Estimating correlation

This is the documentation for a Shiny application that demonstrates the concept of correlation. The application is online here.

The Pearson correlation coefficient refers to the linear dependence between two variables. The correlation coefficient ranges from \(-1\) to \(1\) where \(0\) indicates no correlation. The values of \(-1\) and \(1\) indicate a perfect negative and positive relationship, respectively.

Note that the correlations can represent linear relationship only. Even if there is no correlation between two random variables \(y\) and \(y\), there might be another type of dependency, for example, a polynomial relationship.

In the Shiny application, the slider can be used to choose a correlation value. The relationship between the data points displayed in the scatter plot (first tab) will change accordingly. Try some values and have a look at the results.

The correlation in a population is indicated by the Greek letter \(\rho\). This can be interpreted as the "true" correlation.

The following figure illustrates different correlation coefficients. The five panels display scatter plots of \(y\) values against \(x\) values. The corresponding correlation coefficients are \(-1\), \(-0.7\), \(0\), \(0.7\), and \(1\).

The points in the panels with correlations of \(-1\) and \(1\) lie on a line. This is due to the perfect linear relationship between \(x\) and \(y\). In contrast, there is no linear relationship when the correlation is \(0\).

In the Shiny application, \(x\) and \(y\) are represented as a bivariate normal distribution. This is a combination of two normal (or Gaussian) distributions. In R, the density of the simple normal distribution is implemented in dnorm. You can use the curve function to visualize the distribution.

curve(dnorm(x), -5, 5)

In the bivariate normal distribution, both random variables, \(x\) and \(y\) follow a normal distribution. Additionally, there is a correlation between both variables of the population (see the slider in the Shiny application).

We can sample random values of \(x\) and \(y\) from a population. Due to the population correlation \(\rho\), the realisations of \(x\) are not independent from the realisations of \(y\), although sampling is random.

In the Shiny application, you can use the numeric input field to specify the number of random samples. Each random sample is represented by a point in the scatter plot. Furthermore, the dashed line represents the population correlation \(\rho\) while the solid red line represents the estimated correlation, indicated by \(r\), in the sample data set.

It turns out that the correlation of the sample data set \(r\) may differ from \(\rho\) due to the randomness. Usually, the absolute difference between both values decreases with the number of samples.

This phenomenon is visualized in the plot in the second tab. The red line represents how the sample correlation changes with additional samples.

If you want to have a look at the values of the sample data set, you can find a table view of \(x\) and \(y\) in the third tab. Each sample consists of a pair of two values.

Note that the generation of the second plot may take a while if you choose a very large number of samples.

A final note on the implementation: In R, you can generate random samples from the bivariate normal distribution with the function mvrnorm in the MASS package. The correlation between two variables \(x\) and \(y\) can be calculated with the cor function.

x <- c(0, 2, 4, 6, 2, 5)
y <- c(1, 3, 2, 4, 6, 1)
cor(x, y)

## [1] 0.05395

In the example above, the correlation between vectors x and y is 0.05.