# Pearson’s correlation test¶

This method deals with the modelling of a probability distribution of a random vector . It seeks to find a type of dependency (here a linear correlation) which may exist between two components and .

The Pearson’s correlation coefficient , defined in Pearson’s coefficient, measures the strength of a linear relationship between two random variables and . If we have a sample made up of pairs , we denote to be the estimated coefficient.

Even in the case where two variables and have a Pearson’s coefficient equal to zero, the estimate obtained from the sample may be non-zero: the limited sample size does not provide the perfect image of the real correlation. Pearson’s test nevertheless enables one to determine if the value obtained by is significantly different from zero. More precisely, the user first chooses a probability . From this value the critical value is calculated such that:

• if , one can conclude that the real Pearson’s correlation coefficient is not zero; the risk of error in making this assertion is controlled and equal to ;

• if , there is insufficient evidence to reject the null hypothesis .

An important notion is the so-called “ -value” of the test. This quantity is equal to the limit error probability under which the null correlation hypothesis is rejected. Thus, Pearson’s coefficient is supposed non zero if and only if is greater than the value desired by the user. Note that the higher , the more robust the decision. Examples: