. Biostatistics 627-38,
  2005
• K Dobbin, R Simon. Sample size planning for
  developing classifiers using high dimensional DNA
  microarray data. Biostatistics (In Press)

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Sample Size Planning

• GOAL Identify genes differentially expressed in
  a comparison of two pre-defined classes of
  specimens on dual-label arrays using reference
  design or single label arrays
• Compare classes separately by gene with
  adjustment for multiple comparisons
  
• Approximate expression levels (log ratio or log
  signal) as normally distributed
  
• Determine number of samples n/2 per class to give
  power 1-? for detecting mean difference ? at
  level ?

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Comparing 2 equal size classes

• n 4?2(z?/2 z?)2/?2
• where ? mean log-ratio difference between
  classes
  
• ? within class standard deviation of
  biological replicates
  
• z?/2, z? standard normal percentiles
• Choose ?? small, e.g. ?? .001
• Use percentiles of t distribution for improved
  accuracy
  

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Total Number of Samples for Two Class Comparison
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Dual Label Arrays With Reference DesignPools of
k Biological Samples

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• m number of technical reps per sample
• k number of samples per pool
• n total number of arrays
• ? mean difference between classes in log
  signal
  
• ?2 biological variance within class
• ?2 technical variance
• ? significance level e.g. 0.001
• 1-? power
• z normal percentiles (use t percentiles for
  better accuracy)
  

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?0.001, ?0.05, ?1, ?22?20.25, ?2/?24
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?0.001 ?0.05 ?1 ?22?20.25, ?2/?24
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Number of Events Needed to Detect Gene Specific
Effects on Survival

• ? standard deviation in log2 ratios for each
  gene
  
• ? hazard ratio (1) corresponding to 2-fold
  change in gene expression
  
• ? 1/N for 1 expected false positive gene
  identified per N genes examined
  
• ? 0.05 for 5 false negative rate

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Number of Events Required to Detect Gene Specific
Effects on Survival ?0.001,?0.05
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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(?)

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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
  

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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)

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

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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.
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Optimal significance level cutoffs for gene
selection. 50 differentially expressed genes out
of 22,000 genes on the microarrays
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?0.05, p22,000 genes, gene standard deviation
s0.75.
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a) Power example with 60 samples, 2d/s1/0.71
effect size for differentially expressed genes,
alpha 0.001 cutoffs for gene selection. As the
proportion in the under-represented class gets
smaller, the power to identify differentially
expressed genes decreases.
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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.,
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BRB-ArrayTools

• Integrated software package using Excel-based
  user interface but state-of-the art analysis
  methods programmed in R, Java Fortran
  
• Publicly available for non-commercial use

http//linus.nci.nih.gov/brb
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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
• Time course analysis
• Log intensities for non-reference designs
• Mixed models

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Selected Features of BRB-ArrayTools

• Gene enhancement analysis.
• Find Gene Ontology groups and signaling pathways
  that are differentially expressed among classes
  
• Class prediction
• DLDA, CCP, Nearest Neighbor, Nearest Centroid,
  Shrunken Centroids, SVM, Random Forests
  
• Complete LOOCV, k-fold CV, repeated k-fold,
  .632 bootstrap
  
• permutation significance of cross-validated
  error rate
  

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Selected Features of BRB-ArrayTools

• Clustering tools for class discovery with
  reproducibility statistics on clusters
  
• Internal access to Eisens 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

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Acknowledgements

• Kevin Dobbin
• Michael Radmacher
• Sudhir Varma
• Yingdong Zhao
• BRB-ArrayTools Development Team
• Amy Lam
• Ming-Chung Li
• Supriya Menezes