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Correlation matrix analysis is an important method to find dependence between variables. Computing correlation matrix and drawing correlogram is explained here. The aim of this article is to show you how to get the lower and the upper triangular part of a correlation matrix. We will also use the xtable R package to display a nice correlation table in html or latex formats.

Note that online software is also available here to compute correlation matrix and to plot a correlogram without any installation.

Contents:

## Correlation matrix analysis

The following R code computes a correlation matrix using mtcars data. Click here to read more.

mcor<-round(cor(mtcars),2)
mcor
mpg   cyl  disp    hp  drat    wt  qsec    vs    am  gear  carb
mpg   1.00 -0.85 -0.85 -0.78  0.68 -0.87  0.42  0.66  0.60  0.48 -0.55
cyl  -0.85  1.00  0.90  0.83 -0.70  0.78 -0.59 -0.81 -0.52 -0.49  0.53
disp -0.85  0.90  1.00  0.79 -0.71  0.89 -0.43 -0.71 -0.59 -0.56  0.39
hp   -0.78  0.83  0.79  1.00 -0.45  0.66 -0.71 -0.72 -0.24 -0.13  0.75
drat  0.68 -0.70 -0.71 -0.45  1.00 -0.71  0.09  0.44  0.71  0.70 -0.09
wt   -0.87  0.78  0.89  0.66 -0.71  1.00 -0.17 -0.55 -0.69 -0.58  0.43
qsec  0.42 -0.59 -0.43 -0.71  0.09 -0.17  1.00  0.74 -0.23 -0.21 -0.66
vs    0.66 -0.81 -0.71 -0.72  0.44 -0.55  0.74  1.00  0.17  0.21 -0.57
am    0.60 -0.52 -0.59 -0.24  0.71 -0.69 -0.23  0.17  1.00  0.79  0.06
gear  0.48 -0.49 -0.56 -0.13  0.70 -0.58 -0.21  0.21  0.79  1.00  0.27
carb -0.55  0.53  0.39  0.75 -0.09  0.43 -0.66 -0.57  0.06  0.27  1.00

The result is a table of correlation coefficients between all possible pairs of variables.

## Lower and upper triangular part of a correlation matrix

To get the lower or the upper part of a correlation matrix, the R function lower.tri() or upper.tri() can be used. The formats of the functions are :

lower.tri(x, diag = FALSE)
upper.tri(x, diag = FALSE)

- x : is the correlation matrix - diag : if TRUE the diagonal are not included in the result.

The two functions above, return a matrix of logicals which has the same size of a the correlation matrix. The entries is TRUE in the lower or upper triangle :

upper.tri(mcor)
[,1]  [,2]  [,3]  [,4]  [,5]  [,6]  [,7]  [,8]  [,9] [,10] [,11]
[1,] FALSE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE
[2,] FALSE FALSE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE
[3,] FALSE FALSE FALSE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE
[4,] FALSE FALSE FALSE FALSE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE
[5,] FALSE FALSE FALSE FALSE FALSE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE
[6,] FALSE FALSE FALSE FALSE FALSE FALSE  TRUE  TRUE  TRUE  TRUE  TRUE
[7,] FALSE FALSE FALSE FALSE FALSE FALSE FALSE  TRUE  TRUE  TRUE  TRUE
[8,] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE  TRUE  TRUE  TRUE
[9,] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE  TRUE  TRUE
[10,] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE  TRUE
[11,] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
# Hide upper triangle
upper<-mcor
upper[upper.tri(mcor)]<-""
upper<-as.data.frame(upper)
upper
mpg   cyl  disp    hp  drat    wt  qsec    vs   am gear carb
mpg      1
cyl  -0.85     1
disp -0.85   0.9     1
hp   -0.78  0.83  0.79     1
drat  0.68  -0.7 -0.71 -0.45     1
wt   -0.87  0.78  0.89  0.66 -0.71     1
qsec  0.42 -0.59 -0.43 -0.71  0.09 -0.17     1
vs    0.66 -0.81 -0.71 -0.72  0.44 -0.55  0.74     1
am     0.6 -0.52 -0.59 -0.24  0.71 -0.69 -0.23  0.17    1
gear  0.48 -0.49 -0.56 -0.13   0.7 -0.58 -0.21  0.21 0.79    1
carb -0.55  0.53  0.39  0.75 -0.09  0.43 -0.66 -0.57 0.06 0.27    1
#Hide lower triangle
lower<-mcor
lower[lower.tri(mcor, diag=TRUE)]<-""
lower<-as.data.frame(lower)
lower
mpg   cyl  disp    hp  drat    wt  qsec    vs    am  gear  carb
mpg      -0.85 -0.85 -0.78  0.68 -0.87  0.42  0.66   0.6  0.48 -0.55
cyl              0.9  0.83  -0.7  0.78 -0.59 -0.81 -0.52 -0.49  0.53
disp                  0.79 -0.71  0.89 -0.43 -0.71 -0.59 -0.56  0.39
hp                         -0.45  0.66 -0.71 -0.72 -0.24 -0.13  0.75
drat                             -0.71  0.09  0.44  0.71   0.7 -0.09
wt                                     -0.17 -0.55 -0.69 -0.58  0.43
qsec                                          0.74 -0.23 -0.21 -0.66
vs                                                  0.17  0.21 -0.57
am                                                        0.79  0.06
gear                                                            0.27
carb

## Use xtable R package to display nice correlation table in html format

library(xtable)
print(xtable(upper), type="html")
mpg cyl disp hp drat wt qsec vs am gear carb
mpg 1
cyl -0.85 1
disp -0.85 0.9 1
hp -0.78 0.83 0.79 1
drat 0.68 -0.7 -0.71 -0.45 1
wt -0.87 0.78 0.89 0.66 -0.71 1
qsec 0.42 -0.59 -0.43 -0.71 0.09 -0.17 1
vs 0.66 -0.81 -0.71 -0.72 0.44 -0.55 0.74 1
am 0.6 -0.52 -0.59 -0.24 0.71 -0.69 -0.23 0.17 1
gear 0.48 -0.49 -0.56 -0.13 0.7 -0.58 -0.21 0.21 0.79 1
carb -0.55 0.53 0.39 0.75 -0.09 0.43 -0.66 -0.57 0.06 0.27 1

## Combine matrix of correlation coefficients and significance levels

Custom function corstars() is used to combine the correlation coefficients and the level of significance. The R code of the function is provided at the end of this article. It requires 2 packages :

Before continuing with the following exercises, you should first copy and paste the source code the function corstars(), which you can find at the bottom of this article.

corstars(mtcars[,1:7], result="html")
mpg cyl disp hp drat wt
mpg
cyl -0.85****
disp -0.85**** 0.90****
hp -0.78**** 0.83**** 0.79****
drat 0.68**** -0.70**** -0.71**** -0.45**
wt -0.87**** 0.78**** 0.89**** 0.66**** -0.71****
qsec 0.42* -0.59*** -0.43* -0.71**** 0.09 -0.17

p < .0001 ‘****’; p < .001 ‘***’, p < .01 ‘**’, p < .05 ‘*’

The code of corstars function (The code is adapted from the one posted on this forum and on this blog ):

# x is a matrix containing the data
# method : correlation method. "pearson"" or "spearman"" is supported
# removeTriangle : remove upper or lower triangle
# results :  if "html" or "latex"
# the results will be displayed in html or latex format
corstars <-function(x, method=c("pearson", "spearman"), removeTriangle=c("upper", "lower"),
result=c("none", "html", "latex")){
#Compute correlation matrix
require(Hmisc)
x <- as.matrix(x)
correlation_matrix<-rcorr(x, type=method[1])
R <- correlation_matrix\$r # Matrix of correlation coeficients
p <- correlation_matrix\$P # Matrix of p-value

## Define notions for significance levels; spacing is important.
mystars <- ifelse(p < .0001, "****", ifelse(p < .001, "*** ", ifelse(p < .01, "**  ", ifelse(p < .05, "*   ", "    "))))

## trunctuate the correlation matrix to two decimal
R <- format(round(cbind(rep(-1.11, ncol(x)), R), 2))[,-1]

## build a new matrix that includes the correlations with their apropriate stars
Rnew <- matrix(paste(R, mystars, sep=""), ncol=ncol(x))
diag(Rnew) <- paste(diag(R), " ", sep="")
rownames(Rnew) <- colnames(x)
colnames(Rnew) <- paste(colnames(x), "", sep="")

## remove upper triangle of correlation matrix
if(removeTriangle[1]=="upper"){
Rnew <- as.matrix(Rnew)
Rnew[upper.tri(Rnew, diag = TRUE)] <- ""
Rnew <- as.data.frame(Rnew)
}

## remove lower triangle of correlation matrix
else if(removeTriangle[1]=="lower"){
Rnew <- as.matrix(Rnew)
Rnew[lower.tri(Rnew, diag = TRUE)] <- ""
Rnew <- as.data.frame(Rnew)
}

## remove last column and return the correlation matrix
Rnew <- cbind(Rnew[1:length(Rnew)-1])
if (result[1]=="none") return(Rnew)
else{
if(result[1]=="html") print(xtable(Rnew), type="html")
else print(xtable(Rnew), type="latex")
}
}

## Conclusions

• Use cor() function to compute correlation matrix.
• Use lower.tri() and upper.tri() functions to get the lower or upper part of the correlation matrix
• Use xtable R function to display a nice correlation matrix in latex or html format.

## Infos

This analysis was performed using R (ver. 3.3.2).