Bayesian Models for Gene expression

With DNA Microarray DataJoseph G. Ibrahim, Ming-Hui Chen, and Robert J.

Gray

 

 Presented by Yong Zhang

Goals: 1) To build a model to compare between

normal and tumor tissues and to find the genes that best distinguish between tissue types.

2) to develop model assessment techniques so as to assess the fit of a class of competing models.

Outline

General Model

Gene Selection Algo.

Prior Distributions

L measures(assessment)

example

Data structure

x: the expression level for a given gene

C0: threshold value for which a gene is considered as not expressed

Let p = P(x=c0), then

where y is the continuous part for x.

• j=1, 2 index the tissue type(normal vs. tumor)

• i=1,2,…nj, ith individual

• g=1,…G, gth gene

• xjig : the gene expression mixture random

variable for the jth tissue type for the ith individual and the gth gene.

The General Model

• Assume

• δjig = 1(xjig=c0)

• pjg=P(xjig=c0)=P(δjig = 1)

=(,2,p)

• Data D=(x111,…x2,n2,G, )

• Likelihood function for :

L(|D)=

In order to find which genes best discriminate

between the normal and tumor tissues, let

Then we set

such that we can use g to judge them.

Prior Distributions

•

jg2 ~ Inverse Gamma(aj0,bj0)

j0 ~ N(mj0,vj02), j=1,2

•

• bj0 ~ gamma(qj0,tj0)

• ejg ~ N(uj0,kj0wj02)

Gene Selection Algo.

1) For each gene, compute g and

2) Select a “threshold” value, say r0, to decide which genes are different. If

3) Once the gth genes are declared different,

set 1g 2g, otherwise set 1g = 2g g , where g is treated as unknown.

Gene Selection Algo.

4) Create several submodels using several

values of r0.

5) Use L measure to decide which submodel is the best one(smallest L measure).

The properties of this approach

1) Model the gene expression level as a mixture random variable.

2) Use a lognormal model for the continuous part of the mixture.

3) Use L measure statistic for evaluating models.

L measure for model assessment

• It relies on the notion of an imaginary replicate experiment.

• Let z= (z111, …, z2,n2,G) denote future values of a replicate experiment.

L measure is the expected squared Euclidean distance between x and z,

A more general is

The r.s. of the last formula can be got by MCMC.

Computational Algo.(MCMC)

1.

• For 1–4 and 6, the generation is

straightforward. • For 5, we can use an adaptive

rejection algorithm(Gilks and Wild, 1992) because the corresponding conditional posterior densities are log-concave.

Discussion

• That model development and prior distributions in this paper can be easily extended to handle three or more tissue types.

• More general classes of priors• The gene selection criterions