Multiple testing, correlation and regression, and clustering in R

Multtest packageAnscombe dataset and stats packageCluster package

Multtest package

The multtest package contains a collection of functions for multiple hypothesis testing.

These functions can be used to identify differentially expressed genes in microarray experiments, i.e., genes whose expression levels are associated with a response or covariate of interest.

Multtest package

These procedures are implemented for tests based on t–statistics, F–statistics, paired t–statistics, Wilcoxon statistics. The multtest package implements multiple testing

procedures for controlling different Type I error rates. It includes procedures for controlling the family–wise Type I error rate (FWER): Bonferroni, Hochberg (1988), Holm (1979).

It also includes procedures for controlling the false discovery rate (FDR): Benjamini and Hochberg (1995) and Benjamini and Yekutieli (2001) step–up procedures.

Data Analysis

Leukemia Data by Golub et al. (1999)

> library(multtest, verbose = FALSE)

> data(golub)

Golub Data

Golub et al. (1999) were interested in identifying genes that are differentially expressed in patients with two type of leukemias, acute lymphoblastic leukemia (ALL, class 0) and acute myeloid leukemia (AML, class 1).

Gene expression levels were measured using Affymetrix high–density oligonucleotide chips containing p = 6, 817 human genes.

The learning set comprises n = 38 samples, 27 ALL cases and 11 AML cases (data available at http://www.genome.wi.mit.edu/MPR).

Golub Data

Three preprocessing steps were applied to the normalized matrix of intensity values available on the website: (i) thresholding: floor of 100 and ceiling of 16,000; (ii) filtering: exclusion of genes with max / min <= 5 or (max−min) <=

500, where max and min refer respectively to the maximum and minimum intensities for a particular gene across mRNA samples;

(iii) base 10 logarithmic transformation.

Boxplots of the expression levels for each of the 38 samples revealed the need to standardize the expression levels within arrays before combining data across samples. The data were then summarized by a 3, 051×38 matrix X = (xji), where xji denotes the expression level for gene j in tumor mRNA sample i.

Golub Dataset

The dataset golub contains the gene expression data for the 38 training set tumor mRNA samples and 3,051 genes retained after pre–processing. The dataset includes golub: a 3, 051 × 38 matrix of expression levels; golub.gnames: a 3, 051 × 3 matrix of gene

identifiers; golub.cl: a vector of tumor class labels (0 for

ALL, 1 for AML).

Golub Dataset

> dim(golub)

> [1] 3051 38

Golub Dataset

Golub Dataset

The mt.teststat and mt.teststat.num.denum functions

Used to to compute test statistics for each row of a data frame, e.g., two–sample Welch t–statistics, Wilcoxon statistics, F–statistics, paired t–statistics, block F–statistics.

usage

mt.teststat(X,classlabel,test="t",na=.mt.naNUM,nonpara="n")

mt.teststat.num.denum(X,classlabel,test="t",na=.mt.naNUM,nonpara="n")

'test="wilcoxon"''test="pairt"''test="f"'

two–sample t–statistics

comparing, for each gene, expression in the ALL cases to expression in the AML cases

> teststat <- mt.teststat(golub, golub.cl)

QQ plot

First create an empty .pdf file to plot the QQ plot.

> pdf("mtQQ.pdf") Plot the qqplot> qqnorm(teststat) Plot a diagonal line> qqline(teststat) Shuts down the graphical object (e.g., pdf

file)> dev.off()

What is a QQ plot?

What is a QQ plot?

The quantile-quantile (q-q) plot is a graphical technique for determining if two data sets come from populations with a common distribution (e.g. normal distribution).

A q-q plot is a plot of the quantiles of the first data set (expected) against the quantiles of the second data set (observed).

What is a QQ plot?

By a quantile, we mean the fraction (or percent) of points below the given value. That is, the 0.3 (or 30%) quantile is the point at which 30% percent of the data fall below and 70% fall above that value.

A 45-degree reference line is also plotted. If the two sets come from a population with

the same distribution, the points should fall approximately along this reference line.

Store the numerator and denominator of the test stat.

Create a variable tmp in which you store the numerators and denominators of the test statistic

tmp <- mt.teststat.num.denum(golub, golub.cl, test = "t")

Name the numerator as num (teststat.num atribute of the tmp object)

> num <- tmp$teststat.num

Name the denominator as denum (teststat.denum atribute of the tmp object).

> denum <- tmp$teststat.denum

Plot the num to denum

Create a pdf devise

> pdf("mtNumDen.pdf") Plot

> plot(sqrt(denum), num) Shut off the graphics

> dev.off()

Plot the num to denum

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mt.rawp2adjp function

This function computes adjusted p–values for simple multiple testing procedures from a vector of raw (unadjusted) p–values.

The procedures include the Bonferroni, Holm (1979), Hochberg (1988), and Sidak procedures for strong control of the family–wise Type I error rate (FWER), and the Benjamini and Hochberg (1995) and Benjamini and Yekutieli (2001) procedures for (strong) control of the false discovery rate (FDR).

Raw p-values (unadjusted)

As a first approximation, compute raw nominal two–sided p–values for the 3051 test statistics using the standard Gaussian distribution. Create a variable known as rawp0 that contain the p-values of the 3051 test statistics.

> rawp0 <- 2 * (1 - pnorm(abs(teststat)))

> rawp0[1:5][1] 0.07854436 0.36289759 0.92191171 0.73463771 0.17063542

The order function

> aa<-c(3,5,2,1,9)> ia<-order(aa) #index order

> ia

[1] 4 3 1 2 5> aa[ia] # order aa according to ia

[1] 1 2 3 5 9

Adjusted p-values

Create a vector of character strings containing the names of the multiple testing procedures for which adjusted p-values are to be computed. This vector should include any of the following: '"Bonferroni"', '"Holm"', '"Hochberg"', '"SidakSS"', '"SidakSD"', '"BH"', '"BY"'.

> procs <- c("Bonferroni", "Holm",+ "Hochberg", "SidakSS", "SidakSD",+ "BH", "BY")

Adjusted p-values in the order of the gene list

Adjusted p–values can be stored in the original gene order in adjp using order(res$index)

> res <- mt.rawp2adjp(rawp0, procs)

> adjp <- res$adjp[order(res$index), ]

Adjusted p-values in the order of significance Adjusted p–values can be stored in the

order of significance> adjp <- res$adjp

> adjp[1:5,]

mt.reject function to list number of genes rejected

mt.reject function to list number of genes rejected

Find the genes most significant

Get the data of significantly modulated genes

> which <- mt.reject(cbind(rawp0, adjp), 0.000001)$which[, 2]

> sum(which)[1] 143> gsignificant<-golub[which,]> dim(gsignificant)[1] 143 38> gsignificant[1:5,1:5] [,1] [,2] [,3] [,4] [,5][1,] -1.45769 -0.32639 -1.46227 -0.18983 -0.12402[2,] 0.86019 -0.14271 0.67037 0.70706 0.87697[3,] -1.00702 -0.89365 -1.21154 -1.40715 -1.42668[4,] -1.27427 -0.66834 -0.58252 -1.40715 -0.03531[5,] -0.45670 0.48916 -0.48474 0.02261 0.00704

mt.plot function

The mt.plot function produces a number of graphical summaries for the results of multiple testing procedures and their corresponding adjusted p–values.

mt.plot function

To produce plots of adjusted p–values for the Bonferroni, maxT, Benjamini and Hochberg 7 (1995), and Benjamini and Yekutieli (2001) procedures use

Plot adjp values over number of rejected hypotheses

> res <- mt.rawp2adjp(rawp, c("Bonferroni", "BH", "BY"))> adjp <- res$adjp[order(res$index), ]> allp <- cbind(adjp, maxT)> dimnames(allp)[[2]] <- c(dimnames(adjp)[[2]], "maxT")> procs <- dimnames(allp)[[2]]> procs <- procs[c(1, 2, 5, 3, 4)]> cols <- c(1, 2, 3, 5, 6)> ltypes <- c(1, 2, 2, 3, 3)> mt.plot(adjp,teststat,plottype = "pvsr")

Correlation and Regression in R

Anscombe dataset> data(anscombe)

> ? anscombe

Anscombe dataset

Get summaries

cor function

Find correlation coefficient of all pairwise combinations

Find correlation coefficient of specific pairs

> cor(anscombe[,1],anscombe[,2])[1] 1> cor(anscombe[,1],anscombe[,4])[1] -0.5

cor.test function

cor.test

> cor.test(anscombe[,1],anscombe[,4])

Pearson's product-moment correlation

data: anscombe[, 1] and anscombe[, 4] t = -1.7321, df = 9, p-value = 0.1173alternative hypothesis: true correlation is not

equal to 0 95 percent confidence interval: -0.8460984 0.1426659 sample estimates: cor -0.5

Plot scatterplots

> plot(anscombe[,1],anscombe[,4])

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Application to Golub dataset

Calculate the linear regression coefficients and get a summary> ll<-lm(anscombe[,4] ~ anscombe[,1])> summary(ll)

Get the anova table

Plot the regression line

> plot(anscombe[,1],anscombe[,4])

> abline(lm(anscombe[,4] ~ anscombe[,1]))

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ll$coefficients, ll$residuals

hclust function

hclust for golub data (correlation distance with average linkage)

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hclust (*, "average")as.dist(1 - cor(golub))

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hclust (*, "single")dist(t(golub))

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Cluster Dendrogram

hclust (*, "single")dist(gsignificant)

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heatmap function

heatmap for significant genes

> attributes(clust.euclid.single)$names[1] "merge" "height" "order" "labels" "method" [6] "call" "dist.method"

$class[1] "hclust"

> clust.euclid.single$order [1] 129 10 124 125 54 35 28 59 44 85 110 90 106 15 137 138 139 37 [19] 47 126 63 88 27 134 79 80 76 121 131 81 3 82 11 1 104 78 [37] 140 31 34 98 89 8 26 123 92 113 61 127 16 95 29 96 39 142 [55] 7 57 128 14 32 6 67 9 71 133 52 19 74 72 23 73 132 99 [73] 55 135 56 136 5 97 24 18 25 4 107 84 94 109 114 51 68 69 [91] 45 65 46 41 75 49 102 122 50 100 83 12 86 141 17 116 143 2[109] 77 105 112 22 36 101 111 13 62 21 60 87 38 58 118 115 119 93[127] 30 33 108 43 120 91 70 130 53 40 66 64 20 42 103 48 117> gsignificantordered<-gsignificant[clust.euclid.single$order,]> heatmap(gsignificantordered)

Heatmap for significant genes

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kmeans function

kmeans for samples based on significant genes

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pam function

Pam on samples based on significant genes

Silioutte plot

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Silhouette width si

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Silhouette plot of pam(x = as.dist(1 - cor(gsignificant)), k = 2, diss = TRUE)

Average silhouette width : 0.56

n = 38 2clustersCj

j : nj | aveiCjsi

1 : 25 | 0.5

2 : 13 | 0.66

> plot(clust.pam.2)

Microarray Data Analysis Software

http://rana.lbl.gov/EisenSoftware.htm http://classify.stanford.edu/ http://www.broad.mit.edu/cancer/software/software.html http://homes.esat.kuleuven.be/~dna/Biol/Software.html http://vortex.cs.wayne.edu/Projects.html http://visitor.ics.uci.edu/cgi-bin/genex/rcluster/index.cgi http://www-stat.stanford.edu/%7Etibs/SAM/index.html http://maexplorer.sourceforge.net/

http://ihome.cuhk.edu.hk/~b400559/arraysoft.html http://genome.ws.utk.edu/resources/restech.shtml