*This sixteenth article in the R series will introduce you to correlation.*

In this article, we shall explore correlation. We will use R version 4.2.1 installed on Parabola GNU/Linux-libre (x86-64) for the code snippets.

$ R --version R version 4.2.1 (2022-06-23) -- “Funny-Looking Kid” Copyright (C) 2022 The R Foundation for Statistical Computing Platform: x86_64-pc-linux-gnu (64-bit) R is free software and comes with ABSOLUTELY NO WARRANTY. You are welcome to redistribute it under the terms of the GNU General Public License versions 2 or 3. For more information about these matters see <a href="https://www.gnu.org/licenses/." target="_blank" rel="noopener">https://www.gnu.org/licenses/.</a>

The *cor()* function in R can compute the correlation between two sets of vectors. Its usage is as follows:

cor(x, y, na.rm, use, method)

The correlation function accepts the following arguments:

Argument |
Description |

x | numeric vector, matrix or data frame |

y | vector or NULL |

na.rm | logical value to remove missing values |

use | string to specify the computing method |

method | ‘pearson’, ‘kendall’, or ‘spearman’ coefficient |

Let us create three vectors ‘x’, ‘y’ and ‘z’ for comparison using the* sin()* and *cos()* functions as follows:

> t = seq(0, 10, 0.1) > x = sin(t) > y = sin(t + 0.05) > z = cos(t)

The first few values of the vectors are listed below:

> head(x) [1] 0.00000000 0.09983342 0.19866933 0.29552021 0.38941834 0.47942554 > head(y) [1] 0.04997917 0.14943813 0.24740396 0.34289781 0.43496553 0.52268723 > head(z) [1] 1.0000000 0.9950042 0.9800666 0.9553365 0.9210610 0.8775826

The correlation between the ‘x’ and ‘y’ vectors as well as the ‘x’ and ‘z’ vectors is shown below:

> cor(x, y) [1] 0.9985339 > cor (x, z) [1] 0.05483627

We observe that there is a high correlation of 0.99 between the ‘x’ and ‘y’ vectors as they come from the same sine function. The ‘x’ and ‘z’ vectors have a low correlation of 0.05 as they are defined using sine and cosine functions respectively.

Consider the mtcars data set available in the lattice library:

> 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

A plot comparison between cylinder size and horsepower can be generated using the *plot()* function, as follows:

> plot(mtcars$cyl, mtcars$hp, pch=20)

The correlation coefficient values between the cylinder size and horsepower using the default Pearson’s, Kendall’s and Spearman’s methods are given below:

> cor(mtcars$cyl, mtcars$hp) [1] 0.8324475 > cor(mtcars$cyl, mtcars$hp, method = “kendall”) [1] 0.7851865 > cor(mtcars$cyl, mtcars$hp, method = “spearman”) [1] 0.9017909

The high correlation coefficient signifies that a high horsepower has a positive relation with the number of cylinders. The* cor.test()* function can also be used to test the association between paired samples. The correlation test between the mtcars cylinders and horsepower values is shown below:

> cor.test(mtcars$cyl, mtcars$hp) Pearson’s product-moment correlation data: mtcars$cyl and mtcars$hp t = 8.2286, df = 30, p-value = 3.478e-09 alternative hypothesis: true correlation is not equal to 0 95 percent confidence interval: 0.6816016 0.9154223 sample estimates: cor 0.8324475

The cor.test function accepts the following arguments:

Argument |
Description |

x, y | numeric data vectors |

alternative | ‘two.sided’, ‘greater’ or ‘less’ alternative hypothesis |

method | ‘pearson’, ‘kendall’ or ‘spearman’ |

exact | logical value to be indicated if exact p-value should be computed |

conf.level | confidence level |

continuity | true to use a continuity correction |

data | optional data frame or matrix |

subset | optional vector that specifies subset of observations to be used |

na.action | function to indicate when data has NA values |

We can also handle missing values in the data source vectors by specifying the ‘use’ argument with the* cor()* function. An example is given below:

> a <- c(1, 3, 5) > b <- c(2, 4, NA) > cor(a, b) [1] NA > cor(a, b, use = “complete.obs”) [1] 1

An MxN correlation matrix can be created for a data frame. For example:

> cor(mtcars) mpg cyl disp hp drat wt mpg 1.0000000 -0.8521620 -0.8475514 -0.7761684 0.68117191 -0.8676594 cyl -0.8521620 1.0000000 0.9020329 0.8324475 -0.69993811 0.7824958 disp -0.8475514 0.9020329 1.0000000 0.7909486 -0.71021393 0.8879799 hp -0.7761684 0.8324475 0.7909486 1.0000000 -0.44875912 0.6587479 drat 0.6811719 -0.6999381 -0.7102139 -0.4487591 1.00000000 -0.7124406 wt -0.8676594 0.7824958 0.8879799 0.6587479 -0.71244065 1.0000000 qsec 0.4186840 -0.5912421 -0.4336979 -0.7082234 0.09120476 -0.1747159 vs 0.6640389 -0.8108118 -0.7104159 -0.7230967 0.44027846 -0.5549157 am 0.5998324 -0.5226070 -0.5912270 -0.2432043 0.71271113 -0.6924953 gear 0.4802848 -0.4926866 -0.5555692 -0.1257043 0.69961013 -0.5832870 carb -0.5509251 0.5269883 0.3949769 0.7498125 -0.09078980 0.4276059 qsec vs am gear carb mpg 0.41868403 0.6640389 0.59983243 0.4802848 -0.55092507 cyl -0.59124207 -0.8108118 -0.52260705 -0.4926866 0.52698829 disp -0.43369788 -0.7104159 -0.59122704 -0.5555692 0.39497686 hp -0.70822339 -0.7230967 -0.24320426 -0.1257043 0.74981247 drat 0.09120476 0.4402785 0.71271113 0.6996101 -0.09078980 wt -0.17471588 -0.5549157 -0.69249526 -0.5832870 0.42760594 qsec 1.00000000 0.7445354 -0.22986086 -0.2126822 -0.65624923 vs 0.74453544 1.0000000 0.16834512 0.2060233 -0.56960714 am -0.22986086 0.1683451 1.00000000 0.7940588 0.05753435 gear -0.21268223 0.2060233 0.79405876 1.0000000 0.27407284 carb -0.65624923 -0.5696071 0.05753435 0.2740728 1.00000000

The *corrplot()* function can be used to display a correlation matrix. You can install the same in an R session using the following command:

> install.packages(“corrplot”) Installing package into ‘/home/guest/R/x86_64-pc-linux-gnu-library/4.1’ ... ** testing if installed package can be loaded from temporary location ** testing if installed package can be loaded from final location ** testing if installed package keeps a record of temporary installation path * DONE (corrplot)

After loading the corrplot library, we can view the plot for the mtcars data set as follows:

> library(“corrplot”) corrplot 0.92 loaded > corrplot(cor(mtcars), method = “circle”)

You can also restrict the plot to the upper segment by using the ‘type’ argument. For example:

> corrplot(cor(mtcars), method = “number”, type = “upper”)

The *corrplot()* function accepts the following arguments:

Argument |
Description |

corr | the correlation matrix |

method | ‘circle’, ‘square’, ‘ellipse’, ‘number’, ‘pie’, ‘shade’ and ‘colour’ |

type | ‘full’, ‘upper’, or ‘lower’ |

col | specifies a vector colour of glyphs |

bg | background colour |

title | title of the graph |

add | logical value to add plot to an existing graph |

diag | logical value to display the correlation coefficients |

order | ‘original’, ‘AOE’, ‘FPC’, ‘hclust’, or ‘alphabet’ |

rect.col | colour for the rectangular border |

tl.cex | size of the text label |

t1.col | colour of the text label |

tl.srt | numeric value for text label string rotation |

Another plotting function for the correlation matrix is the *ggcorplot()* function, as illustrated below:

> install.packages(“ggcorrplot”) Installing package into ‘/home/shakthi/R/x86_64-pc-linux-gnu-library/4.1’ ... ** testing if installed package can be loaded from final location ** testing if installed package keeps a record of temporary installation path * DONE (ggcorrplot) > library(“ggcorrplot”) Loading required package: ggplot2 > ggcorrplot(cor(mtcars))

The scatterplots are also useful for visualising the matrix. The* pairs()* function is used to compare the miles per gallon, displacement and horsepower, as shown below:

> pairs(mtcars[, c(“mpg”, “disp”, “hp”)])

The *ggscatterstats()* function accepts a data frame, and produces a combined density and histogram plot. It is provided by the *ggstatsplot* library, which is an extension of the* ggplot2* package. An example is given below:

> install.packages(“ggstatsplot”) ... ** testing if installed package keeps a record of temporary installation path * DONE (ggstatsplot) > library(ggstatsplot) > ggscatterstats(data = mtcars, x = cyl, y = hp)

The *ggscatterstats()* function accepts the following arguments:

Argument |
Description |

data | data frame or matrix, table, array |

x | explanatory variable in the data |

y | response variable in the data |

type | ’parametric’, ’nonparametric’,’robust’,’bayes’ |

bf.prior | prior width for calculating Bayes factors |

bf.message | logical value to display Bayes Factor |

tr | trim level for the mean |

k | significant digits after decimal point |

xfill, yfill | colour fill for x and y axes |

xlab | label for x axis variable |

ylab | label for y axis variable |

title | plot title |

You are encouraged to read the manual pages for the above R functions to learn more on their arguments, options and usage.