Correlation coefficient

lastmod 2018-08-22created 2014-10-234 edits

The correlation coefficient is a numerical value (coefficient) that indicates the degree of correlation between two variates X and Y.

Basic assumptions

Linearity: checks whether the relationship between the two variables X and Y is linear; this assumption can be verified with a scatter plot showing the distribution.
Homoscedasticity: means that the spread of Y is the same regardless of the value of X. Its opposite is heteroscedasticity.
Normality of both variables: the distributions of the measurements of both variables are normal in the population.
Random independent samples: when drawing samples from the population, the sample subjects are selected probabilistically.

There is simple correlation analysis, which simply measures how strongly two variables are related,
and multiple correlation analysis, which measures the strength of the relationship among three or more variables.
In multiple correlation analysis, expressing the strength of the relationship between just two variables while holding their relationships with the other variables fixed is called partial correlation analysis.

Here, if the correlation is \(0 \lt \rho \le +1\), it is called a positive correlation,
if \(-1 \le \rho \lt 0\), a negative correlation,
and if \(\rho = 0\), no correlation. (However, 0 does not mean there is no relationship; it means the relationship is not linear.)