gsMMD.v {MMDvariance}

Gene selection based on variances by using a mixture of marginal distributions

Description

Gene selection based on variances by using the marginal distributions of gene profiles that characterized by a mixture of three-component multivariate distributions. The goal is to detect gene probes having different variances between cases and controls. Input is an object derived from the class ExpressionSet. The function will obtain initial gene cluster membership by its own.

Usage

gsMMD.v(obj.eSet, 
      memSubjects, 
      maxFlag = TRUE, 
      thrshPostProb = 0.5, 
      geneNames = NULL, 
      alpha = 0.05, 
      iniGeneMethod = "myLeveneTest", 
      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 variable 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: (1) genes having higher variance in cases than in controls; (2) genes having equal variance between cases and controls; (3) genes having lower variance in cases than in controls.

Available methods are: “myAWvar”, “myBFTest”, “myFTest”, “myLeveneTest”, “myLevene.TM”, “myiAWvar.BF”, “myiAWvar.Levene”, “myiAWvar.TM”, “myLeveneTest”, “myLeveneTest.TM”.

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

step increase 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 \sum_{k=1}^{3} \pi_k f_k(x|\theta). Each component distribution f_k corresponds to a gene cluster. The 3 components correspond to 3 gene clusters: (1) genes having higher variance in cases than in controls; (2) genes having equal variance between cases and controls; (3) genes having lower variance in cases than in controls. The model parameter vector is \theta=(\pi_1, \pi_2, \pi_3, \sigma^2_{c1}, \sigma^2_{n1}, \mu_{c1}, \rho_{c1}, \mu_{n1}, \rho_{n1}, \sigma^2_2, \mu_{c2}, \rho_{c2}, \mu_{n2}, \rho_{n2}, \sigma^2_{c3}, \sigma^2_{n3}, \mu_{c3}, \rho_{c3}, \mu_{n3}, \rho_{n3}. where \pi_1, \pi_2, and \pi_3 are the mixing proportions; \mu_{c1}, \sigma^2_{c1}, and \rho_{c1} are the marginal mean, variance, and correlation of gene expression levels of cluster 1 (over-variable genes) for diseased subjects; \mu_{n1}, \sigma^2_{n1}, and \rho_{n1} are the marginal mean, variance, and correlation of gene expression levels of cluster 1 (over-variable genes) for non-diseased subjects; \sigma^2_2, \mu_{c2}, \rho_{c2}, \mu_{n2}, and \rho_{n2} are the marginal mean, variance, and correlation of gene expression levels of cluster 2 (equal-variable genes); \mu_{c3}, \sigma^2_{c3}, and \rho_{c3} are the marginal mean, variance, and correlation of gene expression levels of cluster 3 (under-variable genes) for diseased subjects; \mu_{n3}, \sigma^2_{n3}, and \rho_{n3} are the marginal mean, variance, and correlation of gene expression levels of cluster 3 (under-variable) for non-diseased subjects.

Note that genes in cluster 2 are non-differentially variable across abnormal and normal tissue samples. Hence there are only 5 parameters for cluster 2.

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

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

We also has the following constraints for the mixing proportion: \pi_3=1-\pi_1-\pi_2, \pi_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 \theta^*=(\pi_1, \pi_2, s^2_{c1}, \delta_{n1}, \mu_{c1}, r_{c1}, \mu_{n1}, r_{n1}, s^2_2, \mu_{c2}, r_{c2}, \mu_{n2}, r_{n2}, s^2_{c3}, \delta_{n3}, \mu_{c3}, r_{c3}, \mu_{n3}, r_{n3}), where \sigma_{n1}=\sigma_{c1}-\exp(\delta_{n1}), \sigma_{n3}=\sigma_{c3}+\exp(\delta_{n3}), \rho_{c1}=(\exp(r_{c1})-1/(n_c-1))/(1+\exp(r_{c1})), \rho_{n1}=(\exp(r_{n1})-1/(n_n-1))/(1+\exp(r_{n1})), \rho_{2}=(\exp(r_{2})-1/(n-1))/(1+\exp(r_{2})), \rho_{c3}=(\exp(r_{c3})-1/(n_c-1))/(1+\exp(r_{c3})), \rho_{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 over-variable gene; 2 means non-differentially variable gene; 3 means under-variable gene.

memGenes2

an variant of the vector of cluster membership of genes. 1 means differentially variable gene; 0 means non-differentially variable 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 is slow for large data sets.

Author(s)

Xuan Li [e-mail], Yuejiao Fu [e-mail], Xiaogang Wang [e-mail], Dawn L. DeMeo [e-mail], Kelan Tantisira [e-mail], Scott T. Weiss [e-mail], Weiliang Qiu [e-mail]

References

Li X, Fu Y, Wang X, DeMeo DL, Tantisira K, Weiss ST, Qiu W. Detecting Differentially Variable MicroRNAs via Model-Based Clustering. International Journal of Genomics. Article ID 6591634, Volumne 2018 (2018).

Examples

      t1 = proc.time()
      library(ALL)
      data(ALL)
      eSet1 <- ALL[1:50, 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.v <- gsMMD.v(eSet1, memSubjects, transformFlag = FALSE, 
        transformMethod = "boxcox", scaleFlag = FALSE, 
        eps = 1.0e-1, ITMAX = 5, quiet = TRUE)
      print(round(obj.gsMMD.v$para, 3))
      t2=proc.time()-t1
      print(t2)