Measuring correlation

To test the relationship between the Future Orientation Index and the GDP per capita of countries, we are going to measure a correlation coefficient. Here you can find the basics of the concept, take a look even if you study this before, it is always good to have a refresher!

Some univariate statistics notation

• \(X\) is a random variable
• In data: \(X_i\) is the value of the variable for entry \(i\), for example the GDP of a country
• \(E[X]\) is the expected value of \(X\)
  • We estimate the expected value as the mean of \(X\): \(\mu_X = \frac{1}{N}\sum_i X_i\)
  • \(N\) is the number of data points, for example the number of countries
  • In R you can calculate it with the function mean()
• \(V[X]\) is the variance of \(X\)
  • We calculate it as the expected squared difference to the mean X: \(V[X] = \frac{1}{N}\sum_i (X_i - \mu_X)^2\)
  • The variance is measured in the square of units of X (e.g. if X is GDP measured in USD, then V[X] is USD\(^2\))
• \(\sigma_X\) is the standard deviation of X
  • \(\sigma_X = \sqrt{V[X]}\), which is convenient because it measures dispersion in the same units as \(X\)
  • in R you can calculate it with the function sd()

Pearson’s Correlation Coefficient

Correlation: Linear association or dependence between the values of variables X and Y

Pearson’s way to measure correlation between variables \(X\) and \(Y\), denoted as \(\rho(X,Y)\):

• If X and Y are independent, they satisfy that the expectation of the product equals the product of expectations:

\(E[XY] = E[X]E[Y]\)

• The principle: measure correlation as the deviation from \(E[XY] − E [X]E[Y] = 0\)
  
• The absolute value of this difference can be at most \(\sigma_X\sigma_Y\)
  
• \(\rho(X,Y)\) rescales the difference to be between −1 and 1

\(\rho(X,Y) = \frac{E[XY] − E [X]E[Y]}{\sigma_X\sigma_Y}\)

• Can be computed in R with the function cor()

Some examples of Pearson’s Correlation Coefficient:

Independent variables will have a correlation close to zero, but a correlation close to zero does not mean independence

Anscombe’s quartet

Anscombe’s quartet has four examples of scatter plots for two variables. All cases have the same mean and standard deviation for the variables, and the same positive correlation coefficient: 0.816

The first case fits what we expect of linear correlation. The second shows a nonlinear correlation where the upwards trend becomes downwards, the third is a case where an outlier decreases the correlation coefficient and the fourth case is a correlation coefficient generated by a single outlier. Always look at scatter plots to know what your correlations mean!

The Datasaurus dozen

The Datasaurus dozen shows 12 (+1) examples with the same means and standard deviations and the same correlation coefficient of -0.06:

Remember, not everything that has a correlation of zero is independent! There are many kinds of relationships between variables beyond linear ones.