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Hierarchical Bayesian Model Specification
•Model is specified by the Directed Acyclic Network (DAG) and the conditional probability distributions of all nodes given values of its parents•Topology of the DAG defines the conditional dependencies of all variables through the Markov directed Markov property which states that given the values of its parents, a variable in the model is independent of all its non-descendents•DAG and local distributions define the joint probability distribution of data and all parameters in the model•In our case this distribution can not be explicitly characterized but it estimates using Markov Chain Monte Carlo approach (Gibbs sampler)
Uses and Miss-Uses of Clustering
•Define a statistical model that facilitates clustering of genes based on similarities of their expression profiles
•Define the method-selection criteria that allows for estimating the "correct" number of clusters
•Show that inappropriate "pre-filtering" can fool the statistical model in the same way it fools the casual observer
•Show appropriate ways to use cluster analysis and illustrate the importance of using the "best available treatment"
Clustering of gene expression profiles
Microarrays
Diff
eren
tial E
xpre
ssio
n
0 2 4 6 8 10 12
-20
-15
-10
-50
5
1 2 3 4 5 6 7 8 9 10 11
Microarrays
Diff
eren
tial E
xpre
ssio
n
0 2 4 6 8 10 12
-20
-15
-10
-50
5
1 2 3 4 5 6 7 8 9 10 11
Microarrays
Diff
ere
ntia
l Exp
ress
ion
0 2 4 6 8 10 12
-20
-15
-10
-50
5
1 2 3 4 5 6 7 8 9 10 11
0 1 2 3 4 5 6 7 8 9 10 11 12
arrays
-20
-15
-10
-5
0
5
10
0 1 2 3 4 5 6 7 8 9 10 11 12
arrays
-20
-15
-10
-5
0
5
10
“Patterns” of Expression - Finite Mixture Model
0 1 2 3 4 5 6 7 8 9 10 11 12
arrays
-20
-15
-10
-5
0
5
10
Patterni i=(1i, 2i,…, 11i)
Dataik ~ iid N(i, ), k=1,…,ni
ni=number of genes generated
by the Patterni
i=ni/n
“Patterns” of Expression - Finite Mixture Model
Any gene profile x = (x1,x2,…,x11)
),( i
G
1ii σNπ μ
x ~
), ; ( )( iN
G
1ii σpπp μx x
All data x1, x2,…, xn
x xx
n
σpπp1k
ikN
G
1iiG1G1n1 ), ; ( ) ,,..., ,,..., ;,...,( μμμ
} Finite Mixture Model
0 1 2 3 4 5 6 7 8 9 10 11 12
arrays
-20
-15
-10
-5
0
5
10
0 1 2 3 4 5 6 7 8 9 10 11 12
arrays
-20
-15
-10
-5
0
5
10
One-dimensional mixturePattern 1
Pattern 2
Differential Expression on Microarray 1
Differential Expression on Microarray 1
Differential Expression on Microarray 1
N(11, )
N(12, )
),(4
3 ),(
4
12111 σμNσμN
MCLUST
> library(mclust)> SimData<-matrix(rnorm(5000*15),ncol=15)> ColLabels<-c(paste("Tumor_",1:8,sep=""),paste("Control_",1:7,sep=""))> heatmap(SimData,labCol=ColLabels)
> .Mclust$hcModelNames<-c("E","EEI")> .Mclust$emModelNames<-c("EEI")> BIC.emclust<-EMclust(SimData,1:10)> BIC.emclust
BIC: EEI1 -213490.32 -213624.93 -213753.04 -213880.75 -213993.76 -214121.07 -214243.48 -214351.69 -214481.410 -214588.7
> plot(BIC.emclust)EEI "1" >
Determining the number of patterns
log(n) -)ˆ ,ˆ,...,ˆ ,ˆ,...,ˆ ;,...,( parametersG1G1n1 NpBIC xx μμ
1
1
1
1
1
1
1
1
1
1
2 4 6 8 10
-21
46
00
-21
44
00
-21
42
00
-21
40
00
-21
38
00
-21
36
00
number of clusters
BIC
Con
trol
_5
Tum
or_1
Tum
or_2
Con
trol
_1
Tum
or_5
Tum
or_6
Tum
or_8
Con
trol
_4
Con
trol
_3
Tum
or_4
Con
trol
_2
Tum
or_3
Tum
or_7
Con
trol
_6
Con
trol
_7
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MCLUST
> p.value<-apply(SimData,1,function(x) t.test(x[1:8],x[9:15],var.equal=T)$p.value)> > SigData<-SimData[p.value<0.05,]> dim(SigData)[1] 242 15> heatmap(SigData,labCol=ColLabels)>> BIC.emclust<-EMclust(SigData,1:10)> BIC.emclust
BIC: EEI1 -10599.4852 -9647.6453 -9685.8974 -9729.2395 -9796.1196 -9849.1097 -9912.6018 -9973.6459 -10037.43610 -10077.862
> plot(BIC.emclust)EEI "1" >
Determining the number of patterns
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Summary
•The "weak filter" based on selecting "sub-significant" differentially expressed genes created artificial clusters
•When the whole dataset was used, the Bayesian information criteria did the right thing by estimating the correct number of clusters to be equal to one
•Take home message: When "filtering" before clustering make sure that appropriate statistical significance levels have been used
Using clustering to find "patterns" among differentially expressed genes
•Cluster analysis is preceded by a rigorous statistical analysis
•For example-identify genes that were "differentially" expressed on at least one experimental comparison. Among all these genes some will have similar behavior across all experimental conditions
•Clustering is a way of organizing behavior of differentially expressed genes across different experimental conditions
Using clustering to find "patterns" among differentially expressed genes
Up-Regulation
Weak Uniform Up-Regulation
Strong Uniform Up-Regulation
Weak Early Down-Regulation
Strong Uniform Down-Regulation
Down-Regulation
Strong LaterUp-Regulation
Using clustering to find "patterns" among all genes
•No filtering is performed
•You can perform the "quality filtering"
•Trying to identify statistically significant patterns
•Using the best available method becomes extremely important
Does It matter which clustering procedure we use?
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"Complicated" Method(Context-specific Infinite Mixtures)
•5685 Yeast Genes Across Two Experiments (Cell Cycle and Sporulation)•NO VARIABILITY BASED FILTER•135 Genes with closest co-expression partners
"Objective" Performance Assessments Using KEGG as the Gold Standard
•Due to a large imbalance between the total number of negative and positive pairs:There are 17 times more negative pairs than positive pairs - a small FPR can still produce more false positive than true positives
Summary
•Using clustering alone, one can identify "significant" patterns of expression when using appropriate methodology
•For example, Yeast data clustered in this example did not have any replicates so the traditional analysis to identify differentially expressed genes before clustering is not feasable
•Statistical significance of resulting clusters needs to be carefully examined
Infinite Bayesian Mixtures
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•Posterior distribution summarized through
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expression” p(ci=cj|X)
Properties•“Pooling” information from the whole dataset by estimating both “patterns” and “assignments” – similar to K-means (K-means is actually equivalent to a special case of the mixture models with known number of clusters)•Does not require specification of the right number of clusters (unlike K-means)•Gives direct estimates of statistical significance (unlike anything else on the market)•Instead of lamenting which distance measure to use – focus on the appropriate statistical model which is a well-defined problem•Works for any type of data
Finding important functional groups for up-regulated genes
Using the "Ease" annotation tool http://david.niaid.nih.gov/david/
We obtained following significant gene ontologiesUp_DexANDNE2ANDirr_381_GO.htm
Homework:1) Download and install Ease2) Select top 20 most-signficianly up-regulated genes in our W-C dataset and identify significantly over-represented categories (using the three-way ANOVA analysis)3) Repeat the analysis with 30, 40, 50 and 100 up-regulated and down-regulated gene4) Prepare questions for the next class regarding problems you run into