Introduction

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 use also xtable R package to display a nice correlation table.

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")

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 :

• The Hmisc R package to compute the matrix of correlation coefficients and the corresponding p-values.
• The xtable R package for displaying in HTML or Latex format.

corstars(mtcars[,1:7], result="html")

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 < .001, "****", 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

Infos

This analysis was performed using R (ver. 3.2.4).