Pearson Correlation Testing in R Programming
Last Updated : 01 Oct, 2024
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 the direction and strength of a relationship using R Programming Language.
Pearson Correlation Testing in R
There are mainly two types of correlation:
- Parametric Correlation: 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 Correlation: They are rank-based correlation coefficients, and are known as non-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 a 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 Programming 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 the 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
1: Correlation Coefficient Test In R Using cor() method
Here we will discuss Correlation Coefficient Test In R Using cor() method:
R # 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) Output:
Pearson correlation coefficient is: 0.5357143
2: Correlation Coefficient Test In R Using cor.test() method
Now we will discuss Correlation Coefficient Test In R Using cor.test() method:
R # 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)
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).
Testing for Statistical Significance
It's crucial to determine if the observed correlation is statistically significant. We use the rcorr() function from the Hmisc package.
R library(ggplot2) library(corrplot) library(Hmisc) # Load the dataset data("mtcars") # Calculate correlation with significance levels cor_test <- rcorr(as.matrix(mtcars[, c("mpg", "wt", "hp", "disp")]), type = "pearson") # Display the correlation coefficients cor_test$r # Display the p-values cor_test$P Output:
mpg wt hp disp
mpg 1.0000000 -0.8676594 -0.7761684 -0.8475514
wt -0.8676594 1.0000000 0.6587479 0.8879799
hp -0.7761684 0.6587479 1.0000000 0.7909486
disp -0.8475514 0.8879799 0.7909486 1.0000000
mpg wt hp disp
mpg NA 1.293958e-10 1.787835e-07 9.380328e-10
wt 1.293958e-10 NA 4.145827e-05 1.222311e-11
hp 1.787835e-07 4.145827e-05 NA 7.142679e-08
disp 9.380328e-10 1.222311e-11 7.142679e-08 NA
- p<0.05: The correlation is statistically significant.
- p>0.05: The correlation is not statistically significant.
Visualizing Relationships Using ggplot2
It’s helpful to visualize the relationship between two variables using scatter plots.
R # Scatter plot with a regression line ggplot(mtcars, aes(x = wt, y = mpg)) + geom_point(color = "blue", size = 2) + geom_smooth(method = "lm", color = "red", se = FALSE) + labs(title = "Scatter Plot with Pearson Correlation", x = "Weight (wt)", y = "Miles Per Gallon (mpg)") + theme_minimal()
Output:
Pearson Correlation Testing in R Programminggeom_smooth(method = "lm"): Adds a linear regression line to visualize the linear relationship.
Conclusion
The Pearson correlation coefficient is a powerful tool for understanding the linear relationship between two continuous variables. In R, calculating and interpreting Pearson correlation is straightforward with built-in functions and packages like ggplot2 and corrplot. By following the methods outlined in this article, you can perform Pearson correlation testing, visualize relationships, and ensure statistical significance.