The following content is mostly compiled (with some original additions on my side) from the material that can be found at http://www.sthda.com/, as well as in the vignette for the corrplot R package - An Introduction to corrplot Package. The sole purpose of this text is to put all the info into one document in an easy to search format. Since I’m a huge fan of Hadley Wickham’s work I’ll insist on solutions based in “tidyverse” whenewer possible…

### Install and load required R packages

We’ll use the ggpubr R package for an easy ggplot2-based data visualization, corrplot package to plot correlograms, Hmisc to calculate correlation matrices containing both cor. coefs. and p-values,corrplot for plotting correlograms, and of course tidyverse for all the data wrangling, plotting and alike:

require(ggpubr)
require(tidyverse)
require(Hmisc)
require(corrplot)

## 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

## Compute correlation in R

### R functions

Correlation coefficients can be computed in R by using the functions cor() and cor.test():

• cor() computes the correlation coefficient
• cor.test() test for association/correlation between paired samples. It returns both the correlation coefficient and the significance level(or p-value) of the correlation.

The simplified formats are:

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

where:

• x, y: numeric vectors with the same length
• method: correlation method

If the data contain missing values, the following R code can be used to handle missing values by case-wise deletion:

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

## Preliminary considerations

We’ll use the well known built-in mtcars R dataset.

head(mtcars)
##                    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

We’d like to compute the correlation between mpg and wt variables.

First let’s visualise our data by the means of a scatter plot. We’ll be using ggpubr R package

library(ggpubr)

my_data <- mtcars
my_data$cyl <- factor(my_data$cyl)
str(my_data)
## 'data.frame':    32 obs. of  11 variables:
##  $mpg : num 21 21 22.8 21.4 18.7 18.1 14.3 24.4 22.8 19.2 ... ##$ cyl : Factor w/ 3 levels "4","6","8": 2 2 1 2 3 2 3 1 1 2 ...
##  $disp: num 160 160 108 258 360 ... ##$ hp  : num  110 110 93 110 175 105 245 62 95 123 ...
##  $drat: num 3.9 3.9 3.85 3.08 3.15 2.76 3.21 3.69 3.92 3.92 ... ##$ wt  : num  2.62 2.88 2.32 3.21 3.44 ...
##  $qsec: num 16.5 17 18.6 19.4 17 ... ##$ vs  : num  0 0 1 1 0 1 0 1 1 1 ...
##  $am : num 1 1 1 0 0 0 0 0 0 0 ... ##$ gear: num  4 4 4 3 3 3 3 4 4 4 ...
##  $carb: num 4 4 1 1 2 1 4 2 2 4 ... ggscatter(my_data, x = "wt", y = "mpg", add = "reg.line", conf.int = TRUE, cor.coef = TRUE, cor.method = "pearson", xlab = "Weight (1000 lbs)", ylab = "Miles/ (US) gallon") ### Preleminary test to check the test assumptions 1. Is the relation between variables linear? Yes, from the plot above, the relationship can be, closely enough, modeled as 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) following a normal distribution? • Use Shapiro-Wilk normality test $$\rightarrow$$ R function: shapiro.test() • and look at the normality plot $$\rightarrow$$ 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$mpg) # => p = 0.1229
##
##  Shapiro-Wilk normality test
##
## data:  my_data$mpg ## W = 0.94756, p-value = 0.1229 # Shapiro-Wilk normality test for wt shapiro.test(my_data$wt) # => p = 0.09
##
##  Shapiro-Wilk normality test
##
## data:  my_data$wt ## W = 0.94326, p-value = 0.09265 As can be seen from the output, the two p-values are greater than the predetermined significance level of 0.05 implying that the distribution of the data are not significantly different from normal distribution. In other words, we can assume the normality. • One more option for checking the normality of the data distribution is visual inspection of the Q-Q plots (quantile-quantile plots). Q-Q plot draws the correlation between a given sample and the theoretical normal distribution. Again, we’ll use the ggpubr R package to obtain “pretty”, i.e. publishing-ready, Q-Q plots. library("ggpubr") # Check for the normality of "mpg"" ggqqplot(my_data$mpg, ylab = "MPG")

# Check for the normality of "wt""
ggqqplot(my_data$wt, ylab = "WT") From the Q-Q normality plots, we can assume that both samples may come from populations that, closely enough, follow normal distributions. It is important to note that if the data does not follow the normal distribution, at least closely enough, it’s recommended to use the non-parametric correlation, including Spearman and Kendall rank-based correlation tests. ## Pearson correlation test Example: res <- cor.test(my_data$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 So what’s happening here? First of all let’s clarify the meaning of this printout: • t is the t-test statistic value (t = -9.559), • df is the degrees of freedom (df= 30), • p-value is the significance level of the t-test (p-value = $$1.29410^{-10}$$). • conf.int is the confidence interval of the correlation coefficient at 95% (conf.int = [-0.9338, -0.7441]); • sample estimates is the correlation coefficient (Cor.coeff = -0.87). Interpretation of the results: As can be see from the results above the p-value of the test is $$1.29410^{-10}$$, which is less than the significance level $$\alpha = 0.05$$. We can conclude that wt and mpg are significantly correlated with a correlation coefficient of -0.87 and p-value of $$1.29410^{-10}$$. Access to the values returned by cor.test() function The function cor.test() returns a list containing the following components: str(res) ## List of 9 ##$ statistic  : Named num -9.56
##   ..- attr(*, "names")= chr "t"
##  $parameter : Named int 30 ## ..- attr(*, "names")= chr "df" ##$ p.value    : num 1.29e-10
##  $estimate : Named num -0.868 ## ..- attr(*, "names")= chr "cor" ##$ null.value : Named num 0
##   ..- attr(*, "names")= chr "correlation"
##  $alternative: chr "two.sided" ##$ method     : chr "Pearson's product-moment correlation"
##  $data.name : chr "my_data$wt and my_data$mpg" ##$ conf.int   : atomic [1:2] -0.934 -0.744
##   ..- attr(*, "conf.level")= num 0.95
##  - attr(*, "class")= chr "htest"

Of these we are most interested with:

• p.value: the p-value of the test
• estimate: the correlation coefficient
# Extract the p.value
res$p.value ## [1] 1.293959e-10 # Extract the correlation coefficient res$estimate
##        cor
## -0.8676594

## Kendall rank correlation test

The Kendall rank correlation coefficient or Kendall’s $$\tau$$ statistic is used to estimate a rank-based measure of association. This test may be used if the data do not necessarily come from a bivariate normal distribution.

res2 <- cor.test(my_data$mpg, my_data$wt, method = "kendall")
## Warning in cor.test.default(my_data$mpg, my_data$wt, method = "kendall"):
## Cannot compute exact p-value with ties
res2
##
##  Kendall's rank correlation tau
##
## data:  my_data$mpg and my_data$wt
## z = -5.7981, p-value = 6.706e-09
## alternative hypothesis: true tau is not equal to 0
## sample estimates:
##        tau
## -0.7278321

Here tau is the Kendall correlation coefficient, so The correlation coefficient between mpg and wy is -0.7278 and the p-value is $$6.70610^{-9}$$.

## Spearman rank correlation coefficient

Spearman’s $$\rho$$ statistic is also used to estimate a rank-based measure of association. This test may be used if the data do not come from a bivariate normal distribution.

res3 <- cor.test(my_data$wt, my_data$mpg, method = "spearman")
## Warning in cor.test.default(my_data$wt, my_data$mpg, method = "spearman"):
## Cannot compute exact p-value with ties
res3
##
##  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

Here, rho is the Spearman’s correlation coefficient, so the correlation coefficient between mpg and wt is -0.8864 and the p-value is $$1.48810^{-11}$$.

## How to interpret correlation coefficient

Value of the correlation coefficient can vary between -1 and 1:

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

## What is a correlation matrix?

Previously, we described how to perform correlation test between two variables. In the following sections we’ll see how a correlation matrix can be computed and visualized. The correlation matrix is used to investigate the dependence between multiple variables at the same time. The result is a table containing the correlation coefficients between each variable and the others.

## Compute correlation matrix in R

We have already mentioned the cor() function, at the intoductory part of this document dealing with the correlation test for a bivariate case. It be used to compute a correlation matrix. A simplified format of the function is :

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

Here:

• x is numeric matrix or a data frame.
• method: indicates the correlation coefficient to be computed. The default is “pearson”" correlation coefficient which measures the linear dependence between two variables. As already explained “kendall” and “spearman” correlation methods are non-parametric rank-based correlation tests.

If your data contain missing values, the following R code can be used to handle missing values by case-wise deletion:

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

### Plain correlation matrix

Example:

library(dplyr)

my_data <- select(mtcars, mpg, disp, hp, drat, wt, qsec)
head(my_data)
##                    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
#Let's compute the correlation matrix
cor_1 <- round(cor(my_data), 2)
cor_1
##        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

Unfortunately, the function cor() returns only the correlation coefficients between variables. In the next section, we will use Hmisc R package to calculate the correlation p-values.

### Correlation matrix with significance levels (p-value)

The function rcorr() (in Hmisc package) can be used to compute the significance levels for pearson and spearman correlations. It returns both the correlation coefficients and the p-value of the correlation for all possible pairs of columns in the data table.

Simplified format:

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

x should be a matrix. The correlation type can be either pearson or spearman.

Example:

library("Hmisc")

cor_2 <- rcorr(as.matrix(my_data))
cor_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
##
## 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

The output of the function rcorr() is a list containing the following elements :

• r : the correlation matrix
• n : the matrix of the number of observations used in analyzing each pair of variables
• P : the p-values corresponding to the significance levels of correlations.

Extracting the p-values or the correlation coefficients from the output:

str(cor_2)
## List of 3
##  $r: num [1:6, 1:6] 1 -0.848 -0.776 0.681 -0.868 ... ## ..- attr(*, "dimnames")=List of 2 ## .. ..$ : chr [1:6] "mpg" "disp" "hp" "drat" ...
##   .. ..$: chr [1:6] "mpg" "disp" "hp" "drat" ... ##$ n: int [1:6, 1:6] 32 32 32 32 32 32 32 32 32 32 ...
##   ..- attr(*, "dimnames")=List of 2
##   .. ..$: chr [1:6] "mpg" "disp" "hp" "drat" ... ## .. ..$ : chr [1:6] "mpg" "disp" "hp" "drat" ...
##  $P: num [1:6, 1:6] NA 9.38e-10 1.79e-07 1.78e-05 1.29e-10 ... ## ..- attr(*, "dimnames")=List of 2 ## .. ..$ : chr [1:6] "mpg" "disp" "hp" "drat" ...
##   .. ..$: chr [1:6] "mpg" "disp" "hp" "drat" ... ## - attr(*, "class")= chr "rcorr" # As you can see "cor_2" is a list so extracting these values is quite simple... # p-values cor_2$P
##               mpg         disp           hp         drat           wt
## mpg            NA 9.380354e-10 1.787838e-07 1.776241e-05 1.293956e-10
## disp 9.380354e-10           NA 7.142686e-08 5.282028e-06 1.222311e-11
## hp   1.787838e-07 7.142686e-08           NA 9.988768e-03 4.145833e-05
## drat 1.776241e-05 5.282028e-06 9.988768e-03           NA 4.784268e-06
## wt   1.293956e-10 1.222311e-11 4.145833e-05 4.784268e-06           NA
## qsec 1.708199e-02 1.314403e-02 5.766250e-06 6.195823e-01 3.388682e-01
##              qsec
## mpg  1.708199e-02
## disp 1.314403e-02
## hp   5.766250e-06
## drat 6.195823e-01
## wt   3.388682e-01
## qsec           NA
# Correlation matrix
cor_2$r ## mpg disp hp drat wt qsec ## mpg 1.0000000 -0.8475513 -0.7761683 0.68117189 -0.8676594 0.41868404 ## disp -0.8475513 1.0000000 0.7909486 -0.71021390 0.8879799 -0.43369791 ## hp -0.7761683 0.7909486 1.0000000 -0.44875914 0.6587479 -0.70822340 ## drat 0.6811719 -0.7102139 -0.4487591 1.00000000 -0.7124406 0.09120482 ## wt -0.8676594 0.8879799 0.6587479 -0.71244061 1.0000000 -0.17471591 ## qsec 0.4186840 -0.4336979 -0.7082234 0.09120482 -0.1747159 1.00000000 ## Custom function for convinient formatting of the correlation matrix This section provides a simple function for formatting a correlation matrix into a table with 4 columns containing : • Column 1 : row names (variable 1 for the correlation test) • Column 2 : column names (variable 2 for the correlation test) • Column 3 : the correlation coefficients • Column 4 : the p-values of the correlations flat_cor_mat <- function(cor_r, cor_p){ #This function provides a simple formatting of a correlation matrix #into a table with 4 columns containing : # Column 1 : row names (variable 1 for the correlation test) # Column 2 : column names (variable 2 for the correlation test) # Column 3 : the correlation coefficients # Column 4 : the p-values of the correlations library(tidyr) library(tibble) cor_r <- rownames_to_column(as.data.frame(cor_r), var = "row") cor_r <- gather(cor_r, column, cor, -1) cor_p <- rownames_to_column(as.data.frame(cor_p), var = "row") cor_p <- gather(cor_p, column, p, -1) cor_p_matrix <- left_join(cor_r, cor_p, by = c("row", "column")) cor_p_matrix } cor_3 <- rcorr(as.matrix(mtcars[, 1:7])) my_cor_matrix <- flat_cor_mat(cor_3$r, cor_3\$P)
head(my_cor_matrix)
##    row column        cor            p
## 1  mpg    mpg  1.0000000           NA
## 2  cyl    mpg -0.8521619 6.112697e-10
## 3 disp    mpg -0.8475513 9.380354e-10
## 4   hp    mpg -0.7761683 1.787838e-07
## 5 drat    mpg  0.6811719 1.776241e-05
## 6   wt    mpg -0.8676594 1.293956e-10

## Visualization of a correlation matrix

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

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

We’ll run trough all of these, and then go a bit more into deatil with correlograms.

### Use symnum() function: Symbolic number coding

The R function symnum() is used to symbolically encode a given numeric or logical vector or array. It is particularly useful for visualization of structured matrices, e.g., correlation, sparse, or logical ones. In the case of a correlatino matrix it replaces correlation coefficients by symbols according to the level of the correlation.

Simplified format:

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

Here:

• 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 replaced by”.“; etc .
• symbols: the symbols to use.
• abbr.colnames: logical value. If TRUE, colnames are abbreviated.

Example:

cor_4 <- cor(mtcars[1:6])
symnum(cor_4, abbr.colnames = FALSE)
##      mpg cyl disp hp drat wt
## mpg  1
## cyl  +   1
## disp +   *   1
## hp   ,   +   ,    1
## drat ,   ,   ,    .  1
## wt   +   ,   +    ,  ,    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 the 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.

The simplified format of the function is:

corrplot(corr, method="circle")

Here:

• corr: the correlation matrix to be visualized
• method: The visualization method to be used, there are seven different options: “circle”, “square”, “ellipse”, “number”, “shade”, “color”, “pie”.

Example:

M<-cor(mtcars)
head(round(M,2))
##        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
#Visualize the correlation matrix

# method = "circle""
corrplot(M, method = "circle")

# method = "ellipse""
corrplot(M, method = "ellipse")

# method = "pie"
corrplot(M, method = "pie")

# method = "color"
corrplot(M, method = "color")

Display the correlation coefficient:

corrplot(M, method = "number")