• Introduction
• Data
• Correlation coefficient
  • Between two variables
  • Correlation matrix: correlations for all variables
  • Interpretation of a correlation coefficient
• Visualizations
  • A scatterplot for 2 variables
  • Scatterplots for several pairs of variables
  • Another simple correlation matrix
• Correlation test
  • For 2 variables
  • For several pairs of variables
• Combination of correlation coefficients and correlation tests

Introduction

Correlations between variables play an important role in a descriptive analysis. A correlation measures the relationship between two variables, that is, how they are linked to each other. In this sense, a correlation allows to know which variables evolve in the same direction, which ones evolve in the opposite direction, and which ones are independent.

In this article, I show how to compute correlation coefficients, how to perform correlation tests and how to visualize relationships between variables in R.

Correlation is usually computed on two quantitative variables. See the Chi-square test of independence if you need to study the relationship between two qualitative variables.

Data

In this article, we use the mtcars dataset (loaded by default in R):

# display first 5 observations
head(mtcars, 5)
##                    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

The variables vs and am are categorical variables, so they are removed for this article:

# remove vs and am variables
library(tidyverse)
dat <- mtcars %>%
  select(-vs, -am)

# display 5 first obs. of new dataset
head(dat, 5)
##                    mpg cyl disp  hp drat    wt  qsec gear carb
## Mazda RX4         21.0   6  160 110 3.90 2.620 16.46    4    4
## Mazda RX4 Wag     21.0   6  160 110 3.90 2.875 17.02    4    4
## Datsun 710        22.8   4  108  93 3.85 2.320 18.61    4    1
## Hornet 4 Drive    21.4   6  258 110 3.08 3.215 19.44    3    1
## Hornet Sportabout 18.7   8  360 175 3.15 3.440 17.02    3    2

Correlation coefficient

# Pearson correlation between 2 variables
cor(dat$hp, dat$mpg)
## [1] -0.7761684

# Spearman correlation between 2 variables
cor(dat$hp, dat$mpg,
  method = "spearman"
)
## [1] -0.8946646

# correlation for all variables
round(cor(dat),
  digits = 2 # rounded to 2 decimals
)
##        mpg   cyl  disp    hp  drat    wt  qsec  gear  carb
## mpg   1.00 -0.85 -0.85 -0.78  0.68 -0.87  0.42  0.48 -0.55
## cyl  -0.85  1.00  0.90  0.83 -0.70  0.78 -0.59 -0.49  0.53
## disp -0.85  0.90  1.00  0.79 -0.71  0.89 -0.43 -0.56  0.39
## hp   -0.78  0.83  0.79  1.00 -0.45  0.66 -0.71 -0.13  0.75
## drat  0.68 -0.70 -0.71 -0.45  1.00 -0.71  0.09  0.70 -0.09
## wt   -0.87  0.78  0.89  0.66 -0.71  1.00 -0.17 -0.58  0.43
## qsec  0.42 -0.59 -0.43 -0.71  0.09 -0.17  1.00 -0.21 -0.66
## gear  0.48 -0.49 -0.56 -0.13  0.70 -0.58 -0.21  1.00  0.27
## carb -0.55  0.53  0.39  0.75 -0.09  0.43 -0.66  0.27  1.00

Visualizations

# scatterplot
library(ggplot2)

ggplot(dat) +
  aes(x = hp, y = mpg) +
  geom_point(colour = "#0c4c8a") +
  theme_minimal()

plot(dat$hp, dat$mpg)

# multiple scatterplots
pairs(dat[, c(1, 4, 6)])

# improved correlation matrix
library(corrplot)

corrplot(cor(dat),
  method = "number",
  type = "upper" # show only upper side
)

Correlation test

# Pearson correlation test
test <- cor.test(dat$drat, dat$qsec)
test
## 
##  Pearson's product-moment correlation
## 
## data:  dat$drat and dat$qsec
## t = 0.50164, df = 30, p-value = 0.6196
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -0.265947  0.426340
## sample estimates:
##        cor 
## 0.09120476

# correlation tests for whole dataset
library(Hmisc)
res <- rcorr(as.matrix(dat)) # rcorr() accepts matrices only

# display p-values (rounded to 3 decimals)
round(res$P, 3)
##        mpg   cyl  disp    hp  drat    wt  qsec  gear  carb
## mpg     NA 0.000 0.000 0.000 0.000 0.000 0.017 0.005 0.001
## cyl  0.000    NA 0.000 0.000 0.000 0.000 0.000 0.004 0.002
## disp 0.000 0.000    NA 0.000 0.000 0.000 0.013 0.001 0.025
## hp   0.000 0.000 0.000    NA 0.010 0.000 0.000 0.493 0.000
## drat 0.000 0.000 0.000 0.010    NA 0.000 0.620 0.000 0.621
## wt   0.000 0.000 0.000 0.000 0.000    NA 0.339 0.000 0.015
## qsec 0.017 0.000 0.013 0.000 0.620 0.339    NA 0.243 0.000
## gear 0.005 0.004 0.001 0.493 0.000 0.000 0.243    NA 0.129
## carb 0.001 0.002 0.025 0.000 0.621 0.015 0.000 0.129    NA

Combination of correlation coefficients and correlation tests

Now that we covered the concepts of correlation coefficients and correlation tests, let see if we can combine the two concepts.

The correlation function from the easystats {correlation} package allows to combine correlation coefficients and correlation tests in a single table (thanks to krzysiektr for pointing it out to me):

library(correlation)

correlation::correlation(dat,
  include_factors = TRUE, method = "auto"
)
## Parameter1 | Parameter2 |     r |         95% CI |     t | df |      p |  Method | n_Obs
## ----------------------------------------------------------------------------------------
## mpg        |        cyl | -0.85 | [-0.93, -0.72] | -8.92 | 30 | < .001 | Pearson |    32
## mpg        |       disp | -0.85 | [-0.92, -0.71] | -8.75 | 30 | < .001 | Pearson |    32
## mpg        |         hp | -0.78 | [-0.89, -0.59] | -6.74 | 30 | < .001 | Pearson |    32
## mpg        |       drat |  0.68 | [ 0.44,  0.83] |  5.10 | 30 | < .001 | Pearson |    32
## mpg        |         wt | -0.87 | [-0.93, -0.74] | -9.56 | 30 | < .001 | Pearson |    32
## mpg        |       qsec |  0.42 | [ 0.08,  0.67] |  2.53 | 30 | 0.137  | Pearson |    32
## mpg        |       gear |  0.48 | [ 0.16,  0.71] |  3.00 | 30 | 0.065  | Pearson |    32
## mpg        |       carb | -0.55 | [-0.75, -0.25] | -3.62 | 30 | 0.016  | Pearson |    32
## cyl        |       disp |  0.90 | [ 0.81,  0.95] | 11.45 | 30 | < .001 | Pearson |    32
## cyl        |         hp |  0.83 | [ 0.68,  0.92] |  8.23 | 30 | < .001 | Pearson |    32
## cyl        |       drat | -0.70 | [-0.84, -0.46] | -5.37 | 30 | < .001 | Pearson |    32
## cyl        |         wt |  0.78 | [ 0.60,  0.89] |  6.88 | 30 | < .001 | Pearson |    32
## cyl        |       qsec | -0.59 | [-0.78, -0.31] | -4.02 | 30 | 0.007  | Pearson |    32
## cyl        |       gear | -0.49 | [-0.72, -0.17] | -3.10 | 30 | 0.054  | Pearson |    32
## cyl        |       carb |  0.53 | [ 0.22,  0.74] |  3.40 | 30 | 0.027  | Pearson |    32
## disp       |         hp |  0.79 | [ 0.61,  0.89] |  7.08 | 30 | < .001 | Pearson |    32
## disp       |       drat | -0.71 | [-0.85, -0.48] | -5.53 | 30 | < .001 | Pearson |    32
## disp       |         wt |  0.89 | [ 0.78,  0.94] | 10.58 | 30 | < .001 | Pearson |    32
## disp       |       qsec | -0.43 | [-0.68, -0.10] | -2.64 | 30 | 0.131  | Pearson |    32
## disp       |       gear | -0.56 | [-0.76, -0.26] | -3.66 | 30 | 0.015  | Pearson |    32
## disp       |       carb |  0.39 | [ 0.05,  0.65] |  2.35 | 30 | 0.177  | Pearson |    32
## hp         |       drat | -0.45 | [-0.69, -0.12] | -2.75 | 30 | 0.110  | Pearson |    32
## hp         |         wt |  0.66 | [ 0.40,  0.82] |  4.80 | 30 | < .001 | Pearson |    32
## hp         |       qsec | -0.71 | [-0.85, -0.48] | -5.49 | 30 | < .001 | Pearson |    32
## hp         |       gear | -0.13 | [-0.45,  0.23] | -0.69 | 30 | 1.000  | Pearson |    32
## hp         |       carb |  0.75 | [ 0.54,  0.87] |  6.21 | 30 | < .001 | Pearson |    32
## drat       |         wt | -0.71 | [-0.85, -0.48] | -5.56 | 30 | < .001 | Pearson |    32
## drat       |       qsec |  0.09 | [-0.27,  0.43] |  0.50 | 30 | 1.000  | Pearson |    32
## drat       |       gear |  0.70 | [ 0.46,  0.84] |  5.36 | 30 | < .001 | Pearson |    32
## drat       |       carb | -0.09 | [-0.43,  0.27] | -0.50 | 30 | 1.000  | Pearson |    32
## wt         |       qsec | -0.17 | [-0.49,  0.19] | -0.97 | 30 | 1.000  | Pearson |    32
## wt         |       gear | -0.58 | [-0.77, -0.29] | -3.93 | 30 | 0.008  | Pearson |    32
## wt         |       carb |  0.43 | [ 0.09,  0.68] |  2.59 | 30 | 0.132  | Pearson |    32
## qsec       |       gear | -0.21 | [-0.52,  0.15] | -1.19 | 30 | 1.000  | Pearson |    32
## qsec       |       carb | -0.66 | [-0.82, -0.40] | -4.76 | 30 | < .001 | Pearson |    32
## gear       |       carb |  0.27 | [-0.08,  0.57] |  1.56 | 30 | 0.774  | Pearson |    32

As you can see, it gives, among other useful information, the correlation coefficients (column r) and the result of the correlation test (column 95% CI for the confidence interval or p for the \(p\)-value) for all pairs of variables. This table is very useful and informative, but let see if it is possible to combine the concepts of correlation coefficients and correlations test in one single visualization. A visualization that would be easy to read and interpret.

Ideally, we would like to have a concise overview of correlations between all possible pairs of variables present in a dataset, with a clear distinction for correlations that are significantly different from 0.

The figure below, known as a correlogram and adapted from the corrplot() function, does precisely this:

corrplot2 <- function(data,
                      method = "pearson",
                      sig.level = 0.05,
                      order = "original",
                      diag = FALSE,
                      type = "upper",
                      tl.srt = 90,
                      number. = 1,
                      number.cex = 1,
                      mar = c(0, 0, 0, 0)) {
  library(corrplot)
  data_incomplete <- data
  data <- data[complete.cases(data), ]
  mat <- cor(data, method = method)
  cor.mtest <- function(mat, method) {
    mat <- as.matrix(mat)
    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], method = method)
        p.mat[i, j] <- p.mat[j, i] <- tmp$p.value
      }
    }
    colnames(p.mat) <- rownames(p.mat) <- colnames(mat)
    p.mat
  }
  p.mat <- cor.mtest(data, method = method)
  col <- colorRampPalette(c("#BB4444", "#EE9988", "#FFFFFF", "#77AADD", "#4477AA"))
  corrplot(mat,
    method = "color", col = col(200), number. = number.,
    mar = mar, number.cex = number.cex,
    type = type, order = order,
    addCoef.col = "black", # add correlation coefficient
    tl.col = "black", tl.srt = tl.srt, # rotation of text labels
    # combine with significance level
    p.mat = p.mat, sig.level = sig.level, insig = "blank",
    # hide correlation coefficients on the diagonal
    diag = diag
  )
}

corrplot2(
  data = dat,
  method = "pearson",
  sig.level = 0.05,
  order = "original",
  diag = FALSE,
  type = "upper",
  tl.srt = 75
)

The correlogram shows correlation coefficients for all pairs of variables (with more intense colors for more extreme correlations), and correlations not significantly different from 0 are represented by a white box.

To learn more about this plot and the code used, I invite you to read the article entitled “Correlogram in R: how to highlight the most correlated variables in a dataset”.

Thanks for reading. I hope this article helped you to compute correlations and perform correlation tests in R.

As always, if you have a question or a suggestion related to the topic covered in this article, please add it as a comment so other readers can benefit from the discussion.