Package '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 [email protected], Yuejiao Fu [email protected], Xiaogang Wang [email protected], Dawn L. DeMeo [email protected], Kelan Tantisira [email protected], Scott T. Weiss [email protected], Weiliang Qiu [email protected]

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)

---

Plot of histogram and density estimate of the pooled gene expression levels.

Description

Plot of histogram of pooled gene expression levels, composited with density estimate based on the mixture of marginal distributions. The density estimate is based on the assumption that the marginal correlations between subjects are zero.

Usage

plotHistDensity.v(obj.gsMMD,
                plotFlag="case",
                plotComponent=FALSE,
                myxlab="expression level",
                myylab="density",
                mytitle="Histogram (case)",
                x.legend=NULL,
                y.legend=NULL,
                numPoints=500,
                mycol=1:4, 
                mylty=1:4, 
                mylwd=rep(3,4), 
                cex.main=2, 
                cex.lab=1.5, 
                cex.axis=1.5, 
                cex=2,
                bty="n")

Arguments

obj.gsMMD	an object returned by gsMMD.v, gsMMD.default.v, gsMMD2.v, or gsMMD2.default.v
plotFlag	logical. Indicate the plot will based on which type of subjects.
plotComponent	logical. Indicate if components of the mixture of marginal distribution will be plotted.
myxlab	label for x-axis
myylab	label for y-axis
mytitle	title of the plot
x.legend	the x-corrdiates of the legend
y.legend	the y-corrdiates of the legend
numPoints	logical. Indicate how many genes will be plots.
mycol	color for the density estimates (overall and components)
mylty	line styles for the density estimates (overall and components)
mylwd	line width for the density estimates (overall and components)
cex.main	font for main title
cex.lab	font for x- and y-axis labels
cex.axis	font for x- and y-axis
cex	font for texts
bty	the type of box to be drawn around the legend. The allowed values are '"o"' and '"n"' (the default).

Details

For a given type of subjects, we pool their expression levels together if the marginal correlations among subjects are zero. We then draw a histogram of the pooled expression levels. Next, we composite density estimates of gene expression levels for the overal distribution and the 3 component distributions.

Value

A list containing coordinates of the density estimates:

x	sorted pooled gene expression levels for cases or controls.
x2	a subset of x specified by the sequence: seq(from=1,to=len.x, by=delta), where len.x is the length of the vector x, and delta=floor(len.x/numpoints).
y	density estimate corresponding to x2
y1	weighted density estimate for gene cluster 1
y2	weighted density estimate for gene cluster 2
y3	weighted density estimate for gene cluster 3

Note

The density estimate is obtained based on the assumption that the marginal correlation among subjects is zero. If the estimated marginal correlation obtained by gsMMD.v is far from zero, then do not use this plot function.

Author(s)

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

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))
     
  
    plotHistDensity.v(obj.gsMMD.v, plotFlag = "case", 
        mytitle = "Histogram (case)", 
        plotComponent = TRUE, 
        x.legend = c(0.8, 3), 
        y.legend = c(0.3, 0.4), 
        numPoints = 50)
    t2=proc.time()-t1
    print(t2)

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