import numpy as np
data_1 = np.array([1,2,3,4,6,7,8,9])
data_2 = np.array([2,4,6,8,10,12,13,15])
data_3 = np.array([-1,-2,-2,-3,-4,-6,-7,-8])Basic Statistics III: Correlations
Now that we have learned how to compute basic statistics on single variables, we will look at how to measure the relationship between two variables with correlations.
Background on Correlations
A correlation is a measure of the statistical relationship between two variables. Correlation values range from -1 to 1, where the magnitude (a.k.a. absolute value) of the correlation indicates the strength of the relationship and the sign of the correlation represents the direction of the relationship. The correlation value is often denoted with the variable r, so that is what we will use here.
The figure below shows some examples of perfect, strong and weak correlations between two variables in both the positive and negative directions. As you can notice, perfect correlation between two variables corresponds to |r| = 1. Stronger correlations have r values with magnitude closer to 1, and weaker correlations have r values with magnitude closer to 0. When r = 0, there is no linear relationship between the two variables.
What do you notice about the difference between positive correlations and negative correlations?
Computing Correlations
Let’s practice with some test data
Based on how we’ve constructed our variables, what do you expect the correlation values to be?
Visualizing the relationships may help us understand this better:
Now, let’s calculate the actual correlatiton values. We will use the corrcoef function from numpy to calculate correlation values.
r = np.corrcoef([data_1,data_2,data_3])
print(r)[[ 1. 0.99535001 -0.9805214 ]
[ 0.99535001 1. -0.97172394]
[-0.9805214 -0.97172394 1. ]]Does the output of this function make sense to you?
This function returns a correlation matrix, which always has 1’s along the diagonal and is symmetric (i.e. same values above the diagonal as below). This is so you can compute correlations of more than one variable at a time. The correlation values in the matrix above correspond to the following relationships:
Based on these plots, can you figure out why all correlation matrices have: * 1’s on the diagonal? (Talk about how the diagonal is always the correlation of one variable with itself, which will always be perfect correlation) * symmetric entries? (Talk about how the corr(data1, data2) == corr(data2, data1))
So, the output of the corrcoef function from above is a correlation matrix follows the following form:
| _ | data_1 | data_2 | data_3 |
|---|---|---|---|
| data_1 | 1 | 0.995 | -0.980 |
| data_2 | 0.995 | 1 | -0.971 |
| data_3 | -0.980 | -0.971 | 1 |
Now, it should be clear why a correlation matrix always has 1’s along the diagonal - every variable has perfect positive correlation with itself. Furthermore, it is symmetric because the correlation of data_1 & data_2 is the same as the correlation of data_2 & data_1.
Now that we understand our output, let’s check the correlations between the variables in the iris dataset.
# load and preview iris
import pandas as pd
path = 'https://raw.githubusercontent.com/GWC-DCMB/curriculum-notebooks/master/'
iris = pd.read_csv(path + 'SampleData/iris.csv')
iris.head()| sepal_length | sepal_width | petal_length | petal_width | species | |
|---|---|---|---|---|---|
| 0 | 5.1 | 3.5 | 1.4 | 0.2 | setosa |
| 1 | 4.9 | 3.0 | 1.4 | 0.2 | setosa |
| 2 | 4.7 | 3.2 | 1.3 | 0.2 | setosa |
| 3 | 4.6 | 3.1 | 1.5 | 0.2 | setosa |
| 4 | 5.0 | 3.6 | 1.4 | 0.2 | setosa |
# find correlations between sepal_length, sepal_width, petal_length, petal_width
# HINT: Think back to how we subset certain columns in pandas
iris_corrs = np.corrcoef(iris.iloc[:,0:4], rowvar=False)
print(iris_corrs)[[ 1. -0.11756978 0.87175378 0.81794113]
[-0.11756978 1. -0.4284401 -0.36612593]
[ 0.87175378 -0.4284401 1. 0.96286543]
[ 0.81794113 -0.36612593 0.96286543 1. ]]You’ll notice this time we included the rowvar parameter - this is because, by default, the corrcoef function expects that each row represents a variable, with observations in the columns. In our case it is the opposite - each column represents a variable, while the rows contain observations. So here we change the value of rowvar from the default True to False.
In this lesson you learned:
- How to measure the relationship between two variables
- The difference between positive/negative correlations and strong/weak correlations
- How to compute and interpret correlations for multiple variables
Now, lets continue to practice!