Graph of correlation matrix using symnum function

The R function symnum can be used to easily highlight the highly correlated variables. It replaces correlation coefficients by symbols according to the value.

The simplified format of the function is :

symnum(x, cutpoints = c(0.3, 0.6, 0.8, 0.9, 0.95),

       symbols = c(" ", ".", ",", "+", "*", "B"))[/pre]

 - x is the correlation matrix to visualize

- cutpoints : correlation coefficient cutpoints. The correlation coefficients between 0 and 0.3 are replaced by a space (" “ correlation coefficients between 0.3 and 0.6 are replace by”.“; etc …

- symbols : the symbols to use. 


The following R code performs a correlation analysis and displays a graph of the correlation matrix :


## Correlation matrix

corMat<-cor(mtcars)

head(round(corMat,2))[/pre]

       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


## Correlation graph for visualization

## abbr.colnames=FALSE to avoid abbreviation of column names

symnum(corMat, abbr.colnames=FALSE)[/pre]

     mpg cyl disp hp drat wt qsec vs am gear carb

mpg  1                                         

cyl  +   1                                     

disp +   *   1                                 

hp   ,   +   ,    1                            

drat ,   ,   ,    .  1                         

wt   +   ,   +    ,  ,    1                    

qsec .   .   .    ,          1                 

vs   ,   +   ,    ,  .    .  ,    1            

am   .   .   .       ,    ,          1         

gear .   .   .       ,    .          ,  1      

carb .   .   .    ,       .  ,    .          1 

attr(,"legend")

[1] 0 ' ' 0.3 '.' 0.6 ',' 0.8 '+' 0.9 '*' 0.95 'B' 1

Conclusions

Infos

This analysis was performed using R (ver. 3.1.0).[/pre]

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 < .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

Infos

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

Prerequisites

The rquery.cormat function requires the installation of corrplot package. Before proceeding, install it using he following R code :

install.packages("corrplot")

To use the rquery.cormat function, you can source it as follow :

source("https://www.sthda.com/upload/rquery_cormat.r")

The R code of rquery.cormat function is provided at the end of this document.

Example of data

The mtcars data is used in the following examples :

mydata <- mtcars[, c(1,3,4,5,6,7)]
head(mydata)

                   mpg disp  hp drat    wt  qsec
Mazda RX4         21.0  160 110 3.90 2.620 16.46
Mazda RX4 Wag     21.0  160 110 3.90 2.875 17.02
Datsun 710        22.8  108  93 3.85 2.320 18.61
Hornet 4 Drive    21.4  258 110 3.08 3.215 19.44
Hornet Sportabout 18.7  360 175 3.15 3.440 17.02
Valiant           18.1  225 105 2.76 3.460 20.22

Computing the correlation matrix

rquery.cormat(mydata)

$r
        hp  disp    wt  qsec  mpg drat
hp       1                            
disp  0.79     1                      
wt    0.66  0.89     1                
qsec -0.71 -0.43 -0.17     1          
mpg  -0.78 -0.85 -0.87  0.42    1     
drat -0.45 -0.71 -0.71 0.091 0.68    1

$p
          hp    disp      wt  qsec     mpg drat
hp         0                                   
disp 7.1e-08       0                           
wt   4.1e-05 1.2e-11       0                   
qsec 5.8e-06   0.013    0.34     0             
mpg  1.8e-07 9.4e-10 1.3e-10 0.017       0     
drat    0.01 5.3e-06 4.8e-06  0.62 1.8e-05    0

$sym
     hp disp wt qsec mpg drat
hp   1                       
disp ,  1                    
wt   ,  +    1               
qsec ,  .       1            
mpg  ,  +    +  .    1       
drat .  ,    ,       ,   1   
attr(,"legend")
[1] 0 ' ' 0.3 '.' 0.6 ',' 0.8 '+' 0.9 '*' 0.95 'B' 1

Correlogram, Heatmap

Note that in the result above, only the lower triangle of the correlation matrix is shown by default. You can use the following R script to get the upper triangle or the full correlation matrix.

rquery.cormat(mydata, type="upper")

$r
     hp disp   wt  qsec   mpg  drat
hp    1 0.79 0.66 -0.71 -0.78 -0.45
disp       1 0.89 -0.43 -0.85 -0.71
wt              1 -0.17 -0.87 -0.71
qsec                  1  0.42 0.091
mpg                         1  0.68
drat                              1

$p
     hp    disp      wt    qsec     mpg    drat
hp    0 7.1e-08 4.1e-05 5.8e-06 1.8e-07    0.01
disp          0 1.2e-11   0.013 9.4e-10 5.3e-06
wt                    0    0.34 1.3e-10 4.8e-06
qsec                          0   0.017    0.62
mpg                                   0 1.8e-05
drat                                          0

$sym
     hp disp wt qsec mpg drat
hp   1  ,    ,  ,    ,   .   
disp    1    +  .    +   ,   
wt           1       +   ,   
qsec            1    .       
mpg                  1   ,   
drat                     1   
attr(,"legend")
[1] 0 ' ' 0.3 '.' 0.6 ',' 0.8 '+' 0.9 '*' 0.95 'B' 1

rquery.cormat(mydata, type="full")

$r
        hp  disp    wt   qsec   mpg   drat
hp    1.00  0.79  0.66 -0.710 -0.78 -0.450
disp  0.79  1.00  0.89 -0.430 -0.85 -0.710
wt    0.66  0.89  1.00 -0.170 -0.87 -0.710
qsec -0.71 -0.43 -0.17  1.000  0.42  0.091
mpg  -0.78 -0.85 -0.87  0.420  1.00  0.680
drat -0.45 -0.71 -0.71  0.091  0.68  1.000

$p
          hp    disp      wt    qsec     mpg    drat
hp   0.0e+00 7.1e-08 4.1e-05 5.8e-06 1.8e-07 1.0e-02
disp 7.1e-08 0.0e+00 1.2e-11 1.3e-02 9.4e-10 5.3e-06
wt   4.1e-05 1.2e-11 0.0e+00 3.4e-01 1.3e-10 4.8e-06
qsec 5.8e-06 1.3e-02 3.4e-01 0.0e+00 1.7e-02 6.2e-01
mpg  1.8e-07 9.4e-10 1.3e-10 1.7e-02 0.0e+00 1.8e-05
drat 1.0e-02 5.3e-06 4.8e-06 6.2e-01 1.8e-05 0.0e+00

$sym
     hp disp wt qsec mpg drat
hp   1                       
disp ,  1                    
wt   ,  +    1               
qsec ,  .       1            
mpg  ,  +    +  .    1       
drat .  ,    ,       ,   1   
attr(,"legend")
[1] 0 ' ' 0.3 '.' 0.6 ',' 0.8 '+' 0.9 '*' 0.95 'B' 1

col<- colorRampPalette(c("blue", "white", "red"))(20)
cormat<-rquery.cormat(mydata, type="full", col=col)

cormat<-rquery.cormat(mydata, graphType="heatmap")

rquery.cormat(mydata, graph=FALSE)

Format the correlation table

The R code below can be used to format the correlation matrix into a table of four columns containing :

• The names of rows/columns
• The correlation coefficients
• The p-values

For this end, use the argument : type=“flatten”

rquery.cormat(mydata, type="flatten", graph=FALSE)

$r
    row column    cor       p
1    hp   disp  0.790 7.1e-08
2    hp     wt  0.660 4.1e-05
3  disp     wt  0.890 1.2e-11
4    hp   qsec -0.710 5.8e-06
5  disp   qsec -0.430 1.3e-02
6    wt   qsec -0.170 3.4e-01
7    hp    mpg -0.780 1.8e-07
8  disp    mpg -0.850 9.4e-10
9    wt    mpg -0.870 1.3e-10
10 qsec    mpg  0.420 1.7e-02
11   hp   drat -0.450 1.0e-02
12 disp   drat -0.710 5.3e-06
13   wt   drat -0.710 4.8e-06
14 qsec   drat  0.091 6.2e-01
15  mpg   drat  0.680 1.8e-05

$p
NULL

$sym
NULL

Description of rquery.cormat function

A simplified format of the function is :

rquery.cormat(x, type=c('lower', 'upper', 'full', 'flatten'),
              graph=TRUE, graphType=c("correlogram", "heatmap"),
              col=NULL, ...)

Description of the arguments:

• x : matrix of data values
• type : Possible values are “lower” (default), “upper”, “full” or “flatten”. It displays the lower or upper triangular of the matrix, full or flatten matrix.
• graph : if TRUE, a correlogram or heatmap is generated to visualize the correlation matrix.
• graphType : Type of graphs. Possible values are “correlogram” or “heatmap”.
• col: colors to use for the correlogram or the heatmap.
• … : Further arguments to be passed to cor() or cor.test() function.

R code of the rquery.cormat function:

#+++++++++++++++++++++++++
# Computing of correlation matrix
#+++++++++++++++++++++++++
# Required package : corrplot

# x : matrix
# type: possible values are "lower" (default), "upper", "full" or "flatten";
  #display lower or upper triangular of the matrix, full  or flatten matrix.
# graph : if TRUE, a correlogram or heatmap is plotted
# graphType : possible values are "correlogram" or "heatmap"
# col: colors to use for the correlogram
# ... : Further arguments to be passed to cor or cor.test function

# Result is a list including the following components :
  # r : correlation matrix, p :  p-values
  # sym : Symbolic number coding of the correlation matrix
rquery.cormat<-function(x,
                        type=c(&#39;lower&#39;, &#39;upper&#39;, &#39;full&#39;, &#39;flatten&#39;),
                        graph=TRUE,
                        graphType=c("correlogram", "heatmap"),
                        col=NULL, ...)
{
  library(corrplot)
  # Helper functions
  #+++++++++++++++++
  # Compute the matrix of correlation p-values
  cor.pmat <- function(x, ...) {
    mat <- as.matrix(x)
    n <- ncol(mat)
    p.mat<- matrix(NA, n, n)
    diag(p.mat) <- 0
    for (i in 1:(n - 1)) {
      for (j in (i + 1):n) {
        tmp <- cor.test(mat[, i], mat[, j], ...)
        p.mat[i, j] <- p.mat[j, i] <- tmp$p.value
      }
    }
    colnames(p.mat) <- rownames(p.mat) <- colnames(mat)
    p.mat
  }
  # Get lower triangle of the matrix
  getLower.tri<-function(mat){
    upper<-mat
    upper[upper.tri(mat)]<-""
    mat<-as.data.frame(upper)
    mat
  }
  # Get upper triangle of the matrix
  getUpper.tri<-function(mat){
    lt<-mat
    lt[lower.tri(mat)]<-""
    mat<-as.data.frame(lt)
    mat
  }
  # Get flatten matrix
  flattenCorrMatrix <- function(cormat, pmat) {
    ut <- upper.tri(cormat)
    data.frame(
      row = rownames(cormat)[row(cormat)[ut]],
      column = rownames(cormat)[col(cormat)[ut]],
      cor  =(cormat)[ut],
      p = pmat[ut]
    )
  }
  # Define color
  if (is.null(col)) {
    col <- colorRampPalette(
            c("#67001F", "#B2182B", "#D6604D", "#F4A582",
              "#FDDBC7", "#FFFFFF", "#D1E5F0", "#92C5DE", 
             "#4393C3", "#2166AC", "#053061"))(200)
    col<-rev(col)
  }
  
  # Correlation matrix
  cormat<-signif(cor(x, use = "complete.obs", ...),2)
  pmat<-signif(cor.pmat(x, ...),2)
  # Reorder correlation matrix
  ord<-corrMatOrder(cormat, order="hclust")
  cormat<-cormat[ord, ord]
  pmat<-pmat[ord, ord]
  # Replace correlation coeff by symbols
  sym<-symnum(cormat, abbr.colnames=FALSE)
  # Correlogram
  if(graph & graphType[1]=="correlogram"){
    corrplot(cormat, type=ifelse(type[1]=="flatten", "lower", type[1]),
             tl.col="black", tl.srt=45,col=col,...)
  }
  else if(graphType[1]=="heatmap")
    heatmap(cormat, col=col, symm=TRUE)
  # Get lower/upper triangle
  if(type[1]=="lower"){
    cormat<-getLower.tri(cormat)
    pmat<-getLower.tri(pmat)
  }
  else if(type[1]=="upper"){
    cormat<-getUpper.tri(cormat)
    pmat<-getUpper.tri(pmat)
    sym=t(sym)
  }
  else if(type[1]=="flatten"){
    cormat<-flattenCorrMatrix(cormat, pmat)
    pmat=NULL
    sym=NULL
  }
  list(r=cormat, p=pmat, sym=sym)
}

Infos

This analysis has been performed using R (ver. 3.2.4).

What is correlation test?

For instance, if we are interested to know whether there is a relationship between the heights of fathers and sons, a correlation coefficient can be calculated to answer this question.

If there is no relationship between the two variables (father and son heights), the average height of son should be the same regardless of the height of the fathers and vice versa.

Here, we’ll describe the different correlation methods and we’ll provide pratical examples using R software.

Install and load required R packages

We’ll use the ggpubr R package for an easy ggplot2-based data visualization

• Install the latest version from GitHub as follow (recommended):

if(!require(devtools)) install.packages("devtools")
devtools::install_github("kassambara/ggpubr")

• Or, install from CRAN as follow:

install.packages("ggpubr")

• Load ggpubr as follow:

library("ggpubr")

Methods for correlation analyses

There are different methods to perform correlation analysis:

• Pearson correlation (r), which measures a linear dependence between two variables (x and y). It’s also known as a parametric correlation test because it depends to the distribution of the data. It can be used only when x and y are from normal distribution. The plot of y = f(x) is named the linear regression curve.

• Kendall tau and Spearman rho, which are rank-based correlation coefficients (non-parametric)

The most commonly used method is the Pearson correlation method.

Correlation formula

In the formula below,

• x and y are two vectors of length n
• \(m_x\) and \(m_y\) corresponds to the means of x and y, respectively.

Compute correlation in R

cor(x, y, method = c("pearson", "kendall", "spearman"))

cor.test(x, y, method=c("pearson", "kendall", "spearman"))

cor(x, y,  method = "pearson", use = "complete.obs")

# If .txt tab file, use this
my_data <- read.delim(file.choose())

# Or, if .csv file, use this
my_data <- read.csv(file.choose())

my_data <- mtcars
head(my_data, 6)

                   mpg cyl disp  hp drat    wt  qsec vs am gear carb
Mazda RX4         21.0   6  160 110 3.90 2.620 16.46  0  1    4    4
Mazda RX4 Wag     21.0   6  160 110 3.90 2.875 17.02  0  1    4    4
Datsun 710        22.8   4  108  93 3.85 2.320 18.61  1  1    4    1
Hornet 4 Drive    21.4   6  258 110 3.08 3.215 19.44  1  0    3    1
Hornet Sportabout 18.7   8  360 175 3.15 3.440 17.02  0  0    3    2
Valiant           18.1   6  225 105 2.76 3.460 20.22  1  0    3    1

Visualize your data using scatter plots

To use R base graphs, click this link: scatter plot - R base graphs. Here, we’ll use the ggpubr R package.

library("ggpubr")
ggscatter(my_data, x = "mpg", y = "wt", 
          add = "reg.line", conf.int = TRUE, 
          cor.coef = TRUE, cor.method = "pearson",
          xlab = "Miles/(US) gallon", ylab = "Weight (1000 lbs)")

Correlation Test Between Two Variables in R software

Preleminary test to check the test assumptions

1. Is the covariation linear? Yes, form the plot above, the relationship is linear. In the situation where the scatter plots show curved patterns, we are dealing with nonlinear association between the two variables.

2. Are the data from each of the 2 variables (x, y) follow a normal distribution?
   • Use Shapiro-Wilk normality test –> R function: shapiro.test()
   • and look at the normality plot —> R function: ggpubr::ggqqplot()

• Shapiro-Wilk test can be performed as follow:
  • Null hypothesis: the data are normally distributed
  • Alternative hypothesis: the data are not normally distributed

# Shapiro-Wilk normality test for mpg
shapiro.test(my_data$mpg) # => p = 0.1229

# Shapiro-Wilk normality test for wt
shapiro.test(my_data$wt) # => p = 0.09

From the output, the two p-values are greater than the significance level 0.05 implying that the distribution of the data are not significantly different from normal distribution. In other words, we can assume the normality.

• Visual inspection of the data normality using Q-Q plots (quantile-quantile plots). Q-Q plot draws the correlation between a given sample and the normal distribution.

library("ggpubr")
# mpg
ggqqplot(my_data$mpg, ylab = "MPG")
# wt
ggqqplot(my_data$wt, ylab = "WT")

Correlation Test Between Two Variables in R software

From the normality plots, we conclude that both populations may come from normal distributions.

Note that, if the data are not normally distributed, it’s recommended to use the non-parametric correlation, including Spearman and Kendall rank-based correlation tests.

res <- cor.test(my_data$wt, my_data$mpg, 
                    method = "pearson")
res

    Pearson's product-moment correlation

data:  my_data$wt and my_data$mpg
t = -9.559, df = 30, p-value = 1.294e-10
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.9338264 -0.7440872
sample estimates:
       cor 
-0.8676594 

# Extract the p.value
res$p.value

[1] 1.293959e-10

# Extract the correlation coefficient
res$estimate

       cor 
-0.8676594 

res2 <- cor.test(my_data$wt, my_data$mpg,  method="kendall")
res2

    Kendall's rank correlation tau

data:  my_data$wt and my_data$mpg
z = -5.7981, p-value = 6.706e-09
alternative hypothesis: true tau is not equal to 0
sample estimates:
       tau 
-0.7278321 

res2 <-cor.test(my_data$wt, my_data$mpg,  method = "spearman")
res2

    Spearman's rank correlation rho

data:  my_data$wt and my_data$mpg
S = 10292, p-value = 1.488e-11
alternative hypothesis: true rho is not equal to 0
sample estimates:
      rho 
-0.886422 

Interpret correlation coefficient

Correlation coefficient is comprised between -1 and 1:

Correlation Test Between Two Variables in R software

Online correlation coefficient calculator

You can compute correlation test between two variables, online, without any installation by clicking the following link:

Summary

Infos

This analysis has been performed using R software (ver. 3.2.4).

How this chapter is organized?

Correlation Test Between Two Variables in R

Brief outline:

• What is correlation test?
• Methods for correlation analyses
• Correlation formula
  • Pearson correlation formula
  • Spearman correlation formula
  • Kendall correlation formula
• Compute correlation in R
  • R functions
  • Import your data into R
  • Visualize your data using scatter plots
  • Preliminary test to check the test assumptions
  • Pearson correlation test
  • Kendall rank correlation test
  • Spearman rank correlation coefficient
• Interpret correlation coefficient

Read more: —> Correlation Test Between Two Variables in R.

Correlation Matrix: Analyze, Format and Visualize

Correlation matrix is used to analyze the correlation between multiple variables at the same time.

Brief outline:

• What is correlation matrix?
• Compute correlation matrix in R
  • R functions
  • Compute correlation matrix
  • Correlation matrix with significance levels (p-value)
  • A simple function to format the correlation matrix
  • Visualize correlation matrix
    • Use symnum() function: Symbolic number coding
    • Use corrplot() function: Draw a correlogram
    • Use chart.Correlation(): Draw scatter plots
    • Use heatmap()

scatter plot, chart

Read more: —> Correlation Matrix: Analyze, Format and Visualize.

Visualize Correlation Matrix using Correlogram

Correlogram is a graph of correlation matrix. Useful to highlight the most correlated variables in a data table. In this plot, correlation coefficients are colored according to the value. Correlation matrix can be also reordered according to the degree of association between variables.

Brief outline:

• Install R corrplot package
• Data for correlation analysis
• Computing correlation matrix
• Correlogram : Visualizing the correlation matrix
  • Visualization methods
  • Types of correlogram layout
  • Reordering the correlation matrix
  • Changing the color of the correlogram
  • Changing the color and the rotation of text labels
  • Combining correlogram with the significance test
  • Customize the correlogram

library(corrplot)
library(RColorBrewer)
M <-cor(mtcars)
corrplot(M, type="upper", order="hclust",
         col=brewer.pal(n=8, name="RdYlBu"))

Read more: —> Visualize Correlation Matrix using Correlogram.

Elegant Correlation Table using xtable R Package

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.

Brief outline:

• Correlation matrix analysis
• Lower and upper triangular part of a correlation matrix
• Use xtable R package to display nice correlation table in html format
• Combine matrix of correlation coefficients and significance levels

Read more: —> Elegant correlation table using xtable R package.

Correlation Matrix : An R Function to Do All You Need

The goal of this article is to provide you a custom R function, named rquery.cormat(), for calculating and visualizing easily a correlation matrix in a single line R code.

Brief outline:

• Computing the correlation matrix using rquery.cormat()
  • Upper triangle of the correlation matrix
  • Full correlation matrix
  • Change the colors of the correlogram
  • Draw a heatmap
• Format the correlation table
• Description of rquery.cormat() function

source("https://www.sthda.com/upload/rquery_cormat.r")
mydata <- mtcars[, c(1,3,4,5,6,7)]
require("corrplot")
rquery.cormat(mydata)

$r
        hp  disp    wt  qsec  mpg drat
hp       1                            
disp  0.79     1                      
wt    0.66  0.89     1                
qsec -0.71 -0.43 -0.17     1          
mpg  -0.78 -0.85 -0.87  0.42    1     
drat -0.45 -0.71 -0.71 0.091 0.68    1

$p
          hp    disp      wt  qsec     mpg drat
hp         0                                   
disp 7.1e-08       0                           
wt   4.1e-05 1.2e-11       0                   
qsec 5.8e-06   0.013    0.34     0             
mpg  1.8e-07 9.4e-10 1.3e-10 0.017       0     
drat    0.01 5.3e-06 4.8e-06  0.62 1.8e-05    0

$sym
     hp disp wt qsec mpg drat
hp   1                       
disp ,  1                    
wt   ,  +    1               
qsec ,  .       1            
mpg  ,  +    +  .    1       
drat .  ,    ,       ,   1   
attr(,"legend")
[1] 0 ' ' 0.3 '.' 0.6 ',' 0.8 '+' 0.9 '*' 0.95 'B' 1

Read more: —> Correlation Matrix : An R Function to Do All You Need.

See also

• R Basics
• Import and Export Data using R
• Preparing and Reshaping Data in R for Easier Analyses
• Data Manipulation in R
• Data visualization
• Descriptive Statistics and Graphics

Infos

This analysis has been performed using R statistical software (ver. 3.2.4).

What is correlation matrix?

There are different methods for correlation analysis : Pearson parametric correlation test, Spearman and Kendall rank-based correlation analysis. These methods are discussed in the next sections.

The aim of this R tutorial is to show you how to compute and visualize a correlation matrix in R. We provide also an online software for computing and visualizing a correlation matrix.

Compute correlation matrix in R

cor(x, method = c("pearson", "kendall", "spearman"))

cor(x, method = "pearson", use = "complete.obs")

# If .txt tab file, use this
my_data <- read.delim(file.choose())

# Or, if .csv file, use this
my_data <- read.csv(file.choose())

# Load data
data("mtcars")
my_data <- mtcars[, c(1,3,4,5,6,7)]
# print the first 6 rows
head(my_data, 6)

                   mpg disp  hp drat    wt  qsec
Mazda RX4         21.0  160 110 3.90 2.620 16.46
Mazda RX4 Wag     21.0  160 110 3.90 2.875 17.02
Datsun 710        22.8  108  93 3.85 2.320 18.61
Hornet 4 Drive    21.4  258 110 3.08 3.215 19.44
Hornet Sportabout 18.7  360 175 3.15 3.440 17.02
Valiant           18.1  225 105 2.76 3.460 20.22

res <- cor(my_data)
round(res, 2)

       mpg  disp    hp  drat    wt  qsec
mpg   1.00 -0.85 -0.78  0.68 -0.87  0.42
disp -0.85  1.00  0.79 -0.71  0.89 -0.43
hp   -0.78  0.79  1.00 -0.45  0.66 -0.71
drat  0.68 -0.71 -0.45  1.00 -0.71  0.09
wt   -0.87  0.89  0.66 -0.71  1.00 -0.17
qsec  0.42 -0.43 -0.71  0.09 -0.17  1.00

cor(my_data, use = "complete.obs")

rcorr(x, type = c("pearson","spearman"))

install.packages("Hmisc")

library("Hmisc")
res2 <- rcorr(as.matrix(my_data))
res2

       mpg  disp    hp  drat    wt  qsec
mpg   1.00 -0.85 -0.78  0.68 -0.87  0.42
disp -0.85  1.00  0.79 -0.71  0.89 -0.43
hp   -0.78  0.79  1.00 -0.45  0.66 -0.71
drat  0.68 -0.71 -0.45  1.00 -0.71  0.09
wt   -0.87  0.89  0.66 -0.71  1.00 -0.17
qsec  0.42 -0.43 -0.71  0.09 -0.17  1.00

n= 32 


P
     mpg    disp   hp     drat   wt     qsec  
mpg         0.0000 0.0000 0.0000 0.0000 0.0171
disp 0.0000        0.0000 0.0000 0.0000 0.0131
hp   0.0000 0.0000        0.0100 0.0000 0.0000
drat 0.0000 0.0000 0.0100        0.0000 0.6196
wt   0.0000 0.0000 0.0000 0.0000        0.3389
qsec 0.0171 0.0131 0.0000 0.6196 0.3389       

# Extract the correlation coefficients
res2$r

# Extract p-values
res2$P

# ++++++++++++++++++++++++++++
# flattenCorrMatrix
# ++++++++++++++++++++++++++++
# cormat : matrix of the correlation coefficients
# pmat : matrix of the correlation p-values
flattenCorrMatrix <- function(cormat, pmat) {
  ut <- upper.tri(cormat)
  data.frame(
    row = rownames(cormat)[row(cormat)[ut]],
    column = rownames(cormat)[col(cormat)[ut]],
    cor  =(cormat)[ut],
    p = pmat[ut]
    )
}

library(Hmisc)
res2<-rcorr(as.matrix(mtcars[,1:7]))
flattenCorrMatrix(res2$r, res2$P)

    row column         cor            p
1   mpg    cyl -0.85216194 6.112697e-10
2   mpg   disp -0.84755135 9.380354e-10
3   cyl   disp  0.90203285 1.803002e-12
4   mpg     hp -0.77616835 1.787838e-07
5   cyl     hp  0.83244747 3.477856e-09
6  disp     hp  0.79094857 7.142686e-08
7   mpg   drat  0.68117189 1.776241e-05
8   cyl   drat -0.69993812 8.244635e-06
9  disp   drat -0.71021390 5.282028e-06
10   hp   drat -0.44875914 9.988768e-03
11  mpg     wt -0.86765939 1.293956e-10
12  cyl     wt  0.78249580 1.217567e-07
13 disp     wt  0.88797992 1.222311e-11
14   hp     wt  0.65874785 4.145833e-05
15 drat     wt -0.71244061 4.784268e-06
16  mpg   qsec  0.41868404 1.708199e-02
17  cyl   qsec -0.59124213 3.660527e-04
18 disp   qsec -0.43369791 1.314403e-02
19   hp   qsec -0.70822340 5.766250e-06
20 drat   qsec  0.09120482 6.195823e-01
21   wt   qsec -0.17471591 3.388682e-01

Visualize correlation matrix

There are different ways for visualizing a correlation matrix in R software :

• symnum() function
• corrplot() function to plot a correlogram
• scatter plots
• heatmap

Use symnum() function: Symbolic number coding

The R function symnum() replaces correlation coefficients by symbols according to the level of the correlation. It takes the correlation matrix as an argument :

• Simplified format:

symnum(x, cutpoints = c(0.3, 0.6, 0.8, 0.9, 0.95),
       symbols = c(" ", ".", ",", "+", "*", "B"),
       abbr.colnames = TRUE)

• x: the correlation matrix to visualize
• cutpoints: correlation coefficient cutpoints. The correlation coefficients between 0 and 0.3 are replaced by a space (" “); correlation coefficients between 0.3 and 0.6 are replace by”.“; etc …
• symbols : the symbols to use.
• abbr.colnames: logical value. If TRUE, colnames are abbreviated.

• Example of usage:

symnum(res, abbr.colnames = FALSE)

     mpg disp hp drat wt qsec
mpg  1                       
disp +   1                   
hp   ,   ,    1              
drat ,   ,    .  1           
wt   +   +    ,  ,    1      
qsec .   .    ,          1   
attr(,"legend")
[1] 0 ' ' 0.3 '.' 0.6 ',' 0.8 '+' 0.9 '*' 0.95 'B' 1

As indicated in the legend, the correlation coefficients between 0 and 0.3 are replaced by a space (" “); correlation coefficients between 0.3 and 0.6 are replace by”.“; etc …

Use corrplot() function: Draw a correlogram

The function corrplot(), in the package of the same name, creates a graphical display of a correlation matrix, highlighting the most correlated variables in a data table.

In this plot, correlation coefficients are colored according to the value. Correlation matrix can be also reordered according to the degree of association between variables.

• Install corrplot:

install.packages("corrplot")

• Use corrplot() to create a correlogram:

The function corrplot() takes the correlation matrix as the first argument. The second argument (type=“upper”) is used to display only the upper triangular of the correlation matrix.

library(corrplot)
corrplot(res, type = "upper", order = "hclust", 
         tl.col = "black", tl.srt = 45)

Correlation matrix - R software and statistics

Positive correlations are displayed in blue and negative correlations in red color. Color intensity and the size of the circle are proportional to the correlation coefficients. In the right side of the correlogram, the legend color shows the correlation coefficients and the corresponding colors.

• The correlation matrix is reordered according to the correlation coefficient using “hclust” method.
• tl.col (for text label color) and tl.srt (for text label string rotation) are used to change text colors and rotations.
• Possible values for the argument type are : “upper”, “lower”, “full”

Read more : visualize a correlation matrix using corrplot.

It’s also possible to combine correlogram with the significance test. We’ll use the result res.cor2 generated in the previous section with rcorr() function [in Hmisc package]:

# Insignificant correlation are crossed
corrplot(res2$r, type="upper", order="hclust", 
         p.mat = res2$P, sig.level = 0.01, insig = "blank")
# Insignificant correlations are leaved blank
corrplot(res2$r, type="upper", order="hclust", 
         p.mat = res2$P, sig.level = 0.01, insig = "blank")

Correlation matrix - R software and statistics

In the above plot, correlations with p-value > 0.01 are considered as insignificant. In this case the correlation coefficient values are leaved blank or crosses are added.

Use chart.Correlation(): Draw scatter plots

The function chart.Correlation()[ in the package PerformanceAnalytics], can be used to display a chart of a correlation matrix.

• Install PerformanceAnalytics:

install.packages("PerformanceAnalytics")

• Use chart.Correlation():

library("PerformanceAnalytics")
my_data <- mtcars[, c(1,3,4,5,6,7)]
chart.Correlation(my_data, histogram=TRUE, pch=19)

scatter plot, chart

In the above plot:

• The distribution of each variable is shown on the diagonal.
• On the bottom of the diagonal : the bivariate scatter plots with a fitted line are displayed
• On the top of the diagonal : the value of the correlation plus the significance level as stars
• Each significance level is associated to a symbol : p-values(0, 0.001, 0.01, 0.05, 0.1, 1) <=> symbols(“***”, “**”, “*”, “.”, " “)

Use heatmap()

# Get some colors
col<- colorRampPalette(c("blue", "white", "red"))(20)
heatmap(x = res, col = col, symm = TRUE)

Heatmap of correlation matrix

• x : the correlation matrix to be plotted
• col : color palettes
• symm : logical indicating if x should be treated symmetrically; can only be true when x is a square matrix.

Online software to analyze and visualize a correlation matrix

(Designed by Freepik)

Take me to the correlation matrix calculator

The software can be used as follow :

1. Go to the web application : correlation matrix calculator
2. Upload a .txt tab or a CSV file containing your data (columns are variables). The supported file formats are described here. You can use the demo data available on the calculator web page by clicking on the corresponding link.
3. After uploading, an overview of a part of your file is shown to check that the data are correctly imported. If the data are not correctly displayed, please make sure that the format of your file is OK here.
4. Click on the ‘Analyze’ button and select at least 2 variables to calculate the correlation matrix. By default, all variables are selected. Please, deselect the columns containing texts. You can also select the correlation methods (Pearson, Spearman or Kendall). Default is the Pearson method.
5. Click the OK button
6. Results : the output of the software includes :
   • The correlation matrix
   • The visualization of the correlation matrix as a correlogram
   • A web link to export the results as .txt tab file

Summarry

Infos

This analysis has been performed using R software (ver. 3.2.4).

The different types of correlation methods

There are :

• the Pearson correlation method which is a parametric correlation test as it depends on the distribution of the data. This method measures the linear dependence between two variables.
• the Kendall and Spearman rank-based correlation analysis (non-parametric methods). These two methods are recommended if the data do not come from a bivariate normal distribution.

Pearson correlation test is the most commonly used method to calculate the correlation coefficient between two variables. The formula is shown in the next section.

(Designed by Freepik)

Pearson correlation coefficient formula

The Pearson correlation coefficient between two variables, x and y, can be calculated using the formula below :

\[ r = \frac{\sum{(x-m_x)(y-m_y)}}{\sqrt{\sum{(x-mx)^2}\sum{(y-my)^2}}} \]

\(m_x\) and \(m_y\) are the means of x and y variables.

The significance level of the correlation can be determined by reading the table of critical values for the degrees of freedom : \(df = n-2\)

Read more : correlation formula

Calculate the correlation coefficient using R software

Correlation coefficient can be easily computed in R using the function cor() or cor.test(). The simplified formats are :

cor(x, y, method = c("pearson", "kendall", "spearman"))

cor.test(x, y, method=c("pearson", "kendall", "spearman"))

In the R code above x and y are two numeric vectors with the same length.

These functions can be used as follow :

# Define two numeric vectors
x <- c(44.4, 45.9, 41.9, 53.3, 44.7, 44.1, 50.7, 45.2, 60.1)
y <- c( 2.6,  3.1,  2.5,  5.0,  3.6,  4.0,  5.2,  2.8,  3.8)

# Pearson correlation coefficient between x and y
cor(x, y)

[1] 0.5712

# Pearson correlation test 
cor.test(x, y)

    Pearson's product-moment correlation

data:  x and y
t = 1.841, df = 7, p-value = 0.1082
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.1497  0.8956
sample estimates:
   cor 
0.5712 

The correlation coefficient is 0.5712 and the p-value is 0.1082.

Read more : correlation analysis using R software

Spearman and Kendall non-parametric correlation methods can be used as follow :

# Spearman non-parametric correlation test
cor.test(x, y, method="spearman")

# Kendall non-parametric correlation test
cor.test(x, y, method="kendall")

Interpretation of the correlation coefficient

The value of the correlation coefficient is comprised between -1 and 1

• -1 corresponds to a strong negative correlation : this means that every time x increases, y decreases (left panel figure)
• 0 means that there is no association between the two variables (x and y) (middle panel figure)
• 1 corresponds to a strong positive correlation : this means that y increases with x (right panel figure)

Online correlation coefficient calculator

A web application, for computing the different correlation coefficients, is available at this link : correlation coefficient calculator.

It can be used online without any installation to calculate Pearson, Kendall or Spearman correlation coefficient.

(Designed by Freepik)

Go to the correlation coefficient calculator

The correlation coefficient calculator can be used as follow :

1. Go to the application available at this link : correlation coefficient calculator
2. Copy and paste your data from Excel to the calculator. You can use the demo data available in the calculator web page by clicking on the link.
3. Select the correlation methods (Pearson, Spearman or Kendall). Default is the Pearson method.
4. Click the OK button

correlation coefficient definition

Correlation coefficient is a quantity that measures the strength of the association (or dependence) between two variables (x and y). For example if we are interested to know whether there is a relationship between the heights of fathers and son, a correlation coefficient can be calculated.

There are different correlation coefficients :

• Pearson r : linear dependence
• Kendall tau : rank-based correlation coefficient
• Spearman rho : rank-base correlation coefficient

Pearson correlation is the most used one.

correlation formula is described here.

Negative and positive correlation

The value of correlation coefficient can be negative or positive and it is comprised between -1 and 1 (see the plots below).

• -1 means strong negative correlation : In this case y decreases when x increases (left panel figure)
• 0 means that there is no relationship between the two variables (x and y) (middle panel figure)
• 1 means strong positive correlation : In this case y increases when y increases (right panel figure)

Correlation coefficient calculator

Define correlation

Correlation is very helpful to investigate the dependence between two or more variables. As an example we are interested to know whether there is an association between the weights of fathers and son. correlation coefficient can be calculated to answer this question.

If there is no relationship between the two variables (father and son weights), the average weight of son should be the same regardless of the weight of the fathers and vice versa.

There are different methods to perform correlation analysis : Pearson, Kendall and Spearman correlation tests.

The most commonly used is Pearson correlation. The aim of this article is to describe Pearson correlation formula.

Pearson correlation

Pearson correlation measures a linear dependence between two variables (x and y). It’s also known as a parametric correlation test because it depends to the distribution of the data. The plot of y = f(x) is named linear regression curve.

The pearson correlation formula is :

\[ r = \frac{\sum{(x-m_x)(y-m_y)}}{\sqrt{\sum{(x-mx)^2}\sum{(y-my)^2}}} \]

\(m_x\) and \(m_y\) are the means of x and y variables.

the p-value (significance level) of the correlation can be determined :

1. by using the correlation coefficient table for the degrees of freedom : \(df = n-2\)

2. or by calculating the t value : \[ t=\frac{r}{\sqrt{1-r^2}}\sqrt{n-2} \]

In this case the corresponding p-value is determined using t distribution table for \(df = n-2\)

If the p-value is less than 5%, then the correlation is significant.

Interpretation of correlation coefficient

Correlation coefficient is between -1 (strong negative correlation) and 1 (strong positive correlation)

Correlation coefficient calculator