Improved Tumor Marker Validation Success Using Weighted Gene Co-expression Networks and Random Forest Clustering

1
Improved Tumor Marker Validation Success Using
Weighted Gene Co-expression Networks and
Random Forest Clustering

• Steve Horvath
• shorvath_at_mednet.ucla.edu
• Human Genetics Biostatistics
• University of California, Los Angeles

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Contents

• Describe pathway based tumor marker screening
  strategy
• Speculate on the biological reasons why it could
  work.
• Describe 2 empirical success stories for
  identifying tumor markers that validated in
  independent data sets
• Brain cancer survival time
• (Affy) gene expression microarray data
• weighted gene co-expression networks
• Prostate cancer time to PSA recurrence
• tissue microarray data (immunohistochemical
  stainings)
• random forest clustering

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The Embarassing Validation Problem

• A tumor marker is found to be highly predictive
  of a clinical outcome in one data set but fails
  to be validated in an independent data set.
• Bad (analysis) reasons include
• data snooping
• overfitting
• ascertainment issues
• Good (biological) reasons
• genetic heterogeneity
• Little can be done about this.
• Single markers dont capture the essence of the
  whole disease pathway.
• A lot can be done about this?NOVEL STATISTICAL
  METHODS FOR EXTRACTING SIGNAL FROM THE DATA.

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Outline of standard strategy for screening for
markers

• 1) Regress a clinical outcome y on the molecular
  markers (features) X.
• 2) Identify the features that are most
  significant or most predictive of the outcome
  using standard statistical feature selection
  methods
• Empirical finding often poor validation success.

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Pathway Based Strategy for Screening for
Markers

• Find suitably defined clusters in the underlying
  high dimensional feature space X.
• Relate the clusters to clinical outcomes of
  interest. This results in a few disease
  clusters (a.k.a. pathways or modules)
• Use features (markers) that describe the states
  of the disease clusters as final predictors.
• (Limited) Empirical Finding improved validation
  success

6
Motivating why the pathway based screening
strategy may lead to better validation success

• By first clustering the features, one reduces the
  number of multiple comparisons substantially
• By looking at aggregates of features (clusters)
  the feature definition is much more robust and
  more likely to be platform independent.
• Combining the features along pathways is the
  biologically meaningful thing to do.
• Pathways are closer to the clinical phenotype
  than the individual constituents of these
  pathways.
• The whole is more than the sum of its parts

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TEASERValidation success rate of gene
expressions in independent data
300 most significant genes Network based
screening (Cox p-valuelt1.310-3) plt0.05 and
high intramodular connectivity
67
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Weighted Gene Co-Expression Network Analysis.
9

• Novel statistical approach for analyzing
  microarray data weighted network analysis
• Empirical evidence that it matters in practice
• Identification of Brain Cancer Genes that can be
  validated in an independent data set

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Background

• Network based methods have been found useful in
  many domains,
• protein interaction networks
• the world wide web
• social interaction networks
• OUR FOCUS gene co-expression networks

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Does this map tell you which cities are important?
This one does!
The nodes with the largest number of links
(connections) are most important!
Slide courtesy of Paul Mischel and AL Barabasi
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Scale free topology is a fundamental property of
such networks (Barabasi et al)

• It entails the presence of hub nodes that are
  connected to a large number of other nodes
• Such networks are robust with respect to the
  random deletion of nodes but are sensitive to the
  targeted attack on hub nodes
• It has been demonstrated that metabolic networks
  exhibit a scale free topology

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P(k) vs k in scale free networks
P(k)

• Scale Free Topology refers to the frequency
  distribution of the connectivities
• Connectivity k
• p(k)proportion of nodes that have connectivity k

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How to check Scale Free Topology?
Idea Log transformation p(k) and k and look at
scatter plots
Linear Regression model fitting index R2 can be
used to quantify goodness of fit
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Gene Co-expression Networks

• In gene co-expression networks, each gene
  corresponds to a node.
• Two genes are connected by an edge if their
  expression values are highly correlated.
• Definition of high correlation is somewhat
  tricky
• we propose a criterion for picking threshold
  parameter.

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Steps for constructing asimple, unweighted
co-expression network
Overview gene co-expression network analysis

• Hi

• Microarray gene expression data
• Measure concordance of gene expression with a
  Pearson correlation
• C) The Pearson correlation matrix is dichotomized
  to arrive at an adjacency matrix. Binary values
  in the adjacency matrix correspond to an
  unweighted network.
• D) The adjacency matrix can be visualized by a
  graph.

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Our holistic view.

• Weighted Network View Unweighted View
• All genes are connected Some genes are
  connected
• Connection WidthsConnection strenghts All
  connections are equal

We find theoretical and empirical evidence that
the weighted network view is superior to the
simple network view.
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A general frame work for defining weighted gene
co-expression networksBin Zhang, Steve
HorvathTechnical report and R code at
www.genetics.ucla.edu/labs/horvath/CoexpressionNet
work
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Beyond the standard approach

• Dichotomization allows one to easily define
  network-based concepts but it eliminates some
  information regarding the strength of
  interaction.
• To overcome the disadvantage of the
  dichotomization, we generalize the approach
• Measure co-expression by a similarity s(i,j) with
  range 0,1 e.g. absolute value of the Pearson
  correlation
• Define an adjacency matrix A(i,j)AF(s(i,j))
• The adjacency function AF is a monotonic,
  non-negative function defined on 0,1 and
  depends on parameters. The choice of the
  parameters determines the properties of the
  network.
• We consider 2 types of AFs
• Step function AF(s)I(sgttau) with parameter tau
• Power function AF(s)sb with parameter

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Comparing adjacency functions
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How to estimate the parameter values of an
adjacency function?

• We propose to use the following criteria
• A) CONSIDER ONLY THOSE PARAMETER VALUES THAT
  RESULTS IN APPROXIMATE SCALE FREE TOPOLOGY
• B) SELECT THE PARAMETERS THAT RESULT IN THE
  HIGHEST MEAN NUMBER OF CONNECTIONS
• Criterion A is motivated by the finding that most
  metabolic networks (including gene co-expression
  networks, protein-protein interaction networks
  and cellular networks) have been found to exhibit
  a scale free topology
• Criterion B is motivated by our desire to have
  high sensitivity to detect modules (clusters of
  genes) and hub genes.

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Criterion A is measured by the linear model
fitting index R2
Step AF (tau) Power AF (b)
b
tau
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Trade-off between criterion A (R2) and criterion
B (mean no. of connections) when varying the
power b
AF(s)sb
criterion A SFT model fit R2 criterion B mean
connectivity
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Empirical insights for determining the adjacency
function

• For criterion A measure compliance with scale
  free topology by using the adjusted R2 value for
  the linear regression fit between log(p(k)) and
  log(k)
• Usually require R2gt0.8
• For criterion B aim to get a mean(k)50 when
  dealing with 2000 genes.

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Trade-off between criterion A and B when varying
tau
Step Function I(sgttau)
criterion A criterion B
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Mathematical Definition of an Undirected Network
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NetworkAdjacency Matrix

• A network can be represented by an adjacency
  matrix, Aaij, that encodes whether/how a pair
  of nodes is connected.
• A is a symmetric matrix with entries in 0,1.
• For unweighted network, entries are 1 or 0
  depending on whether or not 2 nodes are adjacent
  (connected).
• For weighted networks, the adjacency matrix
  reports the connection strength between gene
  pairs.

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Generalized Connectivity

• Gene connectivity correspond to the row sums of
  the adjacency matrix
• For unweighted networksnumber of direct
  neighbors
• For weighted networks sum of connection
  strengths to other nodes

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Network Analysis Flow Chart
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Define a Gene Co-expression Similarity
Define a Family of Adjacency Functions
Determine the AF Parameters
Define a Measure of Node Dissimilarity
  Identify Network Modules (Clustering)
Relate Network Concepts to Each Other
Relate the Network Concepts to External Gene or
Sample Information
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Network Distance Measure Topological Overlap
Matrix
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How to measure distance in a network?

• Mathematical Answer Geodesics
• length of shortest path connecting 2 nodes
• we have found no empirical evidence that this is
  a biologically meaningful concept in
  co-expression networks
• Biological Answer look at shared neighbors
• Intuition if 2 people share the same friends
  they are close in a social network
• Use the topological overlap measure based
  distance proposed by Ravasz et al 2002 Science)

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Topological Overlap (Ravasz et al) leads to a
network distance measure

• Generalized in Zhang and Horvath (2005) to the
  case of weighted networks
• Generalized in Yip and Horvath (2005) to higher
  order interactions

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Using the TOM matrix to cluster genes

• To group nodes with high topological overlap into
  modules (clusters), we typically use average
  linkage hierarchical clustering coupled with the
  TOM distance measure.
• Once a dendrogram is obtained from a hierarchical
  clustering method, we choose a height cutoff to
  arrive at a clustering.
• Here modules correspond to branches of the
  dendrogram

TOM plot
Genes correspond to rows and columns
TOM matrix
Hierarchical clustering dendrogram
Module Correspond to branches
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More traditional view of module
ColumnsBrain tissue samples
RowsGenes Color band indicates module
membership
Message characteristic vertical bands indicate
tight co-expression of module genes
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Different Ways of Depicting Gene Modules
Topological Overlap Plot Gene
Functions We proposed Multi Dimensional
Scaling Traditional View
1) Rows and columns correspond to genes 2) Red
boxes along diagonal are modules 3) Color
bandsmodules
Idea Use network distance in MDS
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Hub Genes Predict Survival for Brain Cancer
PatientsMischel PS, Zhang B,et al, Horvath S,
Nelson SF.
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Comparing the Module Structure in Cancer and
Normal tissues
55 Brain Tumors
VALIDATION DATA 65 Brain Tumors
Messages 1)Cancer modules can be independently
validated 2) Modules in brain cancer tissue can
also be found in normal, non-brain tissue. --gt
Insights into the biology of cancer
Normal brain (adult fetal)
Normal non-CNS tissues
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Mean Prognostic Significance of Module Genes
Message Focus the attention on the brown module
genes
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Module hub genes predict cancer survival

1. Cox model to regress survival on gene expression
   levels
2. Defined prognostic significance as
   log10(Cox-p-value) the survival association
   between each gene and glioblastoma patient
   survival
3. A module-based measure of gene connectivity
   significantly and reproducibly identifies the
   genes that most strongly predict patient survival

Validation set 65 gbms r 0.55 p-2.2 x 10-16
Test set 55 gbms r 0.56 p-2.2 x 10-16
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The fact that genes with high intramodular
connectivity are more likely to be prognostically
significant facilitates a novel screening
strategy for finding prognostic genes

• Focus on those genes with significant Cox
  regression p-value AND high intramodular
  connectivity.
• It is essential to to take a module centric view
  focus on intramodular connectivity of disease
  related module
• Validation success rate proportion of genes with
  independent test set Cox regression p-valuelt0.05.
  
• Validation success rate of network based
  screening approach (68)
• Standard approach involving top 300 most
  significant genes 26

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Validation success rate of gene expressions in
independent data
300 most significant genes Network based
screening (Cox p-valuelt1.310-3) plt0.05 and
high intramodular connectivity
67
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New ApplicationTissue Microarray Data
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Tissue MicroarrayDNA Microarray
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Tissue Array Section
700 Tissue Samples
0.6 mm 0.2mm
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Ki-67 Expression in Kidney Cancer
High Grade
Low Grade
Message brown staining related to tumor grade
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Multiple measurements per patientSeveral spots
per tumor sample and several scores per spot

• Each patients (tumor sample) is usually
  represented by multiple spots
• 3 tumor spots
• 1 matched normal spot

• Maximum intensity Max
• Percent of cells staining Pos
• Percent of cells staining with the
• maximum intensity PosMax
• Spots have a spot grade NL,1,2,..
• Indicator of missingness

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Properties of TMA Data

• Highly skewed, non-normal,semi-continuous.
• Often a good idea to model as ordinal variables
  with many levels.
• Staining scores of the same markers are highly
  correlated

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Histogram of tumor marker expression scores POS
and MAX
Percent of Cells Staining(POS)
EpCam
P53
CA9
Maximum Intensity (MAX)
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Frequency plot of the same tumor marker in 2
independent data sets
DATA SET 1 Validation Data Set 2
The cut-off corresponds roughly to the 66
percentile. Thresholding this tumor marker allows
one to stratify the cancer patients into high
risk and low risk patients. Although the
distribution looks very different the percentile
threshold can be validated and is clinically
relevant.
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Thresholding methods for tumor marker expressions

• Since clinicians and pathologists prefer
  thresholding tumor marker expressions, it is
  natural to use statistical methods that are based
  on thresholding covariates, e.g. regression
  trees, survival trees, rpart, forest predictors
  etc.
• Dichotomized marker expressions are often fitted
  in a Cox (or alternative) regression model
• Danger Over-fitting due to optimal cut-off
  selection.
• Several thresholding methods and ways for
  adjusting for multiple comparisons are reviewed
  in
• Liu X, Minin V, Huang Y, Seligson DB, Horvath S
  (2004) Statistical Methods for Analyzing Tissue
  Microarray Data. J of Biopharmaceutical
  Statistics. Vol 14(3) 671-685

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Finding tumor markers for predicting clinical
outcomes on the basis of Tissue Microarray Data
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Using the clustering based strategy for finding
tumor markers

• 1) Find distinct patient clusters without regard
  to outcome
• 2) Find whether patient clusters have distinct
  PSA recurrence profiles
• 3) If so, find rules (classifiers) for predicting
  cluster membership
• 4) Validate those rules in independent data.

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Cluster Analysis of Low Gleason Score Prostate
Samples(UCLA data)
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1) Construct a tumor marker rule for predicting
RF cluster membership.2) Validate the rule
predictions in an independent data set
Threshold Rule Validation
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Discussion Prostate TMA Data

• Very weak evidence that individual markers
  predict PSA recurrence
• None of the markers validated individually
• However, cluster membership was highly
  predictive, i.e the rule could be validated in an
  independent data set.

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How to cluster patients on the basis of Tissue
Microarray Data?
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Questions 1)Can TMA data be used for tumor
class discovery, i.e unsupervised learning?2)
If so, what are suitable unsupervised learning
methods?
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Tumor Class Discovery using DNA Microarray Data

• Tumor class discovery entails using a
  unsupervised learning algorithm (i.e.
  hierarchical, k-means, SOM clustering etc.) to
  automatically group tumor samples based on their
  gene expression pattern.

Bullinger et al. N Engl J Med. 2004
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Clusters involving TMA data may have
unconventional shapesLow risk prostate cancer
patients are colored in black.

• Scatter plot involving 2 dependent tumor
  markers. The remaining, less dependent markers
  are not shown.
• Low risk cluster can be described using the
  following rule
• Marker H3K4 gt 45 and H3K18 gt 70.
• The intuition is quite different from that of
  Euclidean distance based clusters.

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Unconventional shape of a clinically meaningful
patient cluster

• 3 dimensional scatter plot along tumor markers
• Low risk patients are colored in black

MARKER 2
MARKER 1
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A dissimilarity measure is an essential input for
tumor class discovery

• Dissimilarities between tumor samples are used in
  clustering and other unsupervised learning
  techniques
• Commonly used dissimilarity measures include
  Euclidean distance, 1 - correlation

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Challenge

• Conventional dissimilarity measures that work for
  DNA microarray data may not be optimal for TMA
  data.
• Dissimilarity measure that are based on the
  intuition of multivariate normal distributions
  (clusters have elliptical shapes) may not be
  optimal
• For tumor marker data, one may want to use a
  different intuition clusters are described using
  thresholding rules involving dependent markers.
• It may be desirable to have a dissimilarity that
  is invariant under monotonic transformations of
  the tumor marker expressions.

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We have found that a random forest (Breiman 2001)
dissimilarity can work well in the unsupervised
analysis of TMA data.Shi et al 2004, Seligson et
al 2005.http//www.genetics.ucla.edu/labs/horvath
/RFclustering/RFclustering.htm
66
Kidney cancerComparing PAM clusters that result
from using the RF dissimilarity vs the Euclidean
distance
Kaplan Meier plots for groups defined by cross
tabulating patients according to their RF and
Euclidean distance cluster memberships.
Message In this application, RF clusters are
more meaningful regarding survival time
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The RF dissimilarity is determined by dependent
tumor markers
Tumor markers

• The RF dissimilarity focuses on the most
  dependent markers (1,2).
• In some applications, it is good to focus on
  markers that are dependent since they may
  constitute a disease pathway.
• The Euclidean distance focuses on the most
  varying marker (4)

Patients sorted by cluster
68
The RF cluster can be described using a
thresholding rule involving the most dependent
markers

• Low risk patient if marker1gtcut1 marker2gt cut2
• This kind of thresholding rule can be used to
  make predictions on independent data sets.
• Validation on independent data set

69
Theoretical reasons for using an RF dissimilarity
for TMA data

• Main reasons
• natural way of weighing tumor marker
  contributions to the dissimilarity
• The more related a tumor marker is to other tumor
  markers the more it contributes to the definition
  of the dissimilarity
• no need to transform the often highly skewed
  features
• based feature ranks
• Chooses cut-off values automatically
• resulting clusters can often be described using
  simple thresholding rules
• Other reasons
• elegant way to deal with missing covariates
• intrinsic proximity matrix handles mixed variable
  types well
• CAVEAT The choice of the dissimilarity should be
  determined by the kind of patterns one hopes to
  find. There will be situations when other
  dissimilarities are preferrable.

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The random forest dissimilarityL. Breiman RF
manualTechnical Report Shi and Horvath
2005http//www.genetics.ucla.edu/labs/horvath/RFc
lustering/RFclustering.htm
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SummaryRandom forest clustering

• Intrinsic variable selection focuses on dependent
  variables
• Depending on the application, this can be
  attractive
• Resulting clusters can often be described using
  thresholding rules?attractive for TMA data.
• RF dissimilarity invariant to monotonic
  transformations of variables
• In some cases, the RF dissimilarity can be
  approximated using a Euclidean distance of ranked
  and scaled features.
• RF clustering was originally suggested by L.
  Breiman (RF manual). Theoretical properties are
  studied as part of the dissertation work of Tao
  Shi. Technical report/code can be found at
  www.genetics.ucla.edu/labs/horvath/RFclustering/R
  Fclustering.htm www.genetics.ucla.edu/labs/horvat
  h/kidneypaper/RCC.htm

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Conclusions

• There is a need to identify/develop appropriate
  data mining methods for TMA data
• highly skewed, semi-continuous, non-normal data
• tree or forest based methods work well
• ALTERNATIVES?

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Acknowledgements

• Former students Postdocs for TMA
• Tao Shi PhD
• Xueli Liu PhD
• Yunda Huang PhD
• Tuyen Hoang PhD

• UCLA
• Tissue Microarray Core
• David Seligson, MD
• Hyung Kim, MD
• Arie Belldegrun, MD
• Robert Figlin, MD
• Siavash Kurdistani, MD

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References RF clustering

• Unsupervised learning tasks in TMA data analysis
• Review random forest predictors (introduced by L.
  Breiman)
• Shi, T. and Horvath, S. (2005) Unsupervised
  learning using random forest predictors Journal
  of Computational and Graphical Statistics
• www.genetics.ucla.edu/labs/horvath/RFclustering/RF
  clustering.htm
• Application to Tissue Array Data
• Shi, T., Seligson, D., Belldegrun, A. S.,
  Palotie, A., Horvath, S. (2004) Tumor Profiling
  of Renal Cell Carcinoma Tissue Microarray Data
• Seligson DB, Horvath S, Shi T, Yu H, Tze S,
  Grunstein M, Kurdistani S (2005) Global histone
  modification patterns predict risk of prostate
  cancer recurrence.