Correlation in R

In this post I am writing about the correlation between two variables, how to find and interpret this concept.

Guilherme Bastos Gomes https://gbastosg.github.io/guilhermesportfolio/
2022-05-15

Hello!

In this post we will work with correlations.

Correlation

Correlation is a statistical measure that takes in consideration the degree of separation (or union) betweeen two variables. It’s important to say that correlation measures the association level of our numbers, and doesn’t show us if one CAUSES the other, or vice-versa. Also doesn’t show us if the association is happening because of another variable from our data.

What can correlation show us?

From two variables we can obtain a numeric value known as the correlation coefficient, which varies from -1 to 1.

This coefficient shows us the relation between two distributions, if it is positive, than it’s closer to 1, otherwise it will show that one variable causes de inverse effect on the other and the coefficient proximates -1. If a variation doesn’t affect the other, then there’s no linear correlation, and the value approximates 0.

There are some methods to obtain the correlation coefficient in R, let’s test of them in the mpg data set

library(tidyverse)

Correlation in R

The correlation coefficient can be calculated using the functions cor() or cor.test(), where:

We can use the function, together with the tests in the following way:

Is there a correlation between the variables displ and cty?

The code:

mpg %>% 
  ggplot(aes(x = displ, y = cty, color = class)) +  
    geom_point() + 
    labs( 
      title = "Motor efficiency of different cars",
      subtitle = "by: Guilherme Bastos Gomes", 
      caption = "Source: mpg data set", 
      x = "Motor Display",
      y = "City miles per gallon",
    ) + 
    theme_classic()

Notice that there is a tendency in out plot, that can be modeled with a line using the function geom_smooth:

Check how the line can explain (in it’s own way) the relation between variables:

mpg %>% 
  ggplot(aes(x = displ, y = cty, color = class)) +  
    geom_point() + 
    labs( 
      title = "Motor efficiency of different cars",
      subtitle = "by: Guilherme Bastos Gomes", 
      caption = "Source: mpg data set", 
      x = "Motor Display",
      y = "City miles per gallon",
    ) + 
    theme_classic()

Line points to the correlation sense, here we have:

As the value of displ increases, the value of cty decreases

Now let’s use correlation tests to obtain the correlation coefficient:

cor(mpg$displ, mpg$cty, method = c("pearson"))
[1] -0.798524

Notice how the correlation coefficient is close to -1, evidencing a negative correlation.

There are 3 most used methods to obtain a correlation coefficient: “Pearson”, “Spearman” or “Kendall”.

To obtain even more evidences on the correlations we can obtain the correlation coefficient for all methods:

cor(mpg$displ, mpg$cty, method = c("spearman"))

cor(mpg$displ, mpg$cty, method = c("kendall"))
[1] -0.8809049
[1] -0.7210828

Notice how the Spearman test gave us a correlation coefficient even closer to -1! It’s important to say that correlation show us a pattern between variables, but it doesn’t tell us that one is CAUSING the other.

In our data we are showing the negative relationship between displ and cty, and as bigger the motor, more gallons are necessary to run a mile. The explanation for this kind of correlation may seem obvious, but it shows us that considered potent motors tend to spend more fuel, what can directly affect who pays for the product.

Pearson’s correlation test

In the last post we have investigated the correlation between the variables Petal.Length and Petal.Width, of each species of the data set iris:

iris %>% 
   ggplot(aes(x = Petal.Length, y = Petal.Width, color = Species)) +  
     geom_point() + 
     labs( 
       title = "Comparison between sizes of Sepals in 3 species of the Iris gender",
       subtitle = "by: Guilherme Bastos Gomes", 
       caption = "Source: Edgar Anderson's Iris Data set", 
       x = "Petal.Length",
       y = "Petal.Width" 
     ) +
     theme_classic() +
   facet_grid(~Species) +
   geom_smooth(method = "lm")

Now let’s do a correlation test between variables Petal.Lengthe Petal.Width:

cor.test(iris$Petal.Length, iris$Petal.Width, 
                    method = "pearson")

Pearson’s product-moment correlation

data: iris$Petal.Length and iris$Petal.Width t = 43.387, df = 148, p-value < 2.2e-16 alternative hypothesis: true correlation is not equal to 0 95 percent confidence interval: 0.9490525 0.9729853 sample estimates: cor 0.9628654

In the results we obtained from the test we can see:

Here we have a strong positive correlation, with strong evidences that suggests that one measure affects the other, allowing us to generate ideas about the petal sizes of these three species we can find in nature.

Interpreting results

Going back to the mpg data set where we have a negative correlation between of variables:

ct <- cor.test(mpg$displ, mpg$cty, 
                    method = "pearson")
                    
ct

Pearson’s product-moment correlation

data: mpg$displ and mpg$cty t = -20.205, df = 232, p-value < 2.2e-16 alternative hypothesis: true correlation is not equal to 0 95 percent confidence interval: -0.8406782 -0.7467508 sample estimates: cor -0.798524

Function cor.test() returns us a list that contains:

ct$p.value

ct$estimate

In statiscs, p-value is the probability of obtaining results as extreme as the observed results in a hypothesis test, assuming that the null hypothesis is true.

I won’t be in details of all these right now, I will write about p-value in another post!

How to perform a correlation test:

cor.test(dataframe\$coluna1, dataframe\$coluna2, 
                    method = "pearson")

Hope it was helpful! Thank you for your time! Next post I will talk about p-values and the null hypothesis.