a quantitative overview to gene expression profiling in animal genetics

A Quantitative Overview to Gene Expression Profiling in Animal Genetics

Armidale Animal Breeding Summer Course, UNE, Feb. 2006

Analysis of(cDNA) Microarray Data:

Part V. Mixtures of Distributions

Model-Based Clusteringvia

Mixtures of Distribution



Mixtures of Distributions

Definition

• The mixture model assumes that each cluster (or component) of the data is generated by an underlying normal distribution.

• Each of the data in y are assumed to be independent observations from a mixture density with k (possibly unknown but finite) components and with probability density function:

k

iiiik Vyyf

1

,;;

Mixing proportions (add to 1)

Normal density function




Introduction

k

jjjjk Vyyf

1

,;;



Mixtures of DistributionsThe Guru http://www.maths.uq.edu.au/~gjm




Software and Resources




EM Algorithm

k

iiiik Vyyf

1

,;;

The EM algorithm obtains the maximum likelihood estimate of by iteration. In the (m+1)th iteration, the estimates of the parameters of interest are updated by:

n

j

mij

mi n

1

)()1( /

n

j

n

j

miji

mij

mi y

1 1

)()()1( /

n

j

mij

Tmii

mii

n

j

mij

mi yyV

1

)()1()1(

1

)()1( /))((

);(/,; )()()()()( mj

mi

mij

mi

mij yfVy Where

Is the Posterior Probability that yj belongs to the i-th component of the mixture (…with a very elegant link to False Discovery Rate).




EM Algorithm

• We proceed for k = 1, 2, 3, …, and so on components.• Criteria for model selection includes the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC):

kkLAIC 2)ˆ(log2

)log()ˆ(log2 nLBIC kk

Where 13 kk Is the number of independent parameters in the mixture.

• Alternatively, the distribution of the likelihood ratio test (LRT) can be estimated by bootstrapping and P-values obtained to contrast a model with k components against a model with k + 1 components.

k

iiiik Vyyf

1

,;;




Simulation 1Consider theseDistribution

N(1,5)N(5,10)

Records

10,000 5,000…and simulate

)10,5(31)5,1(3

2)ˆ;( NNyf The Mixture becomes:

);(,;

j

iijiij yf

Vy Posterior Prob:

LikelihoodN(1,5) N(5,10)

-1 0.120 0.021 0 0.161 4 0.036 1 0.178 0.056 5 0.036 3 0.126 7 0.005 0.103

Weighted average (by mixing proportions)

62




Simulation 2Consider theseDistribution

N(0,1)N(0,10)

Records

9,000 1,000…and simulate

)10,0(1.0)1,0(9.0)ˆ;( NNyf The Mixture becomes:

Microarray

Non-DE Genes DE Genes



Mixtures of DistributionsSimulation 2 )10,0(1.0)1,0(9.0)ˆ;( NNyf

2. Ask EMMIX to fit mixtures with up to 5 components and…

)805.10,010.0(097.0)993.0,006.0(903.0)ˆ;( NNyf

3. EMMIX model of best fit:

1. Simulate:



Mixtures of DistributionsSimulation 2 )10,0(1.0)1,0(9.0)ˆ;( NNyf

)805.10,010.0(097.0)993.0,006.0(903.0)ˆ;( NNyf3. EMMIX best fit:

1. Simulate:

Frequency Post Prob

Posterior Probabilities are “Decision Function” changing at 2.75



Mixtures of DistributionsLinking Posterior Probabilities with False Discovery Rate



Mixtures of DistributionsLinking Posterior Probabilities with False Discovery Rate

Not-DE DESelect the N most extreme genes, and FDR is the average posterior probability of not being in the cluster of extremes.




Simulation 2 )10,0(1.0)1,0(9.0)ˆ;( NNyf

)805.10,010.0(097.0)993.0,006.0(903.0)ˆ;( NNyf3. EMMIX best fit:

1. Simulate:

Post Prob

Select the N most extreme genes, and FDR is the average Post Prob of not being in the cluster of extremes.

FDR by N Genes




Example“Diets”

(only REFERENCE components of the design)

88ii rg

iiHvLi

rgy




Example“Diets”

(only REFERENCE components of the design)

88ii rg

iiHvLi

rgy

k

iiiik Vyyf

1

,;;

)32.2,41.2(366.0)42.10,30.2(590.0)46.67,87.0(044.0)ˆ;(

NNNyf




Example

,)32.2,41.2(366.0)42.10,30.2(590.0)46.67,87.0(044.0)ˆ;( NNNyf

“Diets”(only REFERENCE components of the design)




Example

,)32.2,41.2(366.0)42.10,30.2(590.0)46.67,87.0(044.0)ˆ;( NNNyf

“Diets”(only REFERENCE components of the design)

FDR by N Genes

In Reverter et al. ‘03 (JAS 81:1900), 27 genes were reported as having a PP > 0.95 of being in the extreme cluster.

Now, we can assess that these 27 genes include a FDR < 10%.

a quantitative overview to gene expression profiling in animal genetics

Documents

gene expression profiling

genesa quantitative

mixtures of distributionsmodel

mixture density

simulatethe mixture

finite components

components and3

model selection