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 [url=/wiki/t-distribution-table]t distribution table[/url] 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

R functions

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

The simplified formats are:

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

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: [url=/wiki/best-practices-for-preparing-your-data-set-for-r]Best practices for preparing your data set for R[/url]

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
my_data <- read.delim(file.choose())
# 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: [url=/wiki/scatter-plots-r-base-graphs]scatter plot - R base graphs[/url]. Here, we’ll use the [url=/wiki/ggpubr-r-package-ggplot2-based-publication-ready-plots]ggpubr R package[/url].

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)")

Correlation Test Between Two Variables in R software

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")

Correlation Test Between Two Variables in R software

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 

# 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}.