gsMMD: Gene selection based on a mixture of marginal distributions

Description Usage Arguments Details Value Note Author(s) References See Also Examples

Description

Gene selection based on the marginal distributions of gene profiles that characterized by a mixture of three-component multivariate distributions. Input is an object derived from the class ExpressionSet. The function will obtain initial gene cluster membership by its own.

Usage

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
gsMMD(obj.eSet, 
      memSubjects, 
      maxFlag = TRUE, 
      thrshPostProb = 0.5, 
      geneNames = NULL, 
      alpha = 0.05, 
      iniGeneMethod = "Ttest", 
      transformFlag = FALSE, 
      transformMethod = "boxcox", 
      scaleFlag = TRUE, 
      criterion = c("cor", "skewness", "kurtosis"), 
      minL = -10, 
      maxL = 10, 
      stepL = 0.1, 
      eps = 0.001, 
      ITMAX = 100, 
      plotFlag = FALSE,
      quiet=TRUE)

Arguments

obj.eSet

an object derived from the class ExpressionSet which contains the matrix of gene expression levels. The rows of the matrix are genes. The columns of the matrix are subjects.

memSubjects

a vector of membership of subjects. memSubjects[i]=1 means the i-th subject belongs to diseased group, 0 otherwise.

maxFlag

logical. Indicate how to assign gene class membership. maxFlag=TRUE means that a gene will be assigned to a class in which the posterior probability of the gene belongs to this class is maximum. maxFlag=FALSE means that a gene will be assigned to class 1 if the posterior probability of the gene belongs to class 1 is greater than thrshPostProb. Similarly, a gene will be assigned to class 1 if the posterior probability of the gene belongs to class 1 is greater than thrshPostProb. If the posterior probability is less than thrshPostProb, the gene will be assigned to class 2 (non-differentially expressed gene group).

thrshPostProb

threshold for posterior probabilities. For example, if the posterior probability that a gene belongs to cluster 1 given its gene expression levels is larger than thrshPostProb, then this gene will be assigned to cluster 1.

geneNames

an optional character vector of gene names

alpha

significant level which is equal to 1-conf.level, conf.level is the argument for the function t.test.

iniGeneMethod

method to get initial 3-cluster partition of genes. Available methods are: “Ttest”, “Wilcox”.

transformFlag

logical. Indicate if data transformation is needed

transformMethod

method for transforming data. Available methods include "boxcox", "log2", "log10", "log", "none".

scaleFlag

logical. Indicate if gene profiles are to be scaled to have mean zero and variance one. If transformFlag=TRUE and scaleFlag=TRUE, then scaling is performed after transformation. To avoid linear dependence of tissue samples after scaling gene profiles, we delete one tissue sample after scaling (c.f. details).

criterion

if transformFlag=TRUE, criterion indicates what criterion to determine if data looks like normal. “cor” means using Pearson's correlation. The idea is that the observed quantiles after transformation should be close to theoretical normal quantiles. So we can use Pearson's correlation to check if the scatter plot of theoretical normal quantiles versus observed quantiles is a straightline. “skewness” means using skewness measure to check if the distribution of the transformed data are close to normal distribution; “kurtosis” means using kurtosis measure to check normality.

minL

lower limit for the lambda parameter used in Box-Cox transformation

maxL

upper limit for the lambda parameter used in Box-Cox transformation

stepL

tolerance when searching the optimal lambda parameter used in Box-Cox transformation

eps

a small positive value. If the absolute value of a value is smaller than eps, this value is regarded as zero.

ITMAX

maximum iteration allowed for iterations in the EM algorithm

plotFlag

logical. Indicate if the Box-Cox normality plot should be output.

quiet

logical. Indicate if intermediate results should be printed out.

Details

We assume that the distribution of gene expression profiles is a mixture of 3-component multivariate normal distributions ∑_{k=1}^{3} π_k f_k(x|θ). Each component distribution f_k corresponds to a gene cluster. The 3 components correspond to 3 gene clusters: (1) up-regulated gene cluster, (2) non-differentially expressed gene cluster, and (3) down-regulated gene cluster. The model parameter vector is θ=(π_1, π_2, π_3, μ_{c1}, σ^2_{c1}, ρ_{c1}, μ_{n1}, σ^2_{n1}, ρ_{n1}, μ_2, σ^2_2, ρ_2, μ_{c3}, σ^2_{c3}, ρ_{c3}, μ_{n3}, σ^2_{n3}, ρ_{n3}. where π_1, π_2, and π_3 are the mixing proportions; μ_{c1}, σ^2_{c1}, and ρ_{c1} are the marginal mean, variance, and correlation of gene expression levels of cluster 1 (up-regulated genes) for diseased subjects; μ_{n1}, σ^2_{n1}, and ρ_{n1} are the marginal mean, variance, and correlation of gene expression levels of cluster 1 (up-regulated genes) for non-diseased subjects; μ_2, σ^2_2, and ρ_2 are the marginal mean, variance, and correlation of gene expression levels of cluster 2 (non-differentially expressed genes); μ_{c3}, σ^2_{c3}, and ρ_{c3} are the marginal mean, variance, and correlation of gene expression levels of cluster 3 (up-regulated genes) for diseased subjects; μ_{n3}, σ^2_{n3}, and ρ_{n3} are the marginal mean, variance, and correlation of gene expression levels of cluster 3 (up-regulated genes) for non-diseased subjects.

Note that genes in cluster 2 are non-differentially expressed across abnormal and normal tissue samples. Hence there are only 3 parameters for cluster 2.

To make sure the identifiability, we set the following contraints: μ_{c1}>μ_{n1} and μ_{c3}<μ_{n3}.

To make sure the marginal covariance matrices are poisitive definite, we set the following contraints: -1/(n_c-1)<ρ_{c1}<1, -1/(n_n-1)<ρ_{n1}<1, -1/(n-1)<ρ_{2}<1, -1/(n_c-1)<ρ_{c3}<1, -1/(n_n-1)<ρ_{n3}<1.

We also has the following constraints for the mixing proportion: π_3=1-π_1-π_2, π_k>0, k=1,2,3.

We apply the EM algorithm to estimate the model parameters. We regard the cluster membership of genes as missing values.

To facilitate the estimation of the parameters, we reparametrize the parameter vector as θ^*=(π_1, π_2, μ_{c1}, σ^2_{c1}, r_{c1}, δ_{n1}, σ^2_{n1}, r_{n1}, μ_2, σ^2_2, r_2, μ_{c3}, σ^2_{c3}, r_{c3}, δ_{n3}, σ^2_{n3}, r_{n3}), where μ_{n1}=μ_{c1}-\exp(δ_{n1}), μ_{n3}=μ_{c3}+\exp(δ_{n3}), ρ_{c1}=(\exp(r_{c1})-1/(n_c-1))/(1+\exp(r_{c1})), ρ_{n1}=(\exp(r_{n1})-1/(n_n-1))/(1+\exp(r_{n1})), ρ_{2}=(\exp(r_{2})-1/(n-1))/(1+\exp(r_{2})), ρ_{c3}=(\exp(r_{c3})-1/(n_c-1))/(1+\exp(r_{c3})), ρ_{n3}=(\exp(r_{n3})-1/(n_n-1))/(1+\exp(r_{n3})).

Given a gene, the expression levels of the gene are assumed independent. However, after scaling, the scaled expression levels of the gene are no longer independent and the rank r^*=r-1 of the covariance matrix for the scaled gene profile will be one less than the rank r for the un-scaled gene profile Hence the covariance matrix of the gene profile will no longer be positive-definite. To avoid this problem, we delete a tissue sample after scaling since its information has been incorrporated by other scaled tissue samples. We arbitrarily select the tissue sample, which has the biggest label number, from the tissue sample group that has larger size than the other tissue sample group. For example, if there are 6 cancer tissue samples and 10 normal tissue samples, we delete the 10-th normal tissue sample after scaling.

Value

A list contains 18 elements.

dat

the (transformed) microarray data matrix. If tranformation performed, then dat will be different from the input microarray data matrix.

memSubjects

the same as the input memSubjects.

memGenes

a vector of cluster membership of genes. 1 means up-regulated gene; 2 means non-differentially expressed gene; 3 means down-regulated gene.

memGenes2

an variant of the vector of cluster membership of genes. 1 means differentially expressed gene; 0 means non-differentially expressed gene.

para

parameter estimates (c.f. details).

llkh

value of the loglikelihood function.

wiMat

posterior probability that a gene belongs to a cluster given the expression levels of this gene. Column i is for cluster i.

wiArray

posterior probability matrix for different initial gene selection methods.

memIniMat

a matrix of initial cluster membership of genes.

paraIniMat

a matrix of parameter estimates based on initial gene cluster membership.

llkhIniVec

a vector of values of loglikelihood function.

memMat

a matrix of cluster membership of genes based on the mixture of marginal models with initial parameter estimates obtained initial gene cluster membership.

paraMat

a matrix of parameter estimates based on the mixture of marginal models with initial parameter estimates obtained initial gene cluster membership.

llkhVec

a vector of values of loglikelihood function based on the mixture of marginal models with initial parameter estimates obtained initial gene cluster membership.

lambda

the parameter used to do Box-Cox transformation

paraRP

parameter estimates for reparametrized parameter vector (c.f. details).

paraIniMatRP

a matrix of parameter estimates for reparametrized parameter vector based on initial gene cluster membership.

paraMatRP

a matrix of parameter estimates for reparametrized parameter vector based on the mixture of marginal models with initial parameter estimates obtained initial gene cluster membership.

Note

The speed of the program can be slow for large data sets, however it has been improved using Fortran code.

Author(s)

Jarrett Morrow remdj@channing.harvard.edu, Weiliang Qiu Weiliang.Qiu@gmail.com, Wenqing He whe@stats.uwo.ca, Xiaogang Wang stevenw@mathstat.yorku.ca, Ross Lazarus ross.lazarus@channing.harvard.edu

References

Qiu, W.-L., He, W., Wang, X.-G. and Lazarus, R. (2008). A Marginal Mixture Model for Selecting Differentially Expressed Genes across Two Types of Tissue Samples. The International Journal of Biostatistics. 4(1):Article 20. http://www.bepress.com/ijb/vol4/iss1/20

See Also

gsMMD.default, gsMMD2, gsMMD2.default

Examples

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
      library(ALL)
      data(ALL)
      eSet1 <- ALL[1:100, ALL$BT == "B3" | ALL$BT == "T2"]
      
      mem.str <- as.character(eSet1$BT)
      nSubjects <- length(mem.str)
      memSubjects <- rep(0,nSubjects)
      # B3 coded as 0, T2 coded as 1
      memSubjects[mem.str == "T2"] <- 1
      
      obj.gsMMD <- gsMMD(eSet1, memSubjects, transformFlag = TRUE, 
        transformMethod = "boxcox", scaleFlag = TRUE, quiet = FALSE)
      round(obj.gsMMD$para, 3)

GeneSelectMMD documentation built on Nov. 8, 2020, 6:48 p.m.