One way to quantify the relationship between two variables is to use the Pearson correlation coefficient, which measures the linear association between two variables.

It always takes on a value between -1 and 1 where:

• -1 indicates a perfectly negative linear correlation
• 0 indicates no linear correlation
• 1 indicates a perfectly positive linear correlation

To determine if a correlation coefficient is statistically significant, you can calculate the corresponding t-score and p-value.

The formula to calculate the t-score of a correlation coefficient (r) is:

t = r * √n-2 / √1-r²

The p-value is then calculated as the corresponding two-sided p-value for the t-distribution with n-2 degrees of freedom.

Example: Correlation Test in Python

To determine if the correlation coefficient between two variables is statistically significant, you can perform a correlation test in Python using the pearsonr function from the SciPy library.

This function returns the correlation coefficient between two variables along with the two-tailed p-value.

For example, suppose we have the following two arrays in Python:

#create two arrays
x = [3, 4, 4, 5, 7, 8, 10, 12, 13, 15]
y = [2, 4, 4, 5, 4, 7, 8, 19, 14, 10]

We can import the pearsonr function and calculate the Pearson correlation coefficient between the two arrays:

from scipy.stats.statsimport pearsonr

#calculation correlation coefficient and p-value between x and y
pearsonr(x, y)

(0.8076177030748631, 0.004717255828132089)

Here’s how to interpret the output:

• Pearson correlation coefficient (r): 0.8076
• Two-tailed p-value: 0.0047

Since the correlation coefficient is close to 1, this tells us that there is a strong positive association between the two variables.

Note that we can also extract the individual correlation coefficient and p-value from the pearsonr function as well:

#extract correlation coefficient (rounded to 4 decimal places)
r = round(pearsonr(x, y)[0], 4)

print(r)

0.8076

#extract p-value (rounded to 4 decimal places) 
p = round(pearsonr(x, y)[1], 4)

print(p) 

0.0047

These values are a bit easier to read compared to the output from the original pearsonr function.

Additional Resources

The following tutorials provide additional information about correlation coefficients:

An Introduction to the Pearson Correlation Coefficient