# What is correlation test?

Correlation test is used to evaluate the association between two or more variables.

For instance, if we are 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.

If there is no relationship between the two variables (father and son heights), the average height of son should be the same regardless of the height of the fathers and vice versa.

Here, we’ll describe the different correlation methods and we’ll provide pratical examples using R software.

# Install and load required R packages

We’ll use the ggpubr R package for an easy ggplot2-based data visualization

if(!require(devtools)) install.packages("devtools")
devtools::install_github("kassambara/ggpubr")
• Or, install from CRAN as follow:
install.packages("ggpubr")
library("ggpubr")

# Methods for correlation analyses

There are different methods to perform correlation analysis:

• Pearson correlation (r), which measures a linear dependence between two variables (x and y). It’s also known as a parametric correlation test because it depends to the distribution of the data. It can be used only when x and y are from normal distribution. The plot of y = f(x) is named the linear regression curve.

• Kendall tau and Spearman rho, which are rank-based correlation coefficients (non-parametric)

The most commonly used method is the Pearson correlation method.

# Correlation formula

In the formula below,

• x and y are two vectors of length n
• $$m_x$$ and $$m_y$$ corresponds to the means of x and y, respectively.

## Pearson correlation formula

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

$$m_x$$ and $$m_y$$ are the means of x and y variables.

The p-value (significance level) of the correlation can be determined :

1. by using the correlation coefficient table for the degrees of freedom : $$df = n-2$$, where $$n$$ is the number of observation in x and y variables.

2. or by calculating the t value as follow:

$t = \frac{r}{\sqrt{1-r^2}}\sqrt{n-2}$

In the case 2) the corresponding p-value is determined using t distribution table for $$df = n-2$$

If the p-value is < 5%, then the correlation between x and y is significant.

## Spearman correlation formula

The Spearman correlation method computes the correlation between the rank of x and the rank of y variables.

$rho = \frac{\sum(x' - m_{x'})(y'_i - m_{y'})}{\sqrt{\sum(x' - m_{x'})^2 \sum(y' - m_{y'})^2}}$

Where $$x' = rank(x_)$$ and $$y' = rank(y)$$.

## Kendall correlation formula

The Kendall correlation method measures the correspondence between the ranking of x and y variables. The total number of possible pairings of x with y observations is $$n(n-1)/2$$, where n is the size of x and y.

The procedure is as follow:

• Begin by ordering the pairs by the x values. If x and y are correlated, then they would have the same relative rank orders.

• Now, for each $$y_i$$, count the number of $$y_j > y_i$$ (concordant pairs (c)) and the number of $$y_j < y_i$$ (discordant pairs (d)).

Kendall correlation distance is defined as follow:

$tau = \frac{n_c - n_d}{\frac{1}{2}n(n-1)}$

Where,

• $$n_c$$: total number of concordant pairs
• $$n_d$$: total number of discordant pairs
• $$n$$: size of x and y

# Compute correlation in R

## R functions

Correlation coefficient can be computed using the functions cor() or cor.test():

• cor() computes the correlation coefficient
• cor.test() test for association/correlation between paired samples. It returns both the correlation coefficient and the significance level(or p-value) of the correlation .

The simplified formats are:

cor(x, y, method = c("pearson", "kendall", "spearman"))
cor.test(x, y, method=c("pearson", "kendall", "spearman"))

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

If your data contain missing values, use the following R code to handle missing values by case-wise deletion.

cor(x, y,  method = "pearson", use = "complete.obs")

## Import your data into R

1. Prepare your data as specified here: Best practices for preparing your data set for R

2. Save your data in an external .txt tab or .csv files

3. Import your data into R as follow:

# If .txt tab file, use this
# Or, if .csv file, use this
my_data <- read.csv(file.choose())

Here, we’ll use the built-in R data set mtcars as an example.

The R code below computes the correlation between mpg and wt variables in mtcars data set:

my_data <- mtcars
head(my_data, 6)
                   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

We want to compute the correlation between mpg and wt variables.

## Visualize your data using scatter plots

To use R base graphs, click this link: scatter plot - R base graphs. Here, we’ll use the ggpubr R package.

library("ggpubr")
ggscatter(my_data, x = "mpg", y = "wt",
add = "reg.line", conf.int = TRUE,
cor.coef = TRUE, cor.method = "pearson",
xlab = "Miles/(US) gallon", ylab = "Weight (1000 lbs)")

## Preleminary test to check the test assumptions

1. Is the covariation linear? Yes, form the plot above, the relationship is linear. In the situation where the scatter plots show curved patterns, we are dealing with nonlinear association between the two variables.

2. Are the data from each of the 2 variables (x, y) follow a normal distribution?
• Use Shapiro-Wilk normality test –> R function: shapiro.test()
• and look at the normality plot —> R function: ggpubr::ggqqplot()
• Shapiro-Wilk test can be performed as follow:
• Null hypothesis: the data are normally distributed
• Alternative hypothesis: the data are not normally distributed
# Shapiro-Wilk normality test for mpg
shapiro.test(my_data$mpg) # => p = 0.1229 # Shapiro-Wilk normality test for wt shapiro.test(my_data$wt) # => p = 0.09

From the output, the two p-values are greater than the significance level 0.05 implying that the distribution of the data are not significantly different from normal distribution. In other words, we can assume the normality.

• Visual inspection of the data normality using Q-Q plots (quantile-quantile plots). Q-Q plot draws the correlation between a given sample and the normal distribution.
library("ggpubr")
# mpg
ggqqplot(my_data$mpg, ylab = "MPG") # wt ggqqplot(my_data$wt, ylab = "WT")

From the normality plots, we conclude that both populations may come from normal distributions.

Note that, if the data are not normally distributed, it’s recommended to use the non-parametric correlation, including Spearman and Kendall rank-based correlation tests.

## Pearson correlation test

Correlation test between mpg and wt variables:

res <- cor.test(my_data$wt, my_data$mpg,
method = "pearson")
res

Pearson's product-moment correlation
data:  my_data$wt and my_data$mpg
t = -9.559, df = 30, p-value = 1.294e-10
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
-0.9338264 -0.7440872
sample estimates:
cor
-0.8676594 

In the result above :

• t is the t-test statistic value (t = -9.559),
• df is the degrees of freedom (df= 30),
• p-value is the significance level of the t-test (p-value = 1.29410^{-10}).
• conf.int is the confidence interval of the correlation coefficient at 95% (conf.int = [-0.9338, -0.7441]);
• sample estimates is the correlation coefficient (Cor.coeff = -0.87).

### Interpretation of the result

The p-value of the test is 1.29410^{-10}, which is less than the significance level alpha = 0.05. We can conclude that wt and mpg are significantly correlated with a correlation coefficient of -0.87 and p-value of 1.29410^{-10} .

The function cor.test() returns a list containing the following components:

• p.value: the p-value of the test
• estimate: the correlation coefficient
# Extract the p.value
res$p.value [1] 1.293959e-10 # Extract the correlation coefficient res$estimate
       cor
-0.8676594 

## Kendall rank correlation test

The Kendall rank correlation coefficient or Kendall’s tau statistic is used to estimate a rank-based measure of association. This test may be used if the data do not necessarily come from a bivariate normal distribution.

res2 <- cor.test(my_data$wt, my_data$mpg,  method="kendall")
res2

Kendall's rank correlation tau
data:  my_data$wt and my_data$mpg
z = -5.7981, p-value = 6.706e-09
alternative hypothesis: true tau is not equal to 0
sample estimates:
tau
-0.7278321 

tau is the Kendall correlation coefficient.

The correlation coefficient between x and y are -0.7278 and the p-value is 6.70610^{-9}.

## Spearman rank correlation coefficient

Spearman’s rho statistic is also used to estimate a rank-based measure of association. This test may be used if the data do not come from a bivariate normal distribution.

res2 <-cor.test(my_data$wt, my_data$mpg,  method = "spearman")
res2

Spearman's rank correlation rho
data:  my_data$wt and my_data$mpg
S = 10292, p-value = 1.488e-11
alternative hypothesis: true rho is not equal to 0
sample estimates:
rho
-0.886422 

rho is the Spearman’s correlation coefficient.

The correlation coefficient between x and y are -0.8864 and the p-value is 1.48810^{-11}.

# Interpret correlation coefficient

Correlation coefficient is comprised between -1 and 1:

• -1 indicates a strong negative correlation : this means that every time x increases, y decreases (left panel figure)
• 0 means that there is no association between the two variables (x and y) (middle panel figure)
• 1 indicates a strong positive correlation : this means that y increases with x (right panel figure)

# Online correlation coefficient calculator

You can compute correlation test between two variables, online, without any installation by clicking the following link:

# Summary

• Use the function cor.test(x,y) to analyze the correlation coefficient between two variables and to get significance level of the correlation.
• Three possible correlation methods using the function cor.test(x,y): pearson, kendall, spearman

# Infos

This analysis has been performed using R software (ver. 3.2.4).