Pearson Correlation Testing in R Programming

Correlation is a statistical measure that indicates how strongly two variables are related. It involves the relationship between multiple variables as well. For instance, if one is 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. Generally, it lies between -1 and +1. It is a scaled version of covariance and provides direction and strength of a relationship. There are mainly two types of correlation:

  • Parametric CorrelationPearson correlation(r): It measures a linear dependence between two variables (x and y) is known as a parametric correlation test because it depends on the distribution of the data.
  • Non-Parametric CorrelationKendall(tau) and Spearman(rho): They are rank-based correlation coefficients, are known as non-parametric correlation.

Pearson Rank Correlation Coefficient Formula

Pearson Rank Correlation is a parametric correlation. The Pearson correlation coefficient is probably the most widely used measure for linear relationships between two normal distributed variables and thus often just called “correlation coefficient”. The formula for calculating the Pearson Rank Correlation is as follows:

{{\displaystyle r = \frac { \Sigma(x - m_x)(y - m_y) }{\sqrt{\Sigma(x - m_x)^2 \Sigma(y - m_y)^2}}

where,
r: pearson correlation coefficient
x and y: two vectors of length n
mx and my: corresponds to the means of x and y, respectively.

Note:

  • r takes value between -1 (negative correlation) and 1 (positive correlation).
  • r = 0 means no correlation.
  • Can not be applied to ordinal variables.
  • The sample size should be moderate (20-30) for good estimation.
  • Outliers can lead to misleading values means not robust with outliers.

Implementation in R

R Language provides two methods to calculate the pearson correlation coefficient. By using the functions cor() or cor.test() it can be calculated. It can be noted that cor() computes the correlation coefficient whereas cor.test() computes test for association or correlation between paired samples. It returns both the correlation coefficient and the significance level(or p-value) of the correlation.



Syntax:
cor(x, y, method = “pearson”)
cor.test(x, y, method = “pearson”)

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

Example 1:

# Using cor() method

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# R program to illustrate 
# pearson Correlation Testing 
# Using cor() 
  
# Taking two numeric 
# Vectors with same length 
x = c(1, 2, 3, 4, 5, 6, 7)  
y = c(1, 3, 6, 2, 7, 4, 5
  
# Calculating  
# Correlation coefficient 
# Using cor() method 
result = cor(x, y, method = "pearson"
  
# Print the result 
cat("Pearson correlation coefficient is:", result) 

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Output:

Pearson correlation coefficient is: 0.5357143

# Using cor.test() method

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# R program to illustrate 
# pearson Correlation Testing 
# Using cor.test() 
  
# Taking two numeric 
# Vectors with same length 
x = c(1, 2, 3, 4, 5, 6, 7)  
y = c(1, 3, 6, 2, 7, 4, 5
  
# Calculating  
# Correlation coefficient 
# Using cor.test() method 
result = cor.test(x, y, method = "pearson"
  
# Print the result 
print(result) 

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Output:

Pearson's product-moment correlation

data:  x and y
t = 1.4186, df = 5, p-value = 0.2152
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.3643187  0.9183058
sample estimates:
      cor 
0.5357143 

In the output above:

  • T is the value of the test statistic (T = 1.4186)
  • p-value is the significance level of the test statistic (p-value = 0.2152).
  • alternative hypothesis is a character string describing the alternative hypothesis (true correlation is not equal to 0).
  • sample estimates is the correlation coefficient. For Pearson correlation coefficient it’s named as cor (Cor.coeff = 0.5357).

Example 2:

Data: Download the CSV file here.
Example:

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# R program to illustrate 
# Pearson Correlation Testing 
  
# Import data into RStudio 
df = read.csv("Auto.csv"
  
# Taking two column 
# Vectors with same length 
x = df$mpg 
y = df$weight 
  
  
# Calculating 
# Correlation coefficient 
# Using cor() method 
result = cor(x, y, method = "pearson"
  
# Print the result 
cat("Person correlation coefficient is:", result) 
  
# Using cor.test() method 
res = cor.test(x, y, method = "pearson"
print(res) 

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Output:

Person correlation coefficient is: -0.8782815
    Pearson's product-moment correlation

data:  x and y
t = -31.709, df = 298, p-value < 2.2e-16
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.9018288 -0.8495329
sample estimates:
       cor 
-0.8782815 



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